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arXiv:1312.6405 · Physical Review A

Tunable high-temperature thermodynamics of weakly interacting dipolar gases

K. M. Daily, D. Blume

Published Jan 8, 2014• 1 citation
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Abstract

We consider dilute gases of dipolar bosons or fermions in the high-temperature limit in a spherically symmetric harmonic trapping potential. We examine the system using a virial expansion up to second order in the fugacity. Using the Born approximation and assuming purely dipolar interactions, we find that the second-order virial coefficient for both bosons and fermions depends quadratically on the dipole length and is negative at high temperatures, indicating that to lowest order in the dipole-dipole interactions the dipolar single-component quantum gases are repulsive. If the $s$-wave scattering length for the bosonic system is tunable and its absolute value is made small, then the $s$-wave interactions dominate and the dipolar gas behaves like a weakly interacting Bose gas with isotropic $s$-wave interactions. If the generalized scattering lengths for the fermionic system are tunable, then the dipole length can enter linearly in the virial equation of state, enhancing the dipole-dipole effects in the thermodynamic observables.

DOI
10.1103/physreva.89.013606

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We consider dilute gases of dipolar bosons or fermions in the high-temperature limit in a spherically symmetric harmonic trapping potential. We examine the system using a virial expansion up to second order in the fugacity. Using the Born approximation and assuming purely dipolar interactions, we find that the second-order virial coefficient for both bosons and fermions depends quadratically on the dipole length and is negative at high temperatures, indicating that to lowest order in the dipole-dipole interactions the dipolar single-component quantum gases are repulsive. If the sss-wave scattering length for the bosonic system is tunable and its absolute value is made small, then the sss-wave interactions dominate and the dipolar gas behaves like a weakly interacting Bose gas with isotropic sss-wave interactions. If the generalized scattering lengths for the fermionic system are tunable, then the dipole length can enter linearly in the virial equation of state, enhancing the dipole-dipole effects in the thermodynamic observables.
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.
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 study the virial expansion for three-dimensional Bose and Fermi gases at finite temperature using an approximation that only considers two-body processes and is valid for high temperatures and low densities. The first virial coefficients are computed and the second is exact. The results are obtained for the full range of values of the scattering length, and the unitary limit is recovered as a particular case. A weak coupling expansion is performed and the free case is also obtained as a proper limit. The influence of an anisotropic harmonic trap is considered using the local density approximation (LDA), analytical results are obtained, and the special case of the isotropic trap is discussed in detail.
We characterize the high-temperature thermodynamics of rotating bosons and fermions in two-dimensional (2D) and three-dimensional (3D) isotropic harmonic trapping potentials. We begin by calculating analytically the conventional virial coefficients bn{b}_{n}bn​ for all nnn in the noninteracting case, as functions of the trapping and rotational frequencies. We also report on the virial coefficients for the angular momentum and associated moment of inertia. Using the bn{b}_{n}bn​ coefficients, we analyze the deconfined limit (in which the angular frequency matches the trapping frequency) and derive explicitly the limiting form of the partition function, showing from the thermodynamic standpoint how both the 2D and 3D cases become effectively homogeneous 2D systems. To tackle the virial coefficients in the presence of weak interactions, we implement a coarse temporal lattice approximation and obtain virial coefficients up to third order.
Using a second-order quantum virial expansion, we theoretically study the high-temperature thermodynamics of spin-polarized Fermi gases with ppp-wave interaction confined in two-dimensional harmonic traps. The second virial coefficient is calculated based on the energy spectrum of two harmonically trapped fermions. We find that the two-body spectrum can correctly be evaluated only when the ppp-wave effective range is properly considered, which is referred to as the well-known Wigner's bound. According to the quantum virial expansion, the high-temperature thermodynamics of the system is then discussed based on the two-body solutions. In addition, we investigate the high-temperature behavior of ppp-wave contacts in harmonic traps within the quantum virial expansion, which characterize a set of universal relations of strongly interacting spin-polarized Fermi gases. Our results can provide an accurate benchmark at high temperature for future numerical simulations as well as high-precision thermodynamic measurements of harmonically trapped spin-polarized Fermi gases in two dimensions.
We study the ground state of few bosons with repulsive dipole-dipole interaction in a quasi-one-dimensional harmonic trap by means of the exact diagonalization method. Up to three interaction regimes are found, depending on the strength of the dipolar interaction and the ratio of transverse to axial oscillator lengths: a regime where the dipolar Bose gas resembles a system of weakly $\ensuremath{\delta}$-interacting bosons, a second regime where the bosons are fermionized, and a third regime where the bosons form a Wigner crystal. In the first two regimes, the dipole-dipole potential can be replaced by a $\ensuremath{\delta}$ potential. In the crystalline state, the overlap between the localized wave packets is strongly reduced and all the properties of the boson system equal those of its fermionic counterpart. The transition from the Tonks-Girardeau gas to the solidlike state is accompanied by a rapid increase of the interaction energy and a considerable change of the momentum distribution, which we trace back to the different short-range correlations in the two interaction regimes.