Nonequilibrium states of a quenched Bose gas

Yin and Radzihovsky [X. Yin and L. Radzihovsky, Phys. Rev. A 88, 063611 (2013)] recently developed a self-consistent extension of a Bogoliubov theory, in which the condensate number density $n_c$ is treated as a mean field that changes with time, in order to analyze a JILA experiment by Makotyn et al. [P. Makotyn et al., Nat. Phys. 10, 116 (2014)] on a ${}^{85}\mathrm{Rb}$ Bose gas following a deep quench to a large scattering length. We apply this theory to construct a closed set of equations that highlight the role of ${\dot{n}}_c$, which is to induce an effective interaction between quasiparticles. We show analytically that such a system supports a steady state characterized by a constant condensate density and a steady but periodically changing momentum distribution, whose time average is described exactly by the generalized Gibbs ensemble. We discuss how the ${\dot{n}}_c$-induced effective interaction, which cannot be ignored on the grounds of the adiabatic approximation for modes near the gapless Goldstone mode, can significantly affect condensate populations and Tan's contact for a Bose gas that has undergone a deep quench.

Figure 1

(a) Oscillatory and solid curves represent, respectively, the momentum distribution $n_k\left(t\right)$ and its time average, evaluated long after the scattering length is quenched from $a_i=0.01n^{-1/3}$ to $a_f=0.7n^{-1/3}$. The solid curve also represents the generalized Gibbs ensemble. The dashed curve is the equilibrium distribution for a system with permanent scattering length $a_f$. The inset is the same plot, but on a ${\log }_{10}$ scale. (b) Temperature $T_k$ as a function of the momentum, where $T_k$ is computed using Eq. (27). As in [43], for all curves (in this article), $a_f$ is replaced with Eq. (43) with $\chi =$ 1 so that the physics at unitarity, which is expected to occur in a strongly interacting Bose gas, can be accounted for qualitatively.

Figure 2

Both plots are for the same quench as in Fig. 1. (a) Condensate density $n_c\left(t\right)$ as a function of time. The solid curve is from Eqs. (10) and (20), where the effect of ${\dot{n}}_c$ is taken into full consideration. This may be compared with the dashed curve produced from Eq. (32), where the effect of ${\dot{n}}_c$ is ignored. (b) Contour plot of the adiabaticity parameter ${\eta }_k\left(t\right)$ in Eq. (36) as a function of $k$ and $t$. The regions where ${\eta }_k\left(t\right)$ is larger than one correspond to where the adiabatic approximation fails, which can be seen to occur most strongly in the small-momentum and early-time regions.

Figure 3

Comparison of the short-time $n_c\left(t\right)$ obtained by various methods after a noninteracting BEC is quenched to an interacting one in which atoms interact with an $s$-wave scattering length (a) $a_f=0.3n^{-1/3}$ and (b) $a_f=0.7n^{-1/3}$. The solid black line represents the exact $n_c\left(t\right)$, the solid blue line the adiabatic $n_c\left(t\right)$, the dotted green line the simple perturbation series (39), and finally the dotted red line the self-consistent perturbative series (41). In (a) the quench is shallow enough that all the lines are virtually on top of each other.

Figure 4

Time evolution of the exact $n_c\left(t\right)$ when $a_f=1.175n^{-1/3}$, given by Eq. (44) (dotted line) and the time evolution of the exact $n_c\left(t\right)$ at unitarity $a_f\to \infty$ (solid line). In both cases, the quench starts from a BEC in the noninteracting limit.

Figure 5

Time evolution of $C_k\left(t\right)$ in Eq. (46) at a relatively large momentum after a noninteracting BEC is quenched to an interacting one. In the particular example considered here, $k=14k_F$ and $a_f=1.0n^{-1/3}$.

Figure 6

Contact $C$ as a function of $a_f$ for the adiabatic $n_c^s$ [lower (red) points] and for the exact $n_c^s$ [upper (black) points].