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Perpetual motion and backscattering oscillations of a mobile impurity in a quantum fluid

Oleg Valentinovich Lychkovskiy

Published Mar 28, 2014• 2 citations
OpenAlex

Abstract

We study the dynamics of a mobile impurity in a quantum fluid at zero temperature. Two related settings are considered. In the first setting an impurity is injected in a fluid with some initial velocity ${\mathbf v}_0$, and we are interested in its velocity at infinite time, ${\mathbf v}_\infty$. We derive a rigorous upper bound on $|{\mathbf v}_0-{\mathbf v}_\infty|$ for initial velocities smaller than some critical velocity. In the limit of vanishing impurity-fluid coupling this bound can be regarded as a rigorous proof of the Landau criterion of superfluidity. In the case of a finite coupling the velocity of the impurity can drop, but not to zero; the bound quantifies the maximal possible drop. In the second setting a small constant force acts on the impurity. We identify two possible regimes of the impurity dynamics -- backscattering oscillations of the impurity velocity and saturation of the velocity without oscillations. Which regime is realized depends on the spectrum of the fluid and the mass of the impurity. In particular, backscattering oscillations are possible for sufficiently heavy impurities in fluids with a roton feature of the spectrum. We critically discuss a concept of Bloch oscillations of a mobile impurity in a one-dimensional quantum fluid introduced in [Phys. Rev. Lett. 102, 070402 (2009)] and argue that this type of oscillations is absent in the thermodynamic limit.

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We study the dynamics of a mobile impurity in a quantum fluid at zero temperature. Two related settings are considered. In the first setting an impurity is injected in a fluid with some initial velocity v0{\mathbf v}_0v0​, and we are interested in its velocity at infinite time, v∞{\mathbf v}_\inftyv∞​. We derive a rigorous upper bound on ∣v0−v∞∣|{\mathbf v}_0-{\mathbf v}_\infty|∣v0​−v∞​∣ for initial velocities smaller than some critical velocity. In the limit of vanishing impurity-fluid coupling this bound can be regarded as a rigorous proof of the Landau criterion of superfluidity. In the case of a finite coupling the velocity of the impurity can drop, but not to zero; the bound quantifies the maximal possible drop. In the second setting a small constant force acts on the impurity. We identify two possible regimes of the impurity dynamics -- backscattering oscillations of the impurity velocity and saturation of the velocity without oscillations. Which regime is realized depends on the spectrum of the fluid and the mass of the impurity. In particular, backscattering oscillations are possible for sufficiently heavy impurities in fluids with a roton feature of the spectrum. We critically discuss a concept of Bloch oscillations of a mobile impurity in a one-dimensional quantum fluid introduced in [Phys. Rev. Lett. 102, 070402 (2009)] and argue that this type of oscillations is absent in the thermodynamic limit.
We study the dynamics of a mobile impurity in a quantum fluid at zero temperature. Two related settings are considered. In the first setting, the impurity is injected in the fluid with some initial velocity v0{\mathbf{v}}_{0}v0​, and we are interested in its velocity at infinite time, ${\mathbf{v}}_{\ensuremath{\infty}}$. We derive a rigorous upper bound on $|{\mathbf{v}}_{0}\ensuremath{-}{\mathbf{v}}_{\ensuremath{\infty}}|$ for initial velocities smaller than the generalized critical velocity. In the limit of vanishing impurity-fluid coupling, this bound amounts to ${\mathbf{v}}_{\ensuremath{\infty}}={\mathbf{v}}_{0}$, which can be regarded as a rigorous proof of the Landau criterion of superfluidity. In the case of a finite coupling, the velocity of the impurity can drop, but not to zero; the bound quantifies the maximal possible drop. In the second setting, a small constant force is exerted upon the impurity. We argue that two distinct dynamical regimes exist---backscattering oscillations of the impurity velocity and saturation of the velocity without oscillations. For fluids with vcL=vs{v}_{c\mathrm{L}}={v}_{s}vcL​=vs​ (where vcL{v}_{c\mathrm{L}}vcL​ and vs{v}_{s}vs​ are the Landau critical velocity and sound velocity, respectively), the latter regime is realized. For fluids with vcL<vs{v}_{c\mathrm{L}}<{v}_{s}vcL​<vs​, both regimes are possible. Which regime is realized in this case depends on the mass of the impurity, a nonequilibrium quantum phase transition occurring at some critical mass. Our results are equally valid in one, two, and three dimensions.
We study the notion of superfluid critical velocity in one spatial dimension. It is shown that, for heavy impurities with mass M exceeding a critical mass Mc, the dispersion develops periodic metastable branches resulting in dramatic changes of dynamics in the presence of an external driving force. In contrast to smooth Bloch oscillations for M<Mc, a heavy impurity climbs metastable branches until it reaches a branch termination point or undergoes a random tunneling event, both leading to an abrupt change in velocity and an energy loss. This is predicted to lead to a nonanalytic dependence of the impurity drift velocity on small forces.
We study the dynamics of an impurity of finite mass injected into a one-dimensional quantum liquid at zero temperature, either at finite velocity or at zero velocity with a force driving the impurity. We obtain accurate results using numerical simulations based on matrix product states, and find that in both cases, the impurity undergoes oscillations, however the physical mechanism is different: the driven impurity undergoes Bloch oscillations by following the ground state branch while continuously emitting phonons, whereas the undriven impurity undergoes coherent quantum oscillations at an emergent energy scale, called quantum flutter in previous work, whose amplitude grows with increasing initial velocity. We find these results to be independent of whether the system is integrable or not, and robust to changes in the microscopics of the model, suggesting that they are universal.
Consider an impurity particle injected in a degenerate one-dimensional gas of noninteracting fermions (or, equivalently, Tonks-Girardeau bosons) with some initial momentum p0{p}_{0}p0​. We examine the infinite-time value of the momentum of the impurity, ${p}_{\ensuremath{\infty}}$, as a function of p0{p}_{0}p0​. A lower bound on $|{p}_{\ensuremath{\infty}}({p}_{0})|$ is derived under fairly general conditions. The derivation, based on the existence of the lower edge of the spectrum of the host gas, does not resort to any approximations. The existence of such bound implies the perpetual motion of the impurity in a one-dimensional gas of noninteracting fermions or Tonks-Girardeau bosons at zero temperature. The bound admits an especially simple and useful form when the interaction between the impurity and host particles is everywhere repulsive.
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 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.