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Communications in Theoretical Physics

Quantum Flutter from Nonlinear Luttinger Liquid perspective

Sheng Wang, Zhehao Zhang, Xi‐Wen Guan

Published Nov 18, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1572-9494/ad93b6

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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.
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.
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.
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.
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.
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.
Luttinger liquid theory accounts for the low-energy boson excitations of one-dimensional quantum liquids, but disregards the high-energy excitations. The most important high-energy excitations are holes, which have infinite lifetime at zero temperature. At finite temperatures, they can be scattered by thermally excited bosons. We describe the interaction of a hole with the bosons by treating it as a mobile impurity in a Luttinger liquid. This approach enables us to evaluate the scattering probability at arbitrary interaction strength. In general, the result is expressed in terms of the hole spectrum, its dependence on the density and momentum of the fluid, and the parameters of the Luttinger liquid Hamiltonian. In the special case of Galilean invariant systems, the scattering probability is expressed in terms of only the hole spectrum and its dependence on the fluid density. We apply our results to the problem of equilibration of one-dimensional quantum liquids.