Hydrodynamic attractor in periodically driven ultracold quantum gases

Abstract

Hydrodynamic attractors are a universal phenomenon of strongly interacting systems that describe the hydrodynamic-like evolution far from local equilibrium. In particular, the rapid hydrodynamization of the Quark-Gluon Plasma is behind the remarkable success of hydrodynamic models of high-energy nuclear collisions. So far, hydrodynamic attractors have been explored only in systems undergoing monotonic expansion, such as Bjorken flow. We demonstrate that a system with an oscillating isotropic expansion exhibits a novel cyclic attractor behavior. This phenomenon can be investigated in ultracold quantum gases with externally modulated scattering length, offering a new avenue for experimentally discovering hydrodynamic attractors.

Figures (5)

• Figure 1: Müller-Israel-Stewart hydrodynamics: linear attractor solution for oscillating expansion at different drive frequencies $\omega$. The full dynamical bulk pressure $\Pi(t) \equiv\tfrac{1}{3}T^{i}_{\space i}-P_\text{eq}$ is plotted vs its Navier-Stokes approximation $\Pi_\text{NS}(t)$, such that deviations from Navier-Stokes appear outside the diagonal (dotted line). Different initial conditions (colors) converge toward an asymptotic attractor solution (dashed ellipse) near $t=3\tau_\zeta$, as marked by the vertical bar. For increasing drive frequency (left to right panels) the elliptic shape of the attractor widens; this signifies a growing deviation from Navier-Stokes.
• Figure 2: Massive kinetic theory: nonlinear attractor solution for oscillating expansion at different drive amplitude $A$ and different initial conditions $\alpha$. At small amplitude (left panel) the full kinetic theory solution (solid line) agrees with the linear MIS attractor (dashed line). At larger amplitude (center and right panels) kinetic theory predicts a nonlinear attractor that deviates strongly from the MIS attractor. The asymmetric attractor shape arises because also the equation of state evolves with time, cf. \ref{['fig:temperature']}.
• Figure 3: (top) Temperature evolution for a strong periodic drive. The temperature oscillation has large amplitude, deviates from the sinusoidal shape of the drive, and the peak height grows with each cycle due to dissipative heating. This temperature dependence leads to the asymmetric orbits of the attractor in Fig. \ref{['fig:mKT']}(c), which do not close after one period. (bottom) Bulk pressure evolution for different initial conditions and a strong periodic drive.
• Figure 4: Temperature evolution for different periodic drives in massive kinetic theory. For larger drive amplitude $A$, the temperature oscillations grow in amplitude, they become less sinusoidal, and the peak height grows from one cycle to the next. However, the oscillation phase for different drive amplitudes is approximately the same even after nine periods.
• Figure 5: The bulk susceptibility $\zeta/P_\text{eq}\tau_R$ in massive kinetic theory as a function of $m/T$.