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arXiv:1202.4547 · Physical Review A

BEC-BCS crossover with Feshbach resonance for a three-hyperfine-species model

Guojun Zhu, Anthony J. Leggett

Published Feb 25, 2013• 1 citation
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Abstract

We consider the behavior of an ultracold Fermi gas across a narrow Feshbach resonance, where the occupation of the closed channel may not be negligible. While the corrections to the single-channel formulas associated with the nonzero chemical potential and with particle conservation have been considered in the existing literature, there is a further effect, namely, the ``interchannel Pauli exclusion principle,'' associated with the fact that a single hyperfine species may be common to the two channels. We focus on this effect and show that, as intuitively expected, the resulting corrections are of the order of ${E}_{F}/\ensuremath{\eta}$, where ${E}_{F}$ is the Fermi energy of the gas in the absence of interactions and $\ensuremath{\eta}$ is the Zeeman energy difference between the two channels. We also consider the related corrections to the fermionic excitation spectrum, and briefly discuss the collective modes of the system.

DOI
10.1103/physreva.87.023627

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We consider the behavior of an ultracold Fermi gas across a narrow Feshbach resonance, where the occupation of the closed channel may not be negligible. While the corrections to the single-channel formulas associated with the nonzero chemical potential and with particle conservation have been considered in the existing literature, there is a further effect, namely, the ``interchannel Pauli exclusion principle,'' associated with the fact that a single hyperfine species may be common to the two channels. We focus on this effect and show that, as intuitively expected, the resulting corrections are of the order of EF/\ensuremathη{E}_{F}/\ensuremath{\eta}EF​/\ensuremathη, where EF{E}_{F}EF​ is the Fermi energy of the gas in the absence of interactions and \ensuremathη\ensuremath{\eta}\ensuremathη is the Zeeman energy difference between the two channels. We also consider the related corrections to the fermionic excitation spectrum, and briefly discuss the collective modes of the system.
In a Feshbach resonance, the effective s-wave scattering length grows when one moves toward the resonance point, and eventually diverges at this point. There is one characteristic energy scale, δcδ_cδc​, defined as, in the negative side of the resonance point, the detuning energy at which the weight of the bound state shifts from predominatedly in the open-channel to predominated in the closed-channel. When the many-body energy scale (e.g. the Fermi energy, EFE_{F}EF​) is larger than δcδ_cδc​, the closed-channel weight is significant and has to be included in the many-body theory. Furthermore, when two channels share a hyperfine species, the Pauli exclusion between fermions from two channels also needs to be taken into consideration in the many-body theory. The current thesis addresses the above problem in detail. A set of gap equations and number equations are derived at the mean-field level. The fermionic and bosonic excitation spectra are then derived. Assuming that the uncoupled bound-state of the closed-channel in resonance is much smaller than the inter-particle distance, as well as the s-wave scattering length, asa_sas​, we find that the basic equations in the single-channel crossover model are still valid. The correction first comes from the existing of the finite chemical potential and additional counting complication due to the closed-channel. These two corrections need to be included into the mean-field equations, i.e. the gap equations and the number equations, and be solved self-consistently. Then the correction due to the inter-channel Pauli exclusion is in the order of the ratio of the Fermi energy and the Zeeman energy difference between two channels, EF/ηE_F/ηEF​/η, which can be analyzed perturbatively over the previous corrections. Fermionic and bosonic excitation modes are studied.
Microscopic theories of Feshbach resonance phenomena in fermionic alkali-metal gases have, in the main, relied upon the intuition afforded by a ``Fermi-Bose'' theory which presents the Feshbach molecule as a featureless pointlike Bose particle. While this model may provide a suitable platform to explore the condensation phenomena in the 6Li^{6}\mathrm{Li}6Li system, it has been argued recently that its application to the 40K^{40}\mathrm{K}40K system, where the hyperfine structure is inverted, is inappropriate. Here, the formation of a Feshbach resonance involves the formation of a Feshbach molecule which shares a spin state with the open channel. In such an arrangement, the mechanism of condensate formation cannot escape the effects of Pauli exclusion. Building on a model three-state fermion Hamiltonian recently introduced by Parish et al. to describe the 40K^{40}\mathrm{K}40K system, we elucidate the mean-field structure of the model, investigate the internal structure of the condensate wave function in the crossover region, and develop a field theory of the transition.
Fermion-mediated BCS-BEC crossover in ultracold 40{}^{40}40K gases
Studies of Feshbach resonance phenomena in fermionic alkali gases have drawn heavily on the intuition afforded by a Fermi-Bose theory which presents the Feshbach molecule as a featureless Bose particle. While this model may provide a suitable platform to explore the Li-6 system, we argue that its application to K-40, where the hyperfine structure is inverted, is inappropriate. Introducing a three-state Fermi model, where a spin state is shared by the open and closed channel states, we show that effects of "Pauli blocking" appear in the internal structure of the condensate wave function.
Studies of Feshbach resonance phenomena in fermionic alkali gases have drawn heavily on the intuition afforded by a Fermi-Bose theory which presents the Feshbach molecule as a featureless Bose particle. While this model may provide a suitable platform to explore the 6Li^{6}\mathrm{Li}6Li system, we argue that its application to 40K^{40}\mathrm{K}40K, where the hyperfine structure is inverted, is inappropriate. Introducing a three-state Fermi model, where a spin state is shared by the open and closed channel states, we show that effects of ``Pauli blocking'' appear in the internal structure of the condensate wave function.
We present a detailed analysis of the two-channel atom-molecule effective Hamiltonian for an ultracold two-component homogeneous Fermi gas interacting near a Feshbach resonance. We particularly focus on the two-body and many-body properties of the dressed molecules in such a gas. An exact result for the many-body T matrix of the two-channel theory is derived by both considering coupled vertex equations and functional-integral methods. Making use of this result allows us to incorporate exactly into the many-body theory the two-body physics of the Feshbach scattering by means of simple analytical formulas without any fitting parameters. New interesting many-body effects are discussed in the case of narrow resonances. We give also a description of the BEC-BCS crossover above and below TC{T}_{C}TC​. The effects of different approximations for the self-energy of the dressed molecules are discussed. The single-channel results are derived as a special limit for broad resonances. Moreover, through an analytic analysis of the BEC limit, the relation between the composite boson of the single-channel model and the dressed-molecule of the two-channel model is established.
Abstract By including the effect of a trap with characteristic energy given by the Fermi temperature T F in a two-body two-channel model for Feshbach resonances, we reproduce the measured binding energy of ultracold molecules in a 40 K atomic Fermi gas. We also reproduce the experimental closed-channel fraction Z across the BEC–BCS crossover and into the BCS regime of a 6 Li atomic Fermi gas. We obtain the expected behavior Z ∝ T F at unitarity, together with the recently measured proportionality constant. Our results are also in agreement with recent measurements of the Z dependency on T F on the BCS side, where a significant quantitative discrepancy between experimental data and theory’s predictions has been repeatedly reported. In order to contrast with future experiments we report the proportionality constant at unitarity between Z and T F predicted by our model for a 40 K atomic Fermi gas.