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Physical Review A

Momentum relaxation of strongly coupled impurities in quantum gases: A nonequilibrium Green's function approach

Baihua Gong

Published Sep 11, 2025• 0 citations
Publisher

Abstract

Within the nonequilibrium Green's function framework, we systematically analyze the momentum relaxation dynamics of strongly coupled impurities in three-dimensional nondegenerate quantum gases. Using the Hartree-Fock approximation, we establish that the impurity velocity exhibits exponential decay asymptotically. When the impurity velocity is much lower than the thermal velocity of the gas atoms, the relaxation time $\ensuremath{\tau}$ is governed entirely by the self-energy evaluated at zero momentum and zero frequency, $\mathrm{\ensuremath{\Sigma}}(\mathbf{p}=0;\ensuremath{\omega}=0)$. Furthermore, via virial expansion theory, $\ensuremath{\tau}$ admits a systematic expansion in powers of the fugacity $z\ensuremath{\equiv}{e}^{\ensuremath{\beta}\ensuremath{\mu}}$. The leading-order term precisely recovers the Boltzmann semiclassical result, while higher-order corrections capture quantum many-body effects beyond the semiclassical approximation.

DOI
10.1103/frw1-nsln

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Within the nonequilibrium Green's function framework, we systematically analyze the momentum relaxation dynamics of strongly coupled impurities in three-dimensional nondegenerate quantum gases. Using the Hartree-Fock approximation, we establish that the impurity velocity exhibits exponential decay asymptotically. When the impurity velocity is much lower than the thermal velocity of the gas atoms, the relaxation time \ensuremathτ\ensuremath{\tau}\ensuremathτ is governed entirely by the self-energy evaluated at zero momentum and zero frequency, \ensuremathΣ(p=0;\ensuremathω=0)\mathrm{\ensuremath{\Sigma}}(\mathbf{p}=0;\ensuremath{\omega}=0)\ensuremathΣ(p=0;\ensuremathω=0). Furthermore, via virial expansion theory, \ensuremathτ\ensuremath{\tau}\ensuremathτ admits a systematic expansion in powers of the fugacity z\ensuremath≡e\ensuremathβ\ensuremathμz\ensuremath{\equiv}{e}^{\ensuremath{\beta}\ensuremath{\mu}}z\ensuremath≡e\ensuremathβ\ensuremathμ. The leading-order term precisely recovers the Boltzmann semiclassical result, while higher-order corrections capture quantum many-body effects beyond the semiclassical approximation.
High-temperature virial expansion is a powerful tool in equilibrium statistical mechanics. In this paper we generalize this method to the study of nonequilibrium dynamics based on contour Green's functions and present the diagrammatic rules to calculate the virial expansion coefficients. As an application of our theory, we recover the dynamics of a Bose gas quenched from noninteracting to finite interaction and obtain analytically the long-time limit of the momentum distribution function of the gas, consistent with previous numerical results.
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.
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This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.
This paper extends an earlier quantum kinetics treatment for dilute, weakly interacting, partially Bose-Einstein condensed gases, presented by the author elsewhere (Proukakis N P and Burnett K 1996 J. Res. Natl Inst. Stand. Technol. 101 457), by consistently treating the dynamics of the uncondensed atoms to the same level of approximation as the condensed atoms. Our method is based on a hierarchy of coupled equations of motion for the condensate mean field and fluctuations around this mean field, truncated to second order in the (effective two-body) interatomic potential, and with suitable decoupling approximations for higher-order correlations. By applying perturbation theory in the Hartree-Fock-Bogoliubov basis, we re-derive the quantum kinetic theory of Walser et al (Walser R, Williams J, Cooper J and Holland M 1999 Phys. Rev. A 59 3878), which further indicates the consistency of our treatment with the Kadanoff-Baym non-equilibrium Green function formalism for trapped gases.
A strong-coupling expansion for the Green's functions, self-energies, and correlation functions of the Bose-Hubbard model is developed. We illustrate the general formalism, which includes all possible (normal-phase) inhomogeneous effects in the formalism, such as disorder or a trap potential, as well as effects of thermal excitations. The expansion is then employed to calculate the momentum distribution of the bosons in the Mott phase for an infinite homogeneous periodic system at zero temperature through third order in the hopping. By using scaling theory for the critical behavior at zero momentum and at the critical value of the hopping for the Mott insulator--to--superfluid transition along with a generalization of the random-phase-approximation-like form for the momentum distribution, we are able to extrapolate the series to infinite order and produce very accurate quantitative results for the momentum distribution in a simple functional form for one, two, and three dimensions. The accuracy is better in higher dimensions and is on the order of a few percent relative error everywhere except close to the critical value of the hopping divided by the on-site repulsion. In addition, we find simple phenomenological expressions for the Mott-phase lobes in two and three dimensions which are much more accurate than the truncated strong-coupling expansions and any other analytic approximation we are aware of. The strong-coupling expansions and scaling-theory results are benchmarked against numerically exact quantum Monte Carlo simulations in two and three dimensions and against density-matrix renormalization-group calculations in one dimension. These analytic expressions will be useful for quick comparison of experimental results to theory and in many cases can bypass the need for expensive numerical simulations.