Illustrating the universal relations for a trapped four-fermion system with arbitrary s-wave scattering length

A two-component four-fermion system with equal masses, interspecies $s$ -wave scattering length ${a}_{s}$ , and vanishing intraspecies interactions under external spherically symmetric harmonic confinement is considered. Using a correlated Gaussian basis set expansion approach, we determine the energies and various structural properties of the energetically lowest-lying gaslike state throughout the crossover for various ranges of the underlying two-body potential. Extrapolating to the zero-range limit, our numerical results show explicitly that the total energy, the trap energy, as well as certain aspects of the pair distribution function and of the momentum distribution are related through the so-called integrated contact intensity $I({a}_{s})$ . Furthermore, it is shown explicitly that the total energy and the trap energy are related through a generalized virial theorem that accounts for a nonzero range.A two-component four-fermion system with equal masses, interspecies $s$-wave scattering length ${a}_{s}$, and vanishing intraspecies interactions under external spherically symmetric harmonic confinement is considered. Using a correlated Gaussian basis set expansion approach, we determine the energies and various structural properties of the energetically lowest-lying gaslike state throughout the crossover for various ranges of the underlying two-body potential. Extrapolating to the zero-range limit, our numerical results show explicitly that the total energy, the trap energy, as well as certain aspects of the pair distribution function and of the momentum distribution are related through the so-called integrated contact intensity $I({a}_{s})$. Furthermore, it is shown explicitly that the total energy and the trap energy are related through a generalized virial theorem that accounts for a nonzero range.

arXiv:0909.2701Physical Review ANov 30, 200936 citations

p -wave contacts of quantum gases in quasi-one-dimensional and quasi-two-dimensional traps

Mingyuan He, Qi Zhou

The length scale separation in dilute quantum gases in quasi-one-dimensional or quasi-two-dimensional traps has spatially divided the system into two distinct regimes. Whereas universal relations defined in strict one or two dimensions apply in a scale that is much larger than the characteristic length of the transverse confinements, physical observables in the short distances are inevitably governed by three-dimensional contacts. Here, we show that $p$ -wave contacts defined in different length scales are intrinsically connected by a universal relation, which depends on a simple geometric factor of the transverse confinements. While this universal relation is derived for one of the $p$ -wave contacts, it establishes a concrete example of how dimensional crossover interplays with contacts and universal relations for arbitrary partial wave scatterings.The length scale separation in dilute quantum gases in quasi-one-dimensional or quasi-two-dimensional traps has spatially divided the system into two distinct regimes. Whereas universal relations defined in strict one or two dimensions apply in a scale that is much larger than the characteristic length of the transverse confinements, physical observables in the short distances are inevitably governed by three-dimensional contacts. Here, we show that $p$-wave contacts defined in different length scales are intrinsically connected by a universal relation, which depends on a simple geometric factor of the transverse confinements. While this universal relation is derived for one of the $p$-wave contacts, it establishes a concrete example of how dimensional crossover interplays with contacts and universal relations for arbitrary partial wave scatterings.

arXiv:2106.13158Physical Review AOct 6, 20218 citations

Universal one-dimensional atomic gases near odd-wave resonance

Xiaoling Cui

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.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.

arXiv:1605.04363Physical Review AOct 19, 201652 citations

Pure Pairing Modes in Trapped Fermion Systems

P. Capuzzi, E. Susana Hernández, L. Szybisz

Journal of Low Temperature PhysicsOct 17, 20121 citation

A two-component four-fermion system with equal masses, interspecies

s

-wave scattering length

{a}_{s}

, and vanishing intraspecies interactions under external spherically symmetric harmonic confinement is considered. Using a correlated Gaussian basis set expansion approach, we determine the energies and various structural properties of the energetically lowest-lying gaslike state throughout the crossover for various ranges of the underlying two-body potential. Extrapolating to the zero-range limit, our numerical results show explicitly that the total energy, the trap energy, as well as certain aspects of the pair distribution function and of the momentum distribution are related through the so-called integrated contact intensity

I({a}_{s})

. Furthermore, it is shown explicitly that the total energy and the trap energy are related through a generalized virial theorem that accounts for a nonzero range.

p

-wave contacts defined in different length scales are intrinsically connected by a universal relation, which depends on a simple geometric factor of the transverse confinements. While this universal relation is derived for one of the

p

-wave contacts, it establishes a concrete example of how dimensional crossover interplays with contacts and universal relations for arbitrary partial wave scatterings.

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.