Thermalization and localization of an oscillating Bose-Einstein condensate in a disordered trap

We numerically simulate an oscillating Bose-Einstein condensate in a disordered trap [Phys. Rev. A 82, 033603 (2010)] and the results are in good agreement with the experiment. It shows that the disorder acts as a medium, which results in a relaxation from nonequilibrium to equilibrium, i.e., thermalization. An algebraic localization is realized when the system approaches the equilibrium, and if the system falls into the regime when the healing length of the condensate exceeds the correlation length of the disorder, exponential Anderson localization is to be observed.

Figure 1

(a) Spatial and temporal distribution of an oscillating condensate. The calculated norm of the spatial condensate wave function is potted in $z$ direction at various times. Black dots are the experimentally measured temporal center-of-mass coordinates, taken from Fig. 2 of Ref. [9]. (b),(c) Close examinations of the results for $t<t_c\sim 4\mathrm{s}$ and for $t>t_c$. (d) Blue line corresponds to the temporal center-of-mass positions calculated from the numerical results in (c).

Figure 2

Time evolution of the four energies (in units of $\hslash \omega$) in association with the dynamics shown in Fig. 1. Total energy is conserved for the entire process. The equilibrium temperature $T_{\mathrm{eq}}$ can be extracted from $E_{\text{kin}}(t\to \infty )\equiv k_{\mathrm{B}}{T}_{\mathrm{eq}}/2\simeq 364\hslash \omega$ that yields $T_{\mathrm{eq}}\simeq 192$ nK.

Figure 3

Temporal entropy of the oscillating condensate calculated from the simulation in Fig. 1. The inset shows the Rayleigh-Jeans spectrum for the wave action at $t=7.3\mathrm{s}\gg t_c$. The fitting temperature $T\simeq 192$ nK agrees with the equilibrium temperature $T_{\mathrm{eq}}$ extracted from the equilibrium kinetic energy shown in Fig. 2.

Figure 4

(a) In a semilog plot, for the case $\xi <{\sigma }_D$ the eventual density distribution at $t=7.3\mathrm{s}$ is shown to exhibit an algebraic profile at $20<\left|z\right|<70$. (b) A similar study for the case $\xi >{\sigma }_D$, where an AL exponential profile is obtained for the eventual density distribution at $t=3.7\mathrm{s}$. The fitting Lyapunov exponent is ${\gamma }_{\mathrm{eff}}=0.025$. (c),(d) The goodness of fit in (a),(b) is further confirmed by the log-log plots.

Figure 5

Density profiles at different times obtained from the calculated results in Fig. 1. It clearly distinguishes between (a) the extended states (at $t<t_c$) and (b) the localized states (at $t>t_c$).