Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1210.2006 · Physical Review A

Three s -wave-interacting fermions under anisotropic harmonic confinement: Dimensional crossover of energetics and virial coefficients

Seyed Ebrahim Gharashi, K. M. Daily, D. Blume

Published Oct 4, 2012• 37 citations
arXivHTMLPDF

Abstract

We present highly accurate solutions of the Schr\"odinger equation for three fermions in two different spin states with zero-range $s$-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all $s$-wave scattering lengths.

DOI
10.1103/physreva.86.042702

Related papers

High-temperature thermodynamics of strongly interacting s -wave and p -wave Fermi gases in a harmonic trap

Shi-Guo Peng, Shiqun Li, P. D. Drummond +1 more

We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either $s$-wave or $p$-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either $s$-wave or $p$-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting $s$-wave Fermi gas has already been proved very useful. In the case of $p$-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a $p$-wave Feshbach resonance.

arXiv:1107.2740•Physical Review A•Jun 15, 2011•24 citations

High-temperature thermodynamics of spin-polarized Fermi gases in two-dimensional harmonic traps

Caixia Zhang, Shi-Guo Peng, Kaijun Jiang

Using a second-order quantum virial expansion, we theoretically study the high-temperature thermodynamics of spin-polarized Fermi gases with $p$-wave interaction confined in two-dimensional harmonic traps. The second virial coefficient is calculated based on the energy spectrum of two harmonically trapped fermions. We find that the two-body spectrum can correctly be evaluated only when the $p$-wave effective range is properly considered, which is referred to as the well-known Wigner's bound. According to the quantum virial expansion, the high-temperature thermodynamics of the system is then discussed based on the two-body solutions. In addition, we investigate the high-temperature behavior of $p$-wave contacts in harmonic traps within the quantum virial expansion, which characterize a set of universal relations of strongly interacting spin-polarized Fermi gases. Our results can provide an accurate benchmark at high temperature for future numerical simulations as well as high-precision thermodynamic measurements of harmonically trapped spin-polarized Fermi gases in two dimensions.

Physical Review A•Oct 16, 2018•6 citations

Three attractively interacting fermions in a harmonic trap: Exact solution, ferromagnetism, and high-temperature thermodynamics

Xia-Ji Liu, Hui Hu, P. D. Drummond

Three fermions with strongly repulsive interactions in a spherical harmonic trap constitute the simplest nontrivial system that can exhibit the onset of itinerant ferromagnetism. Here, we present exact solutions for three trapped, attractively interacting fermions near a Feshbach resonance. We analyze energy levels on the upper branch of the resonance where the atomic interaction is effectively repulsive. When the $s$-wave scattering length $a$ is sufficiently positive, three fully polarized fermions are energetically stable against a single spin-flip, indicating the possibility of itinerant ferromagnetism, as inferred in the recent experiment. We also investigate the high-temperature thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion. The second and third virial coefficients are calculated. The resulting equations of state can be tested in future quantitative experimental measurements at high temperatures and can provide a useful benchmark for quantum Monte Carlo simulations.

arXiv:1007.1041•Physical Review A•Aug 30, 2010•91 citations

Analytical solution for the spectrum of two ultracold atoms in a completely anisotropic confinement

Yue Chen, Da-Wu Xiao, Ren Zhang +1 more

We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation ${J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}$ (${J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}$) for the eigenenergy $E$ of this system in the 3D (2D) case, with ${a}_{3\mathrm{D}}$ and ${a}_{2\mathrm{D}}$ being the corresponding $s$-wave scattering lengths, and provide the analytical expressions of the functions ${J}_{3\mathrm{D}}(E)$ and ${J}_{2\mathrm{D}}(E)$. In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.

arXiv:2002.02168•Physical Review A•May 14, 2020•8 citations

Universal relations for a trapped four-fermion system with arbitrary s -wave scattering length

D. Blume, K. M. Daily

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:0909.2701•Physical Review A•Nov 30, 2009•36 citations

Jumping from two and three particles to infinitely many

D. Blume

Three fermions with strongly repulsive interactions in a spherical harmonic trap constitute the simplest nontrivial system that can exhibit the onset of itinerant ferromagnetism.Here, we present exact solutions for three trapped, attractively interacting fermions near a Feshbach resonance.We analyze energy levels on the upper branch of the resonance where the atomic interaction is effectively repulsive.When the s-wave scattering length a is sufficiently positive, three fully polarized fermions are energetically stable against a single spin-flip, indicating the possibility of itinerant ferromagnetism, as inferred in the recent experiment.We also investigate the high-temperature thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion.The second and third virial coefficients are calculated.The resulting equations of state can be tested in future quantitative experimental measurements at high temperatures and can provide a useful benchmark for quantum Monte Carlo simulations.

Physics•Aug 30, 2010•14 citations
We present highly accurate solutions of the Schr"odinger equation for three fermions in two different spin states with zero-range sss-wave interactions under harmonic confinement. Our approach covers spherically symmetric, strictly two-dimensional, strictly one-dimensional, cigar-shaped, and pancake-shaped traps. In particular, we discuss the transition from quasi-one-dimensional to strictly one-dimensional and from quasi-two-dimensional to strictly two-dimensional geometries. We determine and interpret the eigenenergies of the system as a function of the trap geometry and the strength of the zero-range interactions. The eigenenergies are used to investigate the dependence of the second- and third-order virial coefficients, which play an important role in the virial expansion of the thermodynamic potential, on the geometry of the trap. We show that the second- and third-order virial coefficients for anisotropic confinement geometries are, for experimentally relevant temperatures, very well approximated by those for the spherically symmetric confinement for all sss-wave scattering lengths.
We theoretically investigate the high-temperature thermodynamics of a strongly interacting trapped Fermi gas near either sss-wave or ppp-wave Feshbach resonances, using a second-order quantum virial expansion. The second virial coefficient is calculated based on the energy spectrum of two interacting fermions in a harmonic trap. We consider both isotropic and anisotropic harmonic potentials. For the two-fermion interaction, either sss-wave or ppp-wave, we use a pseudopotential parametrized by a scattering length and an effective range. This turns out to be the simplest way of encoding the energy dependence of the low-energy scattering amplitude or phase shift. This treatment of the pseudopotential can be easily generalized to higher partial-wave interactions. We discuss how the second virial coefficient and thermodynamics are affected by the existence of these finite-range interaction effects. The virial expansion result for a strongly interacting sss-wave Fermi gas has already been proved very useful. In the case of ppp-wave interactions, our results for the high-temperature equation of state are applicable to future high-precision thermodynamic measurements for a spin-polarized Fermi gas near a ppp-wave Feshbach resonance.
Using a second-order quantum virial expansion, we theoretically study the high-temperature thermodynamics of spin-polarized Fermi gases with ppp-wave interaction confined in two-dimensional harmonic traps. The second virial coefficient is calculated based on the energy spectrum of two harmonically trapped fermions. We find that the two-body spectrum can correctly be evaluated only when the ppp-wave effective range is properly considered, which is referred to as the well-known Wigner's bound. According to the quantum virial expansion, the high-temperature thermodynamics of the system is then discussed based on the two-body solutions. In addition, we investigate the high-temperature behavior of ppp-wave contacts in harmonic traps within the quantum virial expansion, which characterize a set of universal relations of strongly interacting spin-polarized Fermi gases. Our results can provide an accurate benchmark at high temperature for future numerical simulations as well as high-precision thermodynamic measurements of harmonically trapped spin-polarized Fermi gases in two dimensions.
Three fermions with strongly repulsive interactions in a spherical harmonic trap constitute the simplest nontrivial system that can exhibit the onset of itinerant ferromagnetism. Here, we present exact solutions for three trapped, attractively interacting fermions near a Feshbach resonance. We analyze energy levels on the upper branch of the resonance where the atomic interaction is effectively repulsive. When the sss-wave scattering length aaa is sufficiently positive, three fully polarized fermions are energetically stable against a single spin-flip, indicating the possibility of itinerant ferromagnetism, as inferred in the recent experiment. We also investigate the high-temperature thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion. The second and third virial coefficients are calculated. The resulting equations of state can be tested in future quantitative experimental measurements at high temperatures and can provide a useful benchmark for quantum Monte Carlo simulations.
We study the system of two ultracold atoms in a three-dimensional (3D) or two-dimensional (2D) completely anisotropic harmonic trap. We derive the transcendental equation J3D(E)=1/a3D{J}_{3\mathrm{D}}(E)=1/{a}_{3\mathrm{D}}J3D​(E)=1/a3D​ (J2D(E)=lna2D{J}_{2\mathrm{D}}(E)=ln{a}_{2\mathrm{D}}J2D​(E)=lna2D​) for the eigenenergy EEE of this system in the 3D (2D) case, with a3D{a}_{3\mathrm{D}}a3D​ and a2D{a}_{2\mathrm{D}}a2D​ being the corresponding sss-wave scattering lengths, and provide the analytical expressions of the functions J3D(E){J}_{3\mathrm{D}}(E)J3D​(E) and J2D(E){J}_{2\mathrm{D}}(E)J2D​(E). In previous research, this type of equation was obtained for spherically or axially symmetric harmonic traps [T. Busch et al., Found. Phys. 28, 549 (1998); Z. Idziaszek and T. Calarco, Phys. Rev. A 74, 022712 (2006)]. However, for our cases with a completely anisotropic trap, only the equation for the ground-state energy of some cases has been derived [J.-J. Liang and C. Zhang, Phys. Scr. 77, 025302 (2008)]. Our results in this work are applicable for arbitrary eigenenergy of this system and can be used for the study of dynamics and thermal dynamics of interacting ultracold atoms in this trap, e.g., the calculation of the second virial coefficient or the evolution of two-body wave functions. In addition, our approach for the derivation of the above equations can also be used for other two-body problems of ultracold atoms.
A two-component four-fermion system with equal masses, interspecies sss-wave scattering length as{a}_{s}as​, 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(as)I({a}_{s})I(as​). 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.
Three fermions with strongly repulsive interactions in a spherical harmonic trap constitute the simplest nontrivial system that can exhibit the onset of itinerant ferromagnetism.Here, we present exact solutions for three trapped, attractively interacting fermions near a Feshbach resonance.We analyze energy levels on the upper branch of the resonance where the atomic interaction is effectively repulsive.When the s-wave scattering length a is sufficiently positive, three fully polarized fermions are energetically stable against a single spin-flip, indicating the possibility of itinerant ferromagnetism, as inferred in the recent experiment.We also investigate the high-temperature thermodynamics of a strongly repulsive or attractive Fermi gas using a quantum virial expansion.The second and third virial coefficients are calculated.The resulting equations of state can be tested in future quantitative experimental measurements at high temperatures and can provide a useful benchmark for quantum Monte Carlo simulations.