Dynamically entangled oscillating state in a Bose gas with an attractive polaron

TL;DR

This work analyzes the out-of-equilibrium dynamics of a single attractively interacting impurity in a 1D Bose gas, focusing on finite-momentum quenches and mean-field dynamics in the impurity frame via a Lee-Low-Pines transformation. A finite-momentum attractive polaron is characterized analytically, revealing a periodic energy dispersion and a critical mass delineating different ground-state structures. Most strikingly, for fast and heavy impurities the system forms an entangled, nonstationary state where the depletion cloud localizes near the impurity and drives undamped velocity oscillations, with lifetime and frequency controlled by the impurity- bath coupling and initial conditions. The results highlight a new dynamical regime—the entangled oscillating state—that could be probed with current cold-atom platforms and expands our understanding of impurity problems in low-dimensional quantum fluids.

Abstract

We study the out-of-equilibrium dynamics of an attractively interacting impurity suddenly immersed with a nonzero initial velocity into a system of one-dimensional weakly interacting homogeneous bosons. We uncover and characterize different dynamical regimes in the parameter space. Especially interesting is the relaxation of a fast impurity with a mass close to or exceeding the critical one, where the impurity exhibits undamped temporal long-lived velocity oscillations before reaching a stationary state. The underlying mechanism is the transient localization of a boson depletion cloud near the impurity, that oscillates around the boson density peak situated at the impurity position. The lifetime of this entangled oscillating state increases with the absolute value of the impurity-boson coupling. Cold atomic gases provide an ideal playground where this phenomenon can be probed.

Figures (11)

• Figure 1: Time evolution of the phase $\Theta(x,t)$ and the density $n(x,t) = |\Psi_0(x,t)|^2$ of bosons after a quench of the impurity-boson interaction, for $\tilde{G}=-0.8$, $M=3m$, and $V_0=1.5v$ at (a) $\tilde{t}=t g n_0/\hbar=2$ and (b) $\tilde{t}=40$. The red dashed lines denote the phase and the density of the ground state (\ref{['eq:MFsolution']}) for the numerically obtained final impurity velocity $V_f=0.43 v$. The corresponding impurity velocity evolution in time is shown in Fig. \ref{['fig4']}a.
• Figure 2: Final impurity velocity (\ref{['eq:momentum']}) of the ground state (\ref{['eq:MFsolution']}) as a function of the system momentum for different impurity masses and coupling constants. Here $\gamma=0.1$.
• Figure 3: Ground-state energy dispersion (\ref{['eq:PolaronEnergy']}) for two different values of $\tilde{G}$ and $M/m$ at $\gamma=0.1$. Here, $M=8m>M_c$ for $\tilde{G}=-0.8$.
• Figure 4: (a) Time-evolution of the impurity velocity for $M=3m$ and $\tilde{G}=-0.8$ for different initial velocities. Here $\gamma = 0.1$. (inset) The density profile of emitted soliton for different initial impurity velocities $V_0$ for the aforementioned parameters. (b) Final impurity velocity $V_f$ as a function of the initial impurity momentum $p=M V_0$ for three different sets of parameters. Here, the dotted lines denote the analytic expression (\ref{['eq:momentum']}), while the points denote the numerically obtained values of $V_f$. One can express $p/\hbar n_0=(V_0/v) (M\sqrt{\gamma}/m)$.
• Figure 5: Time evolution of the impurity velocity for $M=3m$ and $V_0=2.5v$ for different dimensionless impurity-bath coupling $\tilde{G}$. Here $\gamma = 0.1$.