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Phase shift and Beth-Uhlenbeck formula for an atomic gas in a harmonic waveguide

Tom Kristensen, X. Leyronas, Ludovic Pricoupenko

Published Aug 22, 2016• 0 citations
OpenAlex

Abstract

In a non-degenerated gas with low density and high temperature, the fugacity is a small parameter. The equation of state can be thus efficiently approached by the virial expansion. Since the n-order virial coefficient is known from the solution of the n-body problem, the virial expansion built a bridge between thermodynamics and few-body physics. In this talk we study the 2nd order virial coefficient for a gas trapped into a 1D or a 2D harmonic waveguide (shape of a cigar or a pancake). Interactions are modeled by a quantitative two-channel model that include the coupling between atoms and Feshbach molecules. We defined in this context the scattering phase-shift and we derive from a diagrammatic approach a Beth-Uhlenbeck formula that takes into account the presence of the molecules and the external waveguide.

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In a non-degenerated gas with low density and high temperature, the fugacity is a small parameter. The equation of state can be thus efficiently approached by the virial expansion. Since the n-order virial coefficient is known from the solution of the n-body problem, the virial expansion built a bridge between thermodynamics and few-body physics. In this talk we study the 2nd order virial coefficient for a gas trapped into a 1D or a 2D harmonic waveguide (shape of a cigar or a pancake). Interactions are modeled by a quantitative two-channel model that include the coupling between atoms and Feshbach molecules. We defined in this context the scattering phase-shift and we derive from a diagrammatic approach a Beth-Uhlenbeck formula that takes into account the presence of the molecules and the external waveguide.
We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either sss-wave or ppp-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either sss-wave or ppp-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting sss-wave Fermi gas has already been proved very useful. In the case of ppp-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a ppp-wave Feshbach resonance.
The virial expansion characterizes the high-temperature approach to the quantum-classical crossover in any quantum many-body system. Here, we calculate the virial coefficients up to the fifth order of Fermi gases in one, two, and three dimensions, with attractive contact interactions, as relevant for a variety of applications in atomic and nuclear physics. To that end, we discretize the imaginary-time direction and calculate the relevant canonical partition functions. In coarse discretizations, we obtain analytic results featuring relationships between the interaction-induced changes $\mathrm{\ensuremath{\Delta}}{b}_{3}, \mathrm{\ensuremath{\Delta}}{b}_{4}$, and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ as functions of $\mathrm{\ensuremath{\Delta}}{b}_{2}$, the latter being exactly known in many cases by virtue of the Beth-Uhlenbeck formula. Using automated-algebra methods, we push our calculations to progressively finer discretizations and extrapolate to the continuous-time limit. We find excellent agreement for $\mathrm{\ensuremath{\Delta}}{b}_{3}$ with previous calculations in all dimensions and we formulate predictions for $\mathrm{\ensuremath{\Delta}}{b}_{4}$ and $\mathrm{\ensuremath{\Delta}}{b}_{5}$ in one and two dimensions. We also provide, for a range of couplings, the subspace contributions $\mathrm{\ensuremath{\Delta}}{b}_{31}, \mathrm{\ensuremath{\Delta}}{b}_{22}, \mathrm{\ensuremath{\Delta}}{b}_{41}$, and $\mathrm{\ensuremath{\Delta}}{b}_{32}$, which determine the equation of state and static response of polarized systems at high temperature. As a performance check, we compare the density equation of state and Tan contact with quantum Monte Carlo calculations, diagrammatic approaches, and experimental data where available. Finally, we apply Pad\'e and Pad\'e-Borel resummation methods to extend the usefulness of the virial coefficients to approach and in some cases go beyond the unit-fugacity point.
We briefly review current research progress in the field of strongly correlated atomic Fermi gases. A deep understanding of strongly correlated quantum gases is plagued by the lack of either exact solutions or a controllable small interaction parameter. In this work, we present a systematic virial expansion method to study the high-temperature thermodynamics of a strongly correlated Fermi gas,where the controllable parameter is given by the fugacity,exp (μkBT),with μ being the chemical potential. A practical way is proposed to obtain the expansion coefficients for both homogeneous and harmonically trapped Fermi gases. We calculate, for the first time accurately, the third-order virial coefficient. The resulting equation of states at high temperatures is compared with a recent thermodynamic measurement and with Monte Carlo simulations.
We theoretically investigate the high-temperature thermodynamics of a harmonically trapped Fermi gas across a narrow Feshbach resonance by using the second-order quantum virial expansion, and point out some new features compared to the broad resonance. The interatomic interaction is modeled by the pseudopotential with an additional parameter, i.e., the effective range, to characterize the narrow-resonance width. Deep inside the width of a narrow Feshbach resonance, we find the second virial coefficient evolves with the effective range from the well-known universal value 1/41/41/4 in the broad-resonance limit to one another value 1/21/21/2 in the narrow-resonance limit. This means the Fermi gas interacts more strongly at the narrow resonance. In addition, far beyond the resonance width, we find the harmonically trapped Fermi gas still manifests an appreciable interaction effect across a narrow Feshbach resonance, which is contrary to our knowledge of the broad Feshbach resonance. All our results can be directly tested in current narrow Feshbach resonance experiments, which are generally carried out in a harmonic trap.