Equilibrium bifurcations and chaotic transitions in coupled microlaser lattices

Analytic and simulation studies for the steady-state equilibria and bifurcations of coupled microlaser arrays are described. Lateral cavity interactions affect the gain in each cavity, leading to active photonic lattice behavior, equivalent to a nonlinear coupled oscillator lattice. The coupled-cavity rate equations are employed to follow the coherent photon and carrier population in each lattice site. Fixed-point-type steady states, of constant lattice phase shift, result for low coupling strengths; the radiation envelope for these states conforms with a periodic Bloch state over the array. Bifurcations to limit cycles of increasing complexity occur at higher coupling via period doubling sequences. The associated spatial patterns of photon and carrier lattice distribution resemble photonic convection cells. Limit cycles of different periods, emanating mathematically from different original fixed points, coexist at high strengths, each one accessible from different initial conditions. The multiplicity of possible limit cycles in systems with many degrees of freedom (number of lattice sites) combined with changes in their accessibility from initial conditions offers new insights to chaotic transitions, compared to low dimensionality paradigms.     

Exact Bloch States of a Spin-orbit Coupled Bose-Einstein Condensate in an Optical Lattice

We study the exact Bloch states of a spin-orbit (SO) coupled Bose-Einstein condensate (BEC) held in an optical lattice. Under a natural condition of the symmetry between the two species, we obtain two different forms of exact solutions corresponding to different existing conditions. Then, we analytically demonstrate that (a) the average atomic number per well can enlarge the region area (consisting of instability and stability parameter regions) existing exact solutions; (b) the sizes of the instability and stability parameter regions exhibit opposite variation trend with the increase in Rabi coupling strength, and the results of different solutions are just opposite. Besides, we find that spin-orbit coupling (SOC) results in the generation of spin-motion entanglement for the Bloch states, the SOC strength and lattice depth can influence the population transfer between two BEC components, and varying the SOC strength and lattice depth can also reveal the dynamical superfluid-insulator transition from the superfluid state to the critical insulating state. These results present a feasible scheme to manipulate the stable superfluid currents, which will be useful to control quantum transport of BEC.

     

Josephson array of mesoscopic objects. Modulation of system properties through the chemical potential

The phase diagram of a two-dimensional Josephson array of mesoscopic objects (superconducting granules, superfluid helium in a porous medium, traps with Bose-condensed atoms, etc.) is examined. Quantum fluctuations in both the modulus and phase of the superconducting order parameter are taken into account within a lattice boson Hubbard model. Modulating the average occupation number n ₀ of the sites in the system (the “number of Cooper pairs” per granule, the number of atoms in a trap, etc.) leads to changes in the state of the array, and the character of these changes depends significantly on the region of the phase diagram being examined. In the region where there are large quantum fluctuations in the phase of the superconducting order parameter, variation of the chemical potential causes oscillations with alternating superconducting (superfluid) and normal states of the array. On the other hand, in the region where the bosons interact weakly, the properties of the system depend monotonically on n ₀. Lowering the temperature and increasing the particle interaction force lead to a reduction in the width of the region of variation in n ₀ within which the system properties depend weakly on the average occupation number. The phase diagram of the array is obtained by mapping this quantum system onto a classical two-dimensional XY model with a renormalized Josephson coupling constant and is consistent with our quantum path-integral Monte Carlo calculations. Zh. éksp. Teor. Fiz. 114, 591–604 (August 1998)     

Novel spatial solitons in light-induced photonic bandgap structures

The study of wave propagation in periodic systems is at the frontiers of physics, from fluids to condensed matter physics, and from photonic crystals to Bose-Einstein condensates. In optics, a typical example of periodic system is a closely-spaced waveguide array, in which collective behavior of wave propagation exhibits many intriguing phenomena that have no counterpart in homogeneous media. Even in a linear waveguide array, the diffraction property of a light beam changes due to evanescent coupling between nearby waveguide sites, leading to normal and anomalous discrete diffraction. In a nonlinear waveguide array, a balance between diffraction and self-action gives rise to novel localized states such as spatial “discrete solitons” in the semi-infinite (or total-internal-reflection) gap or spatial “gap solitons” in the Bragg reflection gaps. Recently, in a series of experiments, we have “fabricated” closely-spaced waveguide arrays (photonic lattices) by optical induction. Such photonic structures have attracted great interest due to their novel physics, link to photonic crystals, as well as potential applications in optical switching and navigation. In this review article, we present a brief overview on our experimental demonstrations of a number of novel spatial soliton phenomena in light-induced photonic bandgap structures, including self-trapping of fundamental discrete solitons and more sophisticated lattice gap solitons. Much of our work has direct impact on the study of similar discrete phenomena in systems beyond optics, including sound waves, water waves, and matter waves (Bose-Einstein condensates) propagating in periodic potentials.      

Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models

The semiclassical quantization of cyclotron orbits for two-dimensional Bloch electrons in a coupled two band model with a particle-hole symmetric spectrum is considered. As concrete examples, we study graphene (both mono and bilayer) and boron nitride. The main focus is on wave effects – such as Berry phase and Maslov index – occurring at order (^h/_2p)\hbar in the semiclassical quantization and producing non-trivial shifts in the resulting Landau levels. Specifically, we show that the index shift appearing in the Landau levels is related to a topological part of the Berry phase – which is basically a winding number of the direction of the pseudo-spin 1/2 associated to the coupled bands – acquired by an electron during a cyclotron orbit and not to the complete Berry phase, as commonly stated. As a consequence, the Landau levels of a coupled band insulator are shifted as compared to a usual band insulator. We also study in detail the Berry curvature in the whole Brillouin zone on a specific example (boron nitride) and show that its computation requires care in defining the “k-dependent Hamiltonian” H(k), where k is the Bloch wavevector.     

Repulsive synchronization in an array of phase oscillators

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.     

Diffusion of a Massive Quantum Particle Coupled to a Quasi-Free Thermal Medium

We consider a heavy quantum particle with an internal degree of freedom moving on the d-dimensional lattice _boxclose^d{{\mathbb Z}^d} (e.g., a heavy atom with finitely many internal states). The particle is coupled to a thermal medium (bath) consisting of free relativistic bosons (photons or Goldstone modes) through an interaction of strength λ linear in creation and annihilation operators. The mass of the quantum particle is assumed to be of order λ⁻², and we assume that the internal degree of freedom is coupled “effectively” to the thermal medium. We prove that the motion of the quantum particle is diffusive in d ≥ 4 and for λ small enough.     

Theoretical analysis of the effects of frequency detuning dependence of photorefractive two- beam coupling coefficient on gain and phase-shifts of the signal beam for a coupled unidirectional photorefractive ring resonators

Dependence of two-beam coupling gain and phase-shift of the signal beams on the frequency detuning for a coupled unidirectional ring resonators based on non-degenerate two-wave mixing in the photorefractive crystals have been investigated in details. The effects of various parameters characterizing the photorefractive medium such as frequency detuning, absorption strength, two-beam energy coupling strength and pump intensity of the external laser beams, on the two-beam coupling gain and phase-shift of the signal beams for a coupled UPRR have also been studied in details. It has been found that the photorefractive gain of the signal beam in the primary cavity of the coupled UPRR can be enhanced to the higher order by taking the lower value of the frequency detuning of the primary cavity which could exist at much lower value of the absorption strength of the crystal B. This higher value of photorefractive gains in the cavities are responsible for the strong coupling between the modes of the oscillations of the coupled UPRR.     

Superpositions and Entanglement of Mesoscopic High-order Squeezed Vacuum States in Cavity QED

We propose a scheme for generating the superpositions and the entanglement between the mesoscopic high-order squeezed vacuum states by considering the multi-photon interaction of N two-level atoms in a cavity with high quality factor, assisted by a strong driving field. In terms of specific choices of the cavity detuning, many multiparty entangled states between the atoms and the mesoscopic high-order squeezed vacuum states and among the high-order squeezed vacuum states of the cavity modes can be generated, including the macroscopic “Schr?dinger cats” of the mesoscopic high-order squeezed vacuum states, the entanged states of the macroscopic “Schr?dinger cats”, and so on. Our scheme is achievable within the current techniques in the cavity QED.     

Small bipolarons in the 2-dimensional Holstein-Hubbard model. I. The adiabatic limit

The spatially localized bound states of two electrons in the adiabatic two-dimensional Holstein-Hubbard model on a square lattice are investigated both numerically and analytically. The interplay between the electron-phonon coupling g, which tends to form bipolarons and the repulsive Hubbard interaction , which tends to break them, generates many different ground-states. There are four domains in the phase diagram delimited by first order transition lines. Except for the domain at weak electron-phonon coupling (small g) where the electrons remain free, the electrons form bipolarons which can 1) be mostly located on a single site (small , large g); 2) be an anisotropic pair of polarons lying on two neighboring sites in the magnetic singlet state (large , large g); or 3) be a “quadrisinglet state” which is the superposition of 4 electronic singlets with a common central site. This quadrisinglet bipolaron is the most stable in a small central domain in between the three other phases. The pinning modes and the Peierls-Nabarro barrier of each of these bipolarons are calculated and the barrier is found to be strongly depressed in the region of stability of the quadrisinglet bipolaron. Received 10 December 1998     

Instabilities and sound speed of a Bose-Einstein condensate in the Kronig-Penney potential

We analyze the full set of Bloch wave stationary solutions for a Bose-Einstein condensate in the Kronig-Penney potential. We investigate the Landau instability and dynamical instability of the Bloch states in the lowest Bloch band, including the loop if it appears. The stability phase diagrams are shown to be similar as in the case of the sinusoidal optical lattice potential. We also compute the speed of sound as a function of the potential strength. The trend is shown to be similar to the sinusoidal case, reflecting our general conclusion that, in any one-dimensional periodic potential, the sound speed always falls monotonically with increasing potential strength, no matter whether the atomic interaction is weak or strong. The Kronig-Penney potential, being analytically tractable and hence more advantageous than the sinusoidal potential, therefore serves as a good model for understanding the phenomena of Bose-Einstein condensation.     

Entanglement Dynamics Induced by a Squeezed Coherent Cavity Coupled Nonlinearly with a Qubit and Filled with a Kerr-Like Medium

An analytical solution for a master equation describing the dynamics of a qubit interacting with a nonlinear Kerr-like cavity through intensity-dependent coupling is established. A superposition of squeezed coherent states is propped as the initial cavity field. The dynamics of the entangled qubit-cavity states are explored by negativity for different deformed function of the intensity-dependent coupling. We have examined the effects of the Kerr-like nonlinearity and the qubit-cavity detuning as well as the phase cavity damping on the generated entanglement. The intensity-dependent coupling increases the sensitivity of the generated entanglement to the phase-damping. The stability and the strength of the entanglement are controlled by the Kerr-like nonlinearity, the qubit-cavity detuning, and the initial cavity non-classicality. These physical parameters enhance the robustness of the qubit-cavity entanglement against the cavity phase-damping. The high initial cavity non-classicality enhances the robustness of the qubit-cavity entanglement against the phase-damping effect.     

Vortex solutions of the discrete Gross-Pitaevskii equation starting from the anti-continuum limit

In this paper, we consider the existence, stability and dynamical evolution of dark vortex states in the two-dimensional defocusing discrete nonlinear Schrödinger model, a model of interest both to atomic physics and to nonlinear optics. Our considerations are chiefly based on initializing such vortex configurations at the anti-continuum limit of zero coupling between adjacent sites, and continuing them to finite values of the coupling. Systematic tools are developed for such continuations based on amplitude-phase decompositions and explicit solvability conditions enforcing the vortex phase structure. Regarding the linear stability of such nonlinear waves, we find that in a way reminiscent of their 1d analogs, i.e., of discrete dark solitons, the discrete defocusing vortices become unstable past a critical coupling strength and, subsequently feature a cascade of alternating stabilization-destabilization windows for any finite lattice. Although the results are mainly geared towards the uniform case, we also consider the effect of harmonic trapping potentials often present in experimental atomic physics settings.     

Phase and frequency shifts of two nonlinearly coupled oscillators

We analyze two nonlinearly phase coupled oscillators with eigenfrequencies ω₁and ω₂, where n\gw₁=m\gw₂+\gp, with integern andm. For \gh=0 there are up to four stable synchronized states which differ from each other only by the difference of the oscillators\rs phases. The number of different synchronized states depends on the coupling constants. If \gh does not vanish phase shifts and frequency shifts may occur givig rise to stable synchronized states which also differ from each other due to the frequencies. By means of the center manifold theorem we calculate these shifts explicitely. Different coupling constants are investigated: symmetrical, homogenously asymmetrical and arbitrary coupling constants. Our results point out the decisive influence of the symmetry of the coupling constants upon the frequency and phase shifts. Moreover the local stability of the unperturbed synchronized state (i.e. for \gh=0) determines the magnitude of the frequency and phase shifts.     

Bloch oscillations and Wannier-Stark ladder in the coupled LC circuits

We present a new scheme for realizing Bloch oscillations and Wannier-Stark ladder based on a lattice of coupled LC circuits. By converting the second order dynamical ODEs of the system into a first order Schrödinger-like equation, we propose an equivalent tight binding Hamiltonian to describe the circuit. We show that a synthesized electric field is produced by introducing a frequency mismatch into the resonant frequency of the adjacent LC resonators. The Wannier-Stark modes are the normal modes of the circuit and the Bloch oscillations can be observed in a coupled LC lattice. By addition of coupling capacitors between nodes of the circuit, we study the Bloch oscillation in the presence of long-range couplings. We also show that the circuit converts to a transmission line simulating synthetic electric fields in the continuum limit. The coupled LC circuit is, in some sense, amongst the simplest physical systems exhibiting Bloch oscillation and Wannier-Stark Ladder.     

Crossovers between superconducting symmetry classes

We study the average density of states in a small metallic grain coupled to two superconductors with the phase difference π, in a magnetic field. The spectrum of the low-energy excitations in the grain is described by the random matrix theory whose symmetry depends on the magnetic field strength and coupling to the superconductors. In the limiting cases, a pure superconducting symmetry class is realized. For intermediate magnetic fields or couplings to the superconductors, the system experiences a crossover between different symmetry classes. With the help of the supersymmetric σ-model we derive the exact expressions for the average density of states in the crossovers between the symmetry classes A-C and CI-C.     

Engineering bands of extended electronic states in a class of topologically disordered and quasiperiodic lattices

We show that a discrete tight-binding model representing either a random or a quasiperiodic array of bonds can have the entire energy spectrum or a substantial part of it absolutely continuous, populated by extended eigenfunctions only, when atomic sites are coupled to the lattice locally, or non-locally from one side. The event can be fine-tuned by controlling only the host–adatom coupling in one case, while in two other cases cited here an additional external magnetic field is necessary. The delocalization of electronic states for the group of systems presented here is sensitive to a subtle correlation between the numerical values of the Hamiltonian parameters – a fact that is not common in the conventional cases of Anderson localization. Our results are analytically exact, and supported by numerical evaluation of the density of states and electronic transmission coefficient.     

Dynamics of spatial structures in arrays of nonlocally coupled bistable units

We present the results of investigation of a dynamical system consisting of nonlocally coupled bistable units. The dynamics of spatial structures in an array with a gradually increasing coupling coefficient is studied. The formation of spatial structures in the case of strong nonlocal coupling between units is examined. This work was presented at the Summer Workshop “Dynamic Days” (Nizhny Novgorod, June 30–July 2, 1998). Volga State Academy of Water Transport, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 12, pp. 1581–1585, December, 1998.