Equilibrium states and chaos in an oscillating double-well potential

Abstract

We investigate numerically parametrically driven coupled nonlinear Schrödinger equations modelling the dynamics of coupled wavefields in a periodically oscillating double-well potential. The equations describe among other things two coupled periodically-curved optical waveguides with Kerr nonlinearity or horizontally shaken Bose-Einstein condensates in a double-well magnetic trap. In particular, we study the persistence of equilibrium states of the undriven system due to the presence of the parametric drive. Using numerical continuations of periodic orbits and calculating the corresponding Floquet multipliers, we find that the drive can (de)stabilize a continuation of an equilibrium state indicated by the change of the (in)stability of the orbit. Hence, we show that parametric drives can provide a powerful control to nonlinear (optical or matter wave) field tunneling. Analytical approximations based on an averaging method are presented. Using perturbation theory the influence of the drive on the symmetry breaking bifurcation point is discussed.

Regular and chaotic behavior of collective atomic motion in two-component Bose-Einstein condensates

“…where the effective parameters are tunneling energy k 0 (15), difference of energies between the wells ∆E (18) and self-interaction energy Λ (19). Using Lagrangian For the center condensate, the equations of motion of ξ and w are obtained as…”

“…More importantly, several experiments have successfully observed coherence oscillations through quantum tunneling of atoms between the condensates trapped in a double well potential [12][13][14], whereas similar phenomena were also seen from attractive to repulsive atomic interactions by experimentally changing atomic scattering lengths through Feshbach resonances [15]. In addition to the regular coherent oscillations mentioned above, the studies on the possibility of the appearance of chaotic motions, by applying an externally time-dependent trap potential, are also a fascinating task with findings that deserve further experimental justification [16][17][18][19].…”

“…The variation of the Gaussian width induces a time-dependent modification to the parameter k + κσ/σ 0 in (11) and (12). In the single BEC experiment [2,3], the similar modification to k can be achieved by the time-dependent laser intensity or the magnetic field with the possibility to induce the Shapiro effect [38], analogue Kaptiza pendulum [39] and even chaotic motions [16][17][18][19]. Here, since σ σ 0 , the time-dependent modification is relatively small compared with k to be ignored in this work.…”

“…One can then reproduce regular oscillations around the stable states studied in [2,3] within some parameter regime by considering the effects from time-averaged trajectories of the central condensate. Additionally, the collective motion of atoms, if departing from the unstable states, could show chaotical behavior, oscillating between bilateral condensates due to the time-dependent driving force given by the dynamics of the condensate centered in the middle [16][17][18][19]. The scattering lengths in the parameter regime, which give the chaotic dynamics of the system, will be useful as a reference for experimentalists to observe the effects.…”

A space-dependent atomic superfluid current in Bose–Einstein condensates

“…[20] Undoubtedly, the superfluids can exert significant influence on BECs, and may even result in chaotic behaviors in the system. It has been reported that for some specific parameters and initial/boundary conditions, spatial chaos [23][24][25][26][27][28] and dynamical chaos [29][30][31][32][33][34][35][36][37][38][39][40][41] can appear in BECs. Due to the diversity of the configurations of BEC systems, there are still open problems to be considered.…”