Equilibrating dynamics in quenched Bose gases: Characterizing multiple time regimes

We address the physics of equilibration in ultracold atomic gases following a quench of the interaction parameter. Our work is based on a bath model which generates damping of the bosonic excitations. We illustrate this dissipative behavior through the momentum distribution of the excitations $n_k$, observing that larger $k$ modes have shorter relaxation times $\tau \left(k\right)$; they will equilibrate faster, as has been claimed in recent experimental work. We identify three time regimes. At short times $n_k$ exhibits oscillations; these are damped out at intermediate times where the system appears to be in a false or slowly converging equilibrium. Finally, at longer times, full equilibration occurs. This false equilibrium is, importantly, associated with the $k$ dependence in $\tau \left(k\right)$ and has implications for experiment.

Figure 1

Characteristic times ${\tau }_{\mathrm{short}}\left(k\right)$ (dashed line) and ${\tau }_{\mathrm{interm}}\left(k\right)$ (solid line) in units of ${\left(g_f{n}_{0,f}\right)}^{-1}$ vs ${\epsilon }_k/g_f{n}_{0,f}$ ($\gamma =0.1$, see text). Inset: Data from Ref. [5] showing the time scale ${\tau }_{\exp }\left(k\right)$ after which the experimental system has attained a steady state for each momentum $k$. ${\epsilon }_n={\left(6{\pi }^{2}n\right)}^{2/3}/2m$ corresponds to the “Fermi energy” (with $n$ the density of atoms), which is the only energy scale for a unitary Bose gas. The specific parameters for these plots are discussed in the context of the subsequent figures. The abscissa units of the graph and the inset are compared in the text.

Figure 2

$n_k\left(t\right)$ vs momentum, in terms of ${\epsilon }_k/g{n}_{0,f}$, for four different times $t=t_0/2$ (solid curve), $t=t_0$ (short dashed curve), $t=2t_0$ (long dashed curve), and $t=5t_0$ (dotted-dashed curve) for a time-independent condensate $n_0\left(t\right)=n_{0,f}$. and $\Gamma =0.1$ The inset plots $n_k\left(t\right)$ vs $t/t_0$ for energy ${\epsilon }_k=g_f{n}_{0,f}$ (top solid curve) and ${\epsilon }_k=3g_f{n}_{0,f}$ (bottom solid curve), as well as Bogoliubov results (i.e., $\Gamma =0$) for the same energies (dashed curves).

Figure 3

Left: $n_k\left(t\right)$ vs $t/t_0$ for a time-dependent condensate [Eq. (4)] with $n_{0,i}=2n_{0,f}$ for ${\epsilon }_k=g_f{n}_{0,f}$ (top solid curve) and ${\epsilon }_k=3g_f{n}_{0,f}$ (bottom solid curve). The arrows show the typical damping time of the oscillations ${\tau }_{\mathrm{interm}}\left(k\right)$ for both energies. Right: Zoom on time evolution of the $n_k\left(t\right)$ for ${\epsilon }_k=3g_f{n}_{0,f}$, showing in more detail the long-time equilibration of $n_k\left(t\right)$ with time-dependent condensate. The dotted line is the equilibrium $n_k^{\mathrm{eq}}\left[n_{0,f}\right]$ and the dashed line corresponds to the quasiadiabatic momentum distribution $n_k^{\mathrm{eq}}\left[n_0\left(t\right)\right]$ [see Eqs. (5) and (4)]. In both figures $\Gamma ={\gamma }_0{t}_0=0.1$.