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arXiv:0905.1836 · Physical Review A

Short-range correlations and entropy in ultracold-atom Fermi gases

Zhenhua Yu, Georg M. Bruun, Gordon Baym

Published Aug 20, 2009• 58 citations
arXivHTMLPDF

Abstract

We relate short-range correlations in ultracold-atom Fermi gases to the entropy of the system over the entire temperature, $T$, vs coupling strength, $\ensuremath{-}1/{k}_{F}a$, plane. In the low-temperature limit the entropy is dominated by phonon excitations and the correlations increase as ${T}^{4}$. In the Bose-Einstein condensate (BEC) limit, we calculate a boson model within the Bogoliubov approximation to show explicitly how phonons enhance the fermion correlations. In the high-temperature limit, we show from the virial expansion that the correlations decrease as $1/T$. The correlations therefore reach a maximum at a finite-temperature. We infer the general structure of the isentropes of the Fermi gas in the $(T,\ensuremath{-}1/{k}_{F}a)$ plane, and the temperature dependence of the correlations in the unitary, BEC, and BCS limits. Our results compare well with measurements of the correlations via photoassociation experiments at higher temperatures.

DOI
10.1103/physreva.80.023615

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We relate short-range correlations in ultracold-atom Fermi gases to the entropy of the system over the entire temperature, TTT, vs coupling strength, \ensuremath−1/kFa\ensuremath{-}1/{k}_{F}a\ensuremath−1/kF​a, plane. In the low-temperature limit the entropy is dominated by phonon excitations and the correlations increase as T4{T}^{4}T4. In the Bose-Einstein condensate (BEC) limit, we calculate a boson model within the Bogoliubov approximation to show explicitly how phonons enhance the fermion correlations. In the high-temperature limit, we show from the virial expansion that the correlations decrease as 1/T1/T1/T. The correlations therefore reach a maximum at a finite-temperature. We infer the general structure of the isentropes of the Fermi gas in the (T,\ensuremath−1/kFa)(T,\ensuremath{-}1/{k}_{F}a)(T,\ensuremath−1/kF​a) plane, and the temperature dependence of the correlations in the unitary, BEC, and BCS limits. Our results compare well with measurements of the correlations via photoassociation experiments at higher temperatures.
We study the short-range correlation strength of three-dimensional spin-half dilute atomic Fermi gases with spin-orbit coupling. The interatomic interaction is modeled by the contact pseudopotential. In the high-temperature limit, we derive the expression for the second-order virial expansion of the thermodynamic potential via the ladder diagrams. We further evaluate the second-order virial expansion in the limit that the spin-orbit coupling constants are small and find that the correlation strength between the fermions increases as the fourth power of the spin-orbit coupling constants. At zero temperature, we consider the cases in which there are symmetric spin-orbit couplings in two or three directions. In such cases, there is always a two-body bound state of zero net momentum. In the limit that the average interparticle distance is much larger than the dimension of the two-body bound state, the system primarily consists of condensed bosonic molecules that fermions pair to form; we find that the correlation strength also becomes bigger compared to that in the absence of spin-orbit coupling. Our results indicate that generic spin-orbit coupling enhances the short-range correlations of the Fermi gases. Measurement of such enhancement by photoassociation experiment is also discussed.
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Ultracold quantum gases are an ideal toolbox for simulating complex condensed or nuclear matter systems and to investigate fundamental quantum properties of matter. In this thesis, we will investigate universal properties connecting the high-momentum tail of the momentum distribution, the short-range correlations and the thermodynamics, encapsulated in Tan's relations. These relations are especially useful in the strongly interacting case, where perturbative approaches usually fail. With the aid of Tan's universal relations, we can still come to general conclusions about strongly interacting quantum gases. In particular, the momentum distribution exhibits a characteristic algebraic decay, unlike the exponential decay of the non-interacting case. The main focus in this thesis is on the one-dimensional, fermionic case, where we study the highly polarized case (the one-dimensional Fermi polaron), verifying Tan's relations using a variety of theoretical tools. In addition, we show that localized systems exhibit a universal, dynamical instability to delocalization when a short-range interaction between particles is switched off rapidly. This delocalization process relies on the algebraic decay of the momentum distribution, which guarantees that at least some of the delocalized single-particle states are occupied with a finite probability. Finally, we investigate the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover for the three-dimensional Fermi gas and develop a novel method to describe the breakdown of the Fermi liquid description in the vicinity of the critical temperature for superfluidity, in good agreement with a recent experiment.
The recently discovered universal thermodynamic behavior of dilute, strongly interacting Fermi gases also implies a universal structure in the many-body pair-correlation function at short distances, as quantified by the contact . Here, we theoretically calculate the temperature dependence of this universal contact for a Fermi gas in free space and in a harmonic trap. At high temperatures above the Fermi degeneracy temperature, T ≳ T F , we obtain a reliable non-perturbative quantum virial expansion up to third order. At low temperatures, we compare different approximate strong-coupling theories. These make different predictions, which need to be tested either by future experiments or by advanced quantum Monte Carlo simulations. We conjecture that in the universal unitarity limit, the contact or correlation decreases monotonically with increasing temperature, unless the temperature is significantly lower than the critical temperature, T ≪ T c ∼0.2 T F . We also discuss briefly how to measure the universal contact in either homogeneous or harmonically trapped Fermi gases.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning
We study the virial relations for ultracold trapped two-component Fermi gases in the case of short finite range interactions. Numerical verifications for such relations are reported through the Bardeen–Cooper–Schrieffer (BCS) Bose–Einstein-condensate (BEC) crossover. As an intermediate step, it is necessary to evaluate the partial derivatives of the many-body energy with respect to the inverse of the scattering length and with respect to the interaction range. Once the binding energy of the formed molecules in the BEC side is subtracted, the corresponding energy derivatives are found to have extreme values at the unitary limit. The value of the derivative with respect to the potential range in that limit is large enough to yield measurable differences between the total energy and twice the trapping energy unless the interacting system is described by extremely short potential ranges. The virial results are used to check the quality of the variational wavefunction involved in the calculations.