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arXiv:2304.14490 · New Journal of Physics

Dynamics of polaron formation in 1D Bose gases in the strong-coupling regime

Martin Will, Michael Fleischhauer

Published Aug 1, 2023• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/acf06a

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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 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.
Bose polarons, quasiparticles composed of mobile impurities surrounded by cold Bose gas, can experience strong interactions mediated by the many-body environment and form bipolaron bound states. Here we present a detailed study of heavy polarons in a one-dimensional Bose gas by formulating a nonperturbative theory and complementing it with exact numerical simulations. We develop an analytic approach for weak boson-boson interactions and arbitrarily strong impurity-boson couplings. Our approach is based on a mean-field theory that accounts for deformations of the superfluid by the impurities and in this way minimizes quantum fluctuations. The mean-field equations are solved exactly in the Born-Oppenheimer approximation, leading to an analytic expression for the interaction potential of heavy polarons, which is found to be in excellent agreement with quantum Monte Carlo (QMC) results. In the strong coupling limit, the potential substantially deviates from the exponential form valid for weak coupling and has a linear shape at short distances. Taking into account the leading-order Born-Huang corrections, we calculate bipolaron binding energies for impurity-boson mass ratios as low as 3 and find excellent agreement with QMC results.
We consider the dynamics of a quantum impurity after a sudden interaction quench into a one-dimensional degenerate Bose gas. We use the Keldysh path-integral formalism to derive a truncated Wigner-like approach that takes the back action of the impurity onto the condensate into account already on the mean-field level and further incorporates thermal and quantum effects up to one-loop accuracy. This framework enables us not only to calculate the real-space trajectory of the impurity but also the absorption spectrum. We find that quantum corrections and thermal effects play a crucial role for the impurity momentum at weak to intermediate impurity-bath couplings. Furthermore, we see the broadening of the absorption spectrum with increasing temperature.
We study a weakly interacting one-dimensional Bose gas with two impurities coupled locally to the boson density. We derive analytical results for the induced interaction between the impurities at arbitrary positive coupling and separation rrr. At $r\ensuremath{\lesssim}\ensuremath{\xi}$, where $\ensuremath{\xi}$ denotes the healing length of the Bose gas, the interaction is well described by the mean-field contribution. Its form changes as the coupling is increased, approaching a linear function of rrr at short distances in the regime of strong coupling. The mean-field contribution decays exponentially at arbitrary coupling for $r\ensuremath{\gg}\ensuremath{\xi}$. At such long distances, however, the effect of quantum fluctuations becomes important, giving rise to a long-ranged quantum contribution to the induced interaction. At longest distances it behaves as 1/r31/{r}^{3}1/r3, while at strong coupling we find an intermediate distance regime with a slower decay, 1/r1/r1/r. The quantum contribution in the crossover regime is also calculated. The induced interaction between impurities (i.e., polarons) is attractive and leads to the formation of their bound state, known as bipolaron. We discuss its binding energy.
We present a semiclassical treatment of a fermionic impurity coupled to Bogolyubov modes of a Bose-Einstein condensate (BEC). In the lowest-order approximation we find a full solution of an initial value problem, which turns out to behave differently in the sub- and supersonic regimes. While in the former case no impurity deceleration is observed, in the latter case non-Markovian dissipation effects kick in, resulting in a slowing down of the fermion. Although this scenario is compatible with the one offered by an elementary field-theoretical picture at weak coupling, the details of the dynamics turn out to be completely different. Fluctuation effects can be taken into account by expanding around the classical path, which leads to a natural cutoff scale for the momentum integrals of the order of the inverse polaron radius. As an application we calculate the drag force which is exerted by the BEC on the impurity moving with constant velocity vvv. Contrary to the perturbative result, according to which the drag force is $\ensuremath{\sim}{v}^{4}$, it turns out to be proportional to 1/v21/{v}^{2}1/v2 in the semiclassical regime.
When a mobile impurity interacts with a surrounding bath of bosons, it forms a polaron. Numerous methods have been developed to calculate how the energy and the effective mass of the polaron are renormalized by the medium for equilibrium situations. Here, we address the much less studied nonequilibrium regime and investigate how polarons form dynamically in time. To this end, we develop a time-dependent renormalization-group approach which allows calculations of all dynamical properties of the system and takes into account the effects of quantum fluctuations in the polaron cloud. We apply this method to calculate trajectories of polarons following a sudden quench of the impurity-boson interaction strength, revealing how the polaronic cloud around the impurity forms in time. Such trajectories provide additional information about the polaron's properties which are challenging to extract directly from the spectral function measured experimentally using ultracold atoms. At strong couplings, our calculations predict the appearance of trajectories where the impurity wavers back at intermediate times as a result of quantum fluctuations. Our method is applicable to a broader class of nonequilibrium problems. As a check, we also apply it to calculate the spectral function and find good agreement with experimental results. At very strong couplings, we predict that quantum fluctuations lead to the appearance of a dark continuum with strongly suppressed spectral weight at low energies. While our calculations start from an effective Fr\"ohlich Hamiltonian describing impurities in a three-dimensional Bose-Einstein condensate, we also calculate the effects of additional terms in the Hamiltonian beyond the Fr\"ohlich paradigm. We demonstrate that the main effect of these additional terms on the attractive side of a Feshbach resonance is to renormalize the coupling strength of the effective Fr\"ohlich model.