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arXiv:1303.3583 · Physical Review Letters

Quantum Flutter: Signatures and Robustness

Michael Knap, Charles Mathy, Martin Ganahl, Mikhail B. Zvonarev, Eugene Demler

Published Jan 7, 2014• 69 citations
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Abstract

We investigate the motion of an impurity particle injected with finite velocity into an interacting one-dimensional quantum gas. Using large-scale numerical simulations based on matrix product states, we observe and quantitatively analyze long-lived oscillations of the impurity momentum around a nonzero saturation value, called quantum flutter. We show that the quantum flutter frequency is equal to the energy difference between two branches of collective excitations of the model. We propose an explanation of the finite saturation momentum of the impurity based on the properties of the edge of the excitation spectrum. Our results indicate that quantum flutter exists away from integrability and provide parameter regions in which it could be observed in experiments with ultracold atoms using currently available technology.

DOI
10.1103/physrevlett.112.015302

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We investigate the motion of an impurity particle injected with finite velocity into an interacting one-dimensional quantum gas. Using large-scale numerical simulations based on matrix product states, we observe and quantitatively analyze long-lived oscillations of the impurity momentum around a nonzero saturation value, called quantum flutter. We show that the quantum flutter frequency is equal to the energy difference between two branches of collective excitations of the model. We propose an explanation of the finite saturation momentum of the impurity based on the properties of the edge of the excitation spectrum. Our results indicate that quantum flutter exists away from integrability and provide parameter regions in which it could be observed in experiments with ultracold atoms using currently available technology.
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.
Abstract Quantum flutter is a ubiquitous phenomenon which can be observed from the fast moving impurity injected into a fermionic or bosonic medium of quantum liquid. In this scenario, one usually consider a medium of a fully polarized state and inject a spin-flipped impurity as the initial state. When the initial velocity of the impurity is beyond the intrinsic sound velocity of the medium, the impurity momentum dramatically exhibits a long-lived periodic oscillation with the periodicity remaining invariant with respect to the initial velocity. In this letter, we show that such a novel phenomenon can be explained by a linear Luttinger liquid coupled to a deep hole in the Fermi sea. Once the deep hole excitations are involved and the impurity momentum surpass the Fermi momentum, the propagator thus displays a periodic oscillation after a quick relaxation decay. The oscillation periodicity is solely determined by the energy of the deepest hole excitation. Our result provides deep insights into the dynamical behavior of quantum impurities immersed into one-dimensional quantum liquids.
A discussion on the momentum evolution of an impurity interacting via a finite delta potential repulsion with a non-interacting fermionic background gas is presented. It has recently been shown that the momentum evolution of this system displays two interesting features, namely a non-zero thermalised value and a long-lived quantum mechanical oscillation around this plateau named "quantum flutter" [Mathy, Zvonarev, Demler, Nat. Phys. 2012]. We discuss revivals in the momentum of the impurity, which have been seen before but not yet thoroughly investigated. Subsequently it is shown the quantum flutter and revivals are caused by disjoint sets of eigenstate transitions, and this fact is used to interpret some of their aspects. This attribution of momentum features to different eigenstate subsets allows quantitative reproduction of these features with much less computational expense than has so far been possible. Finally some results on the distribution of the momentum of eigenstates and their relation to the momentum of the impurity once the system has been thermalised are presented along with a discussion on the time averaged infinite time value of the momentum and its comparison to different eigenstate subsets.
We examine stationary-state properties of an impurity particle injected into a one-dimensional quantum gas. We show that the value of the impurity's end velocity lies between zero and the speed of sound in the gas and is determined by the injection protocol. This way, the impurity's constant motion is a dynamically emergent phenomenon whose description goes beyond accounting for the kinematic constraints of the Landau approach to superfluidity. We provide exact analytic results in the thermodynamic limit and perform finite-size numerical simulations to demonstrate that the predicted phenomena are within the reach of the ultracold gas experiments.
Using the Bethe ansatz (BA), we rigorously obtain non-equilibrium dynamics of an impurity with a large initial momentum QQQ in the one-dimensional (1D) interacting bosonic medium. We show that magnon and exciton-like states obtained from the BA equations drastically determine the oscillation nature of the quantum flutter with the periodicity given by τQF=2π/(∣εc(0)∣−∣εs(0)∣)τ_{\rm QF} = 2π/(|\varepsilon_{\rm c}(0)|- |\varepsilon_{\rm s}(0)|)τQF​=2π/(∣εc​(0)∣−∣εs​(0)∣). Where the charge and spin dressed energies εc,s(0)\varepsilon_{\rm c,s}(0)εc,s​(0) are precisely given by the thermodynamical BA equations. While we further find a persistent revival dynamics of the impurity with a larger periodicity τL=L/(vc(Q−k∗)−vs(k∗))τ_{L} = L/\left(v_{\rm c}(Q-k^*)-v_{\rm s}(k^*)\right)τL​=L/(vc​(Q−k∗)−vs​(k∗)) than τQFτ_{\rm QF}τQF​, manifesting a quantum reflection induced by the periodic boundary conditions of a finite length LLL, here vc,sv_{\rm c,s}vc,s​ are the sound velocities of charge and spin excitations, respectively, and k∗k^*k∗ is a characteristic momentum of the impurity to the Fermi point. Finally, we study the application of such a magnon impurity as a quantum resource for measuring the gravitational force.
Using the Bethe ansatz (BA), we rigorously obtain nonequilibrium dynamics of an impurity with a large initial momentum QQQ in the one-dimensional interacting bosonic medium. We show that magnonlike and excitonlike states obtained from the BA equations drastically determine the oscillation nature of the quantum flutter with the periodicity given by ${\ensuremath{\tau}}_{\mathrm{QF}}=2\ensuremath{\pi}/[|{\ensuremath{\varepsilon}}_{\mathrm{c}}(0)|\ensuremath{-}|{\ensuremath{\varepsilon}}_{\mathrm{s}}(0)|]$, where the charge and spin dressed energies ${\ensuremath{\varepsilon}}_{\mathrm{c},\mathrm{s}}(0)$ are precisely given by the thermodynamical BA equations. While we further find a persistent revival dynamics of the impurity with a larger periodicity ${\ensuremath{\tau}}_{L}=L/[{v}_{\mathrm{c}}(Q\ensuremath{-}{k}^{*})\ensuremath{-}{v}_{\mathrm{s}}({k}^{*})]$ than ${\ensuremath{\tau}}_{\mathrm{QF}}$, manifesting a quantum reflection induced by the periodic boundary conditions of a finite length LLL, here vc,s{v}_{\mathrm{c},\mathrm{s}}vc,s​ are the sound velocities of charge and spin excitations, respectively, and k∗{k}^{*}k∗ is a characteristic momentum of the impurity to the Fermi point. Finally, we study the application of such a magnon impurity as a quantum resource for measuring the gravitational force.