arXiv:0803.3277 · Physical Review A

Virial theorems for trapped cold atoms

Félix Werner

Published Aug 1, 2008• 75 citations

Abstract

We present a general virial theorem for quantum particles with arbitrary zero-range or finite-range interactions in an arbitrary external potential. We deduce virial theorems for several situations relevant to trapped cold atoms: zero-range interactions with and without Efimov effect, hard spheres, narrow Feshbach resonances, and finite-range interactions. If the scattering length $a$ is varied adiabatically in a two-component Fermi gas, we find that the trapping potential energy as a function of $1∕a$ has an inflexion point at unitarity.

DOI: 10.1103/physreva.78.025601

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arXiv:0803.3277 · Physical Review A

Virial theorems for trapped cold atoms

Félix Werner

Published Aug 1, 2008• 75 citations

arXiv HTML PDF

Abstract

a

is varied adiabatically in a two-component Fermi gas, we find that the trapping potential energy as a function of

1∕a

has an inflexion point at unitarity.

DOI: 10.1103/physreva.78.025601

Related papers

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a

is varied adiabatically in a two-component Fermi gas, we find that the trapping potential energy as a function of

1∕a

has an inflexion point at unitarity.

U

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=

, where

ε

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.

U

and any spin mixture, without assuming either the local density approximation or harmonic confinement. A global proof of the virial theorem yields the total energy of the gas at all temperatures in the scale-invariant form $E=⟨ϵ\ensuremath{\partial}U/\ensuremath{\partial}ϵ⟩$, where

ϵ

is an 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 any trap by observing only the cloud profiles, and provides a consistency check for many-body calculations in the universal regime.

We consider mass-imbalanced two-component Fermi gases for which the unequal-mass atoms interact via a zero-range model potential with a diverging

s

-wave scattering length

{a}_{s}

, that is, with

1/{a}_{s}=0

. The high-temperature thermodynamics of the harmonically trapped and homogeneous systems are examined using a virial expansion approach up to third order in the fugacity. We find that the universal part of the third-order virial coefficient associated with two light atoms and one heavy atom is negative, while that associated with two heavy and one light atom changes sign from negative to positive as the mass ratio $\ensuremath{\kappa}$ increases and diverges when Efimov physics sets in at $\ensuremath{\kappa}=13.61$. By examining the Helmholtz free energy, we find that the equilibrium polarization of the trapped and homogeneous systems is 0 for $\ensuremath{\kappa}=1$, but finite for $\ensuremath{\kappa}\ensuremath{\ne}1$ (with a majority of heavy particles). Compared to the equilibrium polarization of the noninteracting system, the equilibrium polarization at unitarity is increased for the trapped system and decreased for the homogeneous system. We find that unequal-mass Fermi gases are stable for all polarizations.

1/4

in the broad-resonance limit to one another value

1/2

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.