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.
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
s-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 (
a,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.
We present a comprehensive derivation of a set of universal relations for spin-orbit-coupled Fermi gases in three or two dimensions, which follow from the short-range behavior of the two-body physics. Besides the adiabatic energy relations, the large-momentum distribution, the grand canonical potential, and pressure relation derived in our previous work for three-dimensional systems [Phys. Rev. Lett. 120, 060408 (2018)], we further derive high-frequency tail of the radio-frequency spectroscopy and the short-range behavior of the pair correlation function. In addition, we also extend the derivation to two-dimensional systems with Rashba-type spin-orbit coupling. To simply demonstrate how the spin-orbit-coupling effect modifies the two-body short-range behavior, we solve the two-body problem in the sub-Hilbert space of zero center-of-mass momentum and zero total angular momentum, and we perturbatively take the spin-orbit-coupling effect into account at short distance, since the strength of the spin-orbit coupling should be much smaller than the corresponding scale of the finite range of interatomic interactions. The universal asymptotic forms of the two-body wave function at short distance are then derived, which are independent on the short-range details of interatomic potentials. We find that new scattering parameters emerge in the universal asymptotic forms, apart from the traditional
s- and
p-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. We show how these two-body parameters characterize the universal relations in the presence of spin-orbit coupling. This work probably sheds light on understanding the profound properties of the many-body quantum systems in the presence of the spin-orbit coupling.
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 develop the contact theory for spin-orbit-coupled Fermi gases. By using a perturbation method, we derive analytically the universal two-body behavior at short distance, which does not depend on the short-range details of interatomic potentials. We find that two new scattering parameters need to be introduced because of spin-orbit coupling, besides the traditional s- and p-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. Consequently, two new adiabatic energy relations with respect to the new scattering parameters are obtained, in which a new contact is involved because of spin-orbit coupling. In addition, we derive the asymptotic behavior of the large-momentum distribution, and find that the subleading tail is corrected by the new contact. This work paves the way for exploring the profound properties of spin-orbit-coupled many-body systems, according to two-body solutions.
Abstract We derive the full set of universal relations for spin-polarized Fermi gases with p -wave interaction in two dimensions, simply using the short-range asymptotic behavior of fermion-pair wave functions. For p -wave interactions, an additional contact related to the effective range needs to be introduced, besides the one related to the scattering volume. Since the subleading tail ( k −4 ) of the large-momentum distribution cannot fully be captured by the contacts defined by the adiabatic relations, an extra term resulted from the center-of-mass motions of the pairs gives rise to an additional divergence in the kinetic energy of the system, besides those related to the contacts defined. We show in Tan’s energy theorem that if only two-body correlations are taken into account, all these divergences are reasonably removed, leading to a finite internal energy of the system. In addition, we find that all the other universal relations, such as the high-frequency behavior of the radio-frequency response, short-range behavior of the pair correlation function, generalized virial theorem, and pressure relation, remain unaffected by the center-of-mass motions of the pairs, and are fully governed by the contacts defined by the adiabatic relations. Our results confirm the feasibility of generalizing the contact theory for higher-partial-wave scatterings, and could readily be confirmed in current experiments with ultracold 40 K and 6 Li atoms.