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arXiv:1907.08324 · Physical Review A

Dynamics of relaxation and dressing of a quenched Bose polaron

D. Boyanovsky, David Jasnow, Xiao-Lun Wu, Rob D. Coalson

Published Oct 30, 2019• 23 citations
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Abstract

We study the nonequilibrium dynamics of relaxation and dressing of a mobile impurity with velocity $v$, suddenly immersed, or quenched, into a zero-temperature homogeneous Bose-Einstein condensate. A many-body generalization of Weisskopf-Wigner theory is implemented to obtain the impurity fidelity, reduced density matrix and entanglement entropy. The dynamics depends crucially on the Mach number $\ensuremath{\beta}=v/c$, with $c$ the speed of sound of superfluid phonons, and features many different timescales. Quantum Zeno behavior at early time is followed by nonequilibrium dynamics determined by Cerenkov emission of long-wavelength phonons for $\ensuremath{\beta}>1$ with a relaxation rate ${\mathrm{\ensuremath{\Gamma}}}_{p}\ensuremath{\propto}{(\ensuremath{\beta}\ensuremath{-}1)}^{3}$. The polaron dressing dynamics slows down as $\ensuremath{\beta}\ensuremath{\rightarrow}1$ and is characterized by power laws ${t}^{\ensuremath{-}\ensuremath{\alpha}}$ with exponents $\ensuremath{\alpha}=3/2,1/2,2$ for $\ensuremath{\beta}>1,=1,<1,$ respectively. The asymptotic entanglement entropy features a sharp discontinuity, and the residue features a cusp at $\ensuremath{\beta}=1$. These nonequilibrium features suggest universal dynamical critical phenomena near $\ensuremath{\beta}\ensuremath{\simeq}1$ and are a direct consequence of the linear dispersion relation of long-wavelength superfluid phonons. We conjecture on the emergence of an asymptotic dynamical attractor with $\ensuremath{\beta}\ensuremath{\le}1$.

DOI
10.1103/physreva.100.043617

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We study the nonequilibrium dynamics of relaxation and dressing of a mobile impurity with velocity vvv, suddenly immersed, or quenched, into a zero-temperature homogeneous Bose-Einstein condensate. A many-body generalization of Weisskopf-Wigner theory is implemented to obtain the impurity fidelity, reduced density matrix and entanglement entropy. The dynamics depends crucially on the Mach number \ensuremathβ=v/c\ensuremath{\beta}=v/c\ensuremathβ=v/c, with ccc the speed of sound of superfluid phonons, and features many different timescales. Quantum Zeno behavior at early time is followed by nonequilibrium dynamics determined by Cerenkov emission of long-wavelength phonons for \ensuremathβ>1\ensuremath{\beta}>1\ensuremathβ>1 with a relaxation rate \ensuremathΓp\ensuremath∝(\ensuremathβ\ensuremath−1)3{\mathrm{\ensuremath{\Gamma}}}_{p}\ensuremath{\propto}{(\ensuremath{\beta}\ensuremath{-}1)}^{3}\ensuremathΓp​\ensuremath∝(\ensuremathβ\ensuremath−1)3. The polaron dressing dynamics slows down as \ensuremathβ\ensuremath→1\ensuremath{\beta}\ensuremath{\rightarrow}1\ensuremathβ\ensuremath→1 and is characterized by power laws t\ensuremath−\ensuremathα{t}^{\ensuremath{-}\ensuremath{\alpha}}t\ensuremath−\ensuremathα with exponents \ensuremathα=3/2,1/2,2\ensuremath{\alpha}=3/2,1/2,2\ensuremathα=3/2,1/2,2 for \ensuremathβ>1,=1,<1,\ensuremath{\beta}>1,=1,<1,\ensuremathβ>1,=1,<1, respectively. The asymptotic entanglement entropy features a sharp discontinuity, and the residue features a cusp at \ensuremathβ=1\ensuremath{\beta}=1\ensuremathβ=1. These nonequilibrium features suggest universal dynamical critical phenomena near \ensuremathβ\ensuremath≃1\ensuremath{\beta}\ensuremath{\simeq}1\ensuremathβ\ensuremath≃1 and are a direct consequence of the linear dispersion relation of long-wavelength superfluid phonons. We conjecture on the emergence of an asymptotic dynamical attractor with \ensuremathβ\ensuremath≤1\ensuremath{\beta}\ensuremath{\le}1\ensuremathβ\ensuremath≤1.
We study the quantum dynamics of a single impurity following its sudden immersion into a Bose–Einstein condensate. The ensuing formation of the Bose polaron in this general setting can be seen as impurity decoherence driven by the condensate, which we describe within a master equation approach. We derive rigorous analytical results for this decoherence dynamics, and thereby reveal distinct stages of its evolution from a universal stretched exponential initial relaxation to the final approach to equilibrium. The associated polaron formation time exhibits a strong dependence on the impurity speed and is found to undergo a critical slowdown around the speed of sound of the condensate. This rich non-equilibrium behavior of quantum impurities is of direct relevance to recent cold atom experiments, in which Bose polarons are created by a sudden quench of the impurity-bath interaction.
We study the quantum dynamics of a single impurity following its sudden immersion into a Bose-Einstein condensate. The formation of the Bose polaron after such a quench stems from decoherence of the impurity, driven by collisions with the condensate. Using a master equation approach, we derive rigorous analytical results for the decoherence dynamics of the impurity, which reveals different stages of its evolution from a universal non-exponential initial relaxation to the final approach of the equilibrium state of the Bose polaron. The associated polaron formation time exhibit a strong dependence of the impurity momentum and is found to undergo a critical slowdown around the Landau critical velocity of the condensate. The rich non-equilibrium behavior of quantum impurities in a Bose gas revealed in this work is of direct relevance to recent cold-atom experiments, in which Bose polarons are created by a sudden quench of the impurity-bath interaction.
Predicting the emergent properties of impurities immersed in a quantum bath is a fundamental challenge that can defy quasiparticle treatments. Here, we measure the spectral properties and real-time dynamics of mobile impurities injected into a weakly interacting homogeneous Bose-Einstein condensate, using two broad Feshbach resonances to tune both the impurity-bath and intrabath interactions. For attractive impurity-bath interactions, the impurity spectrum features a single branch, which away from the resonance corresponds to a well-defined attractive polaron; near the resonance, we observe dramatic broadening of this branch, suggesting a breakdown of the quasiparticle picture. For repulsive impurity-bath interactions, the spectrum features two branches: the attractive branch that is dominated by excitations with energy close to that of the Feshbach dimer, but has a many-body character, and the repulsive polaron branch. Our measurements show that the behavior of impurities in weakly interacting baths is remarkably universal, controlled only by the bath density and a single dimensionless interaction parameter.
The mobile impurity in a Bose-Einstein condensate (BEC) is a paradigmatic many-body problem. For weak interaction between the impurity and the BEC, the impurity deforms the BEC only slightly and it is well described within the Fröhlich model and the Bogoliubov approximation. For strong local attraction this standard approach, however, fails to balance the local attraction with the weak repulsion between the BEC particles and predicts an instability where an infinite number of bosons is attracted toward the impurity. Here we present a solution of the Bose polaron problem beyond the Bogoliubov approximation which includes the local repulsion between bosons and thereby stabilizes the Bose polaron even near and beyond the scattering resonance. We show that the Bose polaron energy remains bounded from below across the resonance and the size of the polaron dressing cloud stays finite. Our results demonstrate how the dressing cloud replaces the attractive impurity potential with an effective many-body potential that excludes binding. We find that at resonance, including the effects of boson repulsion, the polaron energy depends universally on the effective range. Moreover, while the impurity contact is strongly peaked at positive scattering length, it remains always finite. Our solution highlights how Bose polarons are self-stabilized by repulsion, providing a mechanism to understand quench dynamics and nonequilibrium time evolution at strong coupling.
Abstract We discuss the dynamics of the formation of a Bose polaron when an impurity is injected into a weakly interacting one-dimensional Bose condensate. While for small impurity-boson couplings this process can be described within the Froehlich model as generation, emission and binding of Bogoliubov phonons, this is no longer adequate if the coupling becomes strong. To treat this regime we consider a mean-field approach beyond the Froehlich model which accounts for the backaction to the condensate, complemented with Truncated Wigner simulations to include quantum fluctuation. For the stationary polaron we find a periodic energy-momentum relation and non-monotonous relation between impurity velocity and polaron momentum including regions of negative impurity velocity. Studying the polaron formation after turning on the impurity-boson coupling quasi-adiabatically and in a sudden quench, we find a very rich scenario of dynamical regimes. Due to the build-up of an effective mass, the impurity is slowed down even if its initial velocity is below the Landau critical value. For larger initial velocities we find deceleration and even backscattering caused by emission of density waves or grey solitons and subsequent formation of stationary polaron states in different momentum sectors. In order to analyze the effect of quantum fluctuations we consider a trapped condensate to avoid 1D infrared divergencies. Using Truncated Wigner simulations in this case we show under what conditions the influence of quantum fluctuations is small.
We give a detailed account of a stationary impurity in an ideal Bose-Einstein condensate, which we call the ideal Bose polaron, at both zero and nonzero temperatures and arbitrary strength of the impurity-boson coupling. The time evolution is solved exactly and it is found that, surprisingly, many of the features that have been predicted for the real BEC are already present in this simpler setting and can be understood analytically therein. We obtain explicit formulas for the time evolution of the condensate wave function at T=0T=0T=0 and of the one-particle density matrix at T>0T>0T>0. For negative scattering length, the system is found to thermalize even though the dynamics are perfectly coherent. The time evolution and thermal values of the Tan contact are derived and compared to a recent experiment. We find that contrary to the Fermi polaron, the contact is not bounded at unitarity as long as a condensate exists. An explicit formula for the dynamical overlap at T=0T=0T=0 allows us to compute the rf spectrum which can be understood in detail by relating it to the two-body problem of one boson and the impurity.