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arXiv:2507.10402 · Physical Review A

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

Saptarshi Majumdar, Aleksandra Petković

Published Nov 25, 2025• 2 citations
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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.

DOI
10.1103/tgl5-rdn5

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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.
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 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.
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.
The frictionless motion of an object through a fluid medium is commonly viewed as a hallmark of superfluidity. According to Landau, kinematic constraints prohibit superfluid behavior in one-dimensional (1D) bosonic systems. Here, using ultracold atoms, we show how a microscopic impurity can propagate through a strongly interacting 1D Bose gas without any friction, at odds with conventional expectations. We inject the impurity with initial velocities ranging from the subsonic to supersonic regime, and subsequently track its dynamics. For supersonic initial velocities, we observe the formation of a shock wave and a remarkably fast relaxation to a stationary regime, on a time scale that increases with decreasing impurity velocity. After reaching the stationary state, the impurity continues its motion through the system with a finite velocity. Our findings demonstrate how quantum effects can conspire to eliminate dissipation of a microscopic object immersed in a quantum fluid, thereby bringing novel insights into the propagation of matter and information in the quantum realm.
We study the motion of a heavy impurity in a one-dimensional Bose gas. The impurity experiences the friction force due to scattering off thermally excited quasiparticles. We present detailed analysis of an arbitrarily strong impurity-boson coupling in a wide range of temperatures within a microscopic theory. Focusing mostly on weakly interacting bosons, we derive an analytical result for the friction force and uncover new regimes of the impurity dynamics. Particularly interesting is the low-temperature T^{2} dependence of the friction force obtained for a strongly coupled impurity, which should be contrasted with the expected T^{4} scaling. This new regime applies to systems of bosons with an arbitrary repulsion strength. We finally study the evolution of the impurity with a given initial momentum. We evaluate analytically its nonstationary momentum distribution function. The impurity relaxation towards the equilibrium is a realization of the Ornstein-Uhlenbeck process in momentum space.
The challenge of understanding the dynamics of a mobile impurity in an interacting quantum many-body medium comes from the necessity of including entanglement between the impurity and excited states of the environment in a wide range of energy scales. In this Letter, we investigate the motion of a finite mass impurity injected into a three-dimensional quantum Bose fluid as it starts shedding Bogoliubov excitations. We uncover a transition in the dynamics as the impurity's velocity crosses a critical value that depends on the strength of the interaction between the impurity and bosons as well as the impurity's recoil energy. We find that in injection experiments, the two regimes differ not only in the character of the impurity velocity abatement but also exhibit qualitative differences in the Loschmidt echo, density ripples excited in the Bose-Einstein condensate, and momentum distribution of scattered bosonic particles. The transition is a manifestation of a dynamical quantum Cherenkov effect and should be experimentally observable with ultracold atoms using Ramsey interferometry, rf spectroscopy, absorption imaging, and time-of-flight imaging.
Abstract We unravel the nonequilibrium correlated quantum quench dynamics of an impurity traveling through a harmonically confined Bose–Einstein condensate in one-dimension. For weak repulsive interspecies interactions the impurity oscillates within the bosonic gas. At strong repulsions and depending on its prequench position the impurity moves towards an edge of the bosonic medium and subsequently equilibrates. This equilibration being present independently of the initial velocity, the position and the mass of the impurity is inherently related to the generation of entanglement in the many-body system. Focusing on attractive interactions the impurity performs a damped oscillatory motion within the bosonic bath, a behavior that becomes more evident for stronger attractions. To elucidate our understanding of the dynamics an effective potential picture is constructed. The effective mass of the emergent quasiparticle is measured and found to be generically larger than the bare one, especially for strong attractions. In all cases, a transfer of energy from the impurity to the bosonic medium takes place. Finally, by averaging over a sample of simulated in situ single-shot images we expose how the single-particle density distributions and the two-body interspecies correlations can be probed.