Bloch oscillations of a mobile impurity in a one-dimensional Bose gas

TL;DR

This study analyzes the far-from-equilibrium dynamics of an externally driven mobile impurity in a 1D Bose gas with weak interactions. Using a two-step transformation to a co-moving frame and a mean-field treatment for weak boson-boson coupling, it reveals that impurity Bloch oscillations persist across a wide range of forces by periodically emitting nonlinear excitations—dispersive waves, shock waves, and solitons—that transfer momentum to the bath. The oscillations feature a drift velocity $V_d$ and a period close to $T=2\pi\hbar n_0/F$, with the Bloch amplitude $V_B$ and $V_d$ showing rich, parameter-dependent behavior; at strong impurity-bath coupling or large forces, the dynamics cross over to regimes where oscillations weaken or cease. The results highlight the intricate coupling between impurity motion, background density and phase dynamics, and emergent nonlinear excitations, suggesting robust Bloch-like transport in 1D Bose gases under drive and guiding future cold-atom experiments.

Abstract

We study the motion of an impurity under the action of a constant force through a one-dimensional system of weakly-interacting bosons. The interplay of the impurity-boson interaction, the boson-boson interaction, and the driving force gives rise to a rich dynamics. We focus on the influence of a finite external force. Under these far-from-equilibrium conditions, we show that in a wide range of forces, one part of the momentum transferred to the system is periodically channeled into the Bose gas through the emission of dispersive density shock waves, solitons, density waves and the creation of additional phase gradients. As a result, the impurity velocity does not increase indefinitely, but periodically oscillates in time around the drift velocity. We uncover and characterize different dynamical regimes in a wide range of the impurity-boson coupling, the impurity mass and the external force. At a sufficiently large force, the Bloch oscillations cease and the impurity exhibits an unlimited acceleration.

Figures (12)

• Figure 1: Impurity velocity as a function of the system momentum $p=F t$ for $0\leq t\leq 2\pi \hbar n_0/F$ is shown for different values of $F$. In the case of an infinitesimal force, the velocity of the finite-momentum ground state is realized. It is studied in Sec \ref{['sec:Stationary']}. The system parameters are $M=3m,\gamma=1/400$ and $\tilde{G}=G/\hbar v=0.5$. Here, $\tilde{F}=F \xi/g n_0$.
• Figure 2: Phase of the condensate wave function for a) several values of the system momentum in the range $0<p<\pi \hbar n_0$, b) several values of the system momentum in the range $\pi \hbar n_0<p<2\pi \hbar n_0$, c) for $p=7\pi \hbar n_0/2$ and d) $p=11\pi\hbar n_0/2$. For illustrative purposes, the system length is set to $L=50\xi$.
• Figure 3: Time evolution of the impurity velocity for $M=3m$, $\tilde{G}=0.5$ and $\gamma = 1/400$ in the a) small, b) intermediate and c) large-force regime. d) The dimensionless drift velocity $V_d/v$ is shown by the black points and $V_0/v$ by the green squares as a function of the dimensionless force $\tilde{F}$. The linear fit of the drift velocity is represented by the dashed line, $V_d/v = \tilde{\sigma} \tilde{F}$, and gives us the dimensionless mobility $\tilde{\sigma}= 0.094$. e) The dimensionless Bloch oscillation amplitude $V_B/v$ and f) the dimensionless time period $gn_0 T/\hbar$ are shown as a function of the dimensionless force $\tilde{F}$. The dimensionless time period is fitted with the prediction $gn_0 T/\hbar = 2 \pi/(\tilde{F} \sqrt{\gamma})$, shown by the dashed line.
• Figure 4: Time evolution of the boson density in the reference frame defined in Sec. \ref{['sec:model']}, for $\tilde{G}=0.5,M=3m,\tilde{F}=3.5$ and $\gamma=1/400$. The dimensionless time is defines as $\tilde{t}=t g n_0/\hbar$. The ground-state configuration, Eq. (\ref{['densityStationary']}), for the momentum $p=F t$ is shown by the dashed line. The impurity velocity reaches maximal value at approximately $2.4$ and $37.6$, its minimal value at $31.7$ and $66.6$, and zero at $24.1$, $34.4$, $58.9$, and $69.3$.
• Figure 5: Phase (top row) and density profile (bottom row) of the bosons in the reference frame defined in Sec. \ref{['sec:model']} for $\tilde{G}=0.5,M=3m,\tilde{F}=3.5$, and $\gamma=1/400$ at different times: (a) $\tilde{t}=10$, (b) $\tilde{t}=45$, and (c) $\tilde{t}=80$. The time period of the impurity velocity oscillations is $\tilde{T}=34.8$.