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Physical Review A

Microscopic origin of the quantum supersonic phenomenon in one dimension

Z. H. Zhang, Yuzhu Jiang, Hai‐Qing Lin, Xi-Wen Guan

Published Aug 28, 2024• 4 citations
Publisher

Abstract

Using the Bethe ansatz (BA), we rigorously obtain nonequilibrium dynamics of an impurity with a large initial momentum $Q$ 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 $L$, here ${v}_{\mathrm{c},\mathrm{s}}$ are the sound velocities of charge and spin excitations, respectively, and ${k}^{*}$ 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.

DOI
10.1103/physreva.110.023329

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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 \ensuremathτQF=2\ensuremathπ/[∣\ensuremathεc(0)∣\ensuremath−∣\ensuremathεs(0)∣]{\ensuremath{\tau}}_{\mathrm{QF}}=2\ensuremath{\pi}/[|{\ensuremath{\varepsilon}}_{\mathrm{c}}(0)|\ensuremath{-}|{\ensuremath{\varepsilon}}_{\mathrm{s}}(0)|]\ensuremathτQF​=2\ensuremathπ/[∣\ensuremathεc​(0)∣\ensuremath−∣\ensuremathεs​(0)∣], where the charge and spin dressed energies \ensuremathεc,s(0){\ensuremath{\varepsilon}}_{\mathrm{c},\mathrm{s}}(0)\ensuremathεc,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 \ensuremathτL=L/[vc(Q\ensuremath−k∗)\ensuremath−vs(k∗)]{\ensuremath{\tau}}_{L}=L/[{v}_{\mathrm{c}}(Q\ensuremath{-}{k}^{*})\ensuremath{-}{v}_{\mathrm{s}}({k}^{*})]\ensuremathτL​=L/[vc​(Q\ensuremath−k∗)\ensuremath−vs​(k∗)] than \ensuremathτQF{\ensuremath{\tau}}_{\mathrm{QF}}\ensuremathτQF​, 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.
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.
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.
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 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.
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.