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Transition dynamics of single and double Josephson junctions formed by spatially coupled soliton and surface plasmons by G une s Ayd ndo gan A Dissertation Submitted to the Graduate School of Sciences and Engineering in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy in Physics June, 2017 Transition dynamics of single and double Josephson junctions formed by spatially coupled soliton and surface plasmons Ko c University Graduate School of Sciences and Engineering This is to certify that I have examined this copy of a doctoral dissertation by G une s Ayd ndo gan and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the nal examining committee have been made. Committee Members: Assoc. Prof. Kaan G uven Prof. Ozg ur M ustecapl glu Prof. Ali Serpeng uzel Assist. Prof. Yasa Ek sio glu Ozok Assoc. Prof. Ahmet Levent Suba s Date: ABSTRACT This thesis work investigates the crossing dynamics of photons between a spa tially coupled copropagating soliton and a surfaceplasmon, which constitute a type of photonic Josephson junction. By introducing modulations to the spatial coupling, the crossing dynamics exhibit features similar as well as di erent to that of nonlinear LandauZener or RosenZener type transitions. The dependence of the coupling to the soliton amplitude provides an inherent dynamic, which may manifest distinct features in the transition characteristics. The dynamics of the system, which is formulated as a Josephson junction, is investigated by introducing fractional population imbalance and the relative phase variables. A full population conversion between the optical soliton and the surface plasmon is achieved. The governing equations of the system represents a set of nonlinear Schr odinger equations, and the eigenvalue analysis of these equations sheds light on the behaviour of the xed points of the system, here with the stability analysis will be investigated. Under a spatial periodic modulation, this type of Josephson junction may exhibit driven resonance states similar to Shapiro resonances. The stability of these resonances will be investigated in the presence of an external periodic eld. The double Josephson junction is formed by coupling the soliton spatially to two surfaceplasmons, which reside on either side of the soliton propagation axis, respectively. This threestate system may provide rich dynamics with features like collapserevival and plasmonplasmon coupling via a frozen soliton state. By introducing spatial modulations, further dynamical e ects will be explored. Heuristic designs are analyzed for sensor applications by investigating realistic mate rials and associated parameters. iii OZETC E Bu tez cal smas nda uzaysal olarak etkile sim i cinde bulunan soliton ve y uzey plaz monlar aras ndaki foton ge ci sleri ve bu ge ci slerin olu sturdu gu Josephson ekleminin di nami gi incelenmi stir. Bu ge ci sin dinami gi uzaysal etkile sime yap lan kipleme sayesinde lineer olmayan LandauZener ve RosenZener ge ci slerine benzer oldu gu kadar bun lardan farkl ozellikler de g ostermektedir. Soliton ve plazmon aras ndaki etkile simin soliton genli gine olan ba g ml l g sa glanan ge ci sin ozelliklerinde belirgin olgular ortaya c karan i csel bir dinamik sa glamaktad r. Fraksiyonel pop ulasyon orans zl g ve ba g l faz terimlerini kullanarak sistemin dinami gi Josephson eklemi olarak ele al nabilir. Bu sayede optik soliton ve y uzey plazmonu aras nda tam bir pop ulasyon transferi ger cekle sebilir. Sistemi y onlendiren esas denklemler do grusal olmayan Schr odinger denklemi ozellikleri g ostermektedir ve Schr odinger denklemlerinin ozde ger analizi plaz monik sistemin sabit noktalar n n bulunmas na s k tutmaktad r. B oylelikle de sis temin denge analizi yap labilmektedir. Uzaysal kipleme periyodik olarak tasarland g nda bu tarzda olan Josephson eklemleri Shapiro c nlamalar nalar na benzeyen kararl yap da c nlama durumlar n n ortaya c kmas na neden olabilmektedir. Harici bir periyo dik alan n varl g nda bu tarz c nlama durumlar n n denge analizi de yap lacakt r. Solitonun kendi y or ungesinin iki taraf nda konumlanan y uzey plazmonlar yla girdi gi uzaysal etkile sim ikili Josephson eklemini olu sturur. Bu u c katmanl sistem c ok u s canlan s, donuk bir soliton arac l g ile plazmonplazmon etkile simi gibi zengin di nami gi olan ozel durumlara neden olabilir. Tekli Josephson eklemindeki gibi uza ysal kiplemelerin de eklenmesiyle bu kiplemenin sistem dinami gi uzerindeki etki leri incelenecektir. Ayr ca uygulanabilirli gi olan ger cek ci materyallerin ve bunlar n olu sturdu gu sistem parametrelerinin incelenmesiyle alg lay c uygulamalar nda kul lan lmak uzere bulgusal tasar mlar n analizi yap lacakt r. iv TABLE OF CONTENTS List of Figures vii Chapter 1: Introduction 1 Chapter 2: Single Josephson Junction in a MetalDielectric Inter face 5 2.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Surface Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Optical Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.4 LandauZener Transition . . . . . . . . . . . . . . . . . . . . . 18 2.1.5 RosenZener Transition . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Surface PlasmonSpatial Soliton Coupling . . . . . . . . . . . . . . . 21 2.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Hyperbolic Modulation to the Model . . . . . . . . . . . . . . 24 2.2.3 Analogy to LandauZener [LZ] and RosenZener [RZ] Transitions 26 2.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 E ect of relative phase di erence . . . . . . . . . . . . . . . . 36 2.4.2 Fixed Point Analysis . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 50 3.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 50 v 3.1.1 Surface Plasmons in Multilayers . . . . . . . . . . . . . . . . . 50 3.1.2 BoseEinstein Condensates in TripleWell Traps . . . . . . . . 52 3.2 Surface PlasmonSpatial Soliton Coupling in Multilayer Parallel Systems 53 3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.2 Population Trapping . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.3 Classical Collapse and Revival of Coupling Function . . . . . . 59 3.3 Surface PlasmonSpatial Soliton Coupling in Hyperbolically Modulated Multilayer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 4: Shapiro Resonances 77 4.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Periodic Modulation to the Model . . . . . . . . . . . . . . . . . . . . 81 Chapter 5: Applications 85 5.1 Surface PlasmonSoliton Coupling in Realistic Structures . . . . . . . 85 5.2 Proposed Instruments Regarding Surface PlasmonSoliton Coupling . 86 Chapter 6: Conclusion 91 Bibliography 93 vi LIST OF FIGURES 2.1 Illustration of a planar waveguide geometry. z is the propagation direction of the waves. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Geometry for SP propagation at a single interface between a metal and a dielectric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Dispersion relation of SPPs at the interface between a Drude metal with negligible collision frequency and air (gray curves) and silica (black curves) [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Illustration of di erent spatial beam pro les [7]. . . . . . . . . . . . . 11 2.5 The focusing e ect of a simple convex lens. . . . . . . . . . . . . . . . 12 2.6 The shape of the soliton, while propagating with N=1. . . . . . . . . 14 2.7 The basic geometry of a Josephson junction formed between two su perconductors is illustrated. . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 The BEC doublewell trap. E0 1 and E0 2 are the zeropoint energy of each condensate [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9 Transition probability versus external period under di erent nonlinear ity strengths [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.10 Metaldielectricnonlinearity interfaces for plasmonsoliton propagation. 22 2.11 Hyperbolic twolevel system. . . . . . . . . . . . . . . . . . . . . . . . 24 2.12 Eigenenergies of the Hamiltonian (Eq. 2.45) [31]. a) c = 0.1 and v = 0.2, b) c = 0.4 and v = 0.2. . . . . . . . . . . . . . . . . . . . . . . . 28 2.13 Fixed points associated with the eigenenergies in Fig. 2.12 [31]. a) c/v = 0.5, b) c/v = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 vii 2.14 (a) Solid line shows the transition from soliton to surface plasmon am plitude, dashed lines show the reverse transition from surface plasmon to soliton amplitude. = 0:63, d = 5:7, = 0:04, and E = 0:02: (b) Solid line shows the plot of coupling function 'q(Z)' vs. z of the rst transition, and the dashed lines show the plot of 'q(Z)' vs. z of the reverse transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.15 Contour plot of nal Z values with respect to and d for di erent values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (a). Here E is taken 0:02. . . . . . . . . . . . . . . . . . . . . 31 2.16 Error percentage of the values. Red line shows the error. . . . . . . 32 2.17 Contour plot of nal Z values with respect to and d for di erent E values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (d). Here is taken 0:04 respectively. . . . . . . . . . . . . . . 34 2.18 (a) Propagation of surface plasmon and optical soliton wavefunctions. Optical soliton is propagating in the absence of surface plasmon initially and after the transition it transfers almost all of its energy to surface plasmon on the metal surface. Here is 0.63, is 0.04, E is 0.02, and d is 5.7. (b) Propagation of surface plasmon and optical soliton wavefunctions. Surface plasmon is propagating in the absence of optical soliton initially and after the transition it transfers almost all of its energy to optical soliton on the ber. The same parameter values are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 viii 2.19 Signi cant transitions between surface plasmon and soliton under dif ferent d, and values. (a) shows the case that soliton is propagating initially in the absence of surface plasmon, and its energy is divided equally between soliton and surface plasmon. (b) shows the case that soliton transfers almost all of its energy to surface plasmon and then surface plasmon transfers its energy back to soliton. (c) and (d) show the case that in the presence of equal surface plasmon and soliton, the energy can be manipulated to be transferred to surface plasmon or soliton, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.20 The wavefunction pro les of surface plasmon and optical soliton of the same four cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.21 The e ect of di erent initial phase di erences. Solid red line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 1. Dashed lines show the nal Z values for every single di erent 0 value when the system starts with Z = 0. Here the full transition parameter values are used; d = 5:7, = 0:63, = 0:04, and E = 0:02. . . . . . 38 2.22 (a) Solid line shows the case when 0 = 0:2, in which all the energy is transfered to soliton. Dashed lines show the case when 0 = 0:81, in which almost all the energy is transfered to surface plasmon. Here d = 5:6, = 0:4, = 0:04, and E = 0:02. (b) Solid red line shows the nal Z values when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values when the system starts with Zô€€€1 1. Dashed lines show the nal Z values when the system starts with Z = 0. 39 ix 2.23 The evolution of the phasespace trajectories of the system as z changes adiabatically. The red dots indicate the motion of the population im balance in phasespace when Zô€€€1 1. The green dot indicates the motion when Zô€€€1 ô€€€1 Here the parameters are the same as in Fig. 2.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.24 Initial relative phase di erence vs. z graph. (a) Z = ô€€€1, (b) Z = 0, (c) Z = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.25 The evolution of the phasespace trajectories of the system with equal initial amplitude of surface plasmon and optical soliton as z changes adiabatically. The red dot indicate the motion of the population im balance in phasespace with 0 = 0:2, and green squares indicate the motion of the population imbalance in phasespace with 0 = 0:81. The population imbalance changes dramatically from 0 to 1 (red dot) or from 0 to 1 (green dot). Here the parameters are d = 5:6, = 0:4, = 0:04, and E = 0:02. . . . . . . . . . . . . . . . . . . . . . . . . 42 2.26 Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 5:7, = 0:63, = 0:04, and E = 0:02 are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.27 Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 2:15, = 0:32, = 0:1, and E = ô€€€0:04 are used. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.28 Eigenenergies and the corresponding xed points. . . . . . . . . . . . 46 2.29 E ect of E. Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 x 2.30 E ect of . Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; E = 0:02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.31 E ect of E. Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; = 0:02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.32 E ect of . Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; E = 0:03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Illustration of a threelevel system consisting of a middle layer I trapped between two large half spaces II and III. . . . . . . . . . . . . . . . . 51 3.2 Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:04; E1 = 0:03; E2 = ô€€€0:01; Ap0 ' 1; As ' 0 . . . . 56 3.4 Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:05; E1 = 0:03; E2 = 0:03; Ap0 ' 1; As ' 0 . . . . . 56 3.5 Dynamics of amplitudes of soliton and surface plasmons (a) and relative phase di erences (b). Paremeters are d1 = 8; d2 = 4; = 0:06; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:9; As0 0:45. . . . . . . . . . . . . . 57 3.6 Dynamics of amplitudes of soliton and surface plasmons (a) and rel ative phase di erences (b). Paremeters are d1 = 5:7; d2 = 6:3; = 0:06; E1 = 0:02; E2 = 0:02; Ap10 0:8; As0 0:6. . . . . . . . . . 57 xi 3.7 Dynamics of amplitudes of soliton and surface plasmons (a) and rel ative phase di erences (b). Paremeters are d1 = 5:9; d2 = 6:1; = 0:07; E1 = ô€€€0:03; E2 = ô€€€0:03; Ap10 0:7; As0 0:7. . . . . . . . 58 3.8 Quantum collapse and revivals can be seen in BEC atoms [47]. In a) population imbalance (Z) of N = 350 BEC atoms can be seen. Most of the population is trapped in of the states. The relative phase of both states are equal. In b) N = 500 BEC atoms can be seen. Both states are equally populated during the collapse. The relative phases are equal. 60 3.9 Oscillatory part of the z coordinate for di erent absorbed energies n0. Arrows indicate the amplitude collapse of the oscillations [52]. . . . . 61 3.10 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 . . . . . . . . . . . . . . . . . . . 62 3.11 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 . . . . . . . 63 3.12 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 . . . . . . . . . . . . . . . . . . . . . 64 3.13 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 . . . . . . . . . . . . . . 65 3.14 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . . . . . . . . . . . . . 66 3.15 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . 66 xii 3.16 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 . . . . . . . . . . . . . . . . 67 3.17 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 . . . . 67 3.18 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . . . . . . . . . . . 68 3.19 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . . . . 68 3.20 Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.21 A %100 transfer of energy between two surface plasmon channels. Pa rameters are d1 = 8:21; d2 = 3:79; = 0:6; = 0:07; E1 = 0:02; and E2 = 0:04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.22 Population trapping could be observed with Z1 0. The parameters are d1 = 6:5; d2 = 5:5; = 0:51; = 0:062; E1 = 0:033; and E2 = ô€€€0:025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.23 A population trapping could be achieved when the initial soliton ampli tude is di erent than zero. The parameters are d1 = 4:7; d2 = 7:3; = 0:6; = 0:03; E1 = ô€€€0:04; E2 = 0:03; Ap1 = 0:6; As = 0:8. . . . 72 3.24 Fixed points (blue circles are As and green circles are Ap1 points) and corresponding eigenenergies of the system with parameters d1 = 8:21; d2 = 3:79; = 0:6; = 0:07; E1 = 0:02; and E2 = 0:04 for cases 1 = 0; 2 = 0 (a), 1 = 0; 2 = (b), and 1 = ; 2 = (c). 75 xiii 3.25 Fixed points (blue circles are As and green circles are Ap1 points) and corresponding eigenenergies of the system with parameters d1 = 6:5; d2 = 5:5; = 0:51; = 0:062; E1 = 0:033; and E2 = ô€€€0:025 for cases 1 = 0; 2 = 0 (a), 1 = ; 2 = 0 (b), and 1 = ; 2 = (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.26 Fixed points (blue circles are As and green circles are Ap1 points) and corresponding eigenenergies of the system with parameters d1 = 4:7; d2 = 7:3; = 0:6; = 0:03; E1 = ô€€€0:04; E2 = 0:03 for cases 1 = 0; 2 = 0 (a), 1 = ; 2 = 0 (b), and 1 = ; 2 = (c). . . . 76 4.1 Phase space illustration of two di erent pendulums. (a) is = 0, and (b) is = 100. The interaction is = 104, and the driving frequency is = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Population imbalance vs. time when = 104 and = 3x10ô€€€5 with = 100; = 1000. Shapiro resonance can be observed when time averaged population imbalance is trapped around z = 0:1. . . . . . . 80 4.3 Shapiro resonance e ects in weakly coupled BoseEinstein condensates. Idc is the intensity value which is proportional to average population imbalance. is the nonlinearity parameter [37]. . . . . . . . . . . . . 81 4.4 Illustration of the sinusoidally modulated single Josephson junction. . 82 4.5 Shapiro resonance in plasmonic Josephson junction. A drop in the magnitude of average population imbalance can be observed, when the external modulation parameter changes between 10 and 90. System parameters are d = 5; = 0:08; E = 0;A = 0:8;Zô€€€1 = 0:9049: . . . 83 xiv 4.6 Shapiro resonance in plasmonic Josephson junction. Two di erent spikes in the magnitude of average population imbalance can be ob served, when the external modulation parameter changes between 10 and 90. System parameters are d = 3; = 0:07; E = ô€€€0:04;A = 0:6;Zô€€€1 = 0:9049: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1 Geometry of the fourlayer con guration. . . . . . . . . . . . . . . . . 86 5.2 Refractive index sensor based on single Josephson junction formulation of surface plasmonoptical soliton coupling. . . . . . . . . . . . . . . . 87 5.3 A coupling modulator made of metaldielectricnonlinear dielectric dielectricmetal multilayers. . . . . . . . . . . . . . . . . . . . . . . . 89 5.4 The e ect of d2. The other parameters are d1 = 4:7; = 0:6; = 0:03; E1 = ô€€€0:04; E2 = 0:03; Ap1 = 0:6; As = 0:8. . . . . . . . . 90 xv Chapter 1 INTRODUCTION One of the major problems of today's digital information technologies is to carry the signal between ends of a microprocessor, that is in centimeter scale. However, ultrafast transistors are in nanometer scale, and this causes problems regarding speed in signal processing, when copper wires are to be used. On the other hand, ber optic cables are able to carry data much more e cient than electronic circuits, however combining the two counterparts on the same circuit have some signi cant constraints. The proposed solution is an element, that can carry optical signals in nanomater scale, surface plasmons [1]. The improvements and progresses in science and engineering exhibit renewed interest in surface plasmons, that allow metals to be designed in subwavelength scale. This property of surface plasmons has gotten attention from scientist of di erent disciplines. Manipulating the frequency spectrum of lasers, ultra short pulse generation, signal processing, magnetooptic data storage, solar cells, and biosensors for health studies are main topics of surface plasmons, which are being used actively by researchers [2]. The inception of these studies is the transformation of light into surface plasmons, which enables scientist to suppress and channel light using sub wavelength photonic structures. In this way, surface plasmons can be used as control parameters in circuits instead of light, which causes an enhancement in the non linear phenomena. This new subdivision of photonics is called plasmonics, that o ers to combine the advantage of carrying ultrafast signals and operating on small circuits at the same time. The mathematical formulation of surface plasmon propagation lies back to the Maxwell's equations, so that in order to investigate the properties of surface plasmons in nonlinear media, nonlinear Maxwell's equations for the TM 2 Chapter 1: Introduction waves at the metaldielectric interface should be analyzed. The roots of overcoming the di raction limit by surface plasmon polaritons come from the slow wave properties of surface plasmons. The formation of surface plasmons by the interaction between surface charges and the electromagnetic eld has two corollaries. First, the momentum of surface plasmon, ~kSP, is greater than of a photon with the same frequency, ~k0. The second corollary is that, the evanescent tail of the eld, which is perpendicular to the interface decays exponentially throughout the direction perpendicular to the surface, which is caused by the con ned and nonradiative nature of surface plasmons. Anharmonic electromagnetic elds cause nonlinear e ects in optical systems. This anharmonicity can be expanded as power series resulting in a mixture of incident elds and output elds, that oscillate at frequencies of a superposition of incident elds. Second harmonic generation and third harmonic generation are typical examples of these e ects, that result in frequency conversion. Moreover, the thirdorder e ects possess terms at the incident frequencies, which causes a change in the refractive index of the medium, which is called optical Kerr e ect. The optical properties of Kerr e ect enable to create bistability, furthermore di raction caused by linear e ects balances the bistability caused by nonlinear e ects, producing a new type of optical wave, solitons. An optical soliton is a stable optical wave, that propagates without an aberration. The Kerr e ect caused by selfphase modulation creates a red shift at the evanescent tails of the wave, and a blue shift also occurs at the tails due to dispersion. The combination of these two e ects result in a pulse, that maintains its shape while travelling. There are various application areas in which solitons can be used such as surface waves in shallow water, plasma waves, sound waves in 3He, short optical pulses in bers, and optical spatial solitons [3]. Although the physical systems behind these examples are quite di erent, they all obey the same rules, and they are a conse quence of nonlinear partial di erential equations. Between the various soliton kinds, optical solitons are a widespread research hotspot for the recent years, because of its applicability in laboratory environment and extensive control over the parameters. Chapter 1: Introduction 3 The soliton concept was rst discovered in hydrodynamics in the 19th century, and has been used extensively in various disciplines such as optics, condensed mat ter physics, uid mechanics, particle physics, and even astrophysics. However, the most important contribution of solitons between these elds has been made in opti cal communications. Optical solitons have been studied under spatial and temporal domains, however spatial solitons arouse attention of researchers, because of their highdimensional properties, whereas temporal solitons are basicly onedimensional el ements. Moreover, the rich nonlinear properties of spatial solitons which have found application areas in photorefractive materials, liquid crystals, thermophoresis, and colloidal suspensions are important contributing factors. These various nonlinear ef fects enable scientist to study soliton phenomena more e ciently, since it was thought earlier that, solitons were exact solutions of the cubic nonlinear Sch odinger equation. The earlier studies regarding nonlinear waves in metaldielectric interfaces were tak ing into consideration only the transverse direction, however later studies show that, the temporal e ects should also be taken into account, and so the metaldielectric geometry should be studied with a slot width smaller than the wavelength, and the complicated structure of nonlinear localization requires the consideration of boundary conditions. A novel con guration has been proposed, where surface plasmon guided vertically on the metal surface and a soliton selftrapped by nonlinear Kerr e ect combined to yield a plasmonsoliton coupling driven by a lowpower continuouswave optical source [4]. In this thesis work, we investigate the dynamical properties of optical solitons propagating on a Kerr type of nonlinearity and surface plasmons propagating on metaldielectric interface. Matching of surface plasmon and soliton propagation con stants gives rise to coupling of both waves forming a plasmonic Josephson junction. Nonlinear nature of the optical soliton causes the coupling between two channels to be a function of soliton amplitude, which makes the soliton amplitude a driving pa rameter of the system. The Josephson junction formulation can be set by introducing two variables, which have analogues on bosonic Josephson junctions: fractional pop 4 Chapter 1: Introduction ulation imbalance and relative phase di erence. We then consider the interaction between two channels changing adiabatically, hence the soliton trajectory is modu lated spatially such that, the distance between metal surface and Kerr nonlinearity becomes a hyperbolic function. A full population conversion between surface plasmon and optical soliton can be achieved under certain parameter values, indicating that metal/dielectric/Kerr interface can host a RosenZener type of transition. Stability analysis of the system is also studied by inquiring the xed points of the system. We next investigate the transition dynamics in a threelevel system, in which the nonlin earity is sandwiched between metal/dielectric interfaces, which yields the interaction between two surface plasmons via optical soliton propagating between them. This multilayer interface gives rise to di erent phenomena such as complete population transfer, population trapping, and collapse and revival of the coupling function. The xed points and eigenenergies of the system is also analyzed. The following step is the periodic modulation of the system in order to examine Shapiro resonances, which were originally studied in superconductors and BoseEinstein condensates. Finally, a realistic approach into the materials has been made to limit the parameter values, and a few applications of surface plasmonsoliton coupling have been proposed. Chapter 2 SINGLE JOSEPHSON JUNCTION IN A METALDIELECTRIC INTERFACE 2.1 Theoretical Background 2.1.1 Surface Plasmons Surface plasmons(SPs, also called surface plasmonpolaritons) are coherent elec tron oscillations that propagates at the interface between two materials, where the permittivity changes sign across the interface. The most common candidate for SP ex citation is a metaldielectric interface. These electromagnetic surface waves originates from the coupling of the metal's electron plasma with photons. SPs are evanescently con ned in the transverse direction, and they have the ability to propagate within small sized materials. These type of materials have the wavelength much smaller than the wavelength of the surface plasmon. Applying Maxwells equations (Eq.'s 2.1  2.4) to the at interface between a metal and a dielectric leads us to physical prop erties of SPPs. For a more detailed and clearer investigation of these properties, we should start with the derivation of the wave equation, that describes the guiding of electromagnetic waves, hence we start with the Maxwell's equations of macroscopic electromagnetism. r:D = ext (2.1) r:B = 0 (2.2) r E = ô€€€ @B @t (2.3) r H = Jext + @D @t : (2.4) 6 Chapter 2: Single Josephson Junction in a MetalDielectric Interface In the absence of external charge and current densities, Faraday's Law and Am pere's Law (2.3, 2.4) can be put together to generate r r E = ô€€€ 0 @2D @t2 (2.5) Using identities r r E r(r:E)ô€€€r2E and r:( :E) E:r + r:E, and also r:D = 0, equation 2.5 can be rewritten as r ô€€€ 1 E:r ô€€€ r2E = ô€€€ 0 0 @2E @t2 : (2.6) Considering the negligible variation of the permittivity = (r), Eq. (2.6) simpli es to the wave equation, r2E ô€€€ c2 @2E @t2 = 0: (2.7) Since the dielectric pro le, , is constant in the medium, this equation has solutions in both metal and dielectric, and boundary conditions must be used to obtain these solutions. In order to describe the con ned propagating waves with Eq.(2.7), we should rewrite this equation in a di erent form, the wellknown Helmholtz equation. We should rst consider, that there is a harmonic time dependence E(r; t) = E(r)ei!t of the electric eld. If we insert this function into Eq.(2.7), we obtain r2E + k20 E = 0; (2.8) where k0 = ! c is the wave vector of the propagating wave in vacuum. The next step is introducing a speci ed geometry to apply the Helmholtz equation. We choose this geometry such that, depends only on one spatial coordinate, z direction, and therefore does not have any dependence in the perpendicular, y direction (see Fig. 2.1); this is chosen for simplicity, so that = (x). Our electric eld function can be written as E(x; y; z) = E(x)ei z. is de ned as the complex parameter and can be written as = kz, which is the propagation constant of the traveling waves. Inserting this expression into Eq.(2.8) gives the wave equation in the Chapter 2: Single Josephson Junction in a MetalDielectric Interface 7 Figure 2.1: Illustration of a planar waveguide geometry. z is the propagation direction of the waves. expected form @2E(x) @x2 + (k2 0 ô€€€ 2)E = 0: (2.9) The magnetic eld H can be obtained by following similar steps, which will not be shown here. Equation (2.9) will be the key to analyze guided electromagnetic modes in waveg uides. The next step now is to determine explicit expressions for the electric and magnetic elds. These expressions will help us determining the spatial eld pro le and dispersion of propagating waves. Two sets of solutions can be obtained from this type of systems: TM and TE modes. In TM (transverse magnetic) modes Ex;Ez, and Hy are nonzero and all other components are zero. In TE (transverse electric) modes Hx , Hz and Ey are nonzero and all other components are zero. For TM modes, the wave equation is @2Hy @x2 + (k2 0 ô€€€ 2)Hy = 0: (2.10) For TE modes, the corresponding wave equation is @2Ey @x2 + (k2 0 ô€€€ 2)Ey = 0: (2.11) A single, smooth interface (Fig. 2.2) composed of a two halfspaces on both sides, one being a dielectric nonabsorbing material (x > 0) with positive permittivity 2 and the other being a conductive material (x < 0) described via a dielectric function 1(!) is the obvious candidate for a geometry, which sustains SPs. The metallic character 8 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.2: Geometry for SP propagation at a single interface between a metal and a dielectric. requires Re[ 1] < 0. We should look for propagating wave solutions with evanescent decay in the perpendicular xdirection. Let us rst look at TM solutions. Hy(x) = A2ei zeô€€€k2x (2.12) Ez(x) = iA2 1 ! 0 2 k2ei zeô€€€k2x (2.13) Ez(x) = ô€€€A1 ! 0 2 ei zeô€€€k2x (2.14) for x > 0, and Hy(x) = A1ei zek1x (2.15) Ex(x) = ô€€€iA1 1 ! 0 1 k1ei zek1x (2.16) Ez(x) = ô€€€A1 ! 0 1 ei zek1x (2.17) for x < 0. Here ki kx;i(i = 1; 2) is the wave vector, which is perpendicular to the propagation direction. Boundary conditions require the continuity of Hy and iEx at the interface (A1 = A2) and results in k2 k1 = ô€€€ 2 1 : (2.18) Chapter 2: Single Josephson Junction in a MetalDielectric Interface 9 The y component of the magnetic eld, Hy, should satisfy the wave equation 2.18. Plugging in the wave equation gives k2 1 = 2 ô€€€ k2 0 1 (2.19) k2 2 = 2 ô€€€ k2 0 2: (2.20) Combination of these equations along with Eq. (2.18) yields the dispersion relation of SPs propagating at the interface between the two di erent media = k0 r 1 2 1 + 2 : (2.21) We arrived at this result by manipulating the TM solutions of the wave equation. However, no surface modes exist for TE polarization. Thus, surface plasmons only exist for TM polarization. We now investigate dispersion relation of SPs, which will reveal some important properties of SPs. As can be seen in Fig. 2.3, for small wave vectors (low frequencies), the SP propagation constant is close to k0 at the light line. For large wave vectors, the freeelectron dielectric function (!) = 1 ô€€€ !2p !2+i ! can be plugged into the dispersion relation above. This shows that the frequency of the SPs approaches the characteristic surface plasmon frequency !sp = !p p 1 + 2 ; (2.22) If the damping of the conduction electron oscillation is neglected (Im[ 1(!)] = 0), the propagation constant goes to in nity as the frequency approaches !sp. This would allow the group velocity goes to zero, vg ! 0. Then this mode gains electrostatic character, which is known as the surface plasmon. 2.1.2 Optical Solitons Optical spatial solitons are optical beams that travel without the e ects of di raction and show selftrapping properties with nite spatial cross section. The medium has to have a mechanism, that gives a selffocusing response to the light. The strong in teraction between the medium and the electromagnetic wave causes the selffocusing. 10 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.3: Dispersion relation of SPPs at the interface between a Drude metal with negligible collision frequency and air (gray curves) and silica (black curves) [5]. The wave modi es locally the medium and the medium in turn modi es the wave. Therefore, optical beams, whose elds intersect this particular region of the medium, are a ected. This is a typical example of the interaction of solitons with one another, and with surface plasmons as well. We can now say that self trapping can be obtained as a result of a subsequent physical mechanisms. In order to give insight about the SPsoliton interaction, some conceptual ideas should be presented rst. Among these ideas the most fundamental ones are the de nition of optical spatial solitons, structures that can host these type of solitons, and the generality of the interactions between solitons considering the extensively diverse physical origins of the selftrapping. Optical beams broaden with distance with the natural di raction as they propagate, if they are very narrow in shape, which causes them to travel without changing the properties of a medium. If the width of the initial beam is thinner, then the beam diverges and therefore di racts faster. However, in materials that possess nonlinear properties, the presence of light modi es the properties such as refractive index, absorption, or conversion to other Chapter 2: Single Josephson Junction in a MetalDielectric Interface 11 frequencies. The refractive index change forms an optical lens that increases the index in the beams center and leaves it unchanged in the beams tails. This induced lens focuses the beam, what is called selffocusing. Selftrapping can arise, when the self focusing and the di raction stabilize each other. The newly arising beam with a very narrow width is called an optical spatial soliton as shown in Fig. 2.4. Figure 2.4: Illustration of di erent spatial beam pro les [7]. A simple convex lens model has to be explained in order to understand the exis tence of spatial solitons. As shown in the Fig. 2.5, an incident optical eld is focused after getting through the lens, which introduces a xdependent phase change which is a nonuniform function, (x) = k0nL(x), that causes the focusing e ect [8], [9]. Here, L(x) is the width of the lens. Since k0 and n are constants, the width of the lens changes in each point with a shape similar to (x). This is another way of saying that, a focusing e ect could be obtained by introducing a phase change of such a shape without a change of the width [10]. This method should be followed in order to apply this principle to optical solitons. To create the same e ect with a di erent approach, the value of the refractive index n(x) should be changed, with the width L is kept xed in each point. The modulation of the refractive index creates a focusing e ect, that reduces the natural di raction of the eld. When the focusing e ect stabilizes the di raction of the eld, a con ned eld can be observed within the medium. The existence of spatial solitons 12 Chapter 2: Single Josephson Junction in a MetalDielectric Interface are based on this principle. The Kerr e ect creates a selfphase modulation, that alters the refractive index, which is a function of the intensity function, (x) = k0n(x)L, [10]. n(x) = n0 + n2I(x): (2.23) Figure 2.5: The focusing e ect of a simple convex lens. The propagating wave eld develops a berlike guiding structure. Once this guiding is created, and if it is the mode of such a ber at the same time, this indicates nonlinear e ect, which causes focusing and linear e ect, which causes di raction are perfectly balanced, and the eld does not experience a change in its shape. The condition of developing a nonlinear behavior is that n2 must be positive, so that selffocusing can be observed. The mathematical formulation of soliton waves is as follows. Kerr e ect is the main reason for the existence of solitons, hence the refractive index is given by n(I) = n + n2I (2.24) where I = jEj2 2 and = 0= and 0 = p 0= 0 377 . The eld is propagating in the z direction with a phase constant k0n. Any dependence on y axis can be ignored, because the eld is in nite in y direction, which can be expressed as E(x; z; t) = Ama(x; z)ei(k0nzô€€€!t) (2.25) Chapter 2: Single Josephson Junction in a MetalDielectric Interface 13 where Am is the maximum amplitude of the eld and a(x; z) is a dimensionless nor malized function, that represents the shape of the electric eld among x axis [11]. The next step is solving the Helmholtz equation r2E+k2 0n2(I)E = 0 , and we assume that a(x; z) changes slowly, while propagating, i.e. j @2a(x;z) @z2 j j k0 @a(x;z) @z j and the following equation is obtained: @2a @x2 + i2k0n @a @z + k2 0[n2(I) ô€€€ n2]a = 0 (2.26) The e ect of nonlinearity is always much smallar than the e ect of linearity, so the following approximation can be considered valid: [n2(I) ô€€€ n2] = [n(I) ô€€€ n][n(I) + n] = n2I(2n + n2I) 2nn2I (2.27) then we express the intensity in terms of the electric eld: [n2(I) ô€€€ n2] 2nn2 jAmj2ja(x; z)j2 2 0=n = n2n2 jAmj2ja(x; z)j2 0 (2.28) n2 is considered positive, because we assume that selffocusing is created by the nonlinearity. To clear this statement, n2 will be written as jn2j from now on. Some new parameters should be introduced and substituted in the above equation. First parameter, that should be introduced, is = x X0 , so the dependence on the x axis can be expressed with a dimensionless parameter; where X0 is the characteristic length. Second parameter is Ld = X2 0k0n. Beyond the distance X0 on the z axis, the di raction caused by the linear e ects becomes large enough, so they cannot be neglected. The third parameter is = z Ld , a dimensionless variable, that describes the zô€€€dependence. Next parameter is Lnl = 2 0 k0njn2jjAmj2 , which depends upon the electric eld intensity. Beyond the distance X0 on the z axis, the selffocusing caused by the nonlinear e ects becomes large enough, so they cannot be neglected. Final param eter is N2 = Ld Lnl . After the substitutions, Eq. 2.28 yields the nonlinear Schr odinger equation: 1 2 @2a @ 2 + i @a @ + N2jaj2a = 0 (2.29) To interpret this equation in a better way, four di erent regimes should be considered separately. 14 Chapter 2: Single Josephson Junction in a MetalDielectric Interface For N 1, the nonlinear part can be neglected. For N 1, the nonlinear e ect will be dominant over di raction. For N 1, the two e ects balance each other and the equation can be solved. For N = 1, the solution of the equation is the fundamental soliton [10]: a( ; ) = sech( )ei =2 (2.30) Although the solution is a function of z, the shape of the eld will be stationary while it propagates, because the z dependence is in the phase term. The illustration of the soliton can be shown in Fig. 2.6. Figure 2.6: The shape of the soliton, while propagating with N=1. As far as soliton solutions are concerned, N must be an integer, and is called the order of the soliton. For the solitons with higher orders, closed form expressions does not exist. These higer order solitons are all periodic, which have di erent periods. Their shape can be identi ed after generation: a( ; = 0) = Nsech( ): (2.31) The mathematical formulation of the optical waveguide introduced by the prop agating soliton is not only in theory, but also a realistic model. The optical soliton Chapter 2: Single Josephson Junction in a MetalDielectric Interface 15 wave could be utilized to interact with other waves at di erent frequencies, which would otherwise be impossible in linear media. 2.1.3 Josephson Junction The weak coupling of two macroscopic quantum objects features the di erence be tween classical and quantum mechanical dynamics. This fact was predicted by Brian D Josephson in 1962. The system is described as a thin insulator sandwiched be tween two superconductors without an applied external voltage. He predicted that a DC current can ow through these layers. This is called the dcJosephson e ect. Moreover, an external voltage will result in a rapidly oscillating current; this e ect is known as the acJosephson e ect. In general, a Josephson junction can be constructed by a three layer system, one nonsuperconducting layer between two superconducting layers as shown in the gure. Figure 2.7: The basic geometry of a Josephson junction formed between two super conductors is illustrated. In a Josephson junction, the middle nonsuperconducting section should be narrow. If the barrier is made of an insulator, it has to be on the order of 30 angstroms thick or less. If the barrier is made of another kind of metal, its thickness should be in units of micron. A supercurrent can ow across the insulator; so that electron pairs encounter no resistance, while crossing the barrier from one side to the other. However 16 Chapter 2: Single Josephson Junction in a MetalDielectric Interface when the threshold current is exceeded, an AC voltage evolves across the junction. This lowers the junction's threshold current, generates even more normal current to ow and causes a larger AC voltage close to 500 GHz per mV across the junction. The voltage would be zero, until the current through the junction is less than the threshold current. Once the current reaches the threshold current, the voltage would be nonzero and oscillates in time. Detecting and measuring the change between two states is the key to the many applications for Josephson junctions. The Josephson e ects has been used in various applications such as voltage stan dards of the Shaphiro e ect, ultrasensitive magnetic eld sensors, and supercon ducting quantum interference devices (SQUIDS). Furthermore fundamental questions on quantum physics have been studied both theoretically and experimentally with Josephson junctions in various con gurations such as ultra small junctions and long junction arrays. In this thesis study, the discussion of bosonic Josephson junctions (BJJ), generated by con ning a single BoseEinstein condensate (BEC) in a double well potential plays an important role, since we draw an analogy between surface plasmonoptical soliton coupling and bosonic Josephson junctions. The macroscopic quantum phase di erence has been exhibited by setting up a doublewell trap for con ned BoseEinstein condensates, which are divided by a strong laser sheet, that builds an impenetrable barrier between the traps. Turning o the barrier, the two released condensates overlapped, producing a strong twoslit atomic interference pattern, which is a sign for macroscopic phase coherence. The method for observing the phase di erences between two trapped BEC is turning o the bar rier gradually, in other words the intensity of the laser sheet should be lowered to enable atomic tunneling through the barrier, and Josephsonlike currentphase e ects manifest themselves. The structure is shown in Fig. 2.8, [12]. The importance of detection of the phase di erence reveals itself, when monitoring the population of the trapped condansates with phasecontrast microscopy. In this method, the geometry of the wells and the barrier can be adjusted by the location and the strength of the laser sheet that divides the trap. The analogy between bosonic Chapter 2: Single Josephson Junction in a MetalDielectric Interface 17 Figure 2.8: The BEC doublewell trap. E0 1 and E0 2 are the zeropoint energy of each condensate [12]. Josephson junction (BJJ) and superconducting Josephson junction (SJJ) requires some similarities between system components. An external direct current voltage in SJJ could be replaced by the chemical potential between the BECs, which is a function of the initial energy bias, furthermore capacitive SJJ charging energy could be replaced by the initial population imbalance. These replacements between two systems along with the adjusting capability of traps and the nonlinear barrier draw a ful lling analogy between BJJ and SJJ. The similarities between Bosonic Josephson junciton and superconducting Joseph son junctions should be mentioned as well as the di erences between them. The com parison includes some obvious results. In both models the motivation is to obtain a tunneling between two states and to nd the alternating current. However, getting this alternating current requires di erent conditions for both cases. For example, BJJ requires no potential di erence for producing ac, although zero potential di erence produces direct current in SJJ. Another example is that, the ac frequency in BJJ depends on the barrier transmissivity, on the other hand the frequency depends on the external voltage in SJJ. 18 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Experimental realization of Bosonic Josephson junctions has been compared with the theoretical results [13]. A BoseEinstein condensate is trapped in a doublewell, which are separated by a potential barrier. The coupling is constructed weakly so that, the particle can tunnel through the barrier. The population imbalance between the two wells, which is a function of the energy bias shows an agreement with the theoretical expectation. The population imbalance and the relative phase di erence were found to be the key elements, that control the dynamics of the selftrapping regime with the agreement to the solution of the GrossPitaevskii equation. Bosonic Josephson junctions can nd many applications within the eld of BECs and nanophysics, since the rich dynamics of BoseEinstein condensates in double and triplewell traps reveals some particular properties such as the macroscopic quantum self trapping and adiabatic transition between the wells, in which BECs are trapped [14], [15]. These examples encounter some problems regarding the nonlinearity created by the interaction of BECs, and few new approaches have been proposed to overcome these di culties by introducing a modi ed nonlinear LandauZener and RosenZener models [16], [17]. 2.1.4 LandauZener Transition LandauZener [LZ] model [20], [21] is an extensively used twostate model in quantum physics, however its applied elds extend over a large area such as ac current driven Josephson junctions, atomic collision studies, BEC in optical lattices, QED circuits, multiple level crossings, and threelevel systems. There are many reasons for this wide usage of LandauZener model. One of the reasons is that, this model portrays an interaction between a doublewell quantum system and an external eld near the resonance. Another reason is the realistic presumption of the detuning, which is a timedependent function, however the coupling is constant. And the nal reason is the accurate de nition of transition probability, that this model gives. All of these reasons considered, LandauZener model is a promising and satisfying model, that can be adapted to many di erent systems in both applied and theoretical physics. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 19 Despite the wide application elds of LandauZener model, the extention of the model to the nonlinear case urged itself, when the transition probability in the adia batic limit has been started to be investigated. As a result of this investigation, it has been realized, that strong nonlinearity breaks the adiabaticity to enable a tunneling between two states. This breakdown of adiabaticity is caused by the emergence of ex tra xed points and the collision of these xed points, which was throughly explained by analysing the energy levels and quantum eigenstates of the system [22], [23], [24]. The promising applicability of LandauZener model to the two state systems en courages researchers to examine the threelevel systems [25]. Though the triplewell model shares some dynamic properties with its twolevel counterpart such as the breakdown of adiabaticity and nonzero transition probability, it has some distin guished speci cations like selftrapping of populations in a single well, topological change of the system near crossing points, and sensitivity to the initial system pa rameters, which results in chaos. An elaborated analysis has been done on this topic both numerically and analytically [26], and tunneling dynamics of the nonlinear three level system has been exhibited. 2.1.5 RosenZener Transition The the RosenZener [RZ] model [27] was introduced to investigate the double Stern Gerlach experiments regarding the hyper ne Zeeman levels of energy under a rotating magnetic eld. It was proposed at the same times as the LandauZener model with the motivation of encompassing a di erent system structure. The LandauZener model describes the transition properties of two avoidedcrossing levels, in which the coupling is constant and the energy gap between levels are slowly changing. On the contrary, the RosenZener model describes the transition probability between two levels, in which the energy di erence is xed and the coupling is a timedependent function. The structure of this coupling function de nes the driving properties of the system. On of the advantages of this model is to provide both numerical and analytical solutions to two or three level systems such as exchange of ion population in nonres 20 Chapter 2: Single Josephson Junction in a MetalDielectric Interface onant ionatom collision, excitations induced by lasers, nuclear magnetic resonance techniques, and quantum computation. In a recent study [32], the RosenZener model has been enlarged to take the nonlinearity into account, and the e ects of this nonlinearity have been investigated throughly. BoseEinstein condensates have been used as a subject in this investiga tion because of their rich nonlinear properties, which was caused by the interaction between condensates. These nonlinear properties enable to observe various signi cant phenomena such as macroscopic quantum selftrapping, super uidity, and Landau Zener transition in a nonlinear perspective suggesting the bene ts of analysing the nonlinear RosenZener transition. Here a nonlinear Schr odinger equation has been solved numerically to obtain the transition probabilities between traps. i @ @t 0 @ a b 1 A = H(t) 0 @ a b 1 A; (2.32) where the Hamiltonian is given by H(t) = 0 @ 2 + c 2 (jbj2 ô€€€ jaj2) v 2 v 2 ô€€€ 2 ô€€€ c 2 (jbj2 ô€€€ jaj2) 1 A: (2.33) Here, a and b are the probability amplitudes of the condensates, is energy di erence between traps, and c is the nonlinearity parameter indicating the interaction between condensates. The coupling parameter, v, is given as a timedependent function v = 8< : 0; t < 0; t > T; v0sin2( t T ); 0 < t < T; (2.34) where T is the external period of the system. The probability of one of the condensates, which resides in the other trap after the coupling loses its e ect on them is considered as the transition probability. The e ect of system parameters to the transition probability has been investigated, the results for the nonlinearity are presented in Fig. 2.9. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 21 Figure 2.9: Transition probability versus external period under di erent nonlinearity strengths [32]. 2.2 Surface PlasmonSpatial Soliton Coupling 2.2.1 Model Description To design a system, that has both optical soliton and surface plasmon components, that are copropagating along the Kerrdielectric and metaldielectric interfaces re spectively, we adopt the following model in [35]. The system shown in Fig. 2.10 is formed by two interfaces that host optical wave formation: a metal/lineardielectric interface (surface plasmon formation) and a lineardielectric/nonlineardielectric (Kerr) interface (spatial soliton formation). The distance between these two interfaces, in other words the distance from the surface plasmon propagation axis to the soliton center axis is d. The interaction between a surface plasmon guided wave at the metal/dielectric interface, that propagates along the z direction and a spatial soliton in the nonlinear medium propagating along the same direction will be analyzed. It is assumed that, the nonlinearity is selffocusing, and the nonlinear behavior of the soliton is able to provide a propagation constant s 22 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.10: Metaldielectricnonlinearity interfaces for plasmonsoliton propagation. that grows with the soliton peak amplitude. On the other hand, the surface plasmon is formed by two evanescent wave tails both in the metal and in the dielectric with propagation constant p. It is expected that, under appropriate conditions controlled by the soliton power, it is possible to achieve the matching of the propagation con stants of plasmon and soliton. This phasematching condition, s = p, should give rise to a mechanism of nonlinear resonant transfer of energy from soliton to surface plasmon. The total wave eld of the system is represented by the following ansatz, which indicates the superposition of surface plasmon and soliton elds, (x; z) = cp(z) p(x) + cs(z) s(x; jcsj): (2.35) We emphasize that, the transverse pro le of the soliton, s = sech h k p =2jcsj(x ô€€€ d) i , depends on its amplitude jcsj which is the driving parameter of the system, whereas the transverse pro le of the plasmon, p = exp ô€€€ ô€€€x p k2 p ô€€€ k2 at x > 0, represents two exponents decaying away from the metal surface with normalizations p(0) = 1 Chapter 2: Single Josephson Junction in a MetalDielectric Interface 23 and s(d) = 1. We assume that, the soliton amplitude varies slowly enough to use quasistanionary adiabatic approximation. Here is the nonlinearity parameter of the medium. Our motivation is to construct an e ective plasmonsoliton interaction, and in the rst order approximation, we obtain the nonlinear coupled oscillator equations from Ref. [35]: cp + 2 pcp = q(jcsj)cs; cs + 2 s (jcsj)cs = q(jcsj)cp: (2.36) where the derivatives are with respect to the propagation parameter z and q(jcsj) is the coupling function which is the overlap of the tails of the plasmon and soliton waves in the area between metal and dielectric. The rst equation implies that, the righthand side represents an external source, which excites surface plasmons at the metaldielectric interface, and due to weakcoupling approximation it is equal to the soliton eld at the metal surface, q (jcsj) sjx=0 ' exp(ô€€€d p =2jcsj). We need to remind that, in the zeroorder approximation, surface plasmon and soliton amplitudes are independent, because coupling between them is assumed to be zero. However, in the rstorder approximation, they become linearly coupled due to the spatial overlapping of the wave elds of surface plasmon and soliton. These coupled oscillator equations are similar to the system of two weakly coupled nonlinear waveguides. However, there are signi cant di erences between two systems. In the plasmon  soliton coupled system, only one subsystem is nonlinear, while the other is linear. Furthermore, coupling between surface plasmon and soliton is not constant, but soliton dependent. The coupled oscillator equations can be reduced to rst order di erential equa tions by making the substitution cp;s = Cp;seiz and by adapting the slowly varying amplitude approximation, hence we arrive, ô€€€i 0 @ Cp Cs 1 A 0 = 0 @ p ô€€€q(jCsj)=2 ô€€€q(jCsj)=2 s(jCsj) 1 A 0 @ Cp Cs 1 A; (2.37) where Cp;s are the zdependent plasmon and soliton amplitudes, which satisfy the nonlinear Schr odinger equation with a nondiagonal Hamiltonian. p 1 and s = 24 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.11: Hyperbolic twolevel system. jCsj2=4 1 are small deviations of p and s such that p p ô€€€ 1 1 and s s ô€€€ 1 1. The second inequality points the weakness of nonlinearity. 2.2.2 Hyperbolic Modulation to the Model One of the objectives of this study is to obtain a LandauZener type of transition, so we should con gure our system such that, surface plasmon and soliton are uncoupled rst, then, when they enter an e ective interaction area, coupling between them manifests itself via the evanescent tails of the waves. After they leave the e ective interaction area, coupling vanishes, and we again have a system of two uncoupled waveguides. The geometry of this e ective interaction area should be arranged such that, the resonant transfer of energy between soliton and surface plasmon can be observed at maximum e ciency. The optimum candidate for such a geometry is the one, in which the distance between metal surface and nonlinearity is a hyperbolic function of 0z0. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 25 In this modulated geometry the coupling function q(jCsj) can be represented as: q(jCsj) ' jCsjexp(ô€€€ p d2 + z2 2 p =2jCsj); (2.38) where is the parameter of the hyperbolic trajectory, (i.e., ber), followed by the soliton (Fig. 2.11). The dashed lines in Fig. 2.11 represent the asymptotic line and c is the angle between the asymptotic line and the z axis. is de ned as the tangent of this critical angle, i.e. = tan c. Here, p d2 + z2 2 is the distance between the metal surface and the ber de ned by the general hyperbola equation: x2 d2 ô€€€ z2 (d= )2 = 1 (2.39) Since the weakcoupling approximation fails at small soliton amplitudes, jCsj must be inserted before the exponential term to hold the approximation [38]. To draw an analogy to the Josephson junction dynamics, we introduce 'fractional population imbalance' and 'relative phase di erence', respectively Z = jCsj2 ô€€€ jCpj2 jCsj2 + jCpj2 (2.40) = s ô€€€ p (2.41) by applying the substitution Cs;p = jCs;pjei s;p [12], [36], [37]. Using this substitution and the fact that jCsj2 + jCpj2 = N (N is a normalized constant for isolated system and equal to 1), Eq. (2.3) becomes _Z = ô€€€q(Z) p 1 ô€€€ Z2 sin ; (2.42) _ = Z + E + q(Z)Z cos p 1 ô€€€ Z2 : (2.43) Here the parameters is the nonlinearity (i.e., the soliton strength) with = =8, and E parametrizes the asymmetry between the soliton and surface plasmon states occupied by the photons with E = ô€€€ p. We can now express the coupling parameter as a function of Z q(Z) ' r 1 + Z 2 eô€€€ p d2+z2 2 p 2 (1+Z): (2.44) 26 Chapter 2: Single Josephson Junction in a MetalDielectric Interface The range of the parameters in our system is set by the insight of physical cir cumstances. is investigated between 0.01 and 0.1, because the nonlinearity should be weak by de nition s = jCsj2=4 1, E is investigated between 0.04 and 0.04, because p 1, and the range of values allow only this range for E. In a recent study [39], a di erent analytical formulation of the coupling function is stated, revealing that, the equation system is not symmetric in the coupling function, hence two seperate coupling functions are de ned q and q, implying that, a strong soliton drives a weak surface plasmon at a rate q, whereas a strong surface plasmon drives a weak soliton at a much smaller rate q jCsj and q=q 10ô€€€3 [40]. By taking the new considerations into account, it is expected that the new dynamic equations should be di erent from Eq. (2.42) and Eq. (2.43). However, the equations cannot be written in Z and formation, because the presence of two seperate coupling functions breaks the isolation of the system and the equality jCsj2 + jCpj2 = N is no longer obeyed. The reason for this violation is that, the change in total surface plasmon and soliton amplitudes is di erent from zero, since q 6= q in _N = ( q ô€€€ q)CsCpsin . In the following sections, we will investigate the system, which is symmetric in coupling function (q ' q) described by Eq. (2.42)  (2.43). 2.2.3 Analogy to LandauZener [LZ] and RosenZener [RZ] Transitions The original LandauZener model [20],[21] contains a constant coupling coe cient and a level seperation which changes adiabatically with time. The optical analog of the problem is discussed in [28]. In a previous study [29], [30], coupling coe cient has been modi ed such that, the coupling is e ective only for a nite duration and then the transition dynamics have been investigated. The analog of the level seperation in the plasmonsoliton system is the asymmetry between surface plasmon and soliton states ( E), which is a constant coe cient. However the time dependent variable in this system is the coupling function. In [35], a 100% conversion of energy from soliton to plasmon channel could not be achieved, because the adiabadicity of the system fails due to the nonlinear e ects and the system undergoes diabatic evolution, which Chapter 2: Single Josephson Junction in a MetalDielectric Interface 27 causes the energy to be stored in the soliton channel. The reason for this failure is that, the coupling parameter is a function of soliton amplitude, which changes swiftly. However, with the hyperbolic modi cation the velocity of the coupling function can be slowed down, hence almost a full conversion of energy from soliton to plasmon channel can be obtained. In the light of these results, it can be said that, the hyper bolic plasmonsoliton system might be a proper host to LandauZener tunneling. In order to understand the concept of adiabatic transitions which result in LandauZener tunneling, the following system should be analyzed. The socalled nonlinear LandauZener model consists of two levels, in which the energy levels depends on populations of the condensates trapped in the wells. The model can be identi ed by the following Hamiltonian: H( ) = 0 @ 2 + C 2 (jbj2 ô€€€ jaj2) V 2 V 2 ô€€€ 2 ô€€€ C 2 (jbj2 ô€€€ jaj2) 1 A (2.45) where population amplitudes are represented as a and b, respectively. V is the cou pling constant, is the energy di erence, and C is the nonlinear parameter. In BEC trapped wells, the nonlinearity identi es, how much the level energies depend on the populations. Total population and total probability is held constant, jaj2 + jbj2 = 1. The eigenvalues of this Hamiltonian for two di erent nonlinearity values are given in Fig. 2.12. For the weak nonlinearity, C = 0:1, there are two eigenvalues and the system does not experience any unexpected dynamical behaviour. However, for the strong nonlinearity, C = 0:4, two more eigenvalues appear, resulting a loop at the top of the lower level. When a quantum state moves in the lower level until the right end of the loop it has to jump to either upper or lower levels, since there is no further way to move. This jump between states points out that, nonlinear LandauZener tunneling is nonzero in the adiabatic limit. Further analysis on this system can be made by investigating the xed points of the Hamiltonian [31]. First, Eq.2.45 can be converted into a classical Hamiltonian: H(s; ; ) = C 2 s2 + s ô€€€ V p 1 ô€€€ s2cos : (2.46) 28 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.12: Eigenenergies of the Hamiltonian (Eq. 2.45) [31]. a) c = 0.1 and v = 0.2, b) c = 0.4 and v = 0.2. Here s and can be treated as conjugate variables and obey the canonical Hamilton equations: s_ = ô€€€ @H @ ; (2.47) and _ = @H @s : (2.48) Fixed points of the system can be found by setting s_ and _ equal to zero: = 0; or ; (2.49) + Cs + V s p 1 ô€€€ s2 cos = 0: (2.50) The eigenenergies of the system can be found by substituting the xed points into the classical Hamiltonian. This shows a direct correlation between gures 2.12 and 2.13. The loop structure in Fig. 2.12b is caused by the emergence of two additional xed points in Fig. 2.13b which is caused by the strong nonlinearity. The problems with the LandauZener model can be overcome by introducing another tunneling model, which is the RosenZener model. The _Z and _ equations allow us to investigate Chapter 2: Single Josephson Junction in a MetalDielectric Interface 29 Figure 2.13: Fixed points associated with the eigenenergies in Fig. 2.12 [31]. a) c/v = 0.5, b) c/v = 2. the dynamics of RosenZener transition in a nonlinear twolevel system, where the asymmetry between the surface plasmon and soliton states, E, is xed, and the coupling between two channels, q(Z), is a time dependent exponential function. This is a much more closer analogy than the LZ model as far as the mathematical modeling is concerned. A similar study has been done with Bose Einstein condensates [32] by constructing the e ective classical Hamiltonian, that describes the dynamic properties of nonlinear quantum RosenZener system. Although our system cannot be cast into a classical hamiltonian, because the driving parameter is part of the coupling function, Eq. (2.42) and Eq. (2.43) are fully capable of describing the dynamical properties of plasmonsoliton interaction. 2.3 Numerical Analysis In this thesis study, our main objective is to obtain a full population transfer be tween soliton and surface plasmon channels. With the hyperbolic modi cation of the coupling function we implemented in the previous section, we observe that such a transition occur in the neighbourhood of the minimum distance between the metal surface and the nonlinearity. Since the distance is increasing hyperbolically, outside of this e ective interaction area the coupling between soliton and surface plasmon is 30 Chapter 2: Single Josephson Junction in a MetalDielectric Interface negligble. As shown in the Fig. 2.14(a) at certain parameter values a nearly ideal transition between soliton and surface plasmon amplitudes is observed. Figure 2.14: (a) Solid line shows the transition from soliton to surface plasmon am plitude, dashed lines show the reverse transition from surface plasmon to soliton amplitude. = 0:63, d = 5:7, = 0:04, and E = 0:02: (b) Solid line shows the plot of coupling function 'q(Z)' vs. z of the rst transition, and the dashed lines show the plot of 'q(Z)' vs. z of the reverse transition. When the system starts with Zô€€€1 1 , and it approaches 1 implying that, the optical soliton propagating inside the ber in the absence of surface plasmon enters the e ective interaction region, where the population change occurs, it transfers almost all of its energy to the surface plasmon emerging on the metal surface, and the surface plasmon continues propagating in the absence of optical soliton. Since the dissipation in the system is negligible, and the system is symmetric in coupling function (q ' q), total soliton and plasmon amplitudes are conserved as expected from _N = 0. This population transfer is reversible and symmetrical. In the case of Zô€€€1 ô€€€1, it approaches 1 after the transition. The initial Z values can not be taken exactly 1 or 1, because that would make the denominator term zero in Eq. (2.43). Beyond this reason, there is another consideration, why Z should not be chosen 1. It implies that the initial soliton amplitude is zero, which makes the coupling function q(Z) = 0 and Chapter 2: Single Josephson Junction in a MetalDielectric Interface 31 in that case, no transition would occur. Figure 2.14(b) shows the plot of q(Z) vs. z for two di erent cases, and the di er ence can be seen clearly between two coupling parameter plots. In the rst case, where the soliton starts to propagate in the absence of surface plasmon, the rate of change of q(Z) with respect to z is greater than the second case, where the surface plasmon starts to propagate in the absence of soliton. After the transition, the rate of change of q(Z) with respect to z is less than the second case, as expected. This is caused by the direct dependency of coupling function 'q' to the soliton amplitude jCsj. An asymmetrized generalization of the RosenZener model with a similar coupling func tion has been investigated in a previous study [41]. Although this full population transfer is our desired result, some further analysis should be followed, since there are six parameters ( ; d; ; E;Z0, and 0), that e ects the outcome. First we discuss the e ects of d and . Figure 2.15: Contour plot of nal Z values with respect to and d for di erent values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (a). Here E is taken 0:02. 32 Chapter 2: Single Josephson Junction in a MetalDielectric Interface The panels in Fig. 2.15 show the nal values of Z as a function of d and for four di erent values. From these results, we try to determine the optimum system parameters, which yield maximum transition from soliton to surface plasmon. Here E is kept constant at 0:02. The darker territories correspond to strong transitions. A long, thin and dark slit can be seen on each panel, which indicates the optimum parameters for a successful transition. The transition gets weaker when the position on the d plane moves away from the slit, and after some point no transition occurs at all (wide and light area). The reason for this lack of transition is the reverse proportionality of d and to the q(Z) itself. In order to keep the coupling parameter above some threshold value, which would cause a successful transition, we should investigate the area where the Z value is close to 1. It can be seen in panel (a) that, the optimum d and values are 5:7 and 0:63, respectively. The meaning of being small is crucial, because Eq. (2.37) is derived from 1D Maxwell's equations, and in our system we assumed that, the hyperbolicity in the ber is small enough to keep the paraxial approximation valid. For the sake of the validity of our calculations, we do not allow be greater than 0.7, so our modi cation to the coupling parameter does not cause the system to lose its physical meaning. In Fig. 2.16, the error percentage  comparison can be seen. The paraxial Figure 2.16: Error percentage of the values. Red line shows the error. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 33 approximation suggests: tan( ) . is identi ed as the tangent of the angle between z axis and the asymptotic line, called c. Hence, tan = c: (2.51) So the red line in Fig. 2.16 represents the deviation from the approximation, and it is de ned as the di erence between the green and blue lines. The panels in Fig. 2.17 reveal the role of nonlinearity and asymmetry in plasmon soliton coupling. The dark slit moves left in the former, as increases pointing out that, strong nonlinearity requires closer distance, in order to obtain a successful transition. In the latter, the slit expands, when E decreases, which gives us a wider area of transitions. This result comes from the de niton of E: E = ô€€€ p = s 2jCsj2 ô€€€ p (2.52) E can be de ned as the asymmetry between soliton and plasmon wave vectors. As E increases, soliton channel takes over the dominant role of the coupling. Figure 2.18 shows the wavefunction pro les of surface plasmon and optical soliton. It is shown that, soliton initially propagating on the curved ber transfers almost all of its energy to the surface plasmon. The exponent of the surface plasmon pro le on the metal (x < 0) decays away from the metaldielectric interface quicker than the exponent on the dielectric, because of the small skin depth of the metal , which is taken 0:1. As discussed earlier, the decay in the soliton amplitude is greater than the increase in surface plasmon amplitude, which is caused by the 'jCsj' dependency of the coupling parameter. Figure 2.18 also shows the wavefunction pro les of surface plasmon and optical soliton in the case that, surface plasmon propagates initially in the absence of soliton, and transfers almost all of its energy to soliton. The increase in the soliton amplitude is greater than the decrease in the surface plasmon amplitude as expected. The tail of the newly forming soliton wave is seen to be on the metal surface, where the distance between the metal and the ber is at its minimum (around z = 0), which may di er 34 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.17: Contour plot of nal Z values with respect to and d for di erent E values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (d). Here is taken 0:04 respectively. by the choice of the d value. In both cases, the total energy of the surface plasmon and soliton is normalized to 1. The interaction between surface plasmon and optical soliton with a hyperbolic distance between them gives rise to other signi cant results. Changing the system geometry and the strength of the nonlinearity allow us to manipulate the energy transfer between surface plasmon and soliton. Fig. 2.19 shows some of the signi cant transitions under di erent and values. We can divide the energy of the soliton propagating in the absence of the surface plasmon equally between them, and similarly, when we have both soliton and surface plasmon propagating initially, we can manipulate the total energy to form either sur face plasmon or optical soliton only. In the case, where is 0.25, we have subsequent transitions between soliton and surface plasmon. First, soliton transfers almost all of its energy to surface plasmon, then surface plasmon transfers all the energy back Chapter 2: Single Josephson Junction in a MetalDielectric Interface 35 Figure 2.18: (a) Propagation of surface plasmon and optical soliton wavefunctions. Optical soliton is propagating in the absence of surface plasmon initially and after the transition it transfers almost all of its energy to surface plasmon on the metal surface. Here is 0.63, is 0.04, E is 0.02, and d is 5.7. (b) Propagation of surface plasmon and optical soliton wavefunctions. Surface plasmon is propagating in the absence of optical soliton initially and after the transition it transfers almost all of its energy to optical soliton on the ber. The same parameter values are used. to soliton. These subsequent transitions show similar properties of localized surface plasmons. Localized surface plasmons are nonpropagating excitations of the conduction elec trons of subwavelength metallic nanostructures coupled to the electromagnetic eld [5]. These excitations emerge inherently from metallic nanoparticles of dimensions below 100 nm in an oscillating electromagnetic eld. Applying an e ective restoring force by the curved surface of the particle on the driven electrons generates localize surface plasmon resonance, which leads to a eld enhancement in the near eld zone outside the particle. In our system, the small area around the minimum distance from the Kerr nonlinearity to the metal surface can be considered as the nanoparticle men tioned above. Since the nonlinearity in the dielectric is structured as a curved path characterized by the parameter , and this area is in nanometer scale, our system can be counted as a proper host to localized surface plasmons. The wavefunction pro les of surface plasmon and soliton of these four cases can be seen in Fig. 2.20. The distance between nonlinearity and the metal surface d can be arranged as 36 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.19: Signi cant transitions between surface plasmon and soliton under dif ferent d, and values. (a) shows the case that soliton is propagating initially in the absence of surface plasmon, and its energy is divided equally between soliton and surface plasmon. (b) shows the case that soliton transfers almost all of its energy to surface plasmon and then surface plasmon transfers its energy back to soliton. (c) and (d) show the case that in the presence of equal surface plasmon and soliton, the energy can be manipulated to be transferred to surface plasmon or soliton, respectively. desired in order to obtain di erent signi cant results as shown in the gure. The change of d could result in outranging the system from 100 nm scale. In this case, the quasistatic approximation breaks down due to retardation e ects, however localized surface plasmon resonance can still be obtained with damping according to the Mie Theory [5], which will not be discussed here. 2.4 Stability Analysis 2.4.1 E ect of relative phase di erence Up to this point, the relative phase di erence ' ' is chosen zero in our calculations, indicating that, soliton and surface plasmon are propagating in phase. However, our system allows the outcome of the transition to be di erent and signi cant, when soliton and surface plasmon are travelling out of phase. Our previous examples of transitions are immune to e ects of phase change, because the absence of surface Chapter 2: Single Josephson Junction in a MetalDielectric Interface 37 Figure 2.20: The wavefunction pro les of surface plasmon and optical soliton of the same four cases. plasmon or soliton means that, the relative phase between them is irrelevant. However, when initial Z value is di erent from 1 or ô€€€1, we observe that the system is highly sensitive to initial relative phase value. Fig. 2.21 shows the e ect of initial relative phase to di erent initial Z values. The sensibility of the system to the initial relative phase di erence can be used to manipulate the system to give desired outcomes. Starting with equal surface plasmon and soliton amplitudes, it can be arranged to transfer all the energy of the system either to surface plasmon or soliton only. Figure 2.22 shows manipulation of energy to one direction by changing only the initial relative phase di erence. The e ect of relative phase di erence can be best perceived once the phasespace analysis of the full population transfer is discussed [18], [19]. In Fig. 2.23, a compre hensive analysis of the evolution of the population imbalance and the phase di erence is shown. In the bosonic Josepshon junction models the phasespace of the system changes adiabatically, since they change the nonlinearity parameter with constant velocity. This repeated change in the system results in a repeated deformation in the phasespace as expected. However the nonlinearity parameter is determined by the material used as the dielectric in the plasmonic Josephson junction system, therefore, 38 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.21: The e ect of di erent initial phase di erences. Solid red line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 1. Dashed lines show the nal Z values for every single di erent 0 value when the system starts with Z = 0. Here the full transition parameter values are used; d = 5:7, = 0:63, = 0:04, and E = 0:02. it cannot be changed externally along the propagation. In this case, the phasespace of the system would not undergo a deformation, ergo the movement of the population imbalance would be restricted by the current phasespace trajectories indicating that, a LandauZener transition would not manifest itself. In order to overcome this prob lem we modi ed our system spatially, which was the main motivation of this work. When the distance between metal surface and the Kerr nonlinearity decreases adia batically to the minimum distance d and then, starts to increase, in other words, when z changes, the phasespace trajectories also change, and the expected deformations manifest themselves under the control of adiabatic parameter . On every trajec tory the transition probability can be determined explicitly. In Fig. 2.23 some of the signi cant trajectories are shown. Comparing this gure with Fig. 2.14 concurrently would give a good insight to the interpretation of these trajectories. Panel (a) in Fig. 2.23 corresponds to the initial state of the system, z = ô€€€100. The red dot represents the system with Z = 1 and = 0, which means the whole energy is in the soliton channel, and the phase di erence between soliton and surface plasmon Chapter 2: Single Josephson Junction in a MetalDielectric Interface 39 Figure 2.22: (a) Solid line shows the case when 0 = 0:2, in which all the energy is transfered to soliton. Dashed lines show the case when 0 = 0:81, in which almost all the energy is transfered to surface plasmon. Here d = 5:6, = 0:4, = 0:04, and E = 0:02. (b) Solid red line shows the nal Z values when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values when the system starts with Zô€€€1 1. Dashed lines show the nal Z values when the system starts with Z = 0. is zero. When the soliton propagates until z = ô€€€30 the trajectory is still une ected by the geometry of the nonlinearity, so the interaction is still negligible. When the soliton enters the e ective interaction region, z = ô€€€13, a xed point emerges at Z = 1; = which causes a closed trajectory around itself, and surface plasmon soliton coupling falls in to this trajectory and change its course. When z increases, this xed point starts to move downwards along with its trajectories. Surface plasmon soliton coupling has no choice but to follow these trajectories, and ends up trapped by this xed point at Z = ô€€€1; = , which corresponds to transfer of energy to plasmon channel completely. A similar transition can be observed when the system starts with surface plasmon only. The green dot starts at Z = ô€€€1; = 0 and follows its current trajectory. The xed point around Z = ô€€€0:5 a ects its direction, and leads it to a stationarystate at Z = 1. The change in the relative phase can also be clearly seen in Fig. 2.24. Panel (a) shows the phase di erences of the green dot with di erent initial phase di erence 40 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.23: The evolution of the phasespace trajectories of the system as z changes adiabatically. The red dots indicate the motion of the population imbalance in phase space when Zô€€€1 1. The green dot indicates the motion when Zô€€€1 ô€€€1 Here the parameters are the same as in Fig. 2.14. values. Depending on the choice of the initial value, there are two di erent trajecto ries, one of them tends to move left in the phasespace, and the other tends to move right. Whichever the choice might be, the phase di erence increases constantly at the stationary state, indicating the coupling moves constantly right on the uppermost trajectory, since there isn't any xed points emerging at the top. Although the phase di erence increases, the population imbalance reaches its nal value, Z = 1. Panel (c) in Fig. 2.24 shows a similar phase di erence pattern for the red dot. Instead of staying on a stationary state, in which the phase di erence increases with constant velocity, this coupling is trapped at a xed point, and the change in the relative phase is quite small. Panel (b) shows the behaviour of the relative phase, when the system starts from Z = 0. As can be seen from the gure, the nal result is more sensitive to the initial choice of , because population of one of the channels is not zero and coupling is more exible. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 41 Figure 2.24: Initial relative phase di erence vs. z graph. (a) Z = ô€€€1, (b) Z = 0, (c) Z = 1. These two transitions are a clear presentation of the jump from one state to another in the metaldielectricKerr system, which leads us to make a comparison with the LandauZener tunneling [20],[21]. The phasespace analysis of the system with parameters in Fig. 2.22 is also in sightful. The route of the population imbalance in phasespace is shown in Fig. 2.25, such that di erent initial relative phase di erences lead the system to two di erent steadystate solutions. 2.4.2 Fixed Point Analysis The trajectory of the system in the phasespace is radically e ected by the xed points of the system, as seen in the gures 2.23 and 2.25. Since the distance between metal surface and the nonlinearity in the dielectric is not constant, but a function of , it can be said that, the geometry of the system is changing slowly. This geometry change causes a continuous and regular deformation in the phasespace of the system. This is another way of saying that, the xed points can move, disappear, and/or reappear in the e ective interaction region between metal and nonlinearity. The evolution of xed points of the system plays crucial role in the surface plasmon  optical soliton interaction, therefore attention should be given to the analysis of xed points. Fixed points of the system can be found numerically by setting the main dynamic equations equal to zero: _Z = ô€€€q(Z) p 1 ô€€€ Z2sin = 0; (2.53) 42 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.25: The evolution of the phasespace trajectories of the system with equal initial amplitude of surface plasmon and optical soliton as z changes adiabatically. The red dot indicate the motion of the population imbalance in phasespace with 0 = 0:2, and green squares indicate the motion of the population imbalance in phasespace with 0 = 0:81. The population imbalance changes dramatically from 0 to 1 (red dot) or from 0 to 1 (green dot). Here the parameters are d = 5:6, = 0:4, = 0:04, and E = 0:02. _ = Z + E + q(Z)Z p 1 ô€€€ Z2 cos = 0: (2.54) It is easier to solve the rst equation above as sin = 0 would give the solution. q(Z) shouldn't be zero, otherwise no interaction would occur between surface plasmon and soliton, and Z cannot be 1 or 1, which would make the denominator in the second equation zero. Hence, we should plug in = 0, and = solutions in the second equation, and solve for 0Z0 solutions. Since q(Z) is z dependent, for every coordinate on the zô€€€ axis, there may be xed Z points for = 0 and = , seperately. The graph of xed points with respect to z coordinate can be seen in Fig. 2.26. These xed points are retrieved from the system which corresponds to the transition in Fig. 2.14. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 43 Figure 2.26: Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 5:7, = 0:63, = 0:04, and E = 0:02 are used. The evolution of xed points can be seen clearly in Fig. 2.26. The rst xed point, ( = 0), moves between Z ' ô€€€0:5 and Z ' ô€€€0:1. The second xed point, ( = ) however shows a di erent behaviour. It rst starts at Z ' ô€€€1, then around z = ô€€€20 it disappears, then around z = ô€€€10 it reappears at Z ' 1. After z = 0, the movement repeats itself symmetrically. As a matter of fact, this symmetrical evolution of xed points is observed for every combination of parameters in the hyberbolical system, because the mathematical modeling of this geometry points that, the distance between metal surface and the nonlinearity depends on z2, so being on the negative side or positive side of the zô€€€axis does not make a di erence. Another correlation of xed points and phasespace is given in Fig. 2.27. These xed points are retrieved from the system which corresponds to Fig. 2.22. This constraint is the main reason, for why this system cannot host an adiabatic transition. In the previous studies, the authors showed that, a xed point can start 44 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.27: Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 2:15, = 0:32, = 0:1, and E = ô€€€0:04 are used. its journey from one end of the phasespace, and evolve such that at the end the xed point reaches the other end of the phasespace. In the meantime BECs trapped in the xed point would move with it, so a slow external change in the Hamiltonian would result in a 100% transition without a character change in eigenvalues. This is called adiabatic transition. In those studies the adiabatic parameter in the system is the energy gap between traps, however in the plasmonic system the adiabatic parameter is taken as the distance between surface plasmon and soliton. Since the distance cannot be negative, fully adiabatic transition cannot be acquired in the hyperbolically modulated Josephson junction. To understand the properties of adiabatic transition, eigenenergies of the system should be analyzed. Eq. (2.37) can be considered as nonlinear Schr odinger equation Chapter 2: Single Josephson Junction in a MetalDielectric Interface 45 and written as : ô€€€i d dt 0 @ Cp Cs 1 A = H(z) 0 @ Cp Cs 1 A; (2.55) where the Hamiltonian is a function of distance between surface plasmon and soliton: H(z) = 0 @ p ô€€€q(jCsj)=2 ô€€€q(jCsj)=2 s(jCsj) 1 A: (2.56) There are two conventional ways of determining the eigenvalues of this Hamil tonian [34]. The rst way is to construct the classical Hamiltonian for the conju gate variables Z and . These variables satisfy the Hamilton's dynamic equations: _Z = ô€€€@H @ , _ = @H @Z . This way canonical equations (2.42) and (2.43) can be derived and the xed points can be found by setting equations ??eq:2.42) and (2.43) equal to zero. Then the results should be substituted into the classical Hamiltonian to obtain the eigenenergies. These eigenenergies can be considered as the energy levels of the sys tem. However this method cannot be applicable to the plasmonic system, because in the systems, which previously used this method Hamiltonian has the physical mean ing of being total energy of the system. Unfortunately, total energy of the plasmonic system cannot be de ned by the classical Hamiltonian, since Cs and Cp are merely the amplitudes of surface plasmon and soliton. The total energy of this system can be calculated by considering the frequencies of the waves and the momentum, that they carry, while propagating. Moreover, the plasmonic system cannot be cast into classical Hamiltonian analytically, therefore this method is proved to be useless for the plasmonic twolevel system. The second method is to present a quartic equation by substituting ô€€€i d dt in Eq. (2.43) with , which is called the eigenenergy of the Hamiltonian. 0 @ Cp Cs 1 A = H(z) 0 @ Cp Cs 1 A; (2.57) After some elaboration the above equation yields the following quartic equation: ô€€€ 2 p 2 + p 2 + p ô€€€ 2 p ô€€€ 2 p + p 2 exp ô€€€2 p d2 + 2z2 q 2 ô€€€ p + p = 0 46 Chapter 2: Single Josephson Junction in a MetalDielectric Interface (2.58) where = 2 p+4 2ô€€€4 pô€€€ + p . Although the analytical solution of this equation cannot be presented, a numerical approach might be o ered, however this approach would be time ine cient due to the complexity of the equation, so another method is presented here. We start with the Eq.2.57, but instead of constructing it into the quartic equation, we divert it into a relation between and Z: = (1 + Z)2 ô€€€ p(1 ô€€€ Z) 2Z (2.59) Afterwards, we nd the xed points by equating the equations (2.42) and (2.43) equal to zero as previously explained and practiced. Then we plug in the xed points that we have found into the Eq. (2.59) as Z. The resulting values of are considered as the eigenenergies of the plasmonic twolevel system. The eigenenergies of the system with parameters used in Fig. 2.26 and the associated xed points are given in Fig. 2.28. Figure 2.28: Eigenenergies and the corresponding xed points. In order to understand, how the system behaves under di erent parameters, Fig. 2.29 could give insight into the e ect of E to the xed points of the system and the associated eigenvalues. Panels a)c) shows the xed points of the system with parameters d = 2:15; = 0:32, and = 0:1, whereas panels d)f) shows the eigenen ergies of the corresponding xed points. It can be seen that, the choice of E might create three di erent regimes. When E is negative, the rst xed point ( = 0) Chapter 2: Single Josephson Junction in a MetalDielectric Interface 47 always takes positive values. The second xed point, however, starts its evolution on the positive side, when the system is decoupled, then jumps into the negative side, and follows the path shown in panel a). When E is positive, the behaviour of xed points is contrary. Not only the rst xed point is on the negative side, its trajectory is upside down as well. Although the change in the E is symmetric, the ip of the trajectories are not. This is caused by the soliton dominancy in the governing equations of the system. An interesting behaviour manifests itself, when E = 0. The rst xed point becomes zero at all points, and the second xed point follows a trajectory from ô€€€1 to 0, and then to ô€€€1 again. At the e ective transition area, both xed points coincides. In panels d)f) associated eigenenergies of these xed points can be seen. E ect of E to these eigenenergies is similar to the e ect to xed points. Figure 2.29: E ect of E. Upper panels are xed points, lower panels are eigenener gies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; = 0:1. Figure 2.30 shows the behaviour of xed points, when the nonlinearity changes. To understand the e ect of nonlinearity is quite important, since it is directly related to the soliton propagation, s = 2 jCsj2. Unlike E, nonlinearity does not change the pattern of xed point trajectories, but slowly reduces the area, in which the second xed point reappears at Z = 1. Weak nonlinearity, see Fig. 2.30 panel d), causes a more symmetric eigenenergy pattern, whereas the strong nonlinear regime, see panel 48 Chapter 2: Single Josephson Junction in a MetalDielectric Interface f), causes an irregular pattern, in which the evolution along z axis follows the green dots, then around z = 20 there is no path to follow but to jump to upper or lower states. This behaviour, however, is not concluded with an adiabatic transfer of energy, because the system is symmetric with respect to z = 0, so when the evolution along zaxis reaches another dead end, it jumps one of the lower states, which coincides eventually at the value that is exactly equal its inital value at z = ô€€€100. Figure 2.30: E ect of . Upper panels are xed points, lower panels are eigenenergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; E = 0:02. In order to gain more perception about the e ect of and E, another set of parameters is presented in the following gures. Fig. 2.31 and 2.32 show the xed point and eigenenergy plots of the system in Fig. 2.19 d), where the system starts with equal soliton and surface plasmon amplitude, and after the transition all of the population is canalized to the soliton channel. The parameters are d = 14; = 0:297; = 0:02, and E = 0:03. Also note that, the parameters used in gures 2.29 and 2.30 are the same as the system in Fig. 2.19a, where the initial soliton amplitude is divided equally between soliton and surface plasmon channels after the transition. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 49 Figure 2.31: E ect of E. Upper panels are xed points, lower panels are eigenener gies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; = 0:02. Figure 2.32: E ect of . Upper panels are xed points, lower panels are eigenenergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; E = 0:03. Chapter 3 DOUBLE JOSEPHSON JUNCTION IN A METALDIELECTRICMETAL INTERFACE 3.1 Theoretical Background 3.1.1 Surface Plasmons in Multilayers Surface plasmon excitation on a metaldielectric interface has been discussed throughly in Chapter 2. The key property about this subject is that, when the tail of the evanes cent waves encounters the nonlinearity spatial solitons arise, and these two waves obey coupled oscillator equations. When the tail of the soliton wave encounters the metal surface, surface plasmons arise as well. This mechanism can also occur on a three level system, metaldielectricmetal interface, so that, surface plasmons on both sides can interact indirectly via optical soliton. The geometrical condition for coupled modes to occur from the interactions of these optical waves is that, the distance between neighboring interfaces should be comparable to the evanescent tails of the waves. To investigate the general features of coupled surface plasmonsoliton modes, the system in Fig. 3.1 should be analyzed [5], [6]. Surface plasmons occur only in TM modes, so the eld components of TM modes will be discussed. The electric and magnetic elds for z > a region are Hy(x) = Aei zeô€€€k3x; (3.1) Ez(x) = iA 1 ! 0 3 k3ei zeô€€€k3x; (3.2) Ex = ô€€€A ! 0 3 ei zeô€€€k3x; (3.3) for x < ô€€€a region Hy(x) = Bei zek2x; (3.4) Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 51 Figure 3.1: Illustration of a threelevel system consisting of a middle layer I trapped between two large half spaces II and III. Ez(x) = ô€€€iB 1 ! 0 2 k2ei zek2x; (3.5) Ex = ô€€€B ! 0 2 ei zek2x: (3.6) In the center region ô€€€a < x < a, the bottom and top modes couple, yielding Hy = Cei zek1x + Dei zeô€€€k1x; (3.7) Ez = ô€€€iC 1 ! 0 1 k1ei zek1x + iD 1 ! 0 1 k1ei zek1x; (3.8) Ex = C ! 0 1 k1ei zek1x + D ! 0 1 k1ei zek1x: (3.9) The boundary condition requires continuity of Hy and Ez at x = a, which leads to Aeô€€€k3a = Cek1a + Deô€€€k1a; (3.10) A 3 k3eô€€€k3a = ô€€€ C 1 k1ek1a + D 1 k1ek1a; (3.11) and at x = ô€€€a Beô€€€k2a = Ceô€€€k1a + Dek1a; (3.12) B 2 k2eô€€€k2a = ô€€€ C 1 k1eô€€€k1a + D 1 k1ek1a: (3.13) 52 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface We are left with a linear system, that consist of four coupled equations. The solution of these coupled equations comes with Hy ful lling the wave equation in the three speci c regions, via k2 i = 2 ô€€€ k2 0 i (3.14) for i = 1; 2; 3: The desired result of this analytical calculation can be achieved by solv ing this system of linear equations, which is an implicit expression for the dispersion relation relating and ! via eô€€€4k1a = k1= 1 + k2= 2 k1= 1 ô€€€ k2= 2 k1= 1 + k3= 3 k1= 1 ô€€€ k3= 3 : (3.15) In the special case of 2 = 3 and k2 = k3, the dispersion relation can be divided into a couple of equations tanh(k1a) = ô€€€ k2 1 k1 2 ; (3.16) tanh(k1a) = ô€€€ k1 2 k2 1 : (3.17) These equations can be applied to metal/dielectric/Kerr/dielectric/metal struc tures to investigate the surface plasmonsolitonsurface plasmon interaction, which forms a double Josephson junction. 3.1.2 BoseEinstein Condensates in TripleWell Traps BECs in doublewell structures have been discussed previously. The unique proper ties of BEC systems could earn advantages over superconducting Josephson junctions such as the possibility to investigate the nonlinear e ects in doublewell, or the con trol of particle number as it is impracticable in nonlinear waveguide systems. With these advantages many physical phenomena can be investigated such as LandauZener tunneling and macroscopic selftrapping. These phenomena could also be observed in triplewell systems, which have two structural types: chainshaped and ringshaped potentials. In the former, the wells are positioned alongside so the rst and the third well does not interact each other directly [53]. In the latter, the wells are positioned Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 53 on a triangle, so that all wells have a direct interaction. The ringshaped model has been investigated in detail [54], the xed points and the associated eigenstates have been found, and the phasespace of the system have been discussed through Poincare maps to determine chaos in the system. 3.2 Surface PlasmonSpatial Soliton Coupling in Multilayer Parallel Systems 3.2.1 Model Description Figure 3.2: Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. Single Josephson junction model between surface plasmon and soliton can be extended to a threelevel system by adding another metaldielectric interface to the 54 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface other side of the nonlinearity, which forms a double Josephson junction. In this way, the soliton in the middle can act like a bridge between two surface plasmons, resulting in rich dynamical results. The distance from the rst metal to the nonlinearity in the dielectric is held constant at d1, hence 1 = 0. The distance from the second metal to the nonlinearity is taken as both a parallel and a hyperbolical function of 2, and the minimum distance between the two layers is d2, as shown in Fig. 3.2. Both cases will be investigated in detail. In this threelevel system, two di erent coupling functions are de ned correspond ing two di erent surface plasmonspatial soliton interactions: q1(jCsj) = jCsjeô€€€d1 p =2jCsj; (3.18) q2(jCsj) = jCsjeô€€€d2 p =2jCsj: (3.19) The coupled oscillator equations, Eq. 2.36, previously used in twolayer system need to be adapted to threelayer system. The new equations are c 00 p1 + 2 p1cp1 = q1(jcsj)cs (3.20) c 00 p2 + 2 p2cp2 = q2(jcsj)cs (3.21) c 00 s + 2 s cs = q1(jcsj)cp1 + q2(jcsj)cp2 (3.22) Making the substitutions cp1 = Cp1eiz, cp2 = Cp2eiz, and cs = Cseiz, the equations yields the following nonlinear Schr odinger equation: ô€€€i 0 BBB@ _C p1 _C s _C p2 1 CCCA = 0 BBB@ p1 ô€€€q1 2 0 ô€€€q1 2 s ô€€€q2 2 0 ô€€€q2 2 p2 1 CCCA 0 BBB@ Cp1 Cs Cp2 1 CCCA (3.23) After the following substitutions is implemented, Cp1 = jCp1 jei p1 (3.24) Cp2 = jCp2 jei p2 (3.25) Cs = jCsjei s (3.26) Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 55 the governing equations yield _C p1 = q1 2 jCsjsin( s ô€€€ p1); (3.27) _C p2 = q2 2 jCsjsin( s ô€€€ p2); (3.28) _C s = q1 2 jCp1 jsin( p1 ô€€€ s) + q2 2 jCp2 jsin( p2 ô€€€ s); (3.29) _ p1 = ô€€€ E1 ô€€€ q1 2 jCsj jCp1 j cos( s ô€€€ p1); (3.30) _ p2 = ô€€€ E2 ô€€€ q2 2 jCsj jCp2 j cos( s ô€€€ p2); (3.31) _ s = 2 jCsj2 ô€€€ q1 2 jCp1 j jCsj cos( p1 ô€€€ s) ô€€€ q2 2 jCp2 j jCsj cos( p2 ô€€€ s): (3.32) Since solving these equations analytically may seem impossible, numerical ap proach is needed. An ODE solver function in MATLAB ode45 has been used to solve these di erential equations. With no dissipation in the system, subsequent transitions will occur between three channels. Depending on the parameter choice di erent proportions of energy transfer will be observed. Some of them are shown in Fig. 3.3 and Fig. 3.4. 3.2.2 Population Trapping In the metal/dielectric/metal interface without a spatial modi cation, a single tran sition between surface plasmon and soliton amplitudes cannot be achieved, since the distance is unchanged, and there is no energy dissipation, so a stationary state after the transition could not be reached. However, the system can be con gured into a signi cant state, in which the soliton amplitude oscillations can be suppressed into a stationary soliton propagation, so that the soliton channel acts like a bridge be tween two surface plasmon channels. The resonant transfer of energy occurs between surface plasmons via the soliton. This e ect can be obtained for a variety of sys tem parameters as well as the initial surface plasmon and the soliton amplitudes. This phenomenon is called population trapping, which is based on a quantum theory of light propagation in two optical waveguides [43],[44]. A common continuum of 56 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.3: Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:04; E1 = 0:03; E2 = ô€€€0:01; Ap0 ' 1; As ' 0 Figure 3.4: Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:05; E1 = 0:03; E2 = 0:03; Ap0 ' 1; As ' 0 modes coupled to both optical channels enables the tunneling between them [45]. For classical light waves, coupledmode equation analysis indicates that, the emergence Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 57 Figure 3.5: Dynamics of amplitudes of soliton and surface plasmons (a) and relative phase di erences (b). Paremeters are d1 = 8; d2 = 4; = 0:06; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:9; As0 0:45. Figure 3.6: Dynamics of amplitudes of soliton and surface plasmons (a) and relative phase di erences (b). Paremeters are d1 = 5:7; d2 = 6:3; = 0:06; E1 = 0:02; E2 = 0:02; Ap10 0:8; As0 0:6. of a trapped state placed in the continuum is caused by Fano interference between di erent light leakage channels. Results show that, plasmonic structures originally designed to mimic the quantum mechanical phenomena may exhibit themselves as proper analogues of their quantum counterparts. Figure 3.5 shows the rst example of population trapping in nondissipative paral lel metaldielectricmetal interface. As mentioned before, amplitudes of surface plas mons exchange their energies repetitively via the soliton bridge. In this con guration the soliton amplitude is arranged so that surface plasmon in the second metal sur 58 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.7: Dynamics of amplitudes of soliton and surface plasmons (a) and rel ative phase di erences (b). Paremeters are d1 = 5:9; d2 = 6:1; = 0:07; E1 = ô€€€0:03; E2 = ô€€€0:03; Ap10 0:7; As0 0:7. face is not excited initially, and the rst surface plasmon can share half of its energy at most. This makes the nodes and the antinodes of the oscillatory waves full and half populated states respectively. The antinodes are the points where a phaseshift occurs as shown in panel b. Figures 3.6 and 3.7 show the trapping of soliton population for completely di erent set of parameters indicating that, this phenomenon is not bounded to a particular system property. The relative phase di erences in these two examples have the same jumping points, whereas the behaviours of the phase evolution are di erent. The only constraint to this behaviour is that, the asymmetry between the soliton and both surface plasmons, E1 and E2, must be equal. An analytical approach might be insightful for explaining this constraint. In order to have a constant soliton amplitude, the derivative of Cs must equal to zero. _C s = q1 2 jCp1 jsin( p1 ô€€€ s) + q2 2 jCp2 jsin( p2 ô€€€ s) = 0: (3.33) There are a few ways to ensure this. Since coupling functions, q1 and q2, and the amplitudes, jCp1 j and jCp2 j, are nonzero, we must have both sinus functions equal to zero, which yields: 1 = 2 = 0; or : (3.34) Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 59 Unfortunately, we know from the phase di erence plots of the trapped systems, that 1 and 2 are not equal to 0 or at all times. This means that, the only way to have _C s equal to zero, is to obey the following condition: ô€€€q1jCp1 jsin( 1) = q2jCp2 jsin( 2) (3.35) It must be noted that, in the trapped soliton system coupling functions, q1 and q2, are constants, since they are only soliton amplitude dependent. If we look into the dynamic equations of phases and considering the above condition, it can be under stood that, the only di erent term in the two equations are E1 and E2 and why they must be equal in order to have a trapped soliton amplitude. _ 1 = 2 jCsj2ô€€€ q1 2 jCp1 j jCsj cos( p1ô€€€ s)ô€€€ q2 2 jCp2 j jCsj cos( p2ô€€€ s)ô€€€ ô€€€ E1ô€€€ q1 2 jCsj jCp1 j cos( sô€€€ p1); (3.36) _ 2 = 2 jCsj2ô€€€ q1 2 jCp1 j jCsj cos( p1ô€€€ s)ô€€€ q2 2 jCp2 j jCsj cos( p2ô€€€ s)ô€€€ ô€€€ E2ô€€€ q2 2 jCsj jCp2 j cos( sô€€€ p2): (3.37) 3.2.3 Classical Collapse and Revival of Coupling Function The investigation of two quantum states, that enable macroscopic nonlinear tunneling between themselves reveals that, the populations of one of the states might reduce to zero from time to time. However, the mean eld approximation is violated, when one of the populations become negligible, since the following is one of the assump tions of mean eld theory [42]: Large number of particles can be treated as a single averaged particle, which reduces a manybody problem to a onebody problem. This di culty can be seen in systems of twolevel atoms, that mimic the properties of a Josephson junction in which the amplitudes of the states oscillate between each other [47]. Throughout the years, many di erent systems have been used as a host for quantum collapserevival phenomenon such as a singlemode resonant eld prop agating in a Kerrlike medium [48], an e ective giant spin model constructed from 60 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface a coupled twomode BoseEinstein condensate, that possesses adiabatic and cyclic timevarying Raman coupling [49], BoseEinstein condensates trapped in a triplewell [50], or quantum bouncing ball on a truncated lattice [51]. In [47], a twodimensional optical lattice lled with BEC is taken as an example of a twolevel system with equal energy, which allows population oscillations, and this system is realized exper imentally to examine its dynamic behaviours. The quantization of the motion yields collapse and revival of these oscillations. The evolution of BEC atoms and quantum collapses can be seen in Fig. 3.8. Figure 3.8: Quantum collapse and revivals can be seen in BEC atoms [47]. In a) population imbalance (Z) of N = 350 BEC atoms can be seen. Most of the population is trapped in of the states. The relative phase of both states are equal. In b) N = 500 BEC atoms can be seen. Both states are equally populated during the collapse. The relative phases are equal. Some regimes of collapses, such as equal population or disbalanced population oscillations can be useful to predict the tunneling properties, since they don't violate the mean eld approximation. So, we shouldn't look for a collapse in the population of one of the states, but the collapse of the oscillations, in other words, the coupling function. However, the quantization of the motion in the above study still restricts us from obtaining collapse of coupling in the plasmonic Josephson junction, therefore a classical approach should be in order. A classical approach has been proposed in [52], where the analysis of wave packets in weakly anharmonic potentials is given. According to this study, when the wave packets propagate along classical trajectories in harmonic potential, they spread along Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 61 the motion, and regain their shape. Introducing a small anharmonicity to the system leads to a reversible quantum dephasing of wave packets, which is called revival. The amplitude of the oscillation dissapears between these revivals, which is called collapse. A realization of this study has been experimented with coherent A1g phonons in Bismuth. The dynamics of the phonon wave packet in a twoatom system is shown in Fig. 3.9. Figure 3.9: Oscillatory part of the z coordinate for di erent absorbed energies n0. Arrows indicate the amplitude collapse of the oscillations [52]. This study shows that, the system size is the key factor, which a ects the be haviour of the excited phonons, that approach the classical behavior swiftly. This behaviour indicates that, quantum e ects are not included in the revival of the os cillations, which is a substantial discovery, because collapse and revival phenomenon was originally considered a pure quantum e ect with no classical analogy [46]. The physical interpretation of this e ect was given as a linear combination of stationary states of a system. These linear combinations are determined as the coherent oscilla tions, that show collapse and revival properties. Furthermore an important natural speci cation of a stationary state is the number of quanta as a result of being a 62 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface quantum mechanical phenomenon. The changeover from quantum systems without a classical counterpart to quantum optical systems, and nally to optophotonic systems with classical e ects urge us to move to the next step: surface plasmonsoliton Josephson junctions. Unlike the quantum collapserevival phenomenon, the plasmonic system does not lose its total energy, however it is the interaction between these three states, ergo, the coupling function, that collapses as shown in gures 3.10  3.18. Figure 3.10: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 Fig. 3.10 shows an obvious collapserevival behaviour. The oscillations of ampli tudes between two surface plasmons and soliton channels manifest periodic patterns, however the nonoscillating parts, i.e., the length of the collapse time, di er from each other. In order to obtain this behaviour the following set of parameters are used: d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01, and E2 = 0:01. Initial surface plasmon amplitude in the rst metal surface carries the whole energy of the system, Ap10 1, therefore As0 0, and Ap20 0. The outcome of the CR behaviour di ers Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 63 Figure 3.11: Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 from the population trapping, that has been discussed in the previous section. In the population trapping phenomenon, oscillations of the soliton amplitude reduces to zero, stabilizing the soliton population permanently, and in this period of time the amplitudes of surface plasmons continue oscillating. However in the CR phenomenon, not only the oscillations of the soliton amplitude, but the soliton population collapses to zero temporarily, and in this temporal interval of time oscillations of surface plas mons collapses to zero as well. The relative phase di erences between surface plasmons and the soliton should also be discussed. Although the e ect of relative phases to the double Josephson junctions will be explained in the following sections, a brief discourse on relative phase di erences in the CRsystems might be insightful. Fig. 3.11 shows the phases of the system, that was explained in Fig. 3.10. As can be understand from the gure, the relative phases between channels also manifest a nonoscillating behaviour during the collapse time. The fact that, both populations and phases of the optical channels 64 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.12: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 do not change their current state, indicates that, the system may fall into one of the xed points. Figure 3.12 also shows a CR behaviour, but with a more regular pattern. The collapse time between periodic islands seems closer to each other than Fig. 3.10. Both the period of those islands, z 1000, and the shape of the envelopes are similar, although system parameters are quite di erent. This is a sign that CR behaviour is not bound to speci c set of parameters, it can be observed in di erent geometry (d, ) and material properties ( , E). The relative phase di erence plot can be seen in Fig. 3.13. We have de ned collapserevival phenomenon in plasmonic Josephson junction as the collapse of the coupling between surface plasmons and the soliton. More to that, interesting patterns of system dynamics, that does not obey the CR behaviour might be worth mentioning. In Fig. 3.14 and 3.15, the oscillations of soliton draws a regular pattern so that, the oscillations of surface plasmons follows a sequential pattern, which creates a sinusoidal envelope. This sinusoidal envelope can also be Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 65 Figure 3.13: Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 seen in Fig. 3.16 and 3.17 with strong soliton oscillations, and a very steep collapse and revival behaviour. And nally, Fig. 3.18 and 3.19 shows an irregular type of CR oscillations. 3.3 Surface PlasmonSpatial Soliton Coupling in Hyperbolically Mod ulated Multilayer Systems The rich dynamic properties of single Josephson junction formed by surface plasmon and spatial soliton previously discussed in Chapter 2 can also be investigated in double Josephson junctions with one of the metal surfaces being spatially modulated. In this system, nonlinearity is kept straight, indicating a parallel junction on one side and a hyperbolic junction on the other side as can be seen in Fig. 3.20. Dynamic evolution of the system can be described with a similar set of equations as the parallel system. _C p1 = q1 2 jCsjsin( s ô€€€ p1); (3.38) 66 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.14: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 Figure 3.15: Dynamics of relative phase di erences between soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 67 Figure 3.16: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 Figure 3.17: Dynamics of relative phase di erences between soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 68 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.18: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 Figure 3.19: Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 69 Figure 3.20: Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. _C p2 = q2 2 jCsjsin( s ô€€€ p2); (3.39) _C s = q1 2 jCp1 jsin( p1 ô€€€ s) + q2 2 jCp2 jsin( p2 ô€€€ s); (3.40) _ p1 = ô€€€ E1 ô€€€ q1 2 jCsj jCp1 j cos( s ô€€€ p1); (3.41) _ p2 = ô€€€ E2 ô€€€ q2 2 jCsj jCp2 j cos( s ô€€€ p2); (3.42) _ s = 2 jCsj2 ô€€€ q1 2 jCp1 j jCsj cos( p1 ô€€€ s) ô€€€ q2 2 jCp2 j jCsj cos( p2 ô€€€ s): (3.43) with a change in the coupling function q1(jCsj) = jCsjeô€€€d1 p =2jCsj; (3.44) q2(jCsj) = jCsjeô€€€ p
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Title  Transition dynamics of single and double Josephson junctions formed by spatially coupled soliton and surface plasmons 
Author  AydÄ±ndoÄŸan, GÃ¼neÅŸ 
Subject  Plasmons (Physics); Josephson junction; Josephson effect; Superconductors; Surface plasmon resonance 
Faculty Advisor  GÃ¼ven, Kaan 
Institute  KoÃ§ University Graduate School of Sciences & Engineering 
Program  Physics 
Physical Description  xv, 100 leaves : illustrations ; 30 cm. 
Place of Publication  Ä°stanbul 
Publisher  KoÃ§ University 
Resource Type  aydindogan_gunes_diss_2017.pdf 
Date  2017 
Collection  KU Theses and Dissertations 
Transcription  Transition dynamics of single and double Josephson junctions formed by spatially coupled soliton and surface plasmons by G une s Ayd ndo gan A Dissertation Submitted to the Graduate School of Sciences and Engineering in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy in Physics June, 2017 Transition dynamics of single and double Josephson junctions formed by spatially coupled soliton and surface plasmons Ko c University Graduate School of Sciences and Engineering This is to certify that I have examined this copy of a doctoral dissertation by G une s Ayd ndo gan and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the nal examining committee have been made. Committee Members: Assoc. Prof. Kaan G uven Prof. Ozg ur M ustecapl glu Prof. Ali Serpeng uzel Assist. Prof. Yasa Ek sio glu Ozok Assoc. Prof. Ahmet Levent Suba s Date: ABSTRACT This thesis work investigates the crossing dynamics of photons between a spa tially coupled copropagating soliton and a surfaceplasmon, which constitute a type of photonic Josephson junction. By introducing modulations to the spatial coupling, the crossing dynamics exhibit features similar as well as di erent to that of nonlinear LandauZener or RosenZener type transitions. The dependence of the coupling to the soliton amplitude provides an inherent dynamic, which may manifest distinct features in the transition characteristics. The dynamics of the system, which is formulated as a Josephson junction, is investigated by introducing fractional population imbalance and the relative phase variables. A full population conversion between the optical soliton and the surface plasmon is achieved. The governing equations of the system represents a set of nonlinear Schr odinger equations, and the eigenvalue analysis of these equations sheds light on the behaviour of the xed points of the system, here with the stability analysis will be investigated. Under a spatial periodic modulation, this type of Josephson junction may exhibit driven resonance states similar to Shapiro resonances. The stability of these resonances will be investigated in the presence of an external periodic eld. The double Josephson junction is formed by coupling the soliton spatially to two surfaceplasmons, which reside on either side of the soliton propagation axis, respectively. This threestate system may provide rich dynamics with features like collapserevival and plasmonplasmon coupling via a frozen soliton state. By introducing spatial modulations, further dynamical e ects will be explored. Heuristic designs are analyzed for sensor applications by investigating realistic mate rials and associated parameters. iii OZETC E Bu tez cal smas nda uzaysal olarak etkile sim i cinde bulunan soliton ve y uzey plaz monlar aras ndaki foton ge ci sleri ve bu ge ci slerin olu sturdu gu Josephson ekleminin di nami gi incelenmi stir. Bu ge ci sin dinami gi uzaysal etkile sime yap lan kipleme sayesinde lineer olmayan LandauZener ve RosenZener ge ci slerine benzer oldu gu kadar bun lardan farkl ozellikler de g ostermektedir. Soliton ve plazmon aras ndaki etkile simin soliton genli gine olan ba g ml l g sa glanan ge ci sin ozelliklerinde belirgin olgular ortaya c karan i csel bir dinamik sa glamaktad r. Fraksiyonel pop ulasyon orans zl g ve ba g l faz terimlerini kullanarak sistemin dinami gi Josephson eklemi olarak ele al nabilir. Bu sayede optik soliton ve y uzey plazmonu aras nda tam bir pop ulasyon transferi ger cekle sebilir. Sistemi y onlendiren esas denklemler do grusal olmayan Schr odinger denklemi ozellikleri g ostermektedir ve Schr odinger denklemlerinin ozde ger analizi plaz monik sistemin sabit noktalar n n bulunmas na s k tutmaktad r. B oylelikle de sis temin denge analizi yap labilmektedir. Uzaysal kipleme periyodik olarak tasarland g nda bu tarzda olan Josephson eklemleri Shapiro c nlamalar nalar na benzeyen kararl yap da c nlama durumlar n n ortaya c kmas na neden olabilmektedir. Harici bir periyo dik alan n varl g nda bu tarz c nlama durumlar n n denge analizi de yap lacakt r. Solitonun kendi y or ungesinin iki taraf nda konumlanan y uzey plazmonlar yla girdi gi uzaysal etkile sim ikili Josephson eklemini olu sturur. Bu u c katmanl sistem c ok u s canlan s, donuk bir soliton arac l g ile plazmonplazmon etkile simi gibi zengin di nami gi olan ozel durumlara neden olabilir. Tekli Josephson eklemindeki gibi uza ysal kiplemelerin de eklenmesiyle bu kiplemenin sistem dinami gi uzerindeki etki leri incelenecektir. Ayr ca uygulanabilirli gi olan ger cek ci materyallerin ve bunlar n olu sturdu gu sistem parametrelerinin incelenmesiyle alg lay c uygulamalar nda kul lan lmak uzere bulgusal tasar mlar n analizi yap lacakt r. iv TABLE OF CONTENTS List of Figures vii Chapter 1: Introduction 1 Chapter 2: Single Josephson Junction in a MetalDielectric Inter face 5 2.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Surface Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Optical Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Josephson Junction . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.4 LandauZener Transition . . . . . . . . . . . . . . . . . . . . . 18 2.1.5 RosenZener Transition . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Surface PlasmonSpatial Soliton Coupling . . . . . . . . . . . . . . . 21 2.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2 Hyperbolic Modulation to the Model . . . . . . . . . . . . . . 24 2.2.3 Analogy to LandauZener [LZ] and RosenZener [RZ] Transitions 26 2.3 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4.1 E ect of relative phase di erence . . . . . . . . . . . . . . . . 36 2.4.2 Fixed Point Analysis . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 50 3.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 50 v 3.1.1 Surface Plasmons in Multilayers . . . . . . . . . . . . . . . . . 50 3.1.2 BoseEinstein Condensates in TripleWell Traps . . . . . . . . 52 3.2 Surface PlasmonSpatial Soliton Coupling in Multilayer Parallel Systems 53 3.2.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.2 Population Trapping . . . . . . . . . . . . . . . . . . . . . . . 55 3.2.3 Classical Collapse and Revival of Coupling Function . . . . . . 59 3.3 Surface PlasmonSpatial Soliton Coupling in Hyperbolically Modulated Multilayer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 4: Shapiro Resonances 77 4.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2 Periodic Modulation to the Model . . . . . . . . . . . . . . . . . . . . 81 Chapter 5: Applications 85 5.1 Surface PlasmonSoliton Coupling in Realistic Structures . . . . . . . 85 5.2 Proposed Instruments Regarding Surface PlasmonSoliton Coupling . 86 Chapter 6: Conclusion 91 Bibliography 93 vi LIST OF FIGURES 2.1 Illustration of a planar waveguide geometry. z is the propagation direction of the waves. . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Geometry for SP propagation at a single interface between a metal and a dielectric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Dispersion relation of SPPs at the interface between a Drude metal with negligible collision frequency and air (gray curves) and silica (black curves) [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Illustration of di erent spatial beam pro les [7]. . . . . . . . . . . . . 11 2.5 The focusing e ect of a simple convex lens. . . . . . . . . . . . . . . . 12 2.6 The shape of the soliton, while propagating with N=1. . . . . . . . . 14 2.7 The basic geometry of a Josephson junction formed between two su perconductors is illustrated. . . . . . . . . . . . . . . . . . . . . . . . 15 2.8 The BEC doublewell trap. E0 1 and E0 2 are the zeropoint energy of each condensate [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.9 Transition probability versus external period under di erent nonlinear ity strengths [32]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.10 Metaldielectricnonlinearity interfaces for plasmonsoliton propagation. 22 2.11 Hyperbolic twolevel system. . . . . . . . . . . . . . . . . . . . . . . . 24 2.12 Eigenenergies of the Hamiltonian (Eq. 2.45) [31]. a) c = 0.1 and v = 0.2, b) c = 0.4 and v = 0.2. . . . . . . . . . . . . . . . . . . . . . . . 28 2.13 Fixed points associated with the eigenenergies in Fig. 2.12 [31]. a) c/v = 0.5, b) c/v = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 vii 2.14 (a) Solid line shows the transition from soliton to surface plasmon am plitude, dashed lines show the reverse transition from surface plasmon to soliton amplitude. = 0:63, d = 5:7, = 0:04, and E = 0:02: (b) Solid line shows the plot of coupling function 'q(Z)' vs. z of the rst transition, and the dashed lines show the plot of 'q(Z)' vs. z of the reverse transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.15 Contour plot of nal Z values with respect to and d for di erent values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (a). Here E is taken 0:02. . . . . . . . . . . . . . . . . . . . . 31 2.16 Error percentage of the values. Red line shows the error. . . . . . . 32 2.17 Contour plot of nal Z values with respect to and d for di erent E values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (d). Here is taken 0:04 respectively. . . . . . . . . . . . . . . 34 2.18 (a) Propagation of surface plasmon and optical soliton wavefunctions. Optical soliton is propagating in the absence of surface plasmon initially and after the transition it transfers almost all of its energy to surface plasmon on the metal surface. Here is 0.63, is 0.04, E is 0.02, and d is 5.7. (b) Propagation of surface plasmon and optical soliton wavefunctions. Surface plasmon is propagating in the absence of optical soliton initially and after the transition it transfers almost all of its energy to optical soliton on the ber. The same parameter values are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 viii 2.19 Signi cant transitions between surface plasmon and soliton under dif ferent d, and values. (a) shows the case that soliton is propagating initially in the absence of surface plasmon, and its energy is divided equally between soliton and surface plasmon. (b) shows the case that soliton transfers almost all of its energy to surface plasmon and then surface plasmon transfers its energy back to soliton. (c) and (d) show the case that in the presence of equal surface plasmon and soliton, the energy can be manipulated to be transferred to surface plasmon or soliton, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.20 The wavefunction pro les of surface plasmon and optical soliton of the same four cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.21 The e ect of di erent initial phase di erences. Solid red line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 1. Dashed lines show the nal Z values for every single di erent 0 value when the system starts with Z = 0. Here the full transition parameter values are used; d = 5:7, = 0:63, = 0:04, and E = 0:02. . . . . . 38 2.22 (a) Solid line shows the case when 0 = 0:2, in which all the energy is transfered to soliton. Dashed lines show the case when 0 = 0:81, in which almost all the energy is transfered to surface plasmon. Here d = 5:6, = 0:4, = 0:04, and E = 0:02. (b) Solid red line shows the nal Z values when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values when the system starts with Zô€€€1 1. Dashed lines show the nal Z values when the system starts with Z = 0. 39 ix 2.23 The evolution of the phasespace trajectories of the system as z changes adiabatically. The red dots indicate the motion of the population im balance in phasespace when Zô€€€1 1. The green dot indicates the motion when Zô€€€1 ô€€€1 Here the parameters are the same as in Fig. 2.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.24 Initial relative phase di erence vs. z graph. (a) Z = ô€€€1, (b) Z = 0, (c) Z = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.25 The evolution of the phasespace trajectories of the system with equal initial amplitude of surface plasmon and optical soliton as z changes adiabatically. The red dot indicate the motion of the population im balance in phasespace with 0 = 0:2, and green squares indicate the motion of the population imbalance in phasespace with 0 = 0:81. The population imbalance changes dramatically from 0 to 1 (red dot) or from 0 to 1 (green dot). Here the parameters are d = 5:6, = 0:4, = 0:04, and E = 0:02. . . . . . . . . . . . . . . . . . . . . . . . . 42 2.26 Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 5:7, = 0:63, = 0:04, and E = 0:02 are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.27 Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 2:15, = 0:32, = 0:1, and E = ô€€€0:04 are used. . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.28 Eigenenergies and the corresponding xed points. . . . . . . . . . . . 46 2.29 E ect of E. Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 x 2.30 E ect of . Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; E = 0:02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.31 E ect of E. Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; = 0:02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.32 E ect of . Upper panels are xed points, lower panels are eigenen ergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; E = 0:03. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.1 Illustration of a threelevel system consisting of a middle layer I trapped between two large half spaces II and III. . . . . . . . . . . . . . . . . 51 3.2 Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3 Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:04; E1 = 0:03; E2 = ô€€€0:01; Ap0 ' 1; As ' 0 . . . . 56 3.4 Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:05; E1 = 0:03; E2 = 0:03; Ap0 ' 1; As ' 0 . . . . . 56 3.5 Dynamics of amplitudes of soliton and surface plasmons (a) and relative phase di erences (b). Paremeters are d1 = 8; d2 = 4; = 0:06; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:9; As0 0:45. . . . . . . . . . . . . . 57 3.6 Dynamics of amplitudes of soliton and surface plasmons (a) and rel ative phase di erences (b). Paremeters are d1 = 5:7; d2 = 6:3; = 0:06; E1 = 0:02; E2 = 0:02; Ap10 0:8; As0 0:6. . . . . . . . . . 57 xi 3.7 Dynamics of amplitudes of soliton and surface plasmons (a) and rel ative phase di erences (b). Paremeters are d1 = 5:9; d2 = 6:1; = 0:07; E1 = ô€€€0:03; E2 = ô€€€0:03; Ap10 0:7; As0 0:7. . . . . . . . 58 3.8 Quantum collapse and revivals can be seen in BEC atoms [47]. In a) population imbalance (Z) of N = 350 BEC atoms can be seen. Most of the population is trapped in of the states. The relative phase of both states are equal. In b) N = 500 BEC atoms can be seen. Both states are equally populated during the collapse. The relative phases are equal. 60 3.9 Oscillatory part of the z coordinate for di erent absorbed energies n0. Arrows indicate the amplitude collapse of the oscillations [52]. . . . . 61 3.10 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 . . . . . . . . . . . . . . . . . . . 62 3.11 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 . . . . . . . 63 3.12 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 . . . . . . . . . . . . . . . . . . . . . 64 3.13 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 . . . . . . . . . . . . . . 65 3.14 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . . . . . . . . . . . . . 66 3.15 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . 66 xii 3.16 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 . . . . . . . . . . . . . . . . 67 3.17 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 . . . . 67 3.18 Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . . . . . . . . . . . 68 3.19 Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 . . . . . . . . . . . . 68 3.20 Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.21 A %100 transfer of energy between two surface plasmon channels. Pa rameters are d1 = 8:21; d2 = 3:79; = 0:6; = 0:07; E1 = 0:02; and E2 = 0:04. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.22 Population trapping could be observed with Z1 0. The parameters are d1 = 6:5; d2 = 5:5; = 0:51; = 0:062; E1 = 0:033; and E2 = ô€€€0:025. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.23 A population trapping could be achieved when the initial soliton ampli tude is di erent than zero. The parameters are d1 = 4:7; d2 = 7:3; = 0:6; = 0:03; E1 = ô€€€0:04; E2 = 0:03; Ap1 = 0:6; As = 0:8. . . . 72 3.24 Fixed points (blue circles are As and green circles are Ap1 points) and corresponding eigenenergies of the system with parameters d1 = 8:21; d2 = 3:79; = 0:6; = 0:07; E1 = 0:02; and E2 = 0:04 for cases 1 = 0; 2 = 0 (a), 1 = 0; 2 = (b), and 1 = ; 2 = (c). 75 xiii 3.25 Fixed points (blue circles are As and green circles are Ap1 points) and corresponding eigenenergies of the system with parameters d1 = 6:5; d2 = 5:5; = 0:51; = 0:062; E1 = 0:033; and E2 = ô€€€0:025 for cases 1 = 0; 2 = 0 (a), 1 = ; 2 = 0 (b), and 1 = ; 2 = (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.26 Fixed points (blue circles are As and green circles are Ap1 points) and corresponding eigenenergies of the system with parameters d1 = 4:7; d2 = 7:3; = 0:6; = 0:03; E1 = ô€€€0:04; E2 = 0:03 for cases 1 = 0; 2 = 0 (a), 1 = ; 2 = 0 (b), and 1 = ; 2 = (c). . . . 76 4.1 Phase space illustration of two di erent pendulums. (a) is = 0, and (b) is = 100. The interaction is = 104, and the driving frequency is = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Population imbalance vs. time when = 104 and = 3x10ô€€€5 with = 100; = 1000. Shapiro resonance can be observed when time averaged population imbalance is trapped around z = 0:1. . . . . . . 80 4.3 Shapiro resonance e ects in weakly coupled BoseEinstein condensates. Idc is the intensity value which is proportional to average population imbalance. is the nonlinearity parameter [37]. . . . . . . . . . . . . 81 4.4 Illustration of the sinusoidally modulated single Josephson junction. . 82 4.5 Shapiro resonance in plasmonic Josephson junction. A drop in the magnitude of average population imbalance can be observed, when the external modulation parameter changes between 10 and 90. System parameters are d = 5; = 0:08; E = 0;A = 0:8;Zô€€€1 = 0:9049: . . . 83 xiv 4.6 Shapiro resonance in plasmonic Josephson junction. Two di erent spikes in the magnitude of average population imbalance can be ob served, when the external modulation parameter changes between 10 and 90. System parameters are d = 3; = 0:07; E = ô€€€0:04;A = 0:6;Zô€€€1 = 0:9049: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.1 Geometry of the fourlayer con guration. . . . . . . . . . . . . . . . . 86 5.2 Refractive index sensor based on single Josephson junction formulation of surface plasmonoptical soliton coupling. . . . . . . . . . . . . . . . 87 5.3 A coupling modulator made of metaldielectricnonlinear dielectric dielectricmetal multilayers. . . . . . . . . . . . . . . . . . . . . . . . 89 5.4 The e ect of d2. The other parameters are d1 = 4:7; = 0:6; = 0:03; E1 = ô€€€0:04; E2 = 0:03; Ap1 = 0:6; As = 0:8. . . . . . . . . 90 xv Chapter 1 INTRODUCTION One of the major problems of today's digital information technologies is to carry the signal between ends of a microprocessor, that is in centimeter scale. However, ultrafast transistors are in nanometer scale, and this causes problems regarding speed in signal processing, when copper wires are to be used. On the other hand, ber optic cables are able to carry data much more e cient than electronic circuits, however combining the two counterparts on the same circuit have some signi cant constraints. The proposed solution is an element, that can carry optical signals in nanomater scale, surface plasmons [1]. The improvements and progresses in science and engineering exhibit renewed interest in surface plasmons, that allow metals to be designed in subwavelength scale. This property of surface plasmons has gotten attention from scientist of di erent disciplines. Manipulating the frequency spectrum of lasers, ultra short pulse generation, signal processing, magnetooptic data storage, solar cells, and biosensors for health studies are main topics of surface plasmons, which are being used actively by researchers [2]. The inception of these studies is the transformation of light into surface plasmons, which enables scientist to suppress and channel light using sub wavelength photonic structures. In this way, surface plasmons can be used as control parameters in circuits instead of light, which causes an enhancement in the non linear phenomena. This new subdivision of photonics is called plasmonics, that o ers to combine the advantage of carrying ultrafast signals and operating on small circuits at the same time. The mathematical formulation of surface plasmon propagation lies back to the Maxwell's equations, so that in order to investigate the properties of surface plasmons in nonlinear media, nonlinear Maxwell's equations for the TM 2 Chapter 1: Introduction waves at the metaldielectric interface should be analyzed. The roots of overcoming the di raction limit by surface plasmon polaritons come from the slow wave properties of surface plasmons. The formation of surface plasmons by the interaction between surface charges and the electromagnetic eld has two corollaries. First, the momentum of surface plasmon, ~kSP, is greater than of a photon with the same frequency, ~k0. The second corollary is that, the evanescent tail of the eld, which is perpendicular to the interface decays exponentially throughout the direction perpendicular to the surface, which is caused by the con ned and nonradiative nature of surface plasmons. Anharmonic electromagnetic elds cause nonlinear e ects in optical systems. This anharmonicity can be expanded as power series resulting in a mixture of incident elds and output elds, that oscillate at frequencies of a superposition of incident elds. Second harmonic generation and third harmonic generation are typical examples of these e ects, that result in frequency conversion. Moreover, the thirdorder e ects possess terms at the incident frequencies, which causes a change in the refractive index of the medium, which is called optical Kerr e ect. The optical properties of Kerr e ect enable to create bistability, furthermore di raction caused by linear e ects balances the bistability caused by nonlinear e ects, producing a new type of optical wave, solitons. An optical soliton is a stable optical wave, that propagates without an aberration. The Kerr e ect caused by selfphase modulation creates a red shift at the evanescent tails of the wave, and a blue shift also occurs at the tails due to dispersion. The combination of these two e ects result in a pulse, that maintains its shape while travelling. There are various application areas in which solitons can be used such as surface waves in shallow water, plasma waves, sound waves in 3He, short optical pulses in bers, and optical spatial solitons [3]. Although the physical systems behind these examples are quite di erent, they all obey the same rules, and they are a conse quence of nonlinear partial di erential equations. Between the various soliton kinds, optical solitons are a widespread research hotspot for the recent years, because of its applicability in laboratory environment and extensive control over the parameters. Chapter 1: Introduction 3 The soliton concept was rst discovered in hydrodynamics in the 19th century, and has been used extensively in various disciplines such as optics, condensed mat ter physics, uid mechanics, particle physics, and even astrophysics. However, the most important contribution of solitons between these elds has been made in opti cal communications. Optical solitons have been studied under spatial and temporal domains, however spatial solitons arouse attention of researchers, because of their highdimensional properties, whereas temporal solitons are basicly onedimensional el ements. Moreover, the rich nonlinear properties of spatial solitons which have found application areas in photorefractive materials, liquid crystals, thermophoresis, and colloidal suspensions are important contributing factors. These various nonlinear ef fects enable scientist to study soliton phenomena more e ciently, since it was thought earlier that, solitons were exact solutions of the cubic nonlinear Sch odinger equation. The earlier studies regarding nonlinear waves in metaldielectric interfaces were tak ing into consideration only the transverse direction, however later studies show that, the temporal e ects should also be taken into account, and so the metaldielectric geometry should be studied with a slot width smaller than the wavelength, and the complicated structure of nonlinear localization requires the consideration of boundary conditions. A novel con guration has been proposed, where surface plasmon guided vertically on the metal surface and a soliton selftrapped by nonlinear Kerr e ect combined to yield a plasmonsoliton coupling driven by a lowpower continuouswave optical source [4]. In this thesis work, we investigate the dynamical properties of optical solitons propagating on a Kerr type of nonlinearity and surface plasmons propagating on metaldielectric interface. Matching of surface plasmon and soliton propagation con stants gives rise to coupling of both waves forming a plasmonic Josephson junction. Nonlinear nature of the optical soliton causes the coupling between two channels to be a function of soliton amplitude, which makes the soliton amplitude a driving pa rameter of the system. The Josephson junction formulation can be set by introducing two variables, which have analogues on bosonic Josephson junctions: fractional pop 4 Chapter 1: Introduction ulation imbalance and relative phase di erence. We then consider the interaction between two channels changing adiabatically, hence the soliton trajectory is modu lated spatially such that, the distance between metal surface and Kerr nonlinearity becomes a hyperbolic function. A full population conversion between surface plasmon and optical soliton can be achieved under certain parameter values, indicating that metal/dielectric/Kerr interface can host a RosenZener type of transition. Stability analysis of the system is also studied by inquiring the xed points of the system. We next investigate the transition dynamics in a threelevel system, in which the nonlin earity is sandwiched between metal/dielectric interfaces, which yields the interaction between two surface plasmons via optical soliton propagating between them. This multilayer interface gives rise to di erent phenomena such as complete population transfer, population trapping, and collapse and revival of the coupling function. The xed points and eigenenergies of the system is also analyzed. The following step is the periodic modulation of the system in order to examine Shapiro resonances, which were originally studied in superconductors and BoseEinstein condensates. Finally, a realistic approach into the materials has been made to limit the parameter values, and a few applications of surface plasmonsoliton coupling have been proposed. Chapter 2 SINGLE JOSEPHSON JUNCTION IN A METALDIELECTRIC INTERFACE 2.1 Theoretical Background 2.1.1 Surface Plasmons Surface plasmons(SPs, also called surface plasmonpolaritons) are coherent elec tron oscillations that propagates at the interface between two materials, where the permittivity changes sign across the interface. The most common candidate for SP ex citation is a metaldielectric interface. These electromagnetic surface waves originates from the coupling of the metal's electron plasma with photons. SPs are evanescently con ned in the transverse direction, and they have the ability to propagate within small sized materials. These type of materials have the wavelength much smaller than the wavelength of the surface plasmon. Applying Maxwells equations (Eq.'s 2.1  2.4) to the at interface between a metal and a dielectric leads us to physical prop erties of SPPs. For a more detailed and clearer investigation of these properties, we should start with the derivation of the wave equation, that describes the guiding of electromagnetic waves, hence we start with the Maxwell's equations of macroscopic electromagnetism. r:D = ext (2.1) r:B = 0 (2.2) r E = ô€€€ @B @t (2.3) r H = Jext + @D @t : (2.4) 6 Chapter 2: Single Josephson Junction in a MetalDielectric Interface In the absence of external charge and current densities, Faraday's Law and Am pere's Law (2.3, 2.4) can be put together to generate r r E = ô€€€ 0 @2D @t2 (2.5) Using identities r r E r(r:E)ô€€€r2E and r:( :E) E:r + r:E, and also r:D = 0, equation 2.5 can be rewritten as r ô€€€ 1 E:r ô€€€ r2E = ô€€€ 0 0 @2E @t2 : (2.6) Considering the negligible variation of the permittivity = (r), Eq. (2.6) simpli es to the wave equation, r2E ô€€€ c2 @2E @t2 = 0: (2.7) Since the dielectric pro le, , is constant in the medium, this equation has solutions in both metal and dielectric, and boundary conditions must be used to obtain these solutions. In order to describe the con ned propagating waves with Eq.(2.7), we should rewrite this equation in a di erent form, the wellknown Helmholtz equation. We should rst consider, that there is a harmonic time dependence E(r; t) = E(r)ei!t of the electric eld. If we insert this function into Eq.(2.7), we obtain r2E + k20 E = 0; (2.8) where k0 = ! c is the wave vector of the propagating wave in vacuum. The next step is introducing a speci ed geometry to apply the Helmholtz equation. We choose this geometry such that, depends only on one spatial coordinate, z direction, and therefore does not have any dependence in the perpendicular, y direction (see Fig. 2.1); this is chosen for simplicity, so that = (x). Our electric eld function can be written as E(x; y; z) = E(x)ei z. is de ned as the complex parameter and can be written as = kz, which is the propagation constant of the traveling waves. Inserting this expression into Eq.(2.8) gives the wave equation in the Chapter 2: Single Josephson Junction in a MetalDielectric Interface 7 Figure 2.1: Illustration of a planar waveguide geometry. z is the propagation direction of the waves. expected form @2E(x) @x2 + (k2 0 ô€€€ 2)E = 0: (2.9) The magnetic eld H can be obtained by following similar steps, which will not be shown here. Equation (2.9) will be the key to analyze guided electromagnetic modes in waveg uides. The next step now is to determine explicit expressions for the electric and magnetic elds. These expressions will help us determining the spatial eld pro le and dispersion of propagating waves. Two sets of solutions can be obtained from this type of systems: TM and TE modes. In TM (transverse magnetic) modes Ex;Ez, and Hy are nonzero and all other components are zero. In TE (transverse electric) modes Hx , Hz and Ey are nonzero and all other components are zero. For TM modes, the wave equation is @2Hy @x2 + (k2 0 ô€€€ 2)Hy = 0: (2.10) For TE modes, the corresponding wave equation is @2Ey @x2 + (k2 0 ô€€€ 2)Ey = 0: (2.11) A single, smooth interface (Fig. 2.2) composed of a two halfspaces on both sides, one being a dielectric nonabsorbing material (x > 0) with positive permittivity 2 and the other being a conductive material (x < 0) described via a dielectric function 1(!) is the obvious candidate for a geometry, which sustains SPs. The metallic character 8 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.2: Geometry for SP propagation at a single interface between a metal and a dielectric. requires Re[ 1] < 0. We should look for propagating wave solutions with evanescent decay in the perpendicular xdirection. Let us rst look at TM solutions. Hy(x) = A2ei zeô€€€k2x (2.12) Ez(x) = iA2 1 ! 0 2 k2ei zeô€€€k2x (2.13) Ez(x) = ô€€€A1 ! 0 2 ei zeô€€€k2x (2.14) for x > 0, and Hy(x) = A1ei zek1x (2.15) Ex(x) = ô€€€iA1 1 ! 0 1 k1ei zek1x (2.16) Ez(x) = ô€€€A1 ! 0 1 ei zek1x (2.17) for x < 0. Here ki kx;i(i = 1; 2) is the wave vector, which is perpendicular to the propagation direction. Boundary conditions require the continuity of Hy and iEx at the interface (A1 = A2) and results in k2 k1 = ô€€€ 2 1 : (2.18) Chapter 2: Single Josephson Junction in a MetalDielectric Interface 9 The y component of the magnetic eld, Hy, should satisfy the wave equation 2.18. Plugging in the wave equation gives k2 1 = 2 ô€€€ k2 0 1 (2.19) k2 2 = 2 ô€€€ k2 0 2: (2.20) Combination of these equations along with Eq. (2.18) yields the dispersion relation of SPs propagating at the interface between the two di erent media = k0 r 1 2 1 + 2 : (2.21) We arrived at this result by manipulating the TM solutions of the wave equation. However, no surface modes exist for TE polarization. Thus, surface plasmons only exist for TM polarization. We now investigate dispersion relation of SPs, which will reveal some important properties of SPs. As can be seen in Fig. 2.3, for small wave vectors (low frequencies), the SP propagation constant is close to k0 at the light line. For large wave vectors, the freeelectron dielectric function (!) = 1 ô€€€ !2p !2+i ! can be plugged into the dispersion relation above. This shows that the frequency of the SPs approaches the characteristic surface plasmon frequency !sp = !p p 1 + 2 ; (2.22) If the damping of the conduction electron oscillation is neglected (Im[ 1(!)] = 0), the propagation constant goes to in nity as the frequency approaches !sp. This would allow the group velocity goes to zero, vg ! 0. Then this mode gains electrostatic character, which is known as the surface plasmon. 2.1.2 Optical Solitons Optical spatial solitons are optical beams that travel without the e ects of di raction and show selftrapping properties with nite spatial cross section. The medium has to have a mechanism, that gives a selffocusing response to the light. The strong in teraction between the medium and the electromagnetic wave causes the selffocusing. 10 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.3: Dispersion relation of SPPs at the interface between a Drude metal with negligible collision frequency and air (gray curves) and silica (black curves) [5]. The wave modi es locally the medium and the medium in turn modi es the wave. Therefore, optical beams, whose elds intersect this particular region of the medium, are a ected. This is a typical example of the interaction of solitons with one another, and with surface plasmons as well. We can now say that self trapping can be obtained as a result of a subsequent physical mechanisms. In order to give insight about the SPsoliton interaction, some conceptual ideas should be presented rst. Among these ideas the most fundamental ones are the de nition of optical spatial solitons, structures that can host these type of solitons, and the generality of the interactions between solitons considering the extensively diverse physical origins of the selftrapping. Optical beams broaden with distance with the natural di raction as they propagate, if they are very narrow in shape, which causes them to travel without changing the properties of a medium. If the width of the initial beam is thinner, then the beam diverges and therefore di racts faster. However, in materials that possess nonlinear properties, the presence of light modi es the properties such as refractive index, absorption, or conversion to other Chapter 2: Single Josephson Junction in a MetalDielectric Interface 11 frequencies. The refractive index change forms an optical lens that increases the index in the beams center and leaves it unchanged in the beams tails. This induced lens focuses the beam, what is called selffocusing. Selftrapping can arise, when the self focusing and the di raction stabilize each other. The newly arising beam with a very narrow width is called an optical spatial soliton as shown in Fig. 2.4. Figure 2.4: Illustration of di erent spatial beam pro les [7]. A simple convex lens model has to be explained in order to understand the exis tence of spatial solitons. As shown in the Fig. 2.5, an incident optical eld is focused after getting through the lens, which introduces a xdependent phase change which is a nonuniform function, (x) = k0nL(x), that causes the focusing e ect [8], [9]. Here, L(x) is the width of the lens. Since k0 and n are constants, the width of the lens changes in each point with a shape similar to (x). This is another way of saying that, a focusing e ect could be obtained by introducing a phase change of such a shape without a change of the width [10]. This method should be followed in order to apply this principle to optical solitons. To create the same e ect with a di erent approach, the value of the refractive index n(x) should be changed, with the width L is kept xed in each point. The modulation of the refractive index creates a focusing e ect, that reduces the natural di raction of the eld. When the focusing e ect stabilizes the di raction of the eld, a con ned eld can be observed within the medium. The existence of spatial solitons 12 Chapter 2: Single Josephson Junction in a MetalDielectric Interface are based on this principle. The Kerr e ect creates a selfphase modulation, that alters the refractive index, which is a function of the intensity function, (x) = k0n(x)L, [10]. n(x) = n0 + n2I(x): (2.23) Figure 2.5: The focusing e ect of a simple convex lens. The propagating wave eld develops a berlike guiding structure. Once this guiding is created, and if it is the mode of such a ber at the same time, this indicates nonlinear e ect, which causes focusing and linear e ect, which causes di raction are perfectly balanced, and the eld does not experience a change in its shape. The condition of developing a nonlinear behavior is that n2 must be positive, so that selffocusing can be observed. The mathematical formulation of soliton waves is as follows. Kerr e ect is the main reason for the existence of solitons, hence the refractive index is given by n(I) = n + n2I (2.24) where I = jEj2 2 and = 0= and 0 = p 0= 0 377 . The eld is propagating in the z direction with a phase constant k0n. Any dependence on y axis can be ignored, because the eld is in nite in y direction, which can be expressed as E(x; z; t) = Ama(x; z)ei(k0nzô€€€!t) (2.25) Chapter 2: Single Josephson Junction in a MetalDielectric Interface 13 where Am is the maximum amplitude of the eld and a(x; z) is a dimensionless nor malized function, that represents the shape of the electric eld among x axis [11]. The next step is solving the Helmholtz equation r2E+k2 0n2(I)E = 0 , and we assume that a(x; z) changes slowly, while propagating, i.e. j @2a(x;z) @z2 j j k0 @a(x;z) @z j and the following equation is obtained: @2a @x2 + i2k0n @a @z + k2 0[n2(I) ô€€€ n2]a = 0 (2.26) The e ect of nonlinearity is always much smallar than the e ect of linearity, so the following approximation can be considered valid: [n2(I) ô€€€ n2] = [n(I) ô€€€ n][n(I) + n] = n2I(2n + n2I) 2nn2I (2.27) then we express the intensity in terms of the electric eld: [n2(I) ô€€€ n2] 2nn2 jAmj2ja(x; z)j2 2 0=n = n2n2 jAmj2ja(x; z)j2 0 (2.28) n2 is considered positive, because we assume that selffocusing is created by the nonlinearity. To clear this statement, n2 will be written as jn2j from now on. Some new parameters should be introduced and substituted in the above equation. First parameter, that should be introduced, is = x X0 , so the dependence on the x axis can be expressed with a dimensionless parameter; where X0 is the characteristic length. Second parameter is Ld = X2 0k0n. Beyond the distance X0 on the z axis, the di raction caused by the linear e ects becomes large enough, so they cannot be neglected. The third parameter is = z Ld , a dimensionless variable, that describes the zô€€€dependence. Next parameter is Lnl = 2 0 k0njn2jjAmj2 , which depends upon the electric eld intensity. Beyond the distance X0 on the z axis, the selffocusing caused by the nonlinear e ects becomes large enough, so they cannot be neglected. Final param eter is N2 = Ld Lnl . After the substitutions, Eq. 2.28 yields the nonlinear Schr odinger equation: 1 2 @2a @ 2 + i @a @ + N2jaj2a = 0 (2.29) To interpret this equation in a better way, four di erent regimes should be considered separately. 14 Chapter 2: Single Josephson Junction in a MetalDielectric Interface For N 1, the nonlinear part can be neglected. For N 1, the nonlinear e ect will be dominant over di raction. For N 1, the two e ects balance each other and the equation can be solved. For N = 1, the solution of the equation is the fundamental soliton [10]: a( ; ) = sech( )ei =2 (2.30) Although the solution is a function of z, the shape of the eld will be stationary while it propagates, because the z dependence is in the phase term. The illustration of the soliton can be shown in Fig. 2.6. Figure 2.6: The shape of the soliton, while propagating with N=1. As far as soliton solutions are concerned, N must be an integer, and is called the order of the soliton. For the solitons with higher orders, closed form expressions does not exist. These higer order solitons are all periodic, which have di erent periods. Their shape can be identi ed after generation: a( ; = 0) = Nsech( ): (2.31) The mathematical formulation of the optical waveguide introduced by the prop agating soliton is not only in theory, but also a realistic model. The optical soliton Chapter 2: Single Josephson Junction in a MetalDielectric Interface 15 wave could be utilized to interact with other waves at di erent frequencies, which would otherwise be impossible in linear media. 2.1.3 Josephson Junction The weak coupling of two macroscopic quantum objects features the di erence be tween classical and quantum mechanical dynamics. This fact was predicted by Brian D Josephson in 1962. The system is described as a thin insulator sandwiched be tween two superconductors without an applied external voltage. He predicted that a DC current can ow through these layers. This is called the dcJosephson e ect. Moreover, an external voltage will result in a rapidly oscillating current; this e ect is known as the acJosephson e ect. In general, a Josephson junction can be constructed by a three layer system, one nonsuperconducting layer between two superconducting layers as shown in the gure. Figure 2.7: The basic geometry of a Josephson junction formed between two super conductors is illustrated. In a Josephson junction, the middle nonsuperconducting section should be narrow. If the barrier is made of an insulator, it has to be on the order of 30 angstroms thick or less. If the barrier is made of another kind of metal, its thickness should be in units of micron. A supercurrent can ow across the insulator; so that electron pairs encounter no resistance, while crossing the barrier from one side to the other. However 16 Chapter 2: Single Josephson Junction in a MetalDielectric Interface when the threshold current is exceeded, an AC voltage evolves across the junction. This lowers the junction's threshold current, generates even more normal current to ow and causes a larger AC voltage close to 500 GHz per mV across the junction. The voltage would be zero, until the current through the junction is less than the threshold current. Once the current reaches the threshold current, the voltage would be nonzero and oscillates in time. Detecting and measuring the change between two states is the key to the many applications for Josephson junctions. The Josephson e ects has been used in various applications such as voltage stan dards of the Shaphiro e ect, ultrasensitive magnetic eld sensors, and supercon ducting quantum interference devices (SQUIDS). Furthermore fundamental questions on quantum physics have been studied both theoretically and experimentally with Josephson junctions in various con gurations such as ultra small junctions and long junction arrays. In this thesis study, the discussion of bosonic Josephson junctions (BJJ), generated by con ning a single BoseEinstein condensate (BEC) in a double well potential plays an important role, since we draw an analogy between surface plasmonoptical soliton coupling and bosonic Josephson junctions. The macroscopic quantum phase di erence has been exhibited by setting up a doublewell trap for con ned BoseEinstein condensates, which are divided by a strong laser sheet, that builds an impenetrable barrier between the traps. Turning o the barrier, the two released condensates overlapped, producing a strong twoslit atomic interference pattern, which is a sign for macroscopic phase coherence. The method for observing the phase di erences between two trapped BEC is turning o the bar rier gradually, in other words the intensity of the laser sheet should be lowered to enable atomic tunneling through the barrier, and Josephsonlike currentphase e ects manifest themselves. The structure is shown in Fig. 2.8, [12]. The importance of detection of the phase di erence reveals itself, when monitoring the population of the trapped condansates with phasecontrast microscopy. In this method, the geometry of the wells and the barrier can be adjusted by the location and the strength of the laser sheet that divides the trap. The analogy between bosonic Chapter 2: Single Josephson Junction in a MetalDielectric Interface 17 Figure 2.8: The BEC doublewell trap. E0 1 and E0 2 are the zeropoint energy of each condensate [12]. Josephson junction (BJJ) and superconducting Josephson junction (SJJ) requires some similarities between system components. An external direct current voltage in SJJ could be replaced by the chemical potential between the BECs, which is a function of the initial energy bias, furthermore capacitive SJJ charging energy could be replaced by the initial population imbalance. These replacements between two systems along with the adjusting capability of traps and the nonlinear barrier draw a ful lling analogy between BJJ and SJJ. The similarities between Bosonic Josephson junciton and superconducting Joseph son junctions should be mentioned as well as the di erences between them. The com parison includes some obvious results. In both models the motivation is to obtain a tunneling between two states and to nd the alternating current. However, getting this alternating current requires di erent conditions for both cases. For example, BJJ requires no potential di erence for producing ac, although zero potential di erence produces direct current in SJJ. Another example is that, the ac frequency in BJJ depends on the barrier transmissivity, on the other hand the frequency depends on the external voltage in SJJ. 18 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Experimental realization of Bosonic Josephson junctions has been compared with the theoretical results [13]. A BoseEinstein condensate is trapped in a doublewell, which are separated by a potential barrier. The coupling is constructed weakly so that, the particle can tunnel through the barrier. The population imbalance between the two wells, which is a function of the energy bias shows an agreement with the theoretical expectation. The population imbalance and the relative phase di erence were found to be the key elements, that control the dynamics of the selftrapping regime with the agreement to the solution of the GrossPitaevskii equation. Bosonic Josephson junctions can nd many applications within the eld of BECs and nanophysics, since the rich dynamics of BoseEinstein condensates in double and triplewell traps reveals some particular properties such as the macroscopic quantum self trapping and adiabatic transition between the wells, in which BECs are trapped [14], [15]. These examples encounter some problems regarding the nonlinearity created by the interaction of BECs, and few new approaches have been proposed to overcome these di culties by introducing a modi ed nonlinear LandauZener and RosenZener models [16], [17]. 2.1.4 LandauZener Transition LandauZener [LZ] model [20], [21] is an extensively used twostate model in quantum physics, however its applied elds extend over a large area such as ac current driven Josephson junctions, atomic collision studies, BEC in optical lattices, QED circuits, multiple level crossings, and threelevel systems. There are many reasons for this wide usage of LandauZener model. One of the reasons is that, this model portrays an interaction between a doublewell quantum system and an external eld near the resonance. Another reason is the realistic presumption of the detuning, which is a timedependent function, however the coupling is constant. And the nal reason is the accurate de nition of transition probability, that this model gives. All of these reasons considered, LandauZener model is a promising and satisfying model, that can be adapted to many di erent systems in both applied and theoretical physics. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 19 Despite the wide application elds of LandauZener model, the extention of the model to the nonlinear case urged itself, when the transition probability in the adia batic limit has been started to be investigated. As a result of this investigation, it has been realized, that strong nonlinearity breaks the adiabaticity to enable a tunneling between two states. This breakdown of adiabaticity is caused by the emergence of ex tra xed points and the collision of these xed points, which was throughly explained by analysing the energy levels and quantum eigenstates of the system [22], [23], [24]. The promising applicability of LandauZener model to the two state systems en courages researchers to examine the threelevel systems [25]. Though the triplewell model shares some dynamic properties with its twolevel counterpart such as the breakdown of adiabaticity and nonzero transition probability, it has some distin guished speci cations like selftrapping of populations in a single well, topological change of the system near crossing points, and sensitivity to the initial system pa rameters, which results in chaos. An elaborated analysis has been done on this topic both numerically and analytically [26], and tunneling dynamics of the nonlinear three level system has been exhibited. 2.1.5 RosenZener Transition The the RosenZener [RZ] model [27] was introduced to investigate the double Stern Gerlach experiments regarding the hyper ne Zeeman levels of energy under a rotating magnetic eld. It was proposed at the same times as the LandauZener model with the motivation of encompassing a di erent system structure. The LandauZener model describes the transition properties of two avoidedcrossing levels, in which the coupling is constant and the energy gap between levels are slowly changing. On the contrary, the RosenZener model describes the transition probability between two levels, in which the energy di erence is xed and the coupling is a timedependent function. The structure of this coupling function de nes the driving properties of the system. On of the advantages of this model is to provide both numerical and analytical solutions to two or three level systems such as exchange of ion population in nonres 20 Chapter 2: Single Josephson Junction in a MetalDielectric Interface onant ionatom collision, excitations induced by lasers, nuclear magnetic resonance techniques, and quantum computation. In a recent study [32], the RosenZener model has been enlarged to take the nonlinearity into account, and the e ects of this nonlinearity have been investigated throughly. BoseEinstein condensates have been used as a subject in this investiga tion because of their rich nonlinear properties, which was caused by the interaction between condensates. These nonlinear properties enable to observe various signi cant phenomena such as macroscopic quantum selftrapping, super uidity, and Landau Zener transition in a nonlinear perspective suggesting the bene ts of analysing the nonlinear RosenZener transition. Here a nonlinear Schr odinger equation has been solved numerically to obtain the transition probabilities between traps. i @ @t 0 @ a b 1 A = H(t) 0 @ a b 1 A; (2.32) where the Hamiltonian is given by H(t) = 0 @ 2 + c 2 (jbj2 ô€€€ jaj2) v 2 v 2 ô€€€ 2 ô€€€ c 2 (jbj2 ô€€€ jaj2) 1 A: (2.33) Here, a and b are the probability amplitudes of the condensates, is energy di erence between traps, and c is the nonlinearity parameter indicating the interaction between condensates. The coupling parameter, v, is given as a timedependent function v = 8< : 0; t < 0; t > T; v0sin2( t T ); 0 < t < T; (2.34) where T is the external period of the system. The probability of one of the condensates, which resides in the other trap after the coupling loses its e ect on them is considered as the transition probability. The e ect of system parameters to the transition probability has been investigated, the results for the nonlinearity are presented in Fig. 2.9. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 21 Figure 2.9: Transition probability versus external period under di erent nonlinearity strengths [32]. 2.2 Surface PlasmonSpatial Soliton Coupling 2.2.1 Model Description To design a system, that has both optical soliton and surface plasmon components, that are copropagating along the Kerrdielectric and metaldielectric interfaces re spectively, we adopt the following model in [35]. The system shown in Fig. 2.10 is formed by two interfaces that host optical wave formation: a metal/lineardielectric interface (surface plasmon formation) and a lineardielectric/nonlineardielectric (Kerr) interface (spatial soliton formation). The distance between these two interfaces, in other words the distance from the surface plasmon propagation axis to the soliton center axis is d. The interaction between a surface plasmon guided wave at the metal/dielectric interface, that propagates along the z direction and a spatial soliton in the nonlinear medium propagating along the same direction will be analyzed. It is assumed that, the nonlinearity is selffocusing, and the nonlinear behavior of the soliton is able to provide a propagation constant s 22 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.10: Metaldielectricnonlinearity interfaces for plasmonsoliton propagation. that grows with the soliton peak amplitude. On the other hand, the surface plasmon is formed by two evanescent wave tails both in the metal and in the dielectric with propagation constant p. It is expected that, under appropriate conditions controlled by the soliton power, it is possible to achieve the matching of the propagation con stants of plasmon and soliton. This phasematching condition, s = p, should give rise to a mechanism of nonlinear resonant transfer of energy from soliton to surface plasmon. The total wave eld of the system is represented by the following ansatz, which indicates the superposition of surface plasmon and soliton elds, (x; z) = cp(z) p(x) + cs(z) s(x; jcsj): (2.35) We emphasize that, the transverse pro le of the soliton, s = sech h k p =2jcsj(x ô€€€ d) i , depends on its amplitude jcsj which is the driving parameter of the system, whereas the transverse pro le of the plasmon, p = exp ô€€€ ô€€€x p k2 p ô€€€ k2 at x > 0, represents two exponents decaying away from the metal surface with normalizations p(0) = 1 Chapter 2: Single Josephson Junction in a MetalDielectric Interface 23 and s(d) = 1. We assume that, the soliton amplitude varies slowly enough to use quasistanionary adiabatic approximation. Here is the nonlinearity parameter of the medium. Our motivation is to construct an e ective plasmonsoliton interaction, and in the rst order approximation, we obtain the nonlinear coupled oscillator equations from Ref. [35]: cp + 2 pcp = q(jcsj)cs; cs + 2 s (jcsj)cs = q(jcsj)cp: (2.36) where the derivatives are with respect to the propagation parameter z and q(jcsj) is the coupling function which is the overlap of the tails of the plasmon and soliton waves in the area between metal and dielectric. The rst equation implies that, the righthand side represents an external source, which excites surface plasmons at the metaldielectric interface, and due to weakcoupling approximation it is equal to the soliton eld at the metal surface, q (jcsj) sjx=0 ' exp(ô€€€d p =2jcsj). We need to remind that, in the zeroorder approximation, surface plasmon and soliton amplitudes are independent, because coupling between them is assumed to be zero. However, in the rstorder approximation, they become linearly coupled due to the spatial overlapping of the wave elds of surface plasmon and soliton. These coupled oscillator equations are similar to the system of two weakly coupled nonlinear waveguides. However, there are signi cant di erences between two systems. In the plasmon  soliton coupled system, only one subsystem is nonlinear, while the other is linear. Furthermore, coupling between surface plasmon and soliton is not constant, but soliton dependent. The coupled oscillator equations can be reduced to rst order di erential equa tions by making the substitution cp;s = Cp;seiz and by adapting the slowly varying amplitude approximation, hence we arrive, ô€€€i 0 @ Cp Cs 1 A 0 = 0 @ p ô€€€q(jCsj)=2 ô€€€q(jCsj)=2 s(jCsj) 1 A 0 @ Cp Cs 1 A; (2.37) where Cp;s are the zdependent plasmon and soliton amplitudes, which satisfy the nonlinear Schr odinger equation with a nondiagonal Hamiltonian. p 1 and s = 24 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.11: Hyperbolic twolevel system. jCsj2=4 1 are small deviations of p and s such that p p ô€€€ 1 1 and s s ô€€€ 1 1. The second inequality points the weakness of nonlinearity. 2.2.2 Hyperbolic Modulation to the Model One of the objectives of this study is to obtain a LandauZener type of transition, so we should con gure our system such that, surface plasmon and soliton are uncoupled rst, then, when they enter an e ective interaction area, coupling between them manifests itself via the evanescent tails of the waves. After they leave the e ective interaction area, coupling vanishes, and we again have a system of two uncoupled waveguides. The geometry of this e ective interaction area should be arranged such that, the resonant transfer of energy between soliton and surface plasmon can be observed at maximum e ciency. The optimum candidate for such a geometry is the one, in which the distance between metal surface and nonlinearity is a hyperbolic function of 0z0. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 25 In this modulated geometry the coupling function q(jCsj) can be represented as: q(jCsj) ' jCsjexp(ô€€€ p d2 + z2 2 p =2jCsj); (2.38) where is the parameter of the hyperbolic trajectory, (i.e., ber), followed by the soliton (Fig. 2.11). The dashed lines in Fig. 2.11 represent the asymptotic line and c is the angle between the asymptotic line and the z axis. is de ned as the tangent of this critical angle, i.e. = tan c. Here, p d2 + z2 2 is the distance between the metal surface and the ber de ned by the general hyperbola equation: x2 d2 ô€€€ z2 (d= )2 = 1 (2.39) Since the weakcoupling approximation fails at small soliton amplitudes, jCsj must be inserted before the exponential term to hold the approximation [38]. To draw an analogy to the Josephson junction dynamics, we introduce 'fractional population imbalance' and 'relative phase di erence', respectively Z = jCsj2 ô€€€ jCpj2 jCsj2 + jCpj2 (2.40) = s ô€€€ p (2.41) by applying the substitution Cs;p = jCs;pjei s;p [12], [36], [37]. Using this substitution and the fact that jCsj2 + jCpj2 = N (N is a normalized constant for isolated system and equal to 1), Eq. (2.3) becomes _Z = ô€€€q(Z) p 1 ô€€€ Z2 sin ; (2.42) _ = Z + E + q(Z)Z cos p 1 ô€€€ Z2 : (2.43) Here the parameters is the nonlinearity (i.e., the soliton strength) with = =8, and E parametrizes the asymmetry between the soliton and surface plasmon states occupied by the photons with E = ô€€€ p. We can now express the coupling parameter as a function of Z q(Z) ' r 1 + Z 2 eô€€€ p d2+z2 2 p 2 (1+Z): (2.44) 26 Chapter 2: Single Josephson Junction in a MetalDielectric Interface The range of the parameters in our system is set by the insight of physical cir cumstances. is investigated between 0.01 and 0.1, because the nonlinearity should be weak by de nition s = jCsj2=4 1, E is investigated between 0.04 and 0.04, because p 1, and the range of values allow only this range for E. In a recent study [39], a di erent analytical formulation of the coupling function is stated, revealing that, the equation system is not symmetric in the coupling function, hence two seperate coupling functions are de ned q and q, implying that, a strong soliton drives a weak surface plasmon at a rate q, whereas a strong surface plasmon drives a weak soliton at a much smaller rate q jCsj and q=q 10ô€€€3 [40]. By taking the new considerations into account, it is expected that the new dynamic equations should be di erent from Eq. (2.42) and Eq. (2.43). However, the equations cannot be written in Z and formation, because the presence of two seperate coupling functions breaks the isolation of the system and the equality jCsj2 + jCpj2 = N is no longer obeyed. The reason for this violation is that, the change in total surface plasmon and soliton amplitudes is di erent from zero, since q 6= q in _N = ( q ô€€€ q)CsCpsin . In the following sections, we will investigate the system, which is symmetric in coupling function (q ' q) described by Eq. (2.42)  (2.43). 2.2.3 Analogy to LandauZener [LZ] and RosenZener [RZ] Transitions The original LandauZener model [20],[21] contains a constant coupling coe cient and a level seperation which changes adiabatically with time. The optical analog of the problem is discussed in [28]. In a previous study [29], [30], coupling coe cient has been modi ed such that, the coupling is e ective only for a nite duration and then the transition dynamics have been investigated. The analog of the level seperation in the plasmonsoliton system is the asymmetry between surface plasmon and soliton states ( E), which is a constant coe cient. However the time dependent variable in this system is the coupling function. In [35], a 100% conversion of energy from soliton to plasmon channel could not be achieved, because the adiabadicity of the system fails due to the nonlinear e ects and the system undergoes diabatic evolution, which Chapter 2: Single Josephson Junction in a MetalDielectric Interface 27 causes the energy to be stored in the soliton channel. The reason for this failure is that, the coupling parameter is a function of soliton amplitude, which changes swiftly. However, with the hyperbolic modi cation the velocity of the coupling function can be slowed down, hence almost a full conversion of energy from soliton to plasmon channel can be obtained. In the light of these results, it can be said that, the hyper bolic plasmonsoliton system might be a proper host to LandauZener tunneling. In order to understand the concept of adiabatic transitions which result in LandauZener tunneling, the following system should be analyzed. The socalled nonlinear LandauZener model consists of two levels, in which the energy levels depends on populations of the condensates trapped in the wells. The model can be identi ed by the following Hamiltonian: H( ) = 0 @ 2 + C 2 (jbj2 ô€€€ jaj2) V 2 V 2 ô€€€ 2 ô€€€ C 2 (jbj2 ô€€€ jaj2) 1 A (2.45) where population amplitudes are represented as a and b, respectively. V is the cou pling constant, is the energy di erence, and C is the nonlinear parameter. In BEC trapped wells, the nonlinearity identi es, how much the level energies depend on the populations. Total population and total probability is held constant, jaj2 + jbj2 = 1. The eigenvalues of this Hamiltonian for two di erent nonlinearity values are given in Fig. 2.12. For the weak nonlinearity, C = 0:1, there are two eigenvalues and the system does not experience any unexpected dynamical behaviour. However, for the strong nonlinearity, C = 0:4, two more eigenvalues appear, resulting a loop at the top of the lower level. When a quantum state moves in the lower level until the right end of the loop it has to jump to either upper or lower levels, since there is no further way to move. This jump between states points out that, nonlinear LandauZener tunneling is nonzero in the adiabatic limit. Further analysis on this system can be made by investigating the xed points of the Hamiltonian [31]. First, Eq.2.45 can be converted into a classical Hamiltonian: H(s; ; ) = C 2 s2 + s ô€€€ V p 1 ô€€€ s2cos : (2.46) 28 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.12: Eigenenergies of the Hamiltonian (Eq. 2.45) [31]. a) c = 0.1 and v = 0.2, b) c = 0.4 and v = 0.2. Here s and can be treated as conjugate variables and obey the canonical Hamilton equations: s_ = ô€€€ @H @ ; (2.47) and _ = @H @s : (2.48) Fixed points of the system can be found by setting s_ and _ equal to zero: = 0; or ; (2.49) + Cs + V s p 1 ô€€€ s2 cos = 0: (2.50) The eigenenergies of the system can be found by substituting the xed points into the classical Hamiltonian. This shows a direct correlation between gures 2.12 and 2.13. The loop structure in Fig. 2.12b is caused by the emergence of two additional xed points in Fig. 2.13b which is caused by the strong nonlinearity. The problems with the LandauZener model can be overcome by introducing another tunneling model, which is the RosenZener model. The _Z and _ equations allow us to investigate Chapter 2: Single Josephson Junction in a MetalDielectric Interface 29 Figure 2.13: Fixed points associated with the eigenenergies in Fig. 2.12 [31]. a) c/v = 0.5, b) c/v = 2. the dynamics of RosenZener transition in a nonlinear twolevel system, where the asymmetry between the surface plasmon and soliton states, E, is xed, and the coupling between two channels, q(Z), is a time dependent exponential function. This is a much more closer analogy than the LZ model as far as the mathematical modeling is concerned. A similar study has been done with Bose Einstein condensates [32] by constructing the e ective classical Hamiltonian, that describes the dynamic properties of nonlinear quantum RosenZener system. Although our system cannot be cast into a classical hamiltonian, because the driving parameter is part of the coupling function, Eq. (2.42) and Eq. (2.43) are fully capable of describing the dynamical properties of plasmonsoliton interaction. 2.3 Numerical Analysis In this thesis study, our main objective is to obtain a full population transfer be tween soliton and surface plasmon channels. With the hyperbolic modi cation of the coupling function we implemented in the previous section, we observe that such a transition occur in the neighbourhood of the minimum distance between the metal surface and the nonlinearity. Since the distance is increasing hyperbolically, outside of this e ective interaction area the coupling between soliton and surface plasmon is 30 Chapter 2: Single Josephson Junction in a MetalDielectric Interface negligble. As shown in the Fig. 2.14(a) at certain parameter values a nearly ideal transition between soliton and surface plasmon amplitudes is observed. Figure 2.14: (a) Solid line shows the transition from soliton to surface plasmon am plitude, dashed lines show the reverse transition from surface plasmon to soliton amplitude. = 0:63, d = 5:7, = 0:04, and E = 0:02: (b) Solid line shows the plot of coupling function 'q(Z)' vs. z of the rst transition, and the dashed lines show the plot of 'q(Z)' vs. z of the reverse transition. When the system starts with Zô€€€1 1 , and it approaches 1 implying that, the optical soliton propagating inside the ber in the absence of surface plasmon enters the e ective interaction region, where the population change occurs, it transfers almost all of its energy to the surface plasmon emerging on the metal surface, and the surface plasmon continues propagating in the absence of optical soliton. Since the dissipation in the system is negligible, and the system is symmetric in coupling function (q ' q), total soliton and plasmon amplitudes are conserved as expected from _N = 0. This population transfer is reversible and symmetrical. In the case of Zô€€€1 ô€€€1, it approaches 1 after the transition. The initial Z values can not be taken exactly 1 or 1, because that would make the denominator term zero in Eq. (2.43). Beyond this reason, there is another consideration, why Z should not be chosen 1. It implies that the initial soliton amplitude is zero, which makes the coupling function q(Z) = 0 and Chapter 2: Single Josephson Junction in a MetalDielectric Interface 31 in that case, no transition would occur. Figure 2.14(b) shows the plot of q(Z) vs. z for two di erent cases, and the di er ence can be seen clearly between two coupling parameter plots. In the rst case, where the soliton starts to propagate in the absence of surface plasmon, the rate of change of q(Z) with respect to z is greater than the second case, where the surface plasmon starts to propagate in the absence of soliton. After the transition, the rate of change of q(Z) with respect to z is less than the second case, as expected. This is caused by the direct dependency of coupling function 'q' to the soliton amplitude jCsj. An asymmetrized generalization of the RosenZener model with a similar coupling func tion has been investigated in a previous study [41]. Although this full population transfer is our desired result, some further analysis should be followed, since there are six parameters ( ; d; ; E;Z0, and 0), that e ects the outcome. First we discuss the e ects of d and . Figure 2.15: Contour plot of nal Z values with respect to and d for di erent values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (a). Here E is taken 0:02. 32 Chapter 2: Single Josephson Junction in a MetalDielectric Interface The panels in Fig. 2.15 show the nal values of Z as a function of d and for four di erent values. From these results, we try to determine the optimum system parameters, which yield maximum transition from soliton to surface plasmon. Here E is kept constant at 0:02. The darker territories correspond to strong transitions. A long, thin and dark slit can be seen on each panel, which indicates the optimum parameters for a successful transition. The transition gets weaker when the position on the d plane moves away from the slit, and after some point no transition occurs at all (wide and light area). The reason for this lack of transition is the reverse proportionality of d and to the q(Z) itself. In order to keep the coupling parameter above some threshold value, which would cause a successful transition, we should investigate the area where the Z value is close to 1. It can be seen in panel (a) that, the optimum d and values are 5:7 and 0:63, respectively. The meaning of being small is crucial, because Eq. (2.37) is derived from 1D Maxwell's equations, and in our system we assumed that, the hyperbolicity in the ber is small enough to keep the paraxial approximation valid. For the sake of the validity of our calculations, we do not allow be greater than 0.7, so our modi cation to the coupling parameter does not cause the system to lose its physical meaning. In Fig. 2.16, the error percentage  comparison can be seen. The paraxial Figure 2.16: Error percentage of the values. Red line shows the error. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 33 approximation suggests: tan( ) . is identi ed as the tangent of the angle between z axis and the asymptotic line, called c. Hence, tan = c: (2.51) So the red line in Fig. 2.16 represents the deviation from the approximation, and it is de ned as the di erence between the green and blue lines. The panels in Fig. 2.17 reveal the role of nonlinearity and asymmetry in plasmon soliton coupling. The dark slit moves left in the former, as increases pointing out that, strong nonlinearity requires closer distance, in order to obtain a successful transition. In the latter, the slit expands, when E decreases, which gives us a wider area of transitions. This result comes from the de niton of E: E = ô€€€ p = s 2jCsj2 ô€€€ p (2.52) E can be de ned as the asymmetry between soliton and plasmon wave vectors. As E increases, soliton channel takes over the dominant role of the coupling. Figure 2.18 shows the wavefunction pro les of surface plasmon and optical soliton. It is shown that, soliton initially propagating on the curved ber transfers almost all of its energy to the surface plasmon. The exponent of the surface plasmon pro le on the metal (x < 0) decays away from the metaldielectric interface quicker than the exponent on the dielectric, because of the small skin depth of the metal , which is taken 0:1. As discussed earlier, the decay in the soliton amplitude is greater than the increase in surface plasmon amplitude, which is caused by the 'jCsj' dependency of the coupling parameter. Figure 2.18 also shows the wavefunction pro les of surface plasmon and optical soliton in the case that, surface plasmon propagates initially in the absence of soliton, and transfers almost all of its energy to soliton. The increase in the soliton amplitude is greater than the decrease in the surface plasmon amplitude as expected. The tail of the newly forming soliton wave is seen to be on the metal surface, where the distance between the metal and the ber is at its minimum (around z = 0), which may di er 34 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.17: Contour plot of nal Z values with respect to and d for di erent E values. Every position on the d plane corresponds to steadystate 'Z' values after each transition. The maximum energy transfer from soliton to surface plasmon is observed, when d = 5:7, and = 0:63 for panel (d). Here is taken 0:04 respectively. by the choice of the d value. In both cases, the total energy of the surface plasmon and soliton is normalized to 1. The interaction between surface plasmon and optical soliton with a hyperbolic distance between them gives rise to other signi cant results. Changing the system geometry and the strength of the nonlinearity allow us to manipulate the energy transfer between surface plasmon and soliton. Fig. 2.19 shows some of the signi cant transitions under di erent and values. We can divide the energy of the soliton propagating in the absence of the surface plasmon equally between them, and similarly, when we have both soliton and surface plasmon propagating initially, we can manipulate the total energy to form either sur face plasmon or optical soliton only. In the case, where is 0.25, we have subsequent transitions between soliton and surface plasmon. First, soliton transfers almost all of its energy to surface plasmon, then surface plasmon transfers all the energy back Chapter 2: Single Josephson Junction in a MetalDielectric Interface 35 Figure 2.18: (a) Propagation of surface plasmon and optical soliton wavefunctions. Optical soliton is propagating in the absence of surface plasmon initially and after the transition it transfers almost all of its energy to surface plasmon on the metal surface. Here is 0.63, is 0.04, E is 0.02, and d is 5.7. (b) Propagation of surface plasmon and optical soliton wavefunctions. Surface plasmon is propagating in the absence of optical soliton initially and after the transition it transfers almost all of its energy to optical soliton on the ber. The same parameter values are used. to soliton. These subsequent transitions show similar properties of localized surface plasmons. Localized surface plasmons are nonpropagating excitations of the conduction elec trons of subwavelength metallic nanostructures coupled to the electromagnetic eld [5]. These excitations emerge inherently from metallic nanoparticles of dimensions below 100 nm in an oscillating electromagnetic eld. Applying an e ective restoring force by the curved surface of the particle on the driven electrons generates localize surface plasmon resonance, which leads to a eld enhancement in the near eld zone outside the particle. In our system, the small area around the minimum distance from the Kerr nonlinearity to the metal surface can be considered as the nanoparticle men tioned above. Since the nonlinearity in the dielectric is structured as a curved path characterized by the parameter , and this area is in nanometer scale, our system can be counted as a proper host to localized surface plasmons. The wavefunction pro les of surface plasmon and soliton of these four cases can be seen in Fig. 2.20. The distance between nonlinearity and the metal surface d can be arranged as 36 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.19: Signi cant transitions between surface plasmon and soliton under dif ferent d, and values. (a) shows the case that soliton is propagating initially in the absence of surface plasmon, and its energy is divided equally between soliton and surface plasmon. (b) shows the case that soliton transfers almost all of its energy to surface plasmon and then surface plasmon transfers its energy back to soliton. (c) and (d) show the case that in the presence of equal surface plasmon and soliton, the energy can be manipulated to be transferred to surface plasmon or soliton, respectively. desired in order to obtain di erent signi cant results as shown in the gure. The change of d could result in outranging the system from 100 nm scale. In this case, the quasistatic approximation breaks down due to retardation e ects, however localized surface plasmon resonance can still be obtained with damping according to the Mie Theory [5], which will not be discussed here. 2.4 Stability Analysis 2.4.1 E ect of relative phase di erence Up to this point, the relative phase di erence ' ' is chosen zero in our calculations, indicating that, soliton and surface plasmon are propagating in phase. However, our system allows the outcome of the transition to be di erent and signi cant, when soliton and surface plasmon are travelling out of phase. Our previous examples of transitions are immune to e ects of phase change, because the absence of surface Chapter 2: Single Josephson Junction in a MetalDielectric Interface 37 Figure 2.20: The wavefunction pro les of surface plasmon and optical soliton of the same four cases. plasmon or soliton means that, the relative phase between them is irrelevant. However, when initial Z value is di erent from 1 or ô€€€1, we observe that the system is highly sensitive to initial relative phase value. Fig. 2.21 shows the e ect of initial relative phase to di erent initial Z values. The sensibility of the system to the initial relative phase di erence can be used to manipulate the system to give desired outcomes. Starting with equal surface plasmon and soliton amplitudes, it can be arranged to transfer all the energy of the system either to surface plasmon or soliton only. Figure 2.22 shows manipulation of energy to one direction by changing only the initial relative phase di erence. The e ect of relative phase di erence can be best perceived once the phasespace analysis of the full population transfer is discussed [18], [19]. In Fig. 2.23, a compre hensive analysis of the evolution of the population imbalance and the phase di erence is shown. In the bosonic Josepshon junction models the phasespace of the system changes adiabatically, since they change the nonlinearity parameter with constant velocity. This repeated change in the system results in a repeated deformation in the phasespace as expected. However the nonlinearity parameter is determined by the material used as the dielectric in the plasmonic Josephson junction system, therefore, 38 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.21: The e ect of di erent initial phase di erences. Solid red line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values for every single di erent 0 value when the system starts with Zô€€€1 1. Dashed lines show the nal Z values for every single di erent 0 value when the system starts with Z = 0. Here the full transition parameter values are used; d = 5:7, = 0:63, = 0:04, and E = 0:02. it cannot be changed externally along the propagation. In this case, the phasespace of the system would not undergo a deformation, ergo the movement of the population imbalance would be restricted by the current phasespace trajectories indicating that, a LandauZener transition would not manifest itself. In order to overcome this prob lem we modi ed our system spatially, which was the main motivation of this work. When the distance between metal surface and the Kerr nonlinearity decreases adia batically to the minimum distance d and then, starts to increase, in other words, when z changes, the phasespace trajectories also change, and the expected deformations manifest themselves under the control of adiabatic parameter . On every trajec tory the transition probability can be determined explicitly. In Fig. 2.23 some of the signi cant trajectories are shown. Comparing this gure with Fig. 2.14 concurrently would give a good insight to the interpretation of these trajectories. Panel (a) in Fig. 2.23 corresponds to the initial state of the system, z = ô€€€100. The red dot represents the system with Z = 1 and = 0, which means the whole energy is in the soliton channel, and the phase di erence between soliton and surface plasmon Chapter 2: Single Josephson Junction in a MetalDielectric Interface 39 Figure 2.22: (a) Solid line shows the case when 0 = 0:2, in which all the energy is transfered to soliton. Dashed lines show the case when 0 = 0:81, in which almost all the energy is transfered to surface plasmon. Here d = 5:6, = 0:4, = 0:04, and E = 0:02. (b) Solid red line shows the nal Z values when the system starts with Zô€€€1 ô€€€1. Solid blue line shows the nal Z values when the system starts with Zô€€€1 1. Dashed lines show the nal Z values when the system starts with Z = 0. is zero. When the soliton propagates until z = ô€€€30 the trajectory is still une ected by the geometry of the nonlinearity, so the interaction is still negligible. When the soliton enters the e ective interaction region, z = ô€€€13, a xed point emerges at Z = 1; = which causes a closed trajectory around itself, and surface plasmon soliton coupling falls in to this trajectory and change its course. When z increases, this xed point starts to move downwards along with its trajectories. Surface plasmon soliton coupling has no choice but to follow these trajectories, and ends up trapped by this xed point at Z = ô€€€1; = , which corresponds to transfer of energy to plasmon channel completely. A similar transition can be observed when the system starts with surface plasmon only. The green dot starts at Z = ô€€€1; = 0 and follows its current trajectory. The xed point around Z = ô€€€0:5 a ects its direction, and leads it to a stationarystate at Z = 1. The change in the relative phase can also be clearly seen in Fig. 2.24. Panel (a) shows the phase di erences of the green dot with di erent initial phase di erence 40 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.23: The evolution of the phasespace trajectories of the system as z changes adiabatically. The red dots indicate the motion of the population imbalance in phase space when Zô€€€1 1. The green dot indicates the motion when Zô€€€1 ô€€€1 Here the parameters are the same as in Fig. 2.14. values. Depending on the choice of the initial value, there are two di erent trajecto ries, one of them tends to move left in the phasespace, and the other tends to move right. Whichever the choice might be, the phase di erence increases constantly at the stationary state, indicating the coupling moves constantly right on the uppermost trajectory, since there isn't any xed points emerging at the top. Although the phase di erence increases, the population imbalance reaches its nal value, Z = 1. Panel (c) in Fig. 2.24 shows a similar phase di erence pattern for the red dot. Instead of staying on a stationary state, in which the phase di erence increases with constant velocity, this coupling is trapped at a xed point, and the change in the relative phase is quite small. Panel (b) shows the behaviour of the relative phase, when the system starts from Z = 0. As can be seen from the gure, the nal result is more sensitive to the initial choice of , because population of one of the channels is not zero and coupling is more exible. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 41 Figure 2.24: Initial relative phase di erence vs. z graph. (a) Z = ô€€€1, (b) Z = 0, (c) Z = 1. These two transitions are a clear presentation of the jump from one state to another in the metaldielectricKerr system, which leads us to make a comparison with the LandauZener tunneling [20],[21]. The phasespace analysis of the system with parameters in Fig. 2.22 is also in sightful. The route of the population imbalance in phasespace is shown in Fig. 2.25, such that di erent initial relative phase di erences lead the system to two di erent steadystate solutions. 2.4.2 Fixed Point Analysis The trajectory of the system in the phasespace is radically e ected by the xed points of the system, as seen in the gures 2.23 and 2.25. Since the distance between metal surface and the nonlinearity in the dielectric is not constant, but a function of , it can be said that, the geometry of the system is changing slowly. This geometry change causes a continuous and regular deformation in the phasespace of the system. This is another way of saying that, the xed points can move, disappear, and/or reappear in the e ective interaction region between metal and nonlinearity. The evolution of xed points of the system plays crucial role in the surface plasmon  optical soliton interaction, therefore attention should be given to the analysis of xed points. Fixed points of the system can be found numerically by setting the main dynamic equations equal to zero: _Z = ô€€€q(Z) p 1 ô€€€ Z2sin = 0; (2.53) 42 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.25: The evolution of the phasespace trajectories of the system with equal initial amplitude of surface plasmon and optical soliton as z changes adiabatically. The red dot indicate the motion of the population imbalance in phasespace with 0 = 0:2, and green squares indicate the motion of the population imbalance in phasespace with 0 = 0:81. The population imbalance changes dramatically from 0 to 1 (red dot) or from 0 to 1 (green dot). Here the parameters are d = 5:6, = 0:4, = 0:04, and E = 0:02. _ = Z + E + q(Z)Z p 1 ô€€€ Z2 cos = 0: (2.54) It is easier to solve the rst equation above as sin = 0 would give the solution. q(Z) shouldn't be zero, otherwise no interaction would occur between surface plasmon and soliton, and Z cannot be 1 or 1, which would make the denominator in the second equation zero. Hence, we should plug in = 0, and = solutions in the second equation, and solve for 0Z0 solutions. Since q(Z) is z dependent, for every coordinate on the zô€€€ axis, there may be xed Z points for = 0 and = , seperately. The graph of xed points with respect to z coordinate can be seen in Fig. 2.26. These xed points are retrieved from the system which corresponds to the transition in Fig. 2.14. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 43 Figure 2.26: Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 5:7, = 0:63, = 0:04, and E = 0:02 are used. The evolution of xed points can be seen clearly in Fig. 2.26. The rst xed point, ( = 0), moves between Z ' ô€€€0:5 and Z ' ô€€€0:1. The second xed point, ( = ) however shows a di erent behaviour. It rst starts at Z ' ô€€€1, then around z = ô€€€20 it disappears, then around z = ô€€€10 it reappears at Z ' 1. After z = 0, the movement repeats itself symmetrically. As a matter of fact, this symmetrical evolution of xed points is observed for every combination of parameters in the hyberbolical system, because the mathematical modeling of this geometry points that, the distance between metal surface and the nonlinearity depends on z2, so being on the negative side or positive side of the zô€€€axis does not make a di erence. Another correlation of xed points and phasespace is given in Fig. 2.27. These xed points are retrieved from the system which corresponds to Fig. 2.22. This constraint is the main reason, for why this system cannot host an adiabatic transition. In the previous studies, the authors showed that, a xed point can start 44 Chapter 2: Single Josephson Junction in a MetalDielectric Interface Figure 2.27: Evolution of xed points for = 0 and = can be clearly seen in the phasespace diagram on the left. Corresponding xed point vs. z graphs are on the right. Here, d = 2:15, = 0:32, = 0:1, and E = ô€€€0:04 are used. its journey from one end of the phasespace, and evolve such that at the end the xed point reaches the other end of the phasespace. In the meantime BECs trapped in the xed point would move with it, so a slow external change in the Hamiltonian would result in a 100% transition without a character change in eigenvalues. This is called adiabatic transition. In those studies the adiabatic parameter in the system is the energy gap between traps, however in the plasmonic system the adiabatic parameter is taken as the distance between surface plasmon and soliton. Since the distance cannot be negative, fully adiabatic transition cannot be acquired in the hyperbolically modulated Josephson junction. To understand the properties of adiabatic transition, eigenenergies of the system should be analyzed. Eq. (2.37) can be considered as nonlinear Schr odinger equation Chapter 2: Single Josephson Junction in a MetalDielectric Interface 45 and written as : ô€€€i d dt 0 @ Cp Cs 1 A = H(z) 0 @ Cp Cs 1 A; (2.55) where the Hamiltonian is a function of distance between surface plasmon and soliton: H(z) = 0 @ p ô€€€q(jCsj)=2 ô€€€q(jCsj)=2 s(jCsj) 1 A: (2.56) There are two conventional ways of determining the eigenvalues of this Hamil tonian [34]. The rst way is to construct the classical Hamiltonian for the conju gate variables Z and . These variables satisfy the Hamilton's dynamic equations: _Z = ô€€€@H @ , _ = @H @Z . This way canonical equations (2.42) and (2.43) can be derived and the xed points can be found by setting equations ??eq:2.42) and (2.43) equal to zero. Then the results should be substituted into the classical Hamiltonian to obtain the eigenenergies. These eigenenergies can be considered as the energy levels of the sys tem. However this method cannot be applicable to the plasmonic system, because in the systems, which previously used this method Hamiltonian has the physical mean ing of being total energy of the system. Unfortunately, total energy of the plasmonic system cannot be de ned by the classical Hamiltonian, since Cs and Cp are merely the amplitudes of surface plasmon and soliton. The total energy of this system can be calculated by considering the frequencies of the waves and the momentum, that they carry, while propagating. Moreover, the plasmonic system cannot be cast into classical Hamiltonian analytically, therefore this method is proved to be useless for the plasmonic twolevel system. The second method is to present a quartic equation by substituting ô€€€i d dt in Eq. (2.43) with , which is called the eigenenergy of the Hamiltonian. 0 @ Cp Cs 1 A = H(z) 0 @ Cp Cs 1 A; (2.57) After some elaboration the above equation yields the following quartic equation: ô€€€ 2 p 2 + p 2 + p ô€€€ 2 p ô€€€ 2 p + p 2 exp ô€€€2 p d2 + 2z2 q 2 ô€€€ p + p = 0 46 Chapter 2: Single Josephson Junction in a MetalDielectric Interface (2.58) where = 2 p+4 2ô€€€4 pô€€€ + p . Although the analytical solution of this equation cannot be presented, a numerical approach might be o ered, however this approach would be time ine cient due to the complexity of the equation, so another method is presented here. We start with the Eq.2.57, but instead of constructing it into the quartic equation, we divert it into a relation between and Z: = (1 + Z)2 ô€€€ p(1 ô€€€ Z) 2Z (2.59) Afterwards, we nd the xed points by equating the equations (2.42) and (2.43) equal to zero as previously explained and practiced. Then we plug in the xed points that we have found into the Eq. (2.59) as Z. The resulting values of are considered as the eigenenergies of the plasmonic twolevel system. The eigenenergies of the system with parameters used in Fig. 2.26 and the associated xed points are given in Fig. 2.28. Figure 2.28: Eigenenergies and the corresponding xed points. In order to understand, how the system behaves under di erent parameters, Fig. 2.29 could give insight into the e ect of E to the xed points of the system and the associated eigenvalues. Panels a)c) shows the xed points of the system with parameters d = 2:15; = 0:32, and = 0:1, whereas panels d)f) shows the eigenen ergies of the corresponding xed points. It can be seen that, the choice of E might create three di erent regimes. When E is negative, the rst xed point ( = 0) Chapter 2: Single Josephson Junction in a MetalDielectric Interface 47 always takes positive values. The second xed point, however, starts its evolution on the positive side, when the system is decoupled, then jumps into the negative side, and follows the path shown in panel a). When E is positive, the behaviour of xed points is contrary. Not only the rst xed point is on the negative side, its trajectory is upside down as well. Although the change in the E is symmetric, the ip of the trajectories are not. This is caused by the soliton dominancy in the governing equations of the system. An interesting behaviour manifests itself, when E = 0. The rst xed point becomes zero at all points, and the second xed point follows a trajectory from ô€€€1 to 0, and then to ô€€€1 again. At the e ective transition area, both xed points coincides. In panels d)f) associated eigenenergies of these xed points can be seen. E ect of E to these eigenenergies is similar to the e ect to xed points. Figure 2.29: E ect of E. Upper panels are xed points, lower panels are eigenener gies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; = 0:1. Figure 2.30 shows the behaviour of xed points, when the nonlinearity changes. To understand the e ect of nonlinearity is quite important, since it is directly related to the soliton propagation, s = 2 jCsj2. Unlike E, nonlinearity does not change the pattern of xed point trajectories, but slowly reduces the area, in which the second xed point reappears at Z = 1. Weak nonlinearity, see Fig. 2.30 panel d), causes a more symmetric eigenenergy pattern, whereas the strong nonlinear regime, see panel 48 Chapter 2: Single Josephson Junction in a MetalDielectric Interface f), causes an irregular pattern, in which the evolution along z axis follows the green dots, then around z = 20 there is no path to follow but to jump to upper or lower states. This behaviour, however, is not concluded with an adiabatic transfer of energy, because the system is symmetric with respect to z = 0, so when the evolution along zaxis reaches another dead end, it jumps one of the lower states, which coincides eventually at the value that is exactly equal its inital value at z = ô€€€100. Figure 2.30: E ect of . Upper panels are xed points, lower panels are eigenenergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 2:15; = 0:32; E = 0:02. In order to gain more perception about the e ect of and E, another set of parameters is presented in the following gures. Fig. 2.31 and 2.32 show the xed point and eigenenergy plots of the system in Fig. 2.19 d), where the system starts with equal soliton and surface plasmon amplitude, and after the transition all of the population is canalized to the soliton channel. The parameters are d = 14; = 0:297; = 0:02, and E = 0:03. Also note that, the parameters used in gures 2.29 and 2.30 are the same as the system in Fig. 2.19a, where the initial soliton amplitude is divided equally between soliton and surface plasmon channels after the transition. Chapter 2: Single Josephson Junction in a MetalDielectric Interface 49 Figure 2.31: E ect of E. Upper panels are xed points, lower panels are eigenener gies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; = 0:02. Figure 2.32: E ect of . Upper panels are xed points, lower panels are eigenenergies. Black and green dots in the lower panels correspond to blue and red dots in the upper panels, respectively. Here, d = 14; = 0:297; E = 0:03. Chapter 3 DOUBLE JOSEPHSON JUNCTION IN A METALDIELECTRICMETAL INTERFACE 3.1 Theoretical Background 3.1.1 Surface Plasmons in Multilayers Surface plasmon excitation on a metaldielectric interface has been discussed throughly in Chapter 2. The key property about this subject is that, when the tail of the evanes cent waves encounters the nonlinearity spatial solitons arise, and these two waves obey coupled oscillator equations. When the tail of the soliton wave encounters the metal surface, surface plasmons arise as well. This mechanism can also occur on a three level system, metaldielectricmetal interface, so that, surface plasmons on both sides can interact indirectly via optical soliton. The geometrical condition for coupled modes to occur from the interactions of these optical waves is that, the distance between neighboring interfaces should be comparable to the evanescent tails of the waves. To investigate the general features of coupled surface plasmonsoliton modes, the system in Fig. 3.1 should be analyzed [5], [6]. Surface plasmons occur only in TM modes, so the eld components of TM modes will be discussed. The electric and magnetic elds for z > a region are Hy(x) = Aei zeô€€€k3x; (3.1) Ez(x) = iA 1 ! 0 3 k3ei zeô€€€k3x; (3.2) Ex = ô€€€A ! 0 3 ei zeô€€€k3x; (3.3) for x < ô€€€a region Hy(x) = Bei zek2x; (3.4) Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 51 Figure 3.1: Illustration of a threelevel system consisting of a middle layer I trapped between two large half spaces II and III. Ez(x) = ô€€€iB 1 ! 0 2 k2ei zek2x; (3.5) Ex = ô€€€B ! 0 2 ei zek2x: (3.6) In the center region ô€€€a < x < a, the bottom and top modes couple, yielding Hy = Cei zek1x + Dei zeô€€€k1x; (3.7) Ez = ô€€€iC 1 ! 0 1 k1ei zek1x + iD 1 ! 0 1 k1ei zek1x; (3.8) Ex = C ! 0 1 k1ei zek1x + D ! 0 1 k1ei zek1x: (3.9) The boundary condition requires continuity of Hy and Ez at x = a, which leads to Aeô€€€k3a = Cek1a + Deô€€€k1a; (3.10) A 3 k3eô€€€k3a = ô€€€ C 1 k1ek1a + D 1 k1ek1a; (3.11) and at x = ô€€€a Beô€€€k2a = Ceô€€€k1a + Dek1a; (3.12) B 2 k2eô€€€k2a = ô€€€ C 1 k1eô€€€k1a + D 1 k1ek1a: (3.13) 52 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface We are left with a linear system, that consist of four coupled equations. The solution of these coupled equations comes with Hy ful lling the wave equation in the three speci c regions, via k2 i = 2 ô€€€ k2 0 i (3.14) for i = 1; 2; 3: The desired result of this analytical calculation can be achieved by solv ing this system of linear equations, which is an implicit expression for the dispersion relation relating and ! via eô€€€4k1a = k1= 1 + k2= 2 k1= 1 ô€€€ k2= 2 k1= 1 + k3= 3 k1= 1 ô€€€ k3= 3 : (3.15) In the special case of 2 = 3 and k2 = k3, the dispersion relation can be divided into a couple of equations tanh(k1a) = ô€€€ k2 1 k1 2 ; (3.16) tanh(k1a) = ô€€€ k1 2 k2 1 : (3.17) These equations can be applied to metal/dielectric/Kerr/dielectric/metal struc tures to investigate the surface plasmonsolitonsurface plasmon interaction, which forms a double Josephson junction. 3.1.2 BoseEinstein Condensates in TripleWell Traps BECs in doublewell structures have been discussed previously. The unique proper ties of BEC systems could earn advantages over superconducting Josephson junctions such as the possibility to investigate the nonlinear e ects in doublewell, or the con trol of particle number as it is impracticable in nonlinear waveguide systems. With these advantages many physical phenomena can be investigated such as LandauZener tunneling and macroscopic selftrapping. These phenomena could also be observed in triplewell systems, which have two structural types: chainshaped and ringshaped potentials. In the former, the wells are positioned alongside so the rst and the third well does not interact each other directly [53]. In the latter, the wells are positioned Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 53 on a triangle, so that all wells have a direct interaction. The ringshaped model has been investigated in detail [54], the xed points and the associated eigenstates have been found, and the phasespace of the system have been discussed through Poincare maps to determine chaos in the system. 3.2 Surface PlasmonSpatial Soliton Coupling in Multilayer Parallel Systems 3.2.1 Model Description Figure 3.2: Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. Single Josephson junction model between surface plasmon and soliton can be extended to a threelevel system by adding another metaldielectric interface to the 54 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface other side of the nonlinearity, which forms a double Josephson junction. In this way, the soliton in the middle can act like a bridge between two surface plasmons, resulting in rich dynamical results. The distance from the rst metal to the nonlinearity in the dielectric is held constant at d1, hence 1 = 0. The distance from the second metal to the nonlinearity is taken as both a parallel and a hyperbolical function of 2, and the minimum distance between the two layers is d2, as shown in Fig. 3.2. Both cases will be investigated in detail. In this threelevel system, two di erent coupling functions are de ned correspond ing two di erent surface plasmonspatial soliton interactions: q1(jCsj) = jCsjeô€€€d1 p =2jCsj; (3.18) q2(jCsj) = jCsjeô€€€d2 p =2jCsj: (3.19) The coupled oscillator equations, Eq. 2.36, previously used in twolayer system need to be adapted to threelayer system. The new equations are c 00 p1 + 2 p1cp1 = q1(jcsj)cs (3.20) c 00 p2 + 2 p2cp2 = q2(jcsj)cs (3.21) c 00 s + 2 s cs = q1(jcsj)cp1 + q2(jcsj)cp2 (3.22) Making the substitutions cp1 = Cp1eiz, cp2 = Cp2eiz, and cs = Cseiz, the equations yields the following nonlinear Schr odinger equation: ô€€€i 0 BBB@ _C p1 _C s _C p2 1 CCCA = 0 BBB@ p1 ô€€€q1 2 0 ô€€€q1 2 s ô€€€q2 2 0 ô€€€q2 2 p2 1 CCCA 0 BBB@ Cp1 Cs Cp2 1 CCCA (3.23) After the following substitutions is implemented, Cp1 = jCp1 jei p1 (3.24) Cp2 = jCp2 jei p2 (3.25) Cs = jCsjei s (3.26) Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 55 the governing equations yield _C p1 = q1 2 jCsjsin( s ô€€€ p1); (3.27) _C p2 = q2 2 jCsjsin( s ô€€€ p2); (3.28) _C s = q1 2 jCp1 jsin( p1 ô€€€ s) + q2 2 jCp2 jsin( p2 ô€€€ s); (3.29) _ p1 = ô€€€ E1 ô€€€ q1 2 jCsj jCp1 j cos( s ô€€€ p1); (3.30) _ p2 = ô€€€ E2 ô€€€ q2 2 jCsj jCp2 j cos( s ô€€€ p2); (3.31) _ s = 2 jCsj2 ô€€€ q1 2 jCp1 j jCsj cos( p1 ô€€€ s) ô€€€ q2 2 jCp2 j jCsj cos( p2 ô€€€ s): (3.32) Since solving these equations analytically may seem impossible, numerical ap proach is needed. An ODE solver function in MATLAB ode45 has been used to solve these di erential equations. With no dissipation in the system, subsequent transitions will occur between three channels. Depending on the parameter choice di erent proportions of energy transfer will be observed. Some of them are shown in Fig. 3.3 and Fig. 3.4. 3.2.2 Population Trapping In the metal/dielectric/metal interface without a spatial modi cation, a single tran sition between surface plasmon and soliton amplitudes cannot be achieved, since the distance is unchanged, and there is no energy dissipation, so a stationary state after the transition could not be reached. However, the system can be con gured into a signi cant state, in which the soliton amplitude oscillations can be suppressed into a stationary soliton propagation, so that the soliton channel acts like a bridge be tween two surface plasmon channels. The resonant transfer of energy occurs between surface plasmons via the soliton. This e ect can be obtained for a variety of sys tem parameters as well as the initial surface plasmon and the soliton amplitudes. This phenomenon is called population trapping, which is based on a quantum theory of light propagation in two optical waveguides [43],[44]. A common continuum of 56 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.3: Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:04; E1 = 0:03; E2 = ô€€€0:01; Ap0 ' 1; As ' 0 Figure 3.4: Dynamics of plasmonsolitonplasmon interaction. Parameters are d1 = 6; d2 = 6; = 0:05; E1 = 0:03; E2 = 0:03; Ap0 ' 1; As ' 0 modes coupled to both optical channels enables the tunneling between them [45]. For classical light waves, coupledmode equation analysis indicates that, the emergence Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 57 Figure 3.5: Dynamics of amplitudes of soliton and surface plasmons (a) and relative phase di erences (b). Paremeters are d1 = 8; d2 = 4; = 0:06; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:9; As0 0:45. Figure 3.6: Dynamics of amplitudes of soliton and surface plasmons (a) and relative phase di erences (b). Paremeters are d1 = 5:7; d2 = 6:3; = 0:06; E1 = 0:02; E2 = 0:02; Ap10 0:8; As0 0:6. of a trapped state placed in the continuum is caused by Fano interference between di erent light leakage channels. Results show that, plasmonic structures originally designed to mimic the quantum mechanical phenomena may exhibit themselves as proper analogues of their quantum counterparts. Figure 3.5 shows the rst example of population trapping in nondissipative paral lel metaldielectricmetal interface. As mentioned before, amplitudes of surface plas mons exchange their energies repetitively via the soliton bridge. In this con guration the soliton amplitude is arranged so that surface plasmon in the second metal sur 58 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.7: Dynamics of amplitudes of soliton and surface plasmons (a) and rel ative phase di erences (b). Paremeters are d1 = 5:9; d2 = 6:1; = 0:07; E1 = ô€€€0:03; E2 = ô€€€0:03; Ap10 0:7; As0 0:7. face is not excited initially, and the rst surface plasmon can share half of its energy at most. This makes the nodes and the antinodes of the oscillatory waves full and half populated states respectively. The antinodes are the points where a phaseshift occurs as shown in panel b. Figures 3.6 and 3.7 show the trapping of soliton population for completely di erent set of parameters indicating that, this phenomenon is not bounded to a particular system property. The relative phase di erences in these two examples have the same jumping points, whereas the behaviours of the phase evolution are di erent. The only constraint to this behaviour is that, the asymmetry between the soliton and both surface plasmons, E1 and E2, must be equal. An analytical approach might be insightful for explaining this constraint. In order to have a constant soliton amplitude, the derivative of Cs must equal to zero. _C s = q1 2 jCp1 jsin( p1 ô€€€ s) + q2 2 jCp2 jsin( p2 ô€€€ s) = 0: (3.33) There are a few ways to ensure this. Since coupling functions, q1 and q2, and the amplitudes, jCp1 j and jCp2 j, are nonzero, we must have both sinus functions equal to zero, which yields: 1 = 2 = 0; or : (3.34) Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 59 Unfortunately, we know from the phase di erence plots of the trapped systems, that 1 and 2 are not equal to 0 or at all times. This means that, the only way to have _C s equal to zero, is to obey the following condition: ô€€€q1jCp1 jsin( 1) = q2jCp2 jsin( 2) (3.35) It must be noted that, in the trapped soliton system coupling functions, q1 and q2, are constants, since they are only soliton amplitude dependent. If we look into the dynamic equations of phases and considering the above condition, it can be under stood that, the only di erent term in the two equations are E1 and E2 and why they must be equal in order to have a trapped soliton amplitude. _ 1 = 2 jCsj2ô€€€ q1 2 jCp1 j jCsj cos( p1ô€€€ s)ô€€€ q2 2 jCp2 j jCsj cos( p2ô€€€ s)ô€€€ ô€€€ E1ô€€€ q1 2 jCsj jCp1 j cos( sô€€€ p1); (3.36) _ 2 = 2 jCsj2ô€€€ q1 2 jCp1 j jCsj cos( p1ô€€€ s)ô€€€ q2 2 jCp2 j jCsj cos( p2ô€€€ s)ô€€€ ô€€€ E2ô€€€ q2 2 jCsj jCp2 j cos( sô€€€ p2): (3.37) 3.2.3 Classical Collapse and Revival of Coupling Function The investigation of two quantum states, that enable macroscopic nonlinear tunneling between themselves reveals that, the populations of one of the states might reduce to zero from time to time. However, the mean eld approximation is violated, when one of the populations become negligible, since the following is one of the assump tions of mean eld theory [42]: Large number of particles can be treated as a single averaged particle, which reduces a manybody problem to a onebody problem. This di culty can be seen in systems of twolevel atoms, that mimic the properties of a Josephson junction in which the amplitudes of the states oscillate between each other [47]. Throughout the years, many di erent systems have been used as a host for quantum collapserevival phenomenon such as a singlemode resonant eld prop agating in a Kerrlike medium [48], an e ective giant spin model constructed from 60 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface a coupled twomode BoseEinstein condensate, that possesses adiabatic and cyclic timevarying Raman coupling [49], BoseEinstein condensates trapped in a triplewell [50], or quantum bouncing ball on a truncated lattice [51]. In [47], a twodimensional optical lattice lled with BEC is taken as an example of a twolevel system with equal energy, which allows population oscillations, and this system is realized exper imentally to examine its dynamic behaviours. The quantization of the motion yields collapse and revival of these oscillations. The evolution of BEC atoms and quantum collapses can be seen in Fig. 3.8. Figure 3.8: Quantum collapse and revivals can be seen in BEC atoms [47]. In a) population imbalance (Z) of N = 350 BEC atoms can be seen. Most of the population is trapped in of the states. The relative phase of both states are equal. In b) N = 500 BEC atoms can be seen. Both states are equally populated during the collapse. The relative phases are equal. Some regimes of collapses, such as equal population or disbalanced population oscillations can be useful to predict the tunneling properties, since they don't violate the mean eld approximation. So, we shouldn't look for a collapse in the population of one of the states, but the collapse of the oscillations, in other words, the coupling function. However, the quantization of the motion in the above study still restricts us from obtaining collapse of coupling in the plasmonic Josephson junction, therefore a classical approach should be in order. A classical approach has been proposed in [52], where the analysis of wave packets in weakly anharmonic potentials is given. According to this study, when the wave packets propagate along classical trajectories in harmonic potential, they spread along Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 61 the motion, and regain their shape. Introducing a small anharmonicity to the system leads to a reversible quantum dephasing of wave packets, which is called revival. The amplitude of the oscillation dissapears between these revivals, which is called collapse. A realization of this study has been experimented with coherent A1g phonons in Bismuth. The dynamics of the phonon wave packet in a twoatom system is shown in Fig. 3.9. Figure 3.9: Oscillatory part of the z coordinate for di erent absorbed energies n0. Arrows indicate the amplitude collapse of the oscillations [52]. This study shows that, the system size is the key factor, which a ects the be haviour of the excited phonons, that approach the classical behavior swiftly. This behaviour indicates that, quantum e ects are not included in the revival of the os cillations, which is a substantial discovery, because collapse and revival phenomenon was originally considered a pure quantum e ect with no classical analogy [46]. The physical interpretation of this e ect was given as a linear combination of stationary states of a system. These linear combinations are determined as the coherent oscilla tions, that show collapse and revival properties. Furthermore an important natural speci cation of a stationary state is the number of quanta as a result of being a 62 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface quantum mechanical phenomenon. The changeover from quantum systems without a classical counterpart to quantum optical systems, and nally to optophotonic systems with classical e ects urge us to move to the next step: surface plasmonsoliton Josephson junctions. Unlike the quantum collapserevival phenomenon, the plasmonic system does not lose its total energy, however it is the interaction between these three states, ergo, the coupling function, that collapses as shown in gures 3.10  3.18. Figure 3.10: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 Fig. 3.10 shows an obvious collapserevival behaviour. The oscillations of ampli tudes between two surface plasmons and soliton channels manifest periodic patterns, however the nonoscillating parts, i.e., the length of the collapse time, di er from each other. In order to obtain this behaviour the following set of parameters are used: d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01, and E2 = 0:01. Initial surface plasmon amplitude in the rst metal surface carries the whole energy of the system, Ap10 1, therefore As0 0, and Ap20 0. The outcome of the CR behaviour di ers Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 63 Figure 3.11: Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 10:1; d2 = 1:9; = 0:03; E1 = 0:01; E2 = 0:01; Ap10 1; As0 0 from the population trapping, that has been discussed in the previous section. In the population trapping phenomenon, oscillations of the soliton amplitude reduces to zero, stabilizing the soliton population permanently, and in this period of time the amplitudes of surface plasmons continue oscillating. However in the CR phenomenon, not only the oscillations of the soliton amplitude, but the soliton population collapses to zero temporarily, and in this temporal interval of time oscillations of surface plas mons collapses to zero as well. The relative phase di erences between surface plasmons and the soliton should also be discussed. Although the e ect of relative phases to the double Josephson junctions will be explained in the following sections, a brief discourse on relative phase di erences in the CRsystems might be insightful. Fig. 3.11 shows the phases of the system, that was explained in Fig. 3.10. As can be understand from the gure, the relative phases between channels also manifest a nonoscillating behaviour during the collapse time. The fact that, both populations and phases of the optical channels 64 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.12: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 do not change their current state, indicates that, the system may fall into one of the xed points. Figure 3.12 also shows a CR behaviour, but with a more regular pattern. The collapse time between periodic islands seems closer to each other than Fig. 3.10. Both the period of those islands, z 1000, and the shape of the envelopes are similar, although system parameters are quite di erent. This is a sign that CR behaviour is not bound to speci c set of parameters, it can be observed in di erent geometry (d, ) and material properties ( , E). The relative phase di erence plot can be seen in Fig. 3.13. We have de ned collapserevival phenomenon in plasmonic Josephson junction as the collapse of the coupling between surface plasmons and the soliton. More to that, interesting patterns of system dynamics, that does not obey the CR behaviour might be worth mentioning. In Fig. 3.14 and 3.15, the oscillations of soliton draws a regular pattern so that, the oscillations of surface plasmons follows a sequential pattern, which creates a sinusoidal envelope. This sinusoidal envelope can also be Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 65 Figure 3.13: Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 3; d2 = 9; = 0:03; E1 = 0; E2 = 0; Ap10 0:2; As0 0 seen in Fig. 3.16 and 3.17 with strong soliton oscillations, and a very steep collapse and revival behaviour. And nally, Fig. 3.18 and 3.19 shows an irregular type of CR oscillations. 3.3 Surface PlasmonSpatial Soliton Coupling in Hyperbolically Mod ulated Multilayer Systems The rich dynamic properties of single Josephson junction formed by surface plasmon and spatial soliton previously discussed in Chapter 2 can also be investigated in double Josephson junctions with one of the metal surfaces being spatially modulated. In this system, nonlinearity is kept straight, indicating a parallel junction on one side and a hyperbolic junction on the other side as can be seen in Fig. 3.20. Dynamic evolution of the system can be described with a similar set of equations as the parallel system. _C p1 = q1 2 jCsjsin( s ô€€€ p1); (3.38) 66 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.14: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 Figure 3.15: Dynamics of relative phase di erences between soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 5:5; d2 = 6:5; = 0:03; E1 = 0; E2 = 0; Ap10 0:1; As0 0 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 67 Figure 3.16: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 Figure 3.17: Dynamics of relative phase di erences between soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 7:5; d2 = 4:5; = 0:03; E1 = ô€€€0:02; E2 = ô€€€0:02; Ap10 0:2; As0 0 68 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface Figure 3.18: Dynamics of amplitudes of soliton and surface plasmons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 Figure 3.19: Dynamics of relative phase di erences between soliton and surface plas mons in a collapse and revival case. Parameters are d1 = 4; d2 = 8; = 0:03; E1 = ô€€€0:02; E2 = 0; Ap10 0:1; As0 0 Chapter 3: Double Josephson Junction in a MetalDielectricMetal Interface 69 Figure 3.20: Two surface plasmons and a spatial soliton in a metal/dielectric/Kerr/dielectric/metal multilayer. _C p2 = q2 2 jCsjsin( s ô€€€ p2); (3.39) _C s = q1 2 jCp1 jsin( p1 ô€€€ s) + q2 2 jCp2 jsin( p2 ô€€€ s); (3.40) _ p1 = ô€€€ E1 ô€€€ q1 2 jCsj jCp1 j cos( s ô€€€ p1); (3.41) _ p2 = ô€€€ E2 ô€€€ q2 2 jCsj jCp2 j cos( s ô€€€ p2); (3.42) _ s = 2 jCsj2 ô€€€ q1 2 jCp1 j jCsj cos( p1 ô€€€ s) ô€€€ q2 2 jCp2 j jCsj cos( p2 ô€€€ s): (3.43) with a change in the coupling function q1(jCsj) = jCsjeô€€€d1 p =2jCsj; (3.44) q2(jCsj) = jCsjeô€€€ p 
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