Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations

Nonlinear waves in deformed optical honeycomb lattices are investigated. Discrete couple mode equations are used to find higher order continuous nonlinear Dirac systems which are employed to describe key underlying phenomena. For weak deformation and nonlinearity the wave propagation is circular–ellliptical. At strong nonlinearity the diffraction pattern becomes triangular in structure which is traced to appropriate nonequal energy propagation in momentum space. At suitably large deformation the dispersion structure can have nearby Dirac points or small gaps. The effective dynamics of the wave packets is described by two maximally balanced nonlocal nonlinear Schrödinger type equations.

We discuss nonlinear propagation of light beams in anisotropic media, addressing the role of nonlocality and nonlinearity in power-dependent beam self-steering. With specific reference to spatial solitons in positive uniaxial nematic liquid crystals (i.e. nematicons), we describe soliton self-acceleration through reorientational response and nonlinear walk-off.

Self-localized states or dissipative solitons have the freedom of translation in systems with a homogeneous background. When compared to cavity solitons in coherently driven nonlinear optical systems, laser cavity solitons have the additional freedom of the optical phase. We explore the consequences of this additional Goldstone mode and analyze experimentally and numerically frequency and phase locking of laser cavity solitons in a vertical-cavity surface-emitting laser with frequency-selective feedback. Due to growth-related variations of the cavity resonance, the translational symmetry is usually broken in real devices. Pinning to different defects means that separate laser cavity solitons have different frequencies and are mutually incoherent. If two solitons are close to each other, however, their interaction leads to synchronization due to phase and frequency locking with strong similarities to the Adler-scenario of coupled oscillators.

We consider light transmission in 2D photonic crystal waveguide coupled with two identical nonlinear defects positioned symmetrically aside the waveguide. We show that with growth of injected light power there is a breaking of symmetry by two ways. In the first way the symmetry is broken because of different light intensities at the defects. In the second way the intensities at the defects are equaled but phases of complex amplitudes are different. That results in a vortical power flow between the defects similar to the DC Josephson effect if the input power over the waveguide is applied and the defects are coupled. As application of these phenomena we consider the symmetry breaking for the light transmission in a T-shaped photonic waveguide with two nonlinear defects. We demonstrate as this phenomenon can be explored for all-optical switching of light transmission from the left output waveguide to the right one by application of input pulses. Finally we consider the symmetry breaking in the waveguide coupled with single defect presented however by two dipole modes.

In 1998, seminal work [1] had demonstrated that non-Hermitian Hamiltonians (\( \hat{H}^{\dag } \ne \hat{H} \)) can give rise to entirely real eigenvalue spectra (thus being appropriate for physical applications), provided that they obey the condition of the parity-time (PT) symmetry, i.e., \( \hat{H}( - {\mathbf{r}}) = \hat{H}^{\dag } ({\mathbf{r}}) \). This condition implies that the Hermitian and anti-Hermitian parts of the Hamiltonian are spatially even and odd, respectively. Such a Hamiltonian usually features spontaneous breaking of the PT-symmetry at a critical value of the coefficient accounting for its anti-Hermitian part. Above the critical point, the spectrum is no longer completely real [2–6].

We consider spontaneous symmetry breaking (SSB) in nonlinear periodic structures with two embedded identical defects. We focus on Bragg grating (BGs) in which the defects are formed by local phase shifts. The defects are positioned relatively close to each other, so as to allow the light to couple between them. At low optical energies, i.e., in the linear regime, this system supports two symmetric eigenstates, which have identical intensity distributions but different frequencies. At higher energies, the lower-frequency state becomes unstable against symmetry-breaking perturbations, and the light gets predominantly trapped by one of the defects, leading to an asymmetric field distribution. We analyze the SSB effect for different coupling strengths and conclude that, quite naturally, the symmetry-breaking energy threshold increases with the strength. The symmetric state is stable below the SSB threshold, while the emerging asymmetric mode is stable above the threshold.

We review numerous physical phenomena which occur in one- and two-dimensional nonlinear media in the presence of localized gain-defects, and which recently attracted considerable attention. In particular, we discuss stable localized modes in media with linear and nonlinear dissipation; breathers which can be excited in the presence of more than one localized gain channels; vortices. We address the phenomenon of the symmetry breaking in one- and two-dimensional media, resulting in emergence of stable nonsymmetric modes, and analyze possibilities of guiding and switching of waves with help of guiding channels.

In this chapter we review some recent results for the construction of analytical solutions for a class of systems consisting of interlaced linear and nonlinear parts. Periodic waveguide arrays as well as structures consisting of semi-infinite waveguide arrays and their interfaces have been studied, while the method presented here can be applied to even larger classes of systems including combinations of parts of waveguide arrays, homogeneous parts and defects as well as different types of nonlinearities. The method utilizes the phase space description of the system for the construction of analytical solutions. Such solutions can serve as starting points for the exploration of even larger classes of solutions and systems with the utilization of perturbation methods. Moreover, the method provides physical intuition for the formation of solitary waves in such structures.

The main subject of this contribution is the all-optical control over the state of polarization (SOP) of light, understood as the control over the SOP of a signal beam by the SOP of a pump beam. We will show how the possibility of such control arises naturally from a vectorial study of pump-probe Raman interactions in optical fibers. Most studies on the Raman effect in optical fibers assume a scalar model, which is only valid for high-PMD fibers (here, PMD stands for the polarization-mode dispersion). Modern technology enables manufacturing of low-PMD fibers, the description of which requires a full vectorial model. Within this model we gain full control over the SOP of the signal beam. In particular we show how the signal SOP is pulled towards and trapped by the pump SOP. The isotropic symmetry of the fiber is broken by the presence of the polarized pump. This trapping effect is used in experiments for the design of new nonlinear optical devices named Raman polarizers. Along with the property of improved signal amplification, these devices transform an arbitrary input SOP of the signal beam into one and the same SOP towards the output end. This output SOP is fully controlled by the SOP of the pump beam. We overview the state-of-the-art of the subject and introduce the notion of an “ideal Raman polarizer.”

In this chapter, we provide an overview of recent studies of theoretical models which adequately describe the temporal dynamics of circularly polarized few-cycle optical solitons in both long-wave- and short-wave-approximation regimes, beyond the framework of slowly varying envelope approximation. In the long-wave-approximation regime, i.e., when the frequency of the transition is far above the characteristic wave frequency, by using the multiscale analysis (reductive perturbation method), we show that propagation of circularly polarized (vectorial) few-cycle pulses, is described by the nonintegrable complex modified Korteweg–de Vries equation. In the short-wave-approximation regime, i.e., when the frequency of the transition is far below the characteristic wave frequency, by using the multiscale analysis, we derive from the Maxwell-Bloch equations the governing nonlinear evolution equations for the two polarization components of the electric field, in the first order of the perturbation approach. In this latter case we show that propagation of circularly-polarized few-optical-cycle solitons is described by a system of coupled nonlinear evolution equations, which reduces, for the particular case of scalar solitons, to the completely integrable sine-Gordon equation describing the dynamics of linearly polarized few-cycle pulses in the short-wave-approximation regime. It is seen that, from the slowly varying envelope approximation down to a few cycles, circularly polarized solitons are very robust, according to rotation symmetry and conservation of the angular momentum. However, in the sub-cycle regime, they become unstable, showing a spontaneous breaking of the rotation symmetry.

We consider a simplified model of a nonlinear magnetic metamaterial, consisting of a weakly-coupled, periodic split-ring resonator (SRR) array capable of nonlinear capacitive response. We analyze three related problems: (a) The calculation of localized modes around simple magnetoinductive impurities located at the surface or at the bulk of the array, in closed form for both, linear and nonlinear cases. (b) The scattering of magnetoinductive waves across internal (external) capacitive (inductive) defects coupled to the SRR array and the occurrence of Fano resonances, and how to tune them by changing the external parameters of the system. (c) Description of a method for building a stable localized magnetoinductive mode embedded in the linear band of extended states.

The onset of patterns due to a localized gain is studied analytically in the framework of a real or complex amplitude equation. The instability modes are found, and the bifurcation of small localized solutions is analyzed. Some exact solutions are also found.

We study spontaneous pattern formation and symmetry breaking in broad area and pre-patterned (spatially modulated) semiconductor microcavities under lasing conditions. In broad area VCSELs, we observe the spontaneous formation of regular arrays consisting of charge “±1” optical vortices. The formation of these patterns stems from transverse mode locking of almost wavelength degenerated Gauss-Laguerre (GL) modes. The observed patterns in Gain modulated broad area VCSELs and their dynamical behavior depends dramatically on the modulation strength. In ring shaped VCSELs lasers we observe necklace-like pattern formation and switching as a function of the injection current. The formation of the patterns and, in particular, their switching is shown to stem from stability loss of the lasing pattern to perturbations of more complex pattern which, in turn, is stable under similar pumping conditions. Having the advantage of a strong, saturating nonlinear response with an inherent loss compensation mechanism, such lasers are potentially the best microlabortories for studying nonlinear phenomena and for the generation and employment of complex optical fields. Applications can be found in optical data storage, information distribution and processing, laser cooling and more.

In this chapter, we describe a very promising approach to achieve deep sub-wavelength confinement of the optical field guided by plasmonic nanostructures. In the plasmonic nanostructures investigated in our review, namely, one-dimensional (1D) and two-dimensional (2D) arrays of closely spaced parallel metallic nanowires embedded in an optical medium with Kerr nonlinearity, the optical nonlinearity induced by the evanescent component of the guided modes of the nanowires exactly balances the discrete diffraction due to the optical coupling among neighboring metallic nanowires. As a result, nonlinear optical modes, called plasmonic lattice solitons (PLSs), are formed in the plasmonic array. Because the radius of the nanowires and their separation distance could be much smaller than the operating wavelength the size of the PLSs can be deep in the subwavelength regime. We present fundamental (vorticityless) PLSs in both 1D and 2D plasmonic arrays, and also vortical PLSs in 2D arrays, in both focusing and defocusing nonlinear media. We demonstrate that the spatial extent of fundamental and vortical PLSs could be in the deep-subwavelength regime under experimental accessible conditions. Moreover, their existence, stability, and spatial confinement are studied in detail. Our analysis employs a model based on the coupled-mode theory as well as the full set of Maxwell equations, and shows that the predictions of the two models are in excellent agreement for relatively large nanowires separations. We expect that these nonlinear plasmonic modes have important applications to subwavelength nanophotonics. In particular, we demonstrate that the subwavelength PLSs can be used to optically manipulate with nanometer accuracy the power flow in ultra-compact photonic devices.

The process of exciting the gas of trapped bosons from an equilibrium initial state to strongly nonequilibrium states is described as a procedure of symmetry restoration caused by external perturbations. Initially, the trapped gas is cooled down to such low temperatures, when practically all atoms are in Bose–Einstein condensed state, which implies the broken global gauge symmetry. Excitations are realized either by imposing external alternating fields, modulating the trapping potential and shaking the cloud of trapped atoms, or it can be done by varying atomic interactions by means of Feshbach resonance techniques. Gradually increasing the amount of energy pumped into the system, which is realized either by strengthening the modulation amplitude or by increasing the excitation time, produces a series of nonequilibrium states, with the growing fraction of atoms for which the gauge symmetry is restored. In this way, the initial equilibrium system, with the broken gauge symmetry and all atoms condensed, can be excited to the state, where all atoms are in the normal state, with completely restored gauge symmetry. In this process, the system, starting from the regular superfluid state, passes through the states of vortex superfluid, turbulent superfluid, heterophase granular fluid, to the state of normal chaotic fluid in turbulent regime. Both theoretical and experimental studies are presented.

In recent years, bright soliton-like structures composed of gaseous Bose–Einstein condensates have been generated at ultracold temperature. The experimental capacity to precisely engineer the nonlinearity and potential landscape experienced by these solitary waves offers an attractive platform for fundamental study of solitonic structures. The presence of three spatial dimensions and trapping implies that these are strictly distinct objects to the true soliton solutions. Working within the zero-temperature mean-field description, we explore the solutions and stability of bright solitary waves, as well as their interactions. Emphasis is placed on elucidating their similarities and differences to the true bright soliton. The rich behaviour introduced in the bright solitary waves includes the collapse instability and asymmetric collisions. We review the experimental formation and observation of bright solitary matter waves to date, and compare to theoretical predictions. Finally we discuss some topical aspects, including beyond-mean-field descriptions, symmetry breaking, exotic bright solitary waves, and proposals to exploit bright solitary waves in interferometry and as surface probes.

We formulate a semiclassical approach to study the dynamics of coherence loss and revival in a Bose-Josephson dimer. The phase-space structure of the bi-modal system in the Rabi, Josephson, and Fock interaction regimes, is reviewed and the prescription for its WKB quantization is specified. The local density of states (LDOS) is then deduced for any given preparation from its semiclassical projection onto the WKB eigenstates. The LDOS and the non-linear variation of its level-spacing are employed to construct the time evolution of the initial preparation and study the temporal fluctuations of interferometric fringe visibility. The qualitative behavior and characteristic timescales of these fluctuations are set by the pertinent participation number, quantifying the spectral content of the preparation. We employ this methodology to study the Josephson-regime coherence dynamics of several initial state preparations, including a Twin-Fock state and three different coherent states that we denote as ‘Zero’, ‘Pi’, and ‘Edge’ (the latter two are both on-separatrix preparations, while the Zero is the standard ground sate preparation). We find a remarkable agreement between the semiclassical predictions and numerical simulations of the full quantum dynamics. Consequently, a characteristic distinct behavior is implied for each of the different preparations.

We analyze the effects of temperature on the properties of a system of ultracold atoms confined by a double-well potential. We consider the case of repulsive interactions and review the different approximations to the exact many-body results.

Two effectively one-dimensional parallel coupled Bose–Einstein condensates in the presence of external potentials are studied. The system is modelled by linearly coupled Gross–Pitaevskii equations. In particular, the interactions of grey-soliton-like solutions representing analogues of superconducting Josephson fluxons as well as coupled dark solitons are discussed. A theoretical approximation based on variational formulations to calculate the oscillation frequency of the grey-soliton-like solution is derived and a qualitatively good agreement with numerics is obtained.

We study the existence, stability, and dynamics of symmetric and anti-symmetric states of quasi-one-dimensional polariton condensates in double-well potentials, in the presence of nonresonant pumping and nonlinear damping. Some prototypical features of the system, such as the bifurcation of asymmetric solutions, are similar to the Hamiltonian analog of the double-well system considered in the realm of atomic condensates. Nevertheless, there are also some nontrivial differences including, e.g., the unstable nature of both the parent and the daughter branch emerging in the relevant pitchfork bifurcation for slightly larger values of atom numbers. Another interesting feature that does not appear in the atomic condensate case is that the bifurcation for attractive interactions is slightly sub-critical instead of supercritical. These conclusions of the bifurcation analysis are corroborated by direct numerical simulations examining the dynamics of the system in the unstable regime.

The stability analysis of a generalized Dicke model, in the semi-classical limit, describing the interaction of a two-species Bose-Einstein condensate driven by a quantized field in the presence of Kerr and spontaneous parametric processes is presented. The transitions from Rabi to Josephson dynamics are identified depending on the relative value of the involved parameters. Symmetry-breaking dynamics are shown for both types of coherent oscillations due to the quantized field and nonlinear optical processes.

This chapter is devoted to the study of vortex excitations in one-component Bose-Einstein condensates, with a special emphasis on the impact of anisotropic confinement on the existence, stability and dynamical properties of vortices and particularly few-vortex clusters. Symmetry breaking features are pervasive within this system even in its isotropic installment, where cascades of symmetry breaking bifurcations give rise to the multi-vortex clusters, but also within the anisotropic realm which naturally breaks the rotational symmetry of the multi-vortex states. Our first main tool for analyzing the system consists of a weakly nonlinear (bifurcation) approach which starts from the linear states of the problem and examines their continuation and bifurcation into novel symmetry-broken configurations in the nonlinear case. This is first done in the isotropic limit and the modifications introduced by the anisotropy are subsequently presented. The second main tool concerns the highly nonlinear regime where the vortices can be considered as individual topologically charged "particles" which precess within the parabolic trap and interact with each other, similarly to fluid vortices. The conclusions stemming from both the bifurcation and the interacting particle picture are corroborated by numerical computations which are also used to bridge the gap between these two opposite-end regimes.

We study the dynamics of matter waves in an effectively one-dimensional Bose–Einstein condensate in a double well potential. We consider in particular the case when one of the double wells confines excited states. Similarly to the known ground state oscillations, the states can tunnel between the wells experiencing the physics known for electrons in a Josephson junction, or be self-trapped. Numerical existence and stability analysis based on the full equation is performed, where it is shown that such tunneling can be stable. Through a numerical path following method, unstable tunneling is also obtained in different parameter regions. A coupled-mode system is derived and compared to the numerical observations. The validity regions of the two-mode approximation are discussed.

We consider a parametrically driven damped discrete nonlinear Schrödinger (PDDNLS) equation. Analytical and numerical calculations are performed to determine the existence and stability of fundamental discrete bright solitons. We show that there are two types of onsite discrete soliton, namely onsite type I and II. We also show that there are four types of intersite discrete soliton, called intersite type I, II, III, and IV, where the last two types are essentially the same, due to symmetry. Onsite and intersite type I solitons, which can be unstable in the case of no dissipation, are found to be stabilized by the damping, whereas the other types are always unstable. Our further analysis demonstrates that saddle-node and pitchfork (symmetry-breaking) bifurcations can occur. More interestingly, the onsite type I, intersite type I, and intersite type III–IV admit Hopf bifurcations from which emerge periodic solitons (limit cycles). The continuation of the limit cycles as well as the stability of the periodic solitons are computed through the numerical continuation software Matcont. We observe subcritical Hopf bifurcations along the existence curve of the onsite type I and intersite type III–IV. Along the existence curve of the intersite type I we observe both supercritical and subcritical Hopf bifurcations.

Saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions are analyzed. First, general conditions for these bifurcations are derived. Second, it is shown analytically that the linear stability of these solitary waves does not switch at saddle-node bifurcations, which is in stark contrast with finite-dimensional dynamical systems where stability switching takes place. Third, it is shown that this absence of stability switching does not contradict the Vakhitov–Kolokolov stability criterion or the results in finite-dimensional dynamical systems. Fourth, it is shown that this absence of stability switching holds not only for real potentials but also for complex potentials. Lastly, various numerical examples will be given to confirm these analytical findings. In particular, saddle-node bifurcations with both branches of solitary waves being stable will be presented.

In this Chapter we investigate with the methods of signal detection the response of a Josephson junction to a perturbation to decide if the perturbation contains a coherent oscillation embedded in the background noise. When a Josephson Junction is irradiated by an external noisy source, it eventually leaves the static state and reaches a steady voltage state. The appearance of a voltage step allows to measure the time spent in the metastable state before the transition to the running state, thus defining an escape time. The distribution of the escape times depends upon the characteristics of the noise and the Josephson junction. Moreover, the properties of the distribution depends on the features of the signal (amplitude, frequency and phase), which can be therefore inferred through the appropriate signal processing methods. Signal detection with JJ is interesting for practical purposes, inasmuch as the superconductive elements can be (in principle) cooled to the absolute zero and therefore can add (in practice) as little intrinsic noise as refrigeration allows. It is relevant that the escape times bear a hallmark of the noise itself. The spectrum of the fluctuations due to the intrinsic classical (owed to thermal or environmental disturbances) or quantum (due to the tunnel across the barrier) sources are different. Therefore, a careful analysis of the escape times could also assist to discriminate the nature of the noise.

The symmetric and asymmetric buckling of micro beams subjected to distributed electrostatic force is studied. The analysis is carried out for two separate cases: a case of a stress-free beam, which is initially curved by fabrication and a case of a pre-stressed beam buckled due to an axial force. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with vibrational or buckling modes of a straight beam used as the base functions. The criteria of symmetric, limit point, buckling and of non-symmetric bifurcation are derived in terms of the geometric parameters of the beams. While the necessary symmetry breaking criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric limit point buckling curve, the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the symmetric equilibrium path. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through. A comparison between the results provided by the reduced order model, and those obtained by other numerical analyses confirms the accuracy of the symmetry breaking criteria and their applicability for the analysis and design of micro beams.