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arXiv:0801.2580 · Physical Review A

Mean-field stationary state of a Bose gas at a Feshbach resonance

Andrew Carmichael, Juha Javanainen

Published Apr 14, 2008• 5 citations
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Abstract

We study the steady state of a zero-temperature Bose gas near a Feshbach or photoassociation resonance using a two-channel mean-field model that incorporates atomic and molecular condensates, as well as correlated atom pairs originating from dissociation of molecules into pairs of atoms. We start from a many-body Hamiltonian for atom-molecule conversion, and derive the time-dependent version of the mean-field theory. The stationary solution of the time-dependent model is rendered unique with an approximation that entails that all noncondensate atoms are correlated, as if emerging from dissociation of molecules. The steady state is solved numerically, but limiting cases are also found analytically. The system has a phase transition in which the atomic condensate emerges in a nonanalytic fashion. We quantify the scaling of the observable quantities, such as fractions of atomic and molecular condensates, with the detuning and the atom-molecule conversion strength. Qualitatively, the dependence on detuning rounds out with increasing coupling strength. A study of the thermodynamics shows that the pressure of the atom-molecule system is negative, even on the molecule side of the resonance. This indicates the possibility of mechanical instability.

DOI
10.1103/physreva.77.043616

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臺灣大學物理研究所學位論文•Jan 1, 2004•0 citations
We study the steady state of a zero-temperature Bose gas near a Feshbach or photoassociation resonance using a two-channel mean-field model that incorporates atomic and molecular condensates, as well as correlated atom pairs originating from dissociation of molecules into pairs of atoms. We start from a many-body Hamiltonian for atom-molecule conversion, and derive the time-dependent version of the mean-field theory. The stationary solution of the time-dependent model is rendered unique with an approximation that entails that all noncondensate atoms are correlated, as if emerging from dissociation of molecules. The steady state is solved numerically, but limiting cases are also found analytically. The system has a phase transition in which the atomic condensate emerges in a nonanalytic fashion. We quantify the scaling of the observable quantities, such as fractions of atomic and molecular condensates, with the detuning and the atom-molecule conversion strength. Qualitatively, the dependence on detuning rounds out with increasing coupling strength. A study of the thermodynamics shows that the pressure of the atom-molecule system is negative, even on the molecule side of the resonance. This indicates the possibility of mechanical instability.
We construct a simple quantum optics style mean field model to investigate the behaviour of a zero-temperature untrapped quantum degenerate Bose gas close to a Feshbach resonance. The model allows for both atomic and molecular condensates as well as correlated zero-momentum “BCS” pairs whose provenance would be dissociated zero momentum molecules. Beginning with a second quantized (momentum representation) Hamiltonian and equations conserving total (free and bound) atom number and enforcing an assumption that atoms only appear either in the condensates or pairs and the usual Bogoliubov approximation, the system is numerically and in certain limits, analytically, soluble in the steady state and exhibits a second order phase transition to a pure atomic condensate when the controllable parameters of the coupling Rabi frequency and detuning are varied across an (analytically determined) transition line. Analysis of the thermodynamics of the zero-entropy system shows a negative pressure and hence mechanical instability on both sides of the resonance. A mathematical difficulty arising from an ultra-violet divergence due to the assumption of a zero range interaction is resolved with the help of a simpler, exactly analytically soluble two atom version of the problem. ^
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.
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.
We revisit the mean-field treatment of photoassociation and Feshbach resonances in a Bose-Einstein condensate previously used by various authors. Generalizing the Cherny and Shanenko approach [Phys. Rev. E 62, 1646 (2000)] where the finite size of the potentials is explicitly introduced, we develop a two-channel model for a mixed atomic-molecular condensate. Besides the individual dynamics of the condensed and noncondensed atoms, the model also takes into account their pair dynamics by means of pair wave functions. We show that the resulting set of coupled equations can be reduced to the usual coupled Gross-Pitaevski\ifmmode \breve{\imath}\else \u{\i}\fi{} equations when the time scale of the pair dynamics is short compared to that of the individual dynamics. Such time scales are discussed in the case of typical photoassociation experiments with cw lasers. We show that the individual dynamics plays a minor role, demonstrating the validity of the rates predicted by the usual models describing photoassociation in a nondegenerate gas.
We solve a model that describes a stimulated conversion between ultracold bosonic atoms and molecules. The reaction is triggered by a linearly time-dependent transition throughout the Feshbach resonance. Our solution predicts a dependence, with a dynamic phase transition, of the reaction efficiency on the transition rate for both atoms-to-molecule pairing and molecular dissociation processes. We find that for the latter process with a linear energy dispersion of atomic modes, the emerging phase can have a thermalized energy distribution of noninteracting bosons with the temperature defined by the rate of the transition. This provides a simple interpretation of the phase transition in terms of the creation of equilibrium Bose-Einstein condensate.
To study the behavior of atomic Fermi gases in the vicinity of Feshbach resonance, M. Holland et al. [12] and S. J. J. M. F. Kokkelmans et al. [22] proposed resonance mean field theory in which the molecules formed by Feshbach resonance are considered in the many-body Hamiltonian. Motivated by microscopic scattering theory and resonance mean field theory, we develop a thermodynamic method to study this system. In this method we treat atomic Fermi gases near resoance as a atom-molecule mixture. We can use BCS Hamiltonian and Bose- Einstein statistics to describe the behavior of atoms and molecules respectively. Chemical equilibrium can be achieved between atoms and molecules due to hypefine interaction. We use this method to study the system behavior under different temperature and detuning energy and find that at low temperature a smooth BCS-BEC crossever can occure with decreasing detuning energy. We further calculate a measurable quantity, molecular condensation fraction, and find its behavior matches qualitatively with the experimental result [16].