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arXiv:1212.6691

BEC-BCS Crossover with Feshbach Resonance for Three-Hyperfine-Species Model

Zhu, Guojun

Published Dec 30, 2012• 1 citation
arXivHTMLPDF

Abstract

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, $δ_c$, 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, $E_{F}$) is larger than $δ_c$, 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, $a_s$, 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, $E_F/η$, which can be analyzed perturbatively over the previous corrections. Fermionic and bosonic excitation modes are studied.

DOI
10.48550/arxiv.1212.6691

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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.
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 EF{E}_{F}EF​ 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.
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.
We systematically study the BCS-BEC crossover and the quantum phase transition in ultracold 6Li^{6}\mathrm{Li}6Li and 40K^{40}\mathrm{K}40K atoms across a wide Feshbach resonance. The background scattering lengths for 6Li^{6}\mathrm{Li}6Li and 40K^{40}\mathrm{K}40K have opposite signs, which lead to very different behaviors for these two types of atoms. For 40K^{40}\mathrm{K}40K, both the two-body and the many-body calculations show that the system always has two branches of solutions: one corresponds to a deeply bound molecule state; and the other, the one accessed by the current experiments, corresponds to a weakly bound state with population always dominantly in the open channel. For 6Li^{6}\mathrm{Li}6Li, there is only a unique solution with the standard crossover from the weakly bound Cooper pairs to the deeply bound molecules as one sweeps the magnetic field through the crossover region. Because of this difference, for the experimentally accessible state of 40K^{40}\mathrm{K}40K, there is a quantum phase transition at zero temperature from the superfluid to the normal Fermi gas at the positive detuning of the magnetic field where the sss-wave scattering length passes its zero point. For 6Li^{6}\mathrm{Li}6Li, however, the system changes continuously across the zero point of the scattering length. For both types of atoms, we also give a detailed comparison between the results from the two-channel and the single-channel model over the whole region of the magnetic field detuning.
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.
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.
Many-body effects may influence properties, such as scattering parameters, nature of pairing, etc., close to a Feshbach resonance in the fermion BEC-BCS crossover problem. We study effects such as these using a tractable crossing-symmetric approach. This method allow us to include quantum fluctuations, such as density, current, spin, spin-current, and the higher-order fluctuations in a self-consistent fashion. The underlying fermion interaction is reflected in the ``driving'' term. We perform calculations here on both Bose-Einstein condensate (BEC) and BCS sides and taking the driving term to be finite range and of arbitrary strength. These are related to two-body singlet and triplet scattering parameters and can be connected with experimental sss- and ppp-wave Feshbach resonances. We include the $\ensuremath{\ell}=0$ density and spin fluctuations as well as $\ensuremath{\ell}=1$ current and spin-current fluctuations. We calculate renormalized scattering amplitudes, pairing amplitudes, nature of pairing, etc., on both the BEC and BCS sides. We then compare our results qualitatively with experiments.