Relaxation dynamics of a mobile impurity injected into a one-dimensional Bose gas

TL;DR

This work analyzes the nonequilibrium relaxation of a finite-mass impurity injected into a one-dimensional Bose gas at zero temperature, using time-dependent mean-field dynamics in the impurity frame. The authors map the problem to a delta-like obstacle moving with velocity and solve the coupled boson field equations, revealing a rich dynamical landscape including Cherenkov-like density waves, a broad regime where the final impurity velocity $V_f$ remains nearly constant, and distinct heavy-impurity behavior with soliton emission and metastable states. Key findings show that dispersive density shock waves carry away excess momentum, while the impurity locally resides in a stationary state; fast heavy impurities exhibit damped velocity oscillations (quantum flutter) and very heavy impurities can emit solitons near metastable points, slowing relaxation and potentially reversing motion. The results identify a critical mass $M_c$, cusp features in the polaron energy, and a dynamical crossover governed by $M/m$ and the dimensionless coupling $\tilde{G}$, with implications for realizing and probing impurity dynamics in 1D cold-atom experiments.

Abstract

The nonequilibrium dynamics of an impurity immersed with a finite velocity into a one-dimensional system of weakly interacting bosons is studied within the framework of the time-dependent Gross-Pitaevskii equation. We uncover and characterize different regimes of relaxation dynamics. We find that the final impurity velocity remains constant in a large interval of sufficiently big and realistic initial velocities. The underlying physical mechanism is the emission of the dispersive density shock waves that carry away the excess of the initial impurity momentum, while locally the system remains in the same stationary state. In contrast, a heavy impurity with the same coupling constant relaxes differently and the regime of constant final velocity disappears. Furthermore, a fast heavy impurity exhibits damped velocity oscillations in time before reaching a stationary state. This process is accompanied by the oscillations of the local depletion of the boson density around the impurity, until their positions coincide and they continue the motion together. Decreasing the impurity-boson coupling or increasing the strength of repulsion between bosons, the oscillations get amplified. In the case of a heavy impurity with the mass bigger than the critical one, the ground state energy as a function of momentum exhibits cusps and metastable branches. We show that they manifest themselves by a soliton emission, a considerable slowing down of the relaxation, and a change of the impurity direction of motion with respect to the initial one.

Figures (19)

• Figure 1: Final impurity velocity as a function of the initial impurity momentum given by Eq. (\ref{['eq:momentum']}) for $M=m/2$ (solid line) and $M=2m$ (dashed line) for two different values of the dimensionless impurity coupling constant $\tilde{G}=G/\hbar v$. Here $\gamma = 0.1$.
• Figure 2: Polaron energy (\ref{['eq:PolaronEnergy']}) as a function of the initial impurity momentum for $M=m/2$ (solid line) and $M=2m$ (dashed line) for two different values of the dimensionless impurity coupling constant $\tilde{G}=G/\hbar v$. Here $\gamma = 0.1$.
• Figure 3: Energy dispersion (\ref{['eq:PolaronEnergy']}) for $\tilde{G}=1$ and $\gamma=0.1$ for different impurity masses. The blue and red lines denote the first and the second stationary states, respectively.
• Figure 4: Final impurity velocity for the solution of Eq. (\ref{['eq:mean-field2']}) as a function of the initial impurity momentum for $M=m/2$ (solid line), $M=2m$ (dashed line) and $M=10m$ (dotted line) for two different values of the dimensionless impurity coupling $\tilde{G}$. Here $\gamma = 0.1$.
• Figure 5: Energy of the polaron for the solution of Eq. (\ref{['eq:mean-field2']}) as a function of the initial impurity momentum for $M=m/2$ (solid line), $M=10m$ (dotted line), and $M=m/10$ (dashed line) for two different values of the dimensionless impurity coupling $\tilde{G}$. Here $\gamma = 0.1$.