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arXiv:1801.07448 · Physical Review A

High-momentum tail and universal relations of a Fermi gas near a Raman-dressed Feshbach resonance

Fang Qin, Jianwen Jie, Wei Yi, Guang‐Can Guo

Published Mar 16, 2018• 10 citations
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Abstract

In a recent proposal [Jie and Zhang, Phys. Rev. A 95, 060701(R) (2017)], it has been shown that center-of-mass-momentum-dependent two-body interactions can be generated and tuned by Raman coupling the closed-channel bound states in a magnetic Feshbach resonance. Here we investigate the universal relations in a three-dimensional Fermi gas near such a laser modulated $s$-wave Feshbach resonance. Using the operator-product expansion approach, we find that to fully describe the high-momentum tail of the density distribution up to ${q}^{\ensuremath{-}6}$ ($q$ is the relative momentum), four center-of-mass-momentum-dependent parameters are required, which we identify as contacts. These contacts appear in various universal relations connecting microscopic and thermodynamic properties. One contact is related to the variation of energy with respect to the inverse scattering length and determines the leading ${q}^{\ensuremath{-}4}$ tail of the high-momentum distribution. Another vector contact appears in the subleading ${q}^{\ensuremath{-}5}$ tail, which is related to the velocity of closed-channel molecules. The other two contacts emerge in the ${q}^{\ensuremath{-}6}$ tail and are respectively related to the variation of energy with respect to the range parameter and to the kinetic energy of closed-channel molecules. Particularly, we find that the ${q}^{\ensuremath{-}5}$ tail and part of the ${q}^{\ensuremath{-}6}$ tail of the momentum distribution show anisotropic features. We derive the universal relations and, as a concrete example, estimate the contacts for the zero-temperature superfluid ground state of the system using a mean-field approach.

DOI
10.1103/physreva.97.033610

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In a recent proposal [Jie and Zhang, Phys. Rev. A 95, 060701(R) (2017)], it has been shown that center-of-mass-momentum-dependent two-body interactions can be generated and tuned by Raman coupling the closed-channel bound states in a magnetic Feshbach resonance. Here we investigate the universal relations in a three-dimensional Fermi gas near such a laser modulated sss-wave Feshbach resonance. Using the operator-product expansion approach, we find that to fully describe the high-momentum tail of the density distribution up to q\ensuremath−6{q}^{\ensuremath{-}6}q\ensuremath−6 (qqq is the relative momentum), four center-of-mass-momentum-dependent parameters are required, which we identify as contacts. These contacts appear in various universal relations connecting microscopic and thermodynamic properties. One contact is related to the variation of energy with respect to the inverse scattering length and determines the leading q\ensuremath−4{q}^{\ensuremath{-}4}q\ensuremath−4 tail of the high-momentum distribution. Another vector contact appears in the subleading q\ensuremath−5{q}^{\ensuremath{-}5}q\ensuremath−5 tail, which is related to the velocity of closed-channel molecules. The other two contacts emerge in the q\ensuremath−6{q}^{\ensuremath{-}6}q\ensuremath−6 tail and are respectively related to the variation of energy with respect to the range parameter and to the kinetic energy of closed-channel molecules. Particularly, we find that the q\ensuremath−5{q}^{\ensuremath{-}5}q\ensuremath−5 tail and part of the q\ensuremath−6{q}^{\ensuremath{-}6}q\ensuremath−6 tail of the momentum distribution show anisotropic features. We derive the universal relations and, as a concrete example, estimate the contacts for the zero-temperature superfluid ground state of the system using a mean-field approach.
In a recent experiment [Peng et al., Phys. Rev. A 97, 012702 (2018)], it has been shown that the ppp-wave Feshbach resonance can be shifted toward the sss-wave Feshbach resonance by a laser field. Based on this experiment, we study the universal relations and the normal-state properties in an ultracold Fermi gas with coexisting sss- and ppp-wave interactions under optical control of a ppp-wave magnetic Feshbach resonance. Within the operator-product expansion, we derive the high-momentum tail of various observable quantities in terms of contacts. We find that the high-momentum tail becomes anisotropic. Adopting the quantum virial expansion, we calculate the normal-state contacts with and without a laser field for 40K^{40}\mathrm{K}40K atoms using typical experimental parameters. We show that the contacts are dependent on the laser dressing. We also reveal the interplay of laser dressing and different partial-wave interactions on various contacts. In particular, we demonstrate that the impact of the laser dressing in the ppp-wave channel can be probed by measuring the sss-wave contacts, which is a direct manifestation of few-body effects on the many-body level. Our results can be readily checked experimentally.
We present a derivation of the adiabatic energy relations as well as the large momentum distribution of a polarized Fermi gas near ppp-wave Feshbach resonances. The leading asymptotic behavior ${k}^{\ensuremath{-}2}$ and subleading behavior ${k}^{\ensuremath{-}4}$ of the large momentum distribution have recently been predicted by Z. Yu et al. [Phys. Rev. Lett. 115, 135304 (2015)] and by M. Y. He et al. [Phys. Rev. Lett. 116, 045301 (2016)] using two different approaches. Here, we show that the subleading asymptotic behavior ($\ensuremath{\sim}{k}^{\ensuremath{-}4}$) cannot fully be captured by the contact defined from the adiabatic energy relation related to the ppp-wave effective range, and there should be an extra term resulting from the center-of-mass motion of the pairs. The omission of this extra term is perhaps a reasonable approximation at zero temperature. However, it should be taken into account at finite temperature and should have significant importance to understand the recently measured momentum distribution in a resonant ppp-wave Fermi gas of ultracold 40K^{40}\mathrm{K}40K atoms [C. Luciuk et al., Nat. Phys. 12, 599 (2016)].
We study the contacts, large-momentum tail, radio-frequency spectroscopy, and some other universal relations for an ultracold one-dimensional (1D) two-component Fermi gas with spin-orbit coupling (SOC). Different from previous studies, we find that the ${q}^{\ensuremath{-}8}$ tail in the spin-mixing (off-diagonal) terms of the momentum distribution matrix is dependent on the two SOC parameters in the laboratory frame for 1D systems, where qqq is the relative momentum. This tail can be observed through time-of-flight measurement as a direct manifestation of the SOC effects on the many-body level. Besides the traditional 1D even-wave scattering length, we find that two new physical quantities must be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Furthermore, we derive the pressure relation and virial theorem at short distances for this system. To find how the SOC modifies the large-momentum behavior, we take the SOC parameters as perturbations since the strength of the SOC should be much smaller than the corresponding strength scale of the interatomic interactions. In addition, by using the operator product expansion method, we derive the asymptotic behavior of the large-momentum distribution matrix up to the ${q}^{\ensuremath{-}8}$ order and find that the diagonal terms of the distribution matrix include the contact of traditional 1D even-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond the subleading term and is corrected by the SOC parameters. We also find that the momentum distribution matrix shows spin-dependent and anisotropic features. Furthermore, we calculate the momentum distribution matrix in the laboratory frame for the experimental implication. In addition, we calculate the high-frequency tail of the radio-frequency spectroscopy and find that the presence of the contact related to the center-of-mass momentum in the radio-frequency spectral is due to the SOC effects. This paper paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.
We study odd-wave interacting identical fermions in one dimension with finite effective range. We show that to fully describe the high-momentum distribution $\ensuremath{\rho}(k)$ up to ${k}^{\ensuremath{-}4}$, one needs four parameters characterizing the properties when two particles contact each other. Two parameters are related to the variation of energy with respect to the odd-wave scattering length and the effective range, respectively, determining the ${k}^{\ensuremath{-}2}$ tail and part of the ${k}^{\ensuremath{-}4}$ tail in $\ensuremath{\rho}(k)$. The other two parameters are related to the center-of-mass motion of the system, respectively determining the ${k}^{\ensuremath{-}3}$ tail and the other part of the ${k}^{\ensuremath{-}4}$ tail. We point out that the unusual ${k}^{\ensuremath{-}3}$ tail, which has not been discovered before in atomic systems, is an intrinsic component to complete the general form of $\ensuremath{\rho}(k)$ and also is realistically detectable under certain experimental conditions. Various other universal relations are also derived in terms of those contact parameters, and finally the results are confirmed through the exact solution of a two-body problem.
We investigate universal relations in a spinless Fermi gas near a p-wave Feshbach resonance, and show that the momentum distribution n_{k} has an asymptote proportional to k^{-2} with the proportionality constant-the p-wave contact-scaling with the number of closed-channel molecules. We prove the adiabatic sweep theorem for a p-wave resonance which reveals the thermodynamic implication of the p-wave contact. In contrast to the unitary Fermi gas in which Tan's contact is universal, the p-wave contact depends on the short-range details of the interaction.
Synthetic spin-orbit-coupled quantum gases have been widely studied both experimentally and theoretically in the past decade. As shown in previous studies, this modification of single-body dispersion will in general couple different partial waves of the two-body scattering and thus distort the wave function of few-body bound states which determines the short-distance behavior of many-body wave function. In this work, we focus on the two-component Fermi gas with one-dimensional or three-dimensional spin-orbit coupling (SOC) near an sss-wave resonance. Using the method of effective field theory and the operator product expansion, we derive universal relations for both systems, including the adiabatic theorem, viral theorem, and pressure relation, and obtain the momentum distribution matrix $\ensuremath{\langle}{\ensuremath{\psi}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathbf{q}){\ensuremath{\psi}}_{b}(\mathbf{q})\ensuremath{\rangle}$ at large q\mathbf{q}q (a,ba,ba,b are spin indices). The momentum distribution matrix shows both spin-dependent and spatial anisotropic features. And the large momentum tail is modified at the subleading order thanks to the SOC. We also discuss the experimental implication of these results depending on the realization of the SOC.