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arXiv:1605.04363 · Physical Review A

Universal one-dimensional atomic gases near odd-wave resonance

Xiaoling Cui

Published Oct 19, 2016• 52 citations
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Abstract

We show the renormalization of the contact interaction for odd-wave scattering in one-dimension (1D). Based on the renormalized interaction, we exactly solve the two-body problem in a harmonic trap and further explore the universal properties of spin-polarized fermions near odd-wave resonance by using the operator-product-expansion method. It is found that the high-momentum distribution behaves as $C/{k}^{2}$, with $C$ being the odd-wave contact. Various universal relations are derived. Our work suggests a universal system emergent in 1D with large odd-wave scattering length.

DOI
10.1103/physreva.94.043636

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We show the renormalization of the contact interaction for odd-wave scattering in one-dimension (1D). Based on the renormalized interaction, we exactly solve the two-body problem in a harmonic trap and further explore the universal properties of spin-polarized fermions near odd-wave resonance by using the operator-product-expansion method. It is found that the high-momentum distribution behaves as C/k2C/{k}^{2}C/k2, with CCC being the odd-wave contact. Various universal relations are derived. Our work suggests a universal system emergent in 1D with large odd-wave scattering length.
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.
In quasi-one- or quasi-two-dimensional traps with strong transverse confinements, quantum gases behave like strictly one- or two-dimensional systems at large length scales. However, at short distance, the two-body scattering intrinsically has three-dimensional characteristics such that an exact description of any universal thermodynamic relation requires three-dimensional contacts since the range of interaction (a few nm) is orders of magnitude smaller than the harmonic-oscillator length of the transverse confinement ($\ensuremath{\sim}{10}^{2}\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$ for a 100 kHz trap). A fundamental question arises as to whether one- or two-dimensional contacts, which were originally defined for strictly one or two dimensions, are capable of describing quantum gases in quasi-one- or quasi-two-dimensional traps. Here, we point out an exact relation between the three- and low-dimensional contacts in these highly anisotropic traps. Such relation allows us to directly connect physical quantities at different length scales and to characterize the quasi-one- or quasi-two-dimensional traps using universal thermodynamic relations that were derived for strict one or two dimensions.
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.
We renormalize the two-body contact interaction based on the exact solution of two interacting particles in a harmonic trap. This renormalization extends the validity of the contact interaction to large scattering lengths. We apply this renormalized interaction to a degenerate unitary Bose gas to study its stationary properties and elementary excitations using the mean-field theory and the hyperspherical method. Since the scattering length is no longer a relevant length scale at unitarity, universal properties are obtained that depend only on the average particle density. Our treatment shows that the universal relations for the total energy and for the two-body contact are $E/N=12.67{\ensuremath{\hbar}}^{2}\ensuremath{\langle}{n}^{2/3}\ensuremath{\rangle}/2m$ and ${C}_{2}/N=11.8\ensuremath{\langle}{n}^{1/3}\ensuremath{\rangle}$, respectively.
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.
In this work, we present an effective-field theory to describe a two-component Fermi gas near a ddd-wave-interaction resonance. The effective-field theory is renormalizable by matching with the low-energy ddd-wave scattering phase shift. Based on the effective-field theory, we derive universal properties of the Fermi gas by the operator-product-expansion method. We find that beyond the contacts defined by adiabatic theorems, the asymptotic expressions of the momentum distribution and the Raman spectroscopy involve two extra contacts which provide additional information on correlations of the system. Our formalism sets the stage for further explorations of many-body effects in a ddd-wave resonant Fermi gas. Finally, we generalize our effective-field theory for interaction resonances of arbitrary higher partial waves.