Controlling chaos in a weakly coupled array of Bose-Einstein condensates

Interplay between spin-orbit couplings and residual interatomic interactions in the modulational instability of two-component Bose-Einstein condensates

The nonlinear dynamics induced by the modulation instability (MI) of a binary mixture in an atomic Bose-Einstein condensate (BEC) is investigated theoretically under the joint effects of higher-order residual nonlinearities and helicoidal spin-orbit (SO) coupling in a regime of unbalanced chemical potential. The analysis relies on a system of modified coupled Gross-Pitaevskii equations on which the linear stability analysis of plane-wave solutions is performed, from which an expression of the MI gain is obtained. A parametric analysis of regions of instability is carried out, where effects originating from the higher-order interactions and the helicoidal spin-orbit coupling are confronted under different combinations of the signs of the intra- and intercomponent interaction strengths. Direct numerical calculations on the generic model support our analytical predictions and show that the higher-order interspecies interaction and the SO coupling can balance each other suitably for stability to take place. Mainly, it is found that the residual nonlinearity preserves and reinforces the stability of miscible pairs of condensates with SO coupling. Additionally, when a miscible binary mixture of condensates with SO coupling is modulationally unstable, the presence of residual nonlinearity may help soften such instability. Our results finally suggest that MI-induced formation of stable solitons in mixtures of BECs with two-body attraction may be preserved by the residual nonlinearity even though the latter enhances the instability.

Phase-controlled and chaos-assisted or -suppressed quantum entanglement for a spin-orbit coupled Bose-Einstein condensate

It has been demonstrated that the presence of chaos may lead to greater entanglement generation for some physical systems. Here, we find different effects of chaos on the spin-motion entanglement for a two-frequency driven Bose-Einstein condensate with spin-orbit coupling. We analytically and numerically demonstrate that classical chaos can assist or suppress entanglement generation, depending on the initial phase differences between two motional states, which can be manipulated by using the known phase-engineering method. The results could be significant in engineering nonlinear dynamics for quantum information processing with many-body entanglement.

Controlling chaotic spin-motion entanglement of ultracold atoms via spin-orbit coupling

We study the spatially chaoticity-dependent spin-motion entanglement of a spin-orbit (SO) coupled Bose-Einstein condensate with a source of ultracold atoms held in an optical superlattice. In the case of phase synchronization, we analytically demonstrate that (a) the SO coupling (SOC) leads to the generation of spin-motion entanglement; (b) the area of the high-chaoticity parameter region inversely relates to the SOC strength which renormalizes the chemical potential; and (c) the high-chaoticity is associated with the lower chemical potential and the larger ratio of the short-lattice depth to the longer-lattice depth. Then, we numerically generate the Poincaré sections to pinpoint that the chaos probability is enhanced with the decrease in the SOC strength and/or the spin-dependent current components. The existence of chaos is confirmed by computing the corresponding largest Lyapunov exponents. For an appropriate lattice depth ratio, the complete stop of one of (or both) the current components is related to the full chaoticity. The results mean that the weak SOC and/or the small current components can enhance the chaoticity. Based on the insensitivity of chaos probability to initial conditions, we propose a feasible scheme to manipulate the ensemble of chaotic spin-motion entangled states, which may be useful in coherent atom optics with chaotic atom transport.

Models of spin-orbit-coupled oligomers

We address the stability and dynamics of eigenmodes in linearly shaped strings (dimers, trimers, tetramers, and pentamers) built of droplets in a binary Bose-Einstein condensate (BEC). The binary BEC is composed of atoms in two pseudo-spin states with attractive interactions, dressed by properly arranged laser fields, which induce the (pseudo-) spin-orbit (SO) coupling. We demonstrate that the SO-coupling terms help to create eigenmodes of particular types in the strings. Dimer, trimer, and pentamer eigenmodes of the linear system, which correspond to the zero eigenvalue (EV, alias chemical potential) extend into the nonlinear ones, keeping an exact analytical form, while tetramers do not admit such a continuation, because the respective spectrum does not contain a zero EV. Stability areas of these modes shrink with the increasing nonlinearity. Besides these modes, other types of nonlinear states, which are produced by the continuation of their linear counterparts corresponding to some nonzero EVs, are found in a numerical form (including ones for the tetramer system). They are stable in nearly entire existence regions in trimer and pentamer systems, but only in a very small area for the tetramers. Similar results are also obtained, but not displayed in detail, for hexa- and septamers.

Takagi-Sugeno fuzzy modeling and chaos control of partial differential systems

In this paper a unified approach is presented for controlling chaos in nonlinear partial differential systems by a fuzzy control design. First almost all known chaotic partial differential equation systems are represented by Takagi-Sugeno fuzzy model. For investigating design procedure, Kuramoto-Sivashinsky (K-S) equation is selected. Then, all linear subsystems of K-S equation are transformed to ordinary differential equation (ODE) systems by truncated Fourier series of sine-cosine functions. By solving Riccati equation for each ODE systems, parallel stabilizing feedback controllers are determined. Finally, a distributed fuzzy feedback for K-S equation is designed. Numerical simulations are given to show that the distributed fuzzy controller is very easy to design, efficient, and capable to extend.

Dynamics of the central-depleted-well regime in the open Bose-Hubbard trimer

We study the quantum dynamics of the central-depleted-well (CDW) regime in a three-mode Bose Hubbard model subject to a confining parabolic potential. By introducing a suitable set of momentum-like modes we identify the microscopic variables involved in the quantization process and the dynamical algebra of the model. We describe the diagonalization procedure showing that the model reduces to a double oscillator. Interestingly, we find that the parameter-space domain where this scheme entails a discrete spectrum well reproduces the two regions where the classical trimer excludes unstable oscillations. Spectral properties are examined in different limiting cases together with various delocalization effects. These are shown to characterize quantum states of the CDW regime in the proximity of the borderline with classically unstable domains.

Discrete chaotic states of a Bose-Einstein condensate

We examine spatial chaos in a one-dimensional attractive Bose-Einstein condensate interacting with a Gaussian-like laser barrier and perturbed by a weak optical lattice. For a low laser barrier, chaotic regions of the parameters are demonstrated and the chaotic and regular states are illustrated numerically. In the high-barrier case, bounded perturbed solutions that describe a set of discrete chaotic states are constructed for discrete barrier heights and magic numbers of condensed atoms. Chaotic density profiles are exhibited numerically for the lowest quantum number, and analytically bounded but numerically unbounded Gaussian-like configurations are confirmed. It is shown that the chaotic wave packets can be controlled experimentally by adjusting the laser barrier potential.

Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice

We study the chaotic properties of steady-state traveling-wave solutions of the particle number density of a Bose-Einstein condensate with an attractive interatomic interaction loaded into a traveling optical lattice of variable shape. We demonstrate theoretically and numerically that chaotic traveling steady states can be reliably suppressed by small changes of the traveling optical lattice shape while keeping the remaining parameters constant. We find that the regularization route as the optical lattice shape is continuously varied is fairly rich, including crisis phenomena and period-doubling bifurcations. The conditions for a possible experimental realization of the control method are discussed.

Adiabatic condition for nonlinear systems

The adiabatic approximation is an important concept in quantum mechanics. In linear systems, the adiabatic condition is derived with the help of the instantaneous eigenvalues and eigenstates of the Hamiltonian, a procedure that breaks down in the presence of nonlinearity. Using an explicit example relevant to photoassociation of atoms into diatomic molecules, we demonstrate that the proper way to derive the adiabatic condition for nonlinear mean-field (or classical) systems is through a linearization procedure, using which an analytic adiabatic condition is obtained for the nonlinear model under study.

Ground-state properties of small-size nonlinear dynamical lattices

We investigate the ground state of a system of interacting particles in small nonlinear lattices with M >or=3 sites, using as a prototypical example the discrete nonlinear Schrödinger equation that has been recently used extensively in the contexts of nonlinear optics of waveguide arrays and Bose-Einstein condensates in optical lattices. We find that, in the presence of attractive interactions, the dynamical scenario relevant to the ground-state and the lowest-energy modes of such few-site nonlinear lattices reveals a variety of nontrivial features that are absent in the large/infinite lattice limits: the single-pulse solution and the uniform solution are found to coexist in a finite range of the lattice intersite coupling where, depending on the latter, one of them represents the ground state; in addition, the single-pulse mode does not even exist beyond a critical parametric threshold. Finally, the onset of the ground-state (modulational) instability appears to be intimately connected with a nonstandard ("double transcritical") type of bifurcation that, to the best of our knowledge, has not been reported previously in other physical systems.

Dynamic chaos and stability of a weakly open Bose-Einstein condensate in a double-well trap

We investigate the dynamics of a weakly open Bose-Einstein condensate with attractive interaction in a magneto-optical double-well trap. A set of time-dependent ordinary differential equations describing the complex dynamics are derived by using a two-mode approximation. The stability of the stationary solution is analyzed and some stability regions on the parameter space are displayed. In the symmetric well case, the numerical calculations reveal that by adjusting the feeding from the nonequilibrium thermal cloud or the two-body dissipation rate, the system could transit among the periodic motions, chaotic self-trapping states of the Lorenz model, and the steady states with the zero relative atomic population or with the macroscopic quantum self-trapping (MQST). In the asymmetric well case, we find the periodic orbit being a stable two-sided limited cycle with MQST. The results are in good agreement with that of the direct numerical simulations to the Gross-Pitaevskii equation.