Early-time dynamics of Bose gases quenched into the strongly interacting regime

We study the early-time dynamics of a degenerate Bose gas after a sudden quench of the interaction strength, starting from a weakly interacting gas. By making use of a time-dependent generalization of the Nozières-Saint-James variational formalism, we describe the crossover of the early-time dynamics from shallow to deep interaction quenches. We analyze the coherent oscillations that characterize both the density of excited states and the Tan's contact as a function of the final scattering length. For shallow quenches, the oscillatory behavior is negligible and the dynamics is universally governed by the healing length and the mean-field interaction energy. By increasing the final scattering length to intermediate values, we reveal a universal regime where the period of the coherent atom-molecule oscillations is set by the molecule binding energy. For the largest scattering lengths we can numerically simulate in the unitary regime, we find a universal scaling behavior of the typical growth time of the momentum distribution in agreement with recent experimental observations [C. Eigen et al., Nature 563, 221 (2018)].

Figure 1

Noncondensed density $n_{\mathrm{ex}}\left(t\right)$ as a function of time after an instantaneous quench from a noninteracting gas $a_i=0$ to different values of the final scattering length $a_f$. In all panels, solid lines are the results of the numerical integration of the equations of motion (13) and (14). (a) For shallow quenches, density and time are rescaled by the healing length $\xi$ and mean-field time $\tau$ (18), respectively. At intermediate values of $a_f$ (b), it is convenient to rescale the time by the molecular bound state energy $|E_{\text{B}}|$, while, towards unitarity (c), by the energy scale fixed by the gas density ${\epsilon }_n$. In (a) and (b), the dashed lines are obtained via the time-dependent Bogoliubov approximation [27] [see Eq. (A7)].

Figure 2

Amplitude of coherent oscillations for the noncondensed fraction $\mathrm{\Delta }{n}_{\mathrm{ex}}$ as a function of $a_f{k}_n$ ($a_f$ in units of the Bohr radius $a_0$ is given for a density $n={10}^{12}{\mathrm{cm}}^{-3}$). The solid line is a cubic fit to our data (circles), while the dashed line is the analytical estimation deduced in Ref. [21].

Figure 3

Rescaled Tan's contact $C\left(t\right)/C_0$ ($C_0=16{\pi }^2{a}_f^2{n}^2$) for instantaneous quenches from a noninteracting gas $a_i=0$ to different values of the final scattering length $a_f$. (Inset) Momentum distribution $N_k\left(t\right)$ at a fixed time $t$ and scattering length $a_f{k}_n=0.14$.

Figure 4

Period $T$ of coherent oscillations extracted from the excited particle density $n_{\mathrm{ex}}\left(t\right)$ (dots) and from the Tan's contact $C\left(t\right)$ (triangles) as a function of $a_f{k}_n$ ($a_f$ in units of the Bohr radius $a_0$ is given for a density $n={10}^{12}{\mathrm{cm}}^{-3}$). (Top) The period $T$ is plotted in units of the molecular binding energy $|E_B|$. (Bottom) $T$ is plotted in units of the energy scale associated with the gas density ${\epsilon }_n$.

Figure 5

Normalized momentum distribution ${\tilde{N}}_{\mathbf{k}}\left(t\right)$ as a function of time $t$ after a diabatic quench to $a_f{k}_n=4.1$ (corresponding to $a_f=20000a_0$ when $n={10}^{12}{\mathrm{cm}}^{-3}$) for different value of the momentum $k$. The dashed horizontal line represents the value of the mean steady-state ${\tilde{N}}_{\mathrm{ss}}\left(k\right)$ reached after an initial quench transient. The dashed vertical line marks the value of the growth time ${\tau }_{\mathrm{gr}}\left(k\right)$, defined as the time at which ${\tilde{N}}_{\mathbf{k}}\left(t\right)$ reaches $1/4$ of its maximum value (dots) at a given value of $k$.

Figure 6

(Top) Growth time ${\tau }_{\mathrm{gr}}\left(k\right)$ of the normalized momentum distribution ${\tilde{N}}_{\mathbf{k}}\left(t\right)$ (see Fig. 5) as a function of momentum for different values of $a_f$ in the unitary regime and for two densities, $n_1={10}^{12}{\mathrm{cm}}^{-3}$ and $n_2=5\times {10}^{12}{\mathrm{cm}}^{-3}$ (see legend in the bottom panel). (Bottom) Mean steady-state value ${\tilde{N}}_{\mathrm{ss}}\left(k\right)$ reached by the momentum distribution after an initial transient of the quench (see Fig. 5) as a function of momentum for the same values of $a_f$ and densities as the top panel.

Figure 7

Two-step extrapolation process of the period of oscillations $T$ of the noncondensed density $n_{\mathrm{ex}}\left(t\right)$ (Fig. 1) with respect to the two system regularization parameters, $M$ and $\mathrm{\Lambda }$. The period $T$ (dots) is evaluated, at fixed values of $M$ and $\mathrm{\Lambda }$, by averaging over several oscillations of $n_{\mathrm{ex}}\left(t\right)$, with an error given by the standard deviation (not observed in these plots). (a) $T$ is plotted as a function of $M^{-1}$ for a fixed value of $\mathrm{\Lambda }/k_n=513$. The $M\to \infty$ value $T_{M\to \infty }$ is extracted via an exponential fit (solid line). (b) Plot of the extracted period $T_{M\to \infty }$ with respect to ${\mathrm{\Lambda }}^{-1}$ (dots). The extrapolated value $T_{M\to \infty ,\mathrm{\Lambda }\to \infty }$ is obtained via a linear fit (solid line). We have fixed $n={10}^{12}{\mathrm{cm}}^{-3}$ and $a_f=1000a_0$ ($a_f{k}_n=0.21$).