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Physical Review A

Bogoliubov theory for mutually coherent hybrid atomic molecular condensates: Quasiparticles and superchemistry

Chi-Yong Lin, E. J. V. de Passos, A. F. R. de Toledo Piza, Da-Shin Lee, Mahir Saleh Hussein

Published Jan 13, 2006• 5 citations
Publisher

Abstract

We investigate the equilibrium properties of a hybrid atomic and molecular Bose-Einstein condensate in the context of the Feshbach-resonant quantum many-body model. We analyze fully the mean-field solutions of the system for a general choice of interaction parameters. The quantum fluctuations of the system are obtained from the Bogoliubov theory that characterizes the quasiparticle excitations of the system. In particular we find that the excitation energies have two branches: a Goldstone mode and a second one containing a gap. Both gap energy and phonon velocities are functions of the composition, and our analysis indicates that for a large range of parameter space the system is unstable near a Feshbach resonance.

DOI
10.1103/physreva.73.013615

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We investigate the equilibrium properties of a hybrid atomic and molecular Bose-Einstein condensate in the context of the Feshbach-resonant quantum many-body model. We analyze fully the mean-field solutions of the system for a general choice of interaction parameters. The quantum fluctuations of the system are obtained from the Bogoliubov theory that characterizes the quasiparticle excitations of the system. In particular we find that the excitation energies have two branches: a Goldstone mode and a second one containing a gap. Both gap energy and phonon velocities are functions of the composition, and our analysis indicates that for a large range of parameter space the system is unstable near a Feshbach resonance.
We analyze the mean-field equilibrium states of a homogeneous Bose-Einstein condensate of boson particles in different internal states, possibly in the presence of an internal Josephson interaction, and we describe the quantum fluctuations by constructing an explicit form of the Bogoliubov transformation that characterizes the quasiparticle excitations of the system. The derivation of the Bogoliubov transformation is based on a simple geometrical picture which leads to a convenient parametrization of the transformation. Our calculations of physical properties, such as depletions and phonon velocities, as well as our discussion of stability, indicate the sense and the extent to which the internal Josephson interaction controls the many-body physics of multicomponent phase coherent matter.
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.
In the atomic Bose-Einstein condensate, the interactions that bring a binary atom system to an intermediate state molecule in the Feshbach resonance create a second condensate component of molecules. The atomic and molecular condensates coherently exchange pairs of atoms. We discuss a signature of the coherent intercondensate exchange: Josephson-like oscillations of the atomic and molecular populations in response to a sudden change of the energy detuning. The dependence of the many-body ground state energy on volume suggests that the on-resonant ground state is a dilute condensate with the liquidlike property of a self-determined density.
In the wake of successful experiments in Fermi condensates, experimental attention is broadening to include resonant interactions in degenerate Bose-Fermi mixtures. In this thesis we wish to study the equilibrium properties of the fermionic molecules that can be created in such a mixture. To this end, we first discuss the two body properties of the system, and introduce the model Hamiltonian we use to describe the resonant physics, highlighting its virtues, as well as its limitations. We then proceed by analyzing the mean field solution of this model, by studying both the equilibrium problem, and the non-equilibrium equations of motion, thus developing a powerful language to discuss the system. We then highlight the limitations of the mean-field approach, and develop a numerically tractable generalized version of this theory, which is able to correctly describe the two-body properties of the system in the low-density limit. Finally, we study the properties of the system using this generalized mean-field theory, by first analyzing the two-body scattering matrix in the many-body environment, assessing its complex poles in order to understand the stability properties of the Feshbach molecules in the gas. Secondly we solve the equilibrium equations self-consistently, to study the molecular populations and density distributions at equilibrium, as a function of external bias magnetic field.
We discuss the effect of a molecular Bose condensate on the energy of Fermi excitations in a trapped two-component atomic Fermi gas. The single-particle Green's functions can be approximated by the well-known BCS form, in both the BCS (Cooper pairs) and BEC (Feshbach resonance molecules) domains. The composite Bose order parameter Δ~{\tilde Δ}Δ~ describing bound states of two atoms and the Fermi chemical potential μμμ are calculated self-consistently. In the BEC regime characterized by μ<0μ<0μ<0, the Fermi quasiparticle energy gap is given by μ2+Δ~2\sqrt{μ^2+{\tilde Δ}^2}μ2+Δ~2​, instead of ∣Δ~∣|{\tilde Δ}|∣Δ~∣ in the BCS region, where μ>0μ>0μ>0. This shows up in the characteristic energy of atoms from dissociated molecules.
As it was proposed and recently verified experimentally, the mechanism of Feshbach resonance in a condensate can create a second condensate component of molecules that coexists with the atomic condensate. In this work we investigate signatures of the presence of the molecular condensate through the equilibrium properties and collective excitations of the hybrid system of atoms and molecules, subjected to a trap, employing a time-dependent variational ansatz. We show that the shape of the condensate changes significantly by the presence of the molecules and that modes unique to this hybrid system appear at observable frequencies.