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arXiv:0710.5318 · Physical Review A

Phase diagram of a Bose gas near a wide Feshbach resonance

Lan Yin

Published Apr 25, 2008• 12 citations
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Abstract

In this paper, we study the phase diagram of a homogeneous Bose gas with a repulsive interaction near a wide Feshbach resonance at zero temperature. The Bose-Einstein-condensation (BEC) state of atoms is a metastable state. When the scattering length $a$ exceeds a critical value depending on the atom density $n$, $n{a}^{3}>0.035$, the molecular excitation energy is imaginary and the atomic BEC state is dynamically unstable against molecule formation. The BEC state of diatomic molecules has lower energy, where the atomic excitation is gapped and the molecular excitation is gapless. However, when the scattering length is above another critical value, $n{a}^{3}>0.0164$, the molecular BEC state becomes an unstable coherent mixture of atoms and molecules. In both BEC states, the binding energy of diatomic molecules is reduced due to the many-body effect.

DOI
10.1103/physreva.77.043630

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In this paper, we study the phase diagram of a homogeneous Bose gas with a repulsive interaction near a wide Feshbach resonance at zero temperature. The Bose-Einstein-condensation (BEC) state of atoms is a metastable state. When the scattering length aaa exceeds a critical value depending on the atom density nnn, na3>0.035n{a}^{3}>0.035na3>0.035, the molecular excitation energy is imaginary and the atomic BEC state is dynamically unstable against molecule formation. The BEC state of diatomic molecules has lower energy, where the atomic excitation is gapped and the molecular excitation is gapless. However, when the scattering length is above another critical value, na3>0.0164n{a}^{3}>0.0164na3>0.0164, the molecular BEC state becomes an unstable coherent mixture of atoms and molecules. In both BEC states, the binding energy of diatomic molecules is reduced due to the many-body effect.
Bose-Einstein condensation (BEC) of Feshbach molecules in a homogeneous Bose gas is studied at finite temperatures in a single-channel mean-field approach where the Hartree-Fock energy and pairing gap are determined self-consistently. In the molecular-BEC state, the atomic excitation is gapped and the molecular excitation is gapless. The binding energy of Feshbach molecules is shifted from the vacuum value by many-body effects. When the scattering length as{a}_{s}as​ of the atoms is negative, the system is subject to mechanical collapse due to negative compressibility. The system is stable in most regions with positive scattering lengths. However, at low temperatures near the resonance, the molecular-BEC state vanishes, and the coherent mixture of atomic and molecular BEC is subject to mechanical collapse.
We explore the Bose condensed phases of an atomic gas on the molecular side of a Feshbach resonance. In the absence of atom-molecule and molecule-molecule scattering, we show that the atomic condensate is either a saddle point of the free energy with energy always exceeding that of the molecular condensate, or has a negative compressibility (hence unstable to density fluctuations). Therefore no phase transition occurs between the atomic and molecular condensates. We also show that a repulsive molecule-molecule scattering can stabilize a sufficiently dense atomic condensate, leading to the possibility of a phase transition. We caution that 3-body processes may render this transition unobservable.
We explore the Bose condensed phases of an atomic gas on the molecular side of a Feshbach resonance. In the absence of atom-molecule and molecule-molecule scattering, we show that the atomic condensate is either a saddle point of the free energy with energy always exceeding that of the molecular condensate, or has a negative compressibility (hence unstable to density fluctuations). Therefore no phase transition occurs between the atomic and molecular condensates. We also show that a repulsive molecule-molecule scattering can stabilize a sufficiently dense atomic condensate, leading to the possibility of a phase transition. We caution that three-body processes may render this transition unobservable.
In this thesis, we investigated the physics of two- and three-dimensional ultra cold Bose gases in the strongly interacting regime at zero temperature. This regime can be experimentally accessed using a Feshbach resonance. We applied a self-consistent diagrammatic approach to determine the chemical potential of three-dimensional Bose gases for a wide range of interaction values. We showed that such strongly interacting Bose gases become unstable towards the formation of molecules at a finite positive scattering length. In fact, the interaction between atoms becomes effectively attractive and the system looses its metastability before reaching the unitary limit. We also found that such systems are nearly fermionized close to the instability point. Near this critical point, the chemical potential reaches a maximum and the contribution to the system energy due to three-body forces is estimated to be only a few percent. We also studied the same system using a self-consistent renormalization group method. This approach confirms the existence of an instability point towards the formation of molecules as well as fermionization. We showed that the instability and accompanying maximum are precursors of the sign change of the effective two-body interaction strength from repulsive to attractive near resonance. In addition, we examined the physics of two-dimensional Bose gases near resonance using a similar self-consistent diagrammatic approach as the one introduced for three-dimensional Bose gases. We demonstrated that a competition between three-body attractive interactions and two-body repulsive forces results in the chemical potential of two-dimensional Bose gases to exhibit a maximum at a critical scattering length beyond which these quantum gases possess a negative compressibility. For larger scattering lengths, the increasingly prominent role played by three-body attractive interactions leads to an onset instability at a second critical value. The three-body effects studied for these systems are universal, fully characterized by the effective two-dimensional scattering length and are, in comparison to the three-dimensional case, independent of three-body ultraviolet physics.
In this paper, properties of a homogeneous Bose gas with a Feshbach resonance are studied in the dilute region at zero temperature. The stationary state contains condensations of atoms and molecules. The ratio of the molecule density to the atom density is $\ensuremath{\pi}{\mathrm{na}}^{3}.$ There are two types of excitations, molecular excitations and atomic excitations. Atomic excitations are gapless, consistent with the traditional theory of a dilute Bose gas. The molecular excitation energy is finite in the long-wavelength limit as observed in recent experiments on 85Rb.{}^{85}\mathrm{Rb}.85Rb. In addition, the decay process of the condensate is studied. The coefficient of the three-body recombination rate is about 140 times larger than that of a Bose gas without a Feshbach resonance, in reasonably good agreement with the experiment on 23Na.{}^{23}\mathrm{Na}.23Na.
We consider the properties of a gas of bosonic diatomic molecules in the limit when few of the molecules are dissociated. Taking into account the effects of dissociation and scattering among molecules and atoms, we calculate the dispersion relation for a molecule, and the thermal depletion of the condensate. We calculate the dependence of the Bose-Einstein-condensation (BEC) temperature of a uniform gas on the atom-atom scattering length, and conclude that, for a broad Feshbach resonance, the condensation temperature increases as the molecular state becomes less strongly bound, thereby giving rise to a maximum in the transition temperature in the BEC-BCS crossover. We also argue on general grounds that, for a gas in a harmonic trap and for a narrow Feshbach resonance, the condensation temperature will decrease with increasing scattering length.