In the strong-coupling Bose-Einstein condensation (BEC) region where a Feshbach resonance gives rise to tightly bound dimer molecules, we show that a spin-polarized Fermi superfluid reduces to a simple Bose-Fermi mixture of Bose-condensed dimers and the leftover unpaired fermions. Using a many-body functional integral formalism, the Gaussian fluctuations give rise to an induced dimer-dimer interaction mediated by the unpaired fermions, with the dimer-fermion vertex being given by the (mean-field) Born approximation. Treating the pairing fluctuations to quartic order, we show how the action for a spin-polarized Fermi superfluid reduces to one for a Bose-Fermi mixture. This Bose-Fermi action includes an expression for the effective dimer-unpaired fermion interaction in a spin-polarized Fermi superfluid beyond the Born approximation, in the BEC limit. In the low-density limit, we show how this dimer-fermion interaction gives the
s-wave scattering length
aBF=1.18aF (
aF is the
s-wave fermion scattering length), a result first derived by Skorniakov and Ter-Martirosian in 1957 for three interacting fermions.
In this Letter we address the issue of how synthetic spin-orbit (SO) coupling can strongly affect three-body physics in ultracold atomic gases. We consider a system which consists of three fermionic atoms, including two spinless heavy atoms and one spin-1/2 light atom subjected to an isotropic SO coupling. We find that SO coupling can induce universal three-body bound states with a negative s-wave scattering length at a smaller mass ratio, where no trimer bound state can exist if in the absence of SO coupling. The energies of these trimers are independent of the high-energy cutoff, and therefore they are universal ones. Moreover, the resulting atom-dimer resonance can be effectively controlled by SO coupling strength. Our results can be applied to systems like a 6Li and 40K mixture.
We derive the universal relations for an ultracold two-component Fermi gas with a spin-orbit coupling (SOC) ${\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=x,y,z}{\ensuremath{\lambda}}_{\ensuremath{\alpha}\ensuremath{\beta}}{\ensuremath{\sigma}}_{\ensuremath{\alpha}}{p}_{\ensuremath{\beta}}$, where
px,y,z and ${\ensuremath{\sigma}}_{x,y,z}$ are the single-atom momentum and Pauli operators for pseudospin, respectively, and the SOC intensity ${\ensuremath{\lambda}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ could take an arbitrary value. We consider the system with an
s-wave short-range interspecies interaction, and ignore the SOC-induced modification for the value of the scattering length. Using the first-quantized approach developed by Tan [S. Tan, Phys. Rev. Lett. 107, 145302 (2011)], we obtain the short-range and high-momentum expansions for the one-body real-space correlation function and momentum distribution function, respectively. For our system these functions are a $2\ifmmode\times\else\texttimes\fi{}2$ matrix in the pseudospin basis. We find that the leading-order (
1/k4) behavior of the diagonal elements of the momentum distribution function, i.e., ${n}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}(\mathbf{k})$ and ${n}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}(\mathbf{k})$, are not modified by the SOC. However, the SOC can significantly modify the large-
k behaviors of the distribution difference $\ensuremath{\delta}n(\mathbf{k})\ensuremath{\equiv}{n}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}(\mathbf{k})\ensuremath{-}{n}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}(\mathbf{k})$ as well as the nondiagonal elements of the momentum distribution function, i.e., ${n}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}(\mathbf{k})$ and ${n}_{\ensuremath{\downarrow}\ensuremath{\uparrow}}(\mathbf{k})$. In the absence of the SOC, the leading order of $\ensuremath{\delta}n(\mathbf{k})$, ${n}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}(\mathbf{k})$, and ${n}_{\ensuremath{\downarrow}\ensuremath{\uparrow}}(\mathbf{k})$ is
O(1/k6). When SOC appears, it can induce a term on the order of
1/k5 for these elements. We further derive the adiabatic relation and the energy functional. Our results show that the SOC can induce an additional term in the energy functional, which describes the contribution from the SOC to the total energy. In addition, the form of the adiabatic relation for our system is not modified by the SOC. Our results are applicable for the systems with any type of single-atom trapping potential, which could be either diagonal or nondiagonal in the pseudospin basis.
We study the two-body and three-body bound states in ultracold atomic mixtures with one of the atoms subjected to an isotropic spin-orbit (SO) coupling. We consider a system of two identical fermions interacting with one SO-coupled atom. It is found that there can exist two types of three-body bound states, Efimov trimers and universal trimers. The Efimov trimers are energetically less favored by the SO coupling, which will finally merge into the atom-dimer threshold as increasing the SO-coupling strength. Nevertheless, these trimers exhibit a discrete scaling law incorporating the SO-coupling effect. On the other hand, the universal trimers are more favored by the SO coupling. They can be induced at negative
s-wave scattering lengths and with smaller mass ratios than those without SO coupling. These results are obtained by both the Born-Oppenheimer approximation and exact solutions from three-body equations.
We study the contacts, large-momentum tail, and 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−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
q 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
s-wave scattering length, we find that two new contacts need to be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Further, we derive the pressure relation and virial theorem at short distance for this system. To find how the SOC modifies the large-momentun 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−8 order, and find that the diagonal terms of the distribution matrix include the contact of traditional
s-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond 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. This work paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.
Extensive research has been lavished on the effects of spin-orbit couplings (SOCs) in attractively interacting Fermi systems in both neutral cold-atom systems and condensed-matter systems. Recently, it was suggested that a SOC drives a class of BCS to Bose-Einstein condensate (BEC) crossover that is different from the conventional one without a SOC. Here, we explore what are the most relevant physical quantities to describe such a BCS to BEC crossover and their experimental detections. We extend the concepts of the coherence length and ``Cooper-pair size'' in the absence of SOC to Fermi systems with SOC. We investigate the dependence of chemical potential, coherence length, and Cooper-pair size on the SOC strength and the scattering length at three dimensions (3D) (the bound-state energy at 2D) for three attractively interacting Fermi gases with 3D Rashba, 3D Weyl, and 2D Rashba SOC, respectively. We show that only the coherence length can be used to characterize this BCS to BEC crossover. Furthermore, it is the only length which can be directly measured by radio-frequency dissociation spectra type of experiments. We stress crucial differences among the coherence length, Cooper-pair size, and the two-body bound-state size. Our results provide the fundamental and global picture of the BCS to BEC crossover and its experimental detections in various cold-atom and condensed-matter systems.