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

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.

💡 Research Summary

The paper investigates the non‑equilibrium dynamics of a single impurity with attractive interactions that is suddenly injected into a one‑dimensional weakly interacting homogeneous Bose gas. Using a Lee‑Low‑Pines transformation the authors work in the impurity‑centered frame where the total momentum p is conserved. In the limit of weak boson‑boson interactions (γ≪1) the many‑body problem reduces to a Gross‑Pitaevskii‑type equation for the condensate wavefunction Ψ₀(x,t) that contains a drift term proportional to the instantaneous impurity velocity V(t). The velocity itself is self‑consistently defined through the phase gradient of Ψ₀, establishing a feedback loop between impurity motion and the surrounding bosonic cloud.

First, a stationary solution of the mean‑field equation is derived for a finite total momentum. The solution (Eq. 6) is characterized by a density peak at the impurity position, a phase drop θ across the impurity, and two parameters a and b that depend on the final impurity velocity V_f and the mass ratio M/m. For attractive impurity‑boson coupling (G<0) the shift parameter x₀ becomes complex, leading to a hyperbolic‑coth form of the density profile and an increase of the local boson density at the impurity. The critical velocity v_c = v √(1+m/M) bounds the admissible V_f; however, this bound only applies when the impurity‑boson interaction is turned on adiabatically. In a sudden quench, additional excitations—dispersive shock waves and gray solitons—are emitted, carrying away part of the initial momentum.

The authors then perform extensive numerical simulations of the time‑dependent Gross‑Pitaevskii equation using a fully implicit, conservative finite‑difference scheme with an upwind discretization of the drift term. The system size is chosen large enough to avoid finite‑size reflections during the simulated time window. After the quench, the impurity rapidly loses momentum on a timescale of a few ℏ/(g n₀). The lost momentum is transferred to the bath via a density shock front that propagates outward and, for sufficiently large initial velocities, a gray soliton that separates from the shock. As time evolves, both the shock fronts and the soliton move away from the impurity, and the impurity settles to a stationary velocity V_f that is smaller than v_c but remains finite, reflecting the superfluid nature of the zero‑temperature bath.

A particularly striking regime emerges when the impurity mass M exceeds a critical value M_c (determined by the crossing of two energy branches) and the initial impurity velocity is high. In this case, after an initial rapid deceleration the impurity becomes dynamically entangled with a bosonic depletion cloud that remains localized near the impurity. The depletion cloud oscillates back and forth around the density peak that sits at the impurity position. These oscillations are essentially undamped: their amplitude persists for hundreds of ℏ/(g n₀) time units in the simulations. The oscillation period is set by the healing length ξ = ℏ/√(m µ) and the sound speed v, while the lifetime τ grows with the absolute value of the attractive coupling |G|. The phenomenon is termed a “dynamically entangled oscillating state” and represents a non‑trivial, long‑lived non‑equilibrium configuration of the impurity‑polaron system.

The paper provides quantitative analysis of this state: the depth and separation of the depletion holes, the dependence of the oscillation frequency on M, V₀, and G, and the energy‑momentum balance that confirms overall conservation despite the presence of emitted shock waves and solitons. The authors also discuss experimental feasibility. In ultracold‑atom setups, a 1D Bose gas can be realized with a tightly confining transverse optical lattice, while the impurity can be a different atomic species or a different internal state of the same species. A rapid momentum kick can be delivered by a Bragg pulse or a magnetic field gradient, and the subsequent dynamics can be probed with high‑resolution in‑situ imaging or phase‑contrast interferometry, allowing direct observation of the density peak, shock fronts, solitons, and the long‑lived oscillations.

In summary, the work extends the understanding of Bose polarons beyond the static picture, revealing that attractive polarons can support a dynamically entangled, long‑lived oscillatory state when the impurity is sufficiently heavy and fast. This enriches the landscape of impurity physics in low‑dimensional quantum fluids, connects to nonlinear wave phenomena such as solitons and shock waves, and opens new avenues for experimental exploration of non‑equilibrium many‑body dynamics in ultracold gases.