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arXiv:cond-mat/0305692

Cold Bose Gases near Feshbach Resonances

Yeong E. Kim, Alexander L. Zubarev

Published May 29, 2003• 0 citations
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Abstract

The lowest order constrained variational method [Phys. Rev. Lett. 88, 210403 (2002)] has been generalized for a dilute (in the sense that the range of interatomic potential is small compared with inter-particle spacing) uniform gas of bosons near the Feshbach resonance using the multi-channel zero-range potential model. The method has been applied to Na (F=1, m_F=1) atoms near the $B_0=907$G Feshbach resonance. It is shown that at high densities, there are significant differences between our results for the real part of energy per particle and the one-channel zero-range potential approximation. We point out the possibility of stabilization of the uniform con densate for the case of negative scattering length.

DOI
10.48550/arxiv.cond-mat/0305692

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The lowest order constrained variational method [Phys. Rev. Lett. 88, 210403 (2002)] has been generalized for a dilute (in the sense that the range of interatomic potential is small compared with inter-particle spacing) uniform gas of bosons near the Feshbach resonance using the multi-channel zero-range potential model. The method has been applied to Na (F=1, m_F=1) atoms near the B0=907B_0=907B0​=907G Feshbach resonance. It is shown that at high densities, there are significant differences between our results for the real part of energy per particle and the one-channel zero-range potential approximation. We point out the possibility of stabilization of the uniform con densate for the case of negative scattering length.
(2002)] has been generalized for a dilute (in the sense that the range of interatomic potential, R, is small compared with inter-particle spacing r0) uniform gas of bosons near the Feshbach resonance using the multi-channel zero-range potential model. The method has been applied to Na(F = 1, mF = 1) atoms near the B0 = 907G Feshbach resonance. It is shown that at high densities na3 ≫ 1, there are significant differences between our results for the real part of energy per particle and the one-channel zero-range potential approximation. We point out the possibility of stabilization of the uniform condensate for the case of negative scattering length.
Dilute Fermi systems with large sss-wave scattering length as{a}_{s}as​ exhibit universal properties if the interparticle spacing r0{r}_{0}r0​ greatly exceeds the range of the underlying two-body interaction potential. In this regime, r0{r}_{0}r0​ is the only relevant length scale and observables such as the energy per particle depend only on r0{r}_{0}r0​ (or, equivalently, the energy EFG{E}_{\mathrm{FG}}EFG​ of the free Fermi gas). This paper investigates Bose and Fermi systems with nonvanishing angular momentum lll using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of an lll-dependent length scale ${\ensuremath{\xi}}_{l}$, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite lll. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.
In this thesis, we investigate ultracold Bose-Fermi mixtures in both two and three dimensions using a non-perturbative approach based on the Lowest Order Constrained Variational (LOCV) method. This variational technique, originally developed in the context of nuclear matter, is applied here to dilute quantum gases at zero temperature, allowing for an accurate treatment of two-body correlations beyond perturbative limits. We begin by analysing Bose-Fermi mixtures near a broad Fano-Feshbach resonance, which enables precise tuning of the interspecies interaction strength. Particular attention is devoted to the low-dimensional case, where quantum fluctuations are enhanced and experimental realizations have become increasingly accessible. While the three-dimensional problem serves as a useful benchmark, the two-dimensional Bose-Fermi mixture presents unique challenges. Notably, no prior non-perturbative studies exist on its mechanical stability. A central goal of this work is to determine the conditions under which the 2D Bose-Fermi mixture remains mechanically stable as the boson-fermion attraction is increased. Our results demonstrate that the LOCV approach captures the relevant physics from weak to moderate coupling, offering a complementary perspective to perturbative methods or alternative non-perturbative approaches such as the T-matrix formalism. In this coupling range, we identify the critical strength of boson-boson repulsion necessary to prevent phase separation or collapse, thereby providing the first theoretical variational prediction of a stability phase diagram in this two dimensional regime. Overall, this work fills a gap in the literature and provides a useful reference point for future theoretical and experimental studies of strongly interacting Bose-Fermi systems in reduced dimensionality.
Dilute Fermi systems with large s-wave scattering length a_s exhibit universal properties if the interparticle spacing r_o greatly exceeds the range of the underlying two-body interaction potential. In this regime, r_o is the only relevant length scale and observables such as the energy per particle depend only on r_o (or, equivalently, the energy E_{FG} of the free Fermi gas). This paper investigates Bose and Fermi systems with non-vanishing angular momentum l using the lowest order constrained variational method. We focus on the regime where the generalized scattering length becomes large and determine the relevant length scales. For Bose gases with large generalized scattering lengths, we obtain simple expressions for the energy per particle in terms of a l-dependent length scale ξ_l, which depends on the range of the underlying two-body potential and the average interparticle spacing. We discuss possible implications for dilute two-component Fermi systems with finite l. Furthermore, we determine the equation of state of liquid and gaseous bosonic helium.
A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule theory for a Bose gas near a Feshbach resonance. In order to take into account all two-body processes, we have to dress the bare couplings of the atom–molecule model with ladder diagrams. This results in a quantum field theory that exactly reproduces the scattering amplitude of the atoms and the bound-state energy of the molecules. Since these properties are incorporated at the quantum level, the theory can be applied both above and below the critical temperature of the gas. Moreover, making use of the true interatomic potentials ensures that no divergences are encountered at any stage of the calculation. We also present the mean-field theory for the Bose–Einstein condensed phase of the gas.
The well-known results concerning a dilute Bose gas with the short-range repulsive interaction should be reconsidered due to a thermodynamic inconsistency of the method being basic to much of the present understanding of this subject. The aim of our paper is to propose another way of treating the dilute Bose gas with an arbitrary strong interaction. Using the reduced density matrix of the second order and a variational procedure, this way allows us to escape the inconsistency mentioned and operate with singular potentials of the Lennard-Jones type. The derived expansion of the condensate depletion in powers of the boson density n=N/V reproduces the familiar result, while the expansion for the mean energy per particle is of the form epsilon=2 pi planck(2)an/m[1+128/(15 square root of [pi]) square root of [na(3)](1-5b/8a)+...], where a is the scattering length and b> or =0 stands for one more characteristic length depending on the shape of the interaction potential (in particular, for the hard spheres a=b). All the consideration concerns the zero temperature.