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arXiv:1307.4263 · Physical Review A

Universal Bose gases near resonance: A rigorous solution

Shao-Jian Jiang, Wu-Ming Liu, Gordon Walter Semenoff, Fei Zhou

Published Mar 11, 2014• 28 citations
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Abstract

We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an $\ensuremath{\epsilon}$ expansion near four spatial dimensions. In dimension $d=4\ensuremath{-}\ensuremath{\epsilon}$, the chemical potential of Bose gases near resonance is shown to approach the universal value ${\ensuremath{\epsilon}}^{\frac{2}{4\ensuremath{-}\ensuremath{\epsilon}}}{\ensuremath{\epsilon}}_{F}\sqrt{\frac{2}{3}}(1+0.474\ensuremath{\epsilon}\ensuremath{-}i1.217\ensuremath{\epsilon}+\ensuremath{\cdots})$, where ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy defined for a Fermi gas of density $n$, and the condensation fraction is equal to $\frac{2}{3}(1+0.0877\ensuremath{\epsilon}+\ensuremath{\cdots})$. We also discuss the implications on ultracold gases in physical dimensions.

DOI
10.1103/physreva.89.033614

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APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts•May 1, 2002•0 citations
We obtain a rigorous solution of universal Bose gases near resonance and offer an answer to one of the long-standing challenges of quantum gases at large scattering lengths, where the standard dilute theory breaks down. The solution was obtained by using an \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ expansion near four spatial dimensions. In dimension d=4\ensuremath−\ensuremathϵd=4\ensuremath{-}\ensuremath{\epsilon}d=4\ensuremath−\ensuremathϵ, the chemical potential of Bose gases near resonance is shown to approach the universal value \ensuremathϵ24\ensuremath−\ensuremathϵ\ensuremathϵF23(1+0.474\ensuremathϵ\ensuremath−i1.217\ensuremathϵ+\ensuremath⋯){\ensuremath{\epsilon}}^{\frac{2}{4\ensuremath{-}\ensuremath{\epsilon}}}{\ensuremath{\epsilon}}_{F}\sqrt{\frac{2}{3}}(1+0.474\ensuremath{\epsilon}\ensuremath{-}i1.217\ensuremath{\epsilon}+\ensuremath{\cdots})\ensuremathϵ4\ensuremath−\ensuremathϵ2​\ensuremathϵF​32​​(1+0.474\ensuremathϵ\ensuremath−i1.217\ensuremathϵ+\ensuremath⋯), where \ensuremathϵF{\ensuremath{\epsilon}}_{F}\ensuremathϵF​ is the Fermi energy defined for a Fermi gas of density nnn, and the condensation fraction is equal to 23(1+0.0877\ensuremathϵ+\ensuremath⋯)\frac{2}{3}(1+0.0877\ensuremath{\epsilon}+\ensuremath{\cdots})32​(1+0.0877\ensuremathϵ+\ensuremath⋯). We also discuss the implications on ultracold gases in physical dimensions.
The ϵϵϵ expansion (expansion around four spatial dimensions) developed by Nishida and Son for a cold Fermi gas with infinite scattering length is extended to finite scattering length to study the Bose-Einstein condensate (BEC) to BCS crossover. A resummation of higher-order logarithms and a suitable extension of fermion coupling in ddd dimensions are developed in order to apply the theory in the BCS regime. The ratio between the chemical potential and the Fermi energy, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}$, is computed to next-to-leading order in the ϵϵϵ expansion as a function of $\ensuremath{\eta}=1∕(a{k}_{F})$, where aaa is the scattering length and kF{k}_{F}kF​ is the Fermi momentum in a noninteracting system. Near the unitarity limit $\ensuremath{\mid}\ensuremath{\eta}\ensuremath{\mid}\ensuremath{\rightarrow}0$, we found $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.475\ensuremath{-}0.707\ensuremath{\eta}\ensuremath{-}0.5{\ensuremath{\eta}}^{2}$. Near the BEC limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=0.062∕\ensuremath{\eta}\ensuremath{-}{\ensuremath{\eta}}^{2}$, while near the BCS limit $\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$, $\ensuremath{\mu}∕{\ensuremath{\epsilon}}_{F}=1+0.707∕\ensuremath{\eta}$. Overall good agreement with quantum Monte Carlo results is found.
We report in this Letter the results of our investigation of 2D Bose gases beyond the dilute limit emphasizing the role played by three-body scattering events. We demonstrate that a competition between three-body attractive interactions and two-body repulsive forces results in the chemical potential of 2D Bose gases to exhibit a maximum at a critical scattering length beyond which these quantum gases possess a negative compressibility. For larger scattering lengths, the increasingly prominent role played by three-body attractive interactions leads to an onset instability at a second critical value. The three-body effects studied here are universal, fully characterized by the effective 2D scattering length a(2D) (or the size of the 2D bound states) and are, in comparison to the 3D case, independent of three-body ultraviolet physics. We find, within our approach, the ratios of the contribution to the chemical potential due to three-body interactions to the one due to two-body to be 0.27 near the maximum of the chemical potential and 0.73 in the vicinity of the onset instability.
In this paper, we explore the nature of three-dimensional Bose gases at large positive scattering lengths via resummation of dominating processes involving a minimum number of virtual atoms. We focus on the energetics of the nearly fermionized Bose gases beyond the usual dilute limit. We also find that an onset instability sets in at a critical scattering length, beyond which the near-resonance Bose gases become strongly coupled to molecules and lose the metastability. Near the point of instability, the chemical potential reaches a maximum, and the effect of the three-body forces can be estimated to be around a few percent.
In this thesis, we investigated the physics of two- and three-dimensional ultra cold Bose gases in the strongly interacting regime at zero temperature. This regime can be experimentally accessed using a Feshbach resonance. We applied a self-consistent diagrammatic approach to determine the chemical potential of three-dimensional Bose gases for a wide range of interaction values. We showed that such strongly interacting Bose gases become unstable towards the formation of molecules at a finite positive scattering length. In fact, the interaction between atoms becomes effectively attractive and the system looses its metastability before reaching the unitary limit. We also found that such systems are nearly fermionized close to the instability point. Near this critical point, the chemical potential reaches a maximum and the contribution to the system energy due to three-body forces is estimated to be only a few percent. We also studied the same system using a self-consistent renormalization group method. This approach confirms the existence of an instability point towards the formation of molecules as well as fermionization. We showed that the instability and accompanying maximum are precursors of the sign change of the effective two-body interaction strength from repulsive to attractive near resonance. In addition, we examined the physics of two-dimensional Bose gases near resonance using a similar self-consistent diagrammatic approach as the one introduced for three-dimensional Bose gases. We demonstrated that a competition between three-body attractive interactions and two-body repulsive forces results in the chemical potential of two-dimensional Bose gases to exhibit a maximum at a critical scattering length beyond which these quantum gases possess a negative compressibility. For larger scattering lengths, the increasingly prominent role played by three-body attractive interactions leads to an onset instability at a second critical value. The three-body effects studied for these systems are universal, fully characterized by the effective two-dimensional scattering length and are, in comparison to the three-dimensional case, independent of three-body ultraviolet physics.
We extend previous work on applying the ϵϵϵ expansion to universal properties of a cold, dilute Fermi gas in the unitary regime of infinite scattering length. We compute the ratio $\ensuremath{\xi}=\ensuremath{\mu}∕{ϵ}_{\mathrm{F}}$ of chemical potential to ideal gas Fermi energy to next-to-next-to-leading-order (NNLO) in $ϵ=4\ensuremath{-}d$, where ddd is the number of spatial dimensions. We also explore the nature of corrections from the order after NNLO.
We derive a resonance field-theory for ultra-cold Bose or Fermi gases that is valid when scattering resonances are present. Experiments with a ^85Rb condensate have shown new interesting physics close to resonance(N.R. Claussen, E.A. Donley, S.T. Thompson, C.E. Wieman, e-print cond-mat/0201400.). It is also expected that utilizing a Feshbach resonance can greatly enhance the critical temperature for superfluidity in a Fermi gas(M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. 87), 120406 (2001).. Therefore it is important to treat the rich scattering physics properly inside a many-body theory. We show that the scattering length only is insufficient to describe a system close to resonance. Our many-body equations have build-in the two-body scattering equations, and we include the full scattering solution over the relevant energy range by means of a renormalized resonance scattering theory(S.J.J.M.F. Kokkelmans, J.N. Milstein, M.L. Chiofalo, R. Walser, and M.J. Holland, e-print cond-mat/0112283.).