Universal Behavior on the Relaxation Dynamics of Far-From-Equilibrium Quantum Fluids

Abstract

Investigating the initial conditions that lead many-body quantum systems to an out-of-equilibrium state is fundamental for understanding their thermalization dynamics. In this work we observe the relaxation for two regimes of excitation that can drive the turbulent Bose-Einstein condensate into two distinct final states, and are defined by the amount of energy injected into the system. The subcritical regime is characterized by a lower injection of energy, which can lead to an inverse particle cascade and, consequently, to the BEC mode repopulation during the relaxation process. The supercritical regime is marked by a higher energy injection, that may lead to the BEC dissolution and a final thermal state. In both cases we observe relaxation stages that exhibit the same key features: a direct cascade, a non-thermal fixed point with the same exponents, a prethermalization region and, finally, the thermalization of the system. In the final thermalization stage, universal scaling is observed for both regimes, even though their final states are completely different. By analyzing the coherence length of our turbulent cloud, we clearly visualize the recovery and the loss of the coherence for the subcritical and supercritical regimes after relaxation. These results indicate that the evolution of turbulence occurs independent of its initial conditions and of the final state achieved.

Figures (7)

• Figure 1: (a) Temperature measured at $t \approx 400$ ms for different excitation amplitudes. The horizontal blue line indicates the estimated critical temperature, and the shaded region denotes its standard deviation. For excitation amplitudes below (above) $\approx 1.00,\mu_0$, the system lies in the subcritical (supercritical) regime. (b) Time evolution of the low-momentum population $n(k\rightarrow0,t)$, where $t_0$ indicates the end of the excitation stage. The green dots in the top panel correspond to the supercritical regime, while the dashed curve in the bottom panel represents the subcritical regime. Error bars denote one standard deviation SupMat. The orange curves show the power-law fits $n(k\rightarrow0,t) \propto t^\alpha$ associated with the direct energy cascade. The blue curves show the power-law fits $n(k\rightarrow0,t) \propto (t^*-t)^\lambda$, corresponding to a second-kind self-similar solution that characterizes the BEC dissolution (top panel) and the BEC repopulation (bottom panel). (c) Absorption images taken during the supercritical regime for different holding times; each squared marker in panel (b) corresponds to one of these images.
• Figure 2: Momentum distributions $n(k,t)$ for different relaxation stages of the supercritical case. (a)-(c) are data from $t_2$ to $t_3$ where we observe a direct cascade, while (d) to (f) are the data from the thermalization stage. In (a) the momentum distribution for $t = 130.1$ ms shows a direct cascade with a power law $k^\gamma$ with $\gamma=-2.34(2)$, represented by the the red dashed line. The vertical dashed lines represent the momentum region where we made the fitting. In the termalization stage, shown in (d) for $t = 355.3$ ms, we can also observe a power law $k^{-\nu}$ from which we obtain $\nu=0.54(1)$, the dashed vertical lines represent the region where we made the fitting. In (b) we show the momentum distribution for different hold times and (c) its respective scaling following Eq. (\ref{['eq: Eq1']}), with universal exponents $\alpha = -0.57(9)$ and $\beta = -0.25(8)$. In (e) we show the momentum distribution for different hold times and its scaling following Eq. (\ref{['eq: Eq2']}), with $\lambda=0.6(2)$ and $\mu = 0.8(3)$. The error bars correspond to the standard deviations SupMat.
• Figure 3: Coherent length as a function of the hold time for subcritical and supercritical regimes. In interval from $t_2$ to $t_3$, corresponding to the prethermalization stage, $\ell(t)$ is approximately constant. From $t_3$ to $t_4$ we have the thermalization stage. The orange curves represent the fitting of the power law $\ell(t) \propto (t^*-t)^{\lambda'}$, from which we obtained $\lambda^{\prime}_{\mathrm{sub}}=-0.49(5)$ for the recovery of coherence and $\lambda^{\prime}_{\mathrm{sup}}=0.19(1)$ for the continued loss of coherence in subcritical and supercritical regimes, respectively. The dashed vertical lines represents $t_2$, $t_3$ and $t_4$. The error bars correspond to the standard deviations SupMat.
• Figure S1: Total number of atoms during hold time for three amplitudes. The number is approximately constant over time, with small experimental fluctuations. The error bars correspond to the standard deviation of the mean value of all measurements.
• Figure S2: Single shot illustrations of the angular averaging procedure to compute the momentum distributions for the excitation amplitude $A=1.054\mu_0$ at hold times (a) $t=6.03$ ms and (c) $t=107.6$ ms. In both cases, the dashed lines delimit two $30^{\circ}$ angle regions centered around the major axis of expansion of the cloud, where $n(k)$ is computed. Images (b) and (d) represent the same information as (a) and (c) respectively, but in polar coordinates.