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Effective dynamics for the Bose polaron in the large-volume mean-field limit

Jonas Lampart, Peter Pickl, Siegfried Spruck

Published Jan 1, 2026• 0 citations
Publisher

Abstract

We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of $N$ bosons evolving in $\mathbb{R}^3$ in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density $\rho$ and large volume $\Lambda$ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint $\Lambda^3\ll\rho$, the effective description by the translation-invariant Bogoliubov-Fröhlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.

DOI
10.5445/ir/1000193527

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Dérivation de la Dynamique Effective du Polaron Bosonique Nous considérons la dynamique d'un gaz quantique dense et de grande taille composé de N bosons évoluant dans ⁠{\displaystyle \mathbb {R} }^3 en présence d'une particule d'impureté dans la limite de champs moyen. Dans l'état initial du système, presque tous les bosons sont dans le condensat de Bose-Einstein, avec quelques particules hors du condensat appelées excitations. Notre objectif est de prouver la persistance de la condensation et la formation d'un polaron bosonique, une quasi-particule composée de l'impureté et des bosons environnants.Dans les expériences, les particules d'impureté, qui se distinguent du gaz bosonique, sont utilisées pour sonder les propriétés du gaz, telles que la superfluidité ou sa distribution de densité. Le gaz de Bose est mathématiquement intriguant car la condensation de Bose-Einstein permet une analyse rigoureuse, donnant une compréhension profonde des systèmes quantiques à nombreux corps.Dans notre résultat principal, nous dérivons une description effective de la théorie quantique des champs pour de tels systèmes à partir de la dynamique microscopique. Cette description est régie par l'hamiltonien de Bogoliubov-Fröhlich dans la limite des grandes densités de gaz et des volumes, modélisant la dynamique de l'impureté et des excitations, qui peuvent former une quasi-particule connue sous le nom de polaron bosonique.Les travaux précédents ont traité le problème du champ moyen correspondant sur un volume unitaire avec des conditions aux limites périodiques. La principale nouveauté de cette thèse réside dans la prise en compte d'un volume initial important par rapport à la plage d'interaction des bosons et dans la suppression des conditions aux limites périodiques, ce qui conduit à un condensat non constant. Ces adaptations rapprochent le modèle de la réalité physique.Pour prouver notre résultat principal, qui est formulé dans la dynamique effective de Bogoliubov, nous contrôlons le nombre d'excitations afin d'assurer la persistance de la condensation. De plus, comme l'impureté doit rester dans le nuage de gaz pour la formation d'une quasi-particule, nous prouvons sa localisation. Pour cette deuxième partie en particulier, un contrôle raffiné de l'état initial du condensat est nécessaire afin d'éviter d'importants transferts d'énergie du condensat vers l'impureté. Le contrôle des quantités susmentionnées est établi par des estimations de propagation et par le lemme de Grönwall.En combinant le contrôle du nombre d'excitations et la localisation de l'impureté, nous validons la dynamique de Bogoliubov-Fröhlich.

•Jul 1, 2025•0 citations
We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of NNN bosons evolving in R3\mathbb{R}^3R3 in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density ρ\rhoρ and large volume Λ\LambdaΛ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint Λ3≪ρ\Lambda^3\ll\rhoΛ3≪ρ, the effective description by the translation-invariant Bogoliubov-Fröhlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.
We consider the dynamics of the Bose polaron system, a dense quantum gas consisting of NNN bosons evolving in R3\mathbb{R}^3R3 in the presence of an impurity particle. The system is studied in the mean-field scaling with initially high density ρρρ and large volume ΛΛΛ of the gas. In the initial state, almost all bosons are in the Bose-Einstein condensate, with a few excitations. We derive from the microscopic dynamics, in the joint limit of large densities and volumes, with the constraint Λ3≪ρΛ^3 \ll ρΛ3≪ρ, the effective description by the translation-invariant Bogoliubov-Fröhlich Hamiltonian, which couples the quantum field of excitations linearly to the impurity particle.
Abstract We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle on a three-dimensional box with periodic boundary condition. For large Fermi momentum kFk_{\text {F}}kF​ k F , we prove that the effective dynamics is generated by a Fröhlich-type polaron Hamiltonian, which linearly couples the impurity particle to an almost-bosonic excitation field. Moreover, we prove that the effective dynamics can be approximated by an explicit coupled coherent state. Our method is applicable to a range of interaction couplings, in particular including interaction couplings of order 1 and time scales of the order kF−1k_{\text {F}}^{-1}kF−1​ k F - 1 .
We study the quantum motion of an impurity atom immersed in a Bose-Einstein condensate in arbitrary dimensions. It was shown, for all dimensions, that the Bogoliubov excitations of the Bose-Einstein condensate act as a bosonic bath for the impurity, where linear coupling is possible for a certain regime of validity, which was assessed only in one dimension. Here we present the detailed derivation of the ddd-dimensional Langevin equations that describe the quantum dynamics of the system, and of the associated generalized tensor that describes the spectral density in the full generality, and assesses the linear assumption in all dimensions. As results, we obtain, when the impurity is not trapped, the mean square displacement in all dimensions, showing that the motion is superdiffusive. We obtain also explicit expressions for the superdiffusive coefficient in the small and large temperature limits. We find that, in the latter case, the maximal value of this coefficient is the same in all dimensions, but is only reachable in one dimension, within the validity of the assumptions. We study also the behavior of the average energy and compare the results for various dimensions. In the trapped case, we study squeezing and find that the stronger position squeezing can be obtained in lower dimensions. We quantify the non-Markovianity of the particle's motion and find that it increases with dimensionality.
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.
Dérivation de la Dynamique Effective du Polaron Bosonique Nous considérons la dynamique d'un gaz quantique dense et de grande taille composé de N bosons évoluant dans ⁠{\displaystyle \mathbb {R} }^3 en présence d'une particule d'impureté dans la limite de champs moyen. Dans l'état initial du système, presque tous les bosons sont dans le condensat de Bose-Einstein, avec quelques particules hors du condensat appelées excitations. Notre objectif est de prouver la persistance de la condensation et la formation d'un polaron bosonique, une quasi-particule composée de l'impureté et des bosons environnants.Dans les expériences, les particules d'impureté, qui se distinguent du gaz bosonique, sont utilisées pour sonder les propriétés du gaz, telles que la superfluidité ou sa distribution de densité. Le gaz de Bose est mathématiquement intriguant car la condensation de Bose-Einstein permet une analyse rigoureuse, donnant une compréhension profonde des systèmes quantiques à nombreux corps.Dans notre résultat principal, nous dérivons une description effective de la théorie quantique des champs pour de tels systèmes à partir de la dynamique microscopique. Cette description est régie par l'hamiltonien de Bogoliubov-Fröhlich dans la limite des grandes densités de gaz et des volumes, modélisant la dynamique de l'impureté et des excitations, qui peuvent former une quasi-particule connue sous le nom de polaron bosonique.Les travaux précédents ont traité le problème du champ moyen correspondant sur un volume unitaire avec des conditions aux limites périodiques. La principale nouveauté de cette thèse réside dans la prise en compte d'un volume initial important par rapport à la plage d'interaction des bosons et dans la suppression des conditions aux limites périodiques, ce qui conduit à un condensat non constant. Ces adaptations rapprochent le modèle de la réalité physique.Pour prouver notre résultat principal, qui est formulé dans la dynamique effective de Bogoliubov, nous contrôlons le nombre d'excitations afin d'assurer la persistance de la condensation. De plus, comme l'impureté doit rester dans le nuage de gaz pour la formation d'une quasi-particule, nous prouvons sa localisation. Pour cette deuxième partie en particulier, un contrôle raffiné de l'état initial du condensat est nécessaire afin d'éviter d'importants transferts d'énergie du condensat vers l'impureté. Le contrôle des quantités susmentionnées est établi par des estimations de propagation et par le lemme de Grönwall.En combinant le contrôle du nombre d'excitations et la localisation de l'impureté, nous validons la dynamique de Bogoliubov-Fröhlich.