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arXiv:cond-mat/0508290

Effective potential for the second virial coefficient at Feshbach resonance

Rajat K. Bhaduri, M. K. Srivastava

Published Aug 11, 2005• 0 citations
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Abstract

In a recent paper entitled "High temperature expansion applied to fermions near Feshbach resonance" (Phys. Rev. Lett. 92, 160404 (2004)), Ho and Muller have demonstrated a remarkable similarity between its high and low temperature properties at resonance. The quantum second virial coefficient plays a crucial role in their analysis, and has a universal value at resonance. In this paper, we explore the connection between the quantum and classical second virial coefficients, and show that near a Feshbach resonance, an exact mapping from the quantum to classical form is possible. This gives rise to a scale independent inverse square potential for the classical second virial coefficient. It is suggested that it may be tested by measuring the isothermal compressibility of the gas on the repulsive side of the resonance.

DOI
10.48550/arxiv.cond-mat/0508290

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In a recent paper entitled "High temperature expansion applied to fermions near Feshbach resonance" (Phys. Rev. Lett. 92, 160404 (2004)), Ho and Muller have demonstrated a remarkable similarity between its high and low temperature properties at resonance. The quantum second virial coefficient plays a crucial role in their analysis, and has a universal value at resonance. In this paper, we explore the connection between the quantum and classical second virial coefficients, and show that near a Feshbach resonance, an exact mapping from the quantum to classical form is possible. This gives rise to a scale independent inverse square potential for the classical second virial coefficient. It is suggested that it may be tested by measuring the isothermal compressibility of the gas on the repulsive side of the resonance.
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.
The authors calculate the second virial coefficient of the quantum Lorentz gas with point interaction. It can be evaluated in closed form for arbitrary strength of the interaction. Particular attention is given to the critical coupling which gives rise to a zero-energy resonance. Point interactions satisfy a modified Levinson's theorem; as a consequence they show that there is no cancellation between the bound state and the scattering contributions in the high-temperature expansion of the second virial coefficient, contrary to the regular potentials case. They also derive high- and low-energy sum rules for the point interaction. As an application of the sum rules they rederive a high-temperature expansion for the second virial coefficient.
We use the virial expansion to investigate the behavior of the two-component, attractive Fermi gas in the high-temperature limit, where the system smoothly evolves from weakly attractive fermions to weakly repulsive bosonic dimers as the short-range attraction is increased. We present a formalism for computing the virial coefficients that employs a diagrammatic approach to the grand potential and allows one to easily include an effective range R∗{R}^{*}R∗ in the interaction. In the limit where the thermal wavelength $\ensuremath{\lambda}\ensuremath{\ll}{R}^{*}$, the calculation of the virial coefficients is perturbative even at unitarity and the system resembles a weakly interacting Bose-Fermi mixture for all scattering lengths aaa. By interpolating from the perturbative limits $\ensuremath{\lambda}/|a|\ensuremath{\gg}1$ and ${R}^{*}/\ensuremath{\lambda}\ensuremath{\gg}1$, we estimate the value of the fourth virial coefficient at unitarity for R∗=0{R}^{*}=0R∗=0 and we find that it is close to the value obtained in recent experiments. We also derive the equations of state for the pressure, density, and entropy, as well as the spectral function at high temperatures.
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.
Optically-trapped two-component Fermi gases near a broad Feshbach resonance exhibit universal thermodynamics, where the properties of the gas are independent of the details of the two-body scattering interactions. We present a global proof that such a universal gas obeys the virial theorem for {\it any} trapping potential UUU and any spin mixture, without assuming either the local density approximation or harmonic confinement. The total energy of the gas is given in scale invariant form by E=E=E=, where εεε is an {\it arbitrary} energy scale in terms of which all length and energy scales that appear in the confining potential are written. This result enables model-independent energy measurement in traps that are anharmonic as well as anisotropic by observing only the cloud profile, and provides a consistency check for many-body calculations in the universal regime.