Dynamics and thermalization of a Bose-Einstein condensate in a Sinai-oscillator trap

Abstract

We study numerically the evolution of Bose-Einstein condensate in the Sinai oscillator trap described by the Gross-Pitaevskii equation in two dimensions. In the absence of interactions this trap mimics the properties of Sinai billiards where the classical dynamics is chaotic and the quantum evolution is described by generic properties of quantum chaos and random matrix theory. We show that, above a certain border, the nonlinear interactions between atoms lead to the emergence of dynamical thermalization which generates the statistical Bose-Einstein distribution over eigenmodes of the system without interactions. Below the thermalization border the evolution remains quasi-integrable. Such a Sinai oscillator trap, formed by the oscillator potential and a repulsive disk located in the vicinity of the center, had been already realized in first experiments with the Bose-Einstein condensate formation by Ketterle group in 1995 and we argue that it can form a convenient test bed for experimental investigations of dynamical of thermalization. Possible links and implications for Kolmogorov turbulence in absence of noise are also discussed.

Thermalization, condensate growth, and defect formation in an out-of-equilibrium Bose gas

“…Thermalization processes depend largely on the microscopic properties of the system involved, such as the s-wave scattering length. Dimensionality also plays a critical role in defining these processes, and work has been done to investigate the properties of ultracold Bose gases in one- [11][12][13], two- [14], and three-dimensions [3,15]. Despite this, there is currently no universally accepted theoretical description detailing the full growth, relaxation, and thermalization * dylan.brown@auckland.ac.nz † Present address: Department of Applied Physics, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan properties of ultracold atomic gases [3].…”

Dynamical thermalization in isolated quantum dots and black holes

“…However, the fluctuations are significant and also the DTC should be verified for all eigen- states at a given set of parameters. Due to that we test the DTC validity using the approach developed for bosons in [38][39][40] based on the numerical computation of the dependence S(E) from eigenstates of Hamiltonian (2).…”

Dynamical Thermalization of Interacting Fermionic Atoms in a Sinai Oscillator Trap

“…The disk center is located at (x d , y d ) = (−1/2, −1/2) so that the disk bungs a hole in the center as it was the case in the experiments [49]. The Poincare sections at different energies are presented in [48] showing that the phase space is almost fully chaotic (see Figure 1 there). The quantum evolution is described by the Schrödinger equation with the quantized Hamiltonian (2) where the conjugate momentum and coordinate variables become operators with the commutation relation [x, p x ] = [y, p y ] = i [48].…”

“…The Poincare sections at different energies are presented in [48] showing that the phase space is almost fully chaotic (see Figure 1 there). The quantum evolution is described by the Schrödinger equation with the quantized Hamiltonian (2) where the conjugate momentum and coordinate variables become operators with the commutation relation [x, p x ] = [y, p y ] = i [48]. For the quantum problem we use the value of the dimensionless Planck constant = 1 so that the ground state energy is E g = 1.685.…”

“…For the quantum problem we use the value of the dimensionless Planck constant = 1 so that the ground state energy is E g = 1.685. In the following the energies are expressed in atomic like units of energy E u = ω x (for our choice of Sinai oscillator parameters we also have E u = ω x = 2 /(mr d 2 )) [48] with the typical size of oscillator ground state being equal to the disk radius: a 0 = ∆x osc = ( /mω x ) 1/2 = r d . In [48] it is shown that the classical dynamics of this system is almost fully chaotic.…”