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Physical Review A

Bogoliubov theory for mutually coherent condensates

Paolo Tommasini, E. J. V. de Passos, A. F. R. de Toledo Piza, Mahir Saleh Hussein, Eddy Timmermans

Published Feb 14, 2003• 91 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.67.023606

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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.
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.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
The one-body density matrix of weakly interacting condensed bosons in external potentials is calculated using inhomogeneous Bogoliubov theory. We determine the condensate deformation caused by weak external potentials at the mean-field level. The momentum distribution of quantum fluctuations around the deformed ground state is obtained analytically and finally, the resulting quantum depletion is calculated. The depletion due to the external potential, or potential depletion for short, is a small correction to the homogeneous depletion, validating our inhomogeneous Bogoliubov theory. Analytical results are derived for weak lattices and spatially correlated random potentials, with simple, universal results in the Thomas–Fermi limit of very smooth potentials.
A general theory is formulated for the fluctuations in the condensate of a degenerate boson fluid in homogeneous thermodynamic equilibrium. The point of view adopted is based on Bogoliubov's description of the condensate in terms of quasiaverages and on his "Principle of Diminishing Correlation." An essential aspect of the theory is the consistent treatment of the fluctuation operators ccc and ${c}^{\ifmmode\dagger\else\textdagger\fi{}}$, where $c={a}_{0}\ensuremath{-}\ensuremath{\alpha}$. a0{a}_{0}a0​ is the annihilation operator for the zero momentum state, and the order parameter $\ensuremath{\alpha}$ is the quasiaverage of a0{a}_{0}a0​. After the Hamiltonian for the fluctuation operators is obtained, a stability condition is derived, which we then demonstrate is satisfied under very general conditions. The energy spectrum of the condensate fluctuations is shown to be discrete, and explicit expressions are derived for the quantum states of the condensate. The ground state of the condensate is not a coherent state of the type studied by Glauber. A detailed analysis is given of the fluctuations in the order parameter and in the number of particles in the condensate. The physical properties of the fluctuations are determined by two parameters, which are the averages of the second derivatives of the Hamiltonian with respect to the order parameter. These quantities are shown to be related to the one-particle Green's functions introduced by Beliaev. The implications of this general theory are also worked out for Bogoliubov's low-density, weak-coupling model. The limit of the ideal Bose gas is then shown to possess pathological features, namely an unnormalizable ground state and infinite fluctuations, in contrast to the interacting fluid where all properties are well-behaved.
1. Define Bose-Einstein condensation in an interacting boson system in term of macroscopic occupation. 2. Introduce the Gross-Pitaevskii equation and its applicable conditions. 3. Derive the hydrodynamic equations and various physical consequences of the hydrodynamics with or without harmonic trap. 4. Introduce sound wave and emphasize the difference between the wave behavior and the diffusive behavior. 5. Introduce the criterion for superfluidity and determine the superfluid critical velocity. 6. Introduce two-fluid hydrodynamics. 7. Introduce the Bogoliubov theory. 8. Introduce various concepts related to quasi-particle, including ground state as vacuum of quasi-particles, the zero-point energy of quasi-particle, quasi-particle lifetime and vacuum fluctuation. 9. Introduce healing length as the characteristic length scale for low-energy physics of an interacting Bose condensate. 10. Summarize the difference between an interacting and non-interacting Bose condensate. 11. Introduce the Bragg spectroscopy. 12. Show that the Bogoliubov theory fails in one-dimension due to the infrared divergence. 13. Discuss the basic idea of the Bethe-ansatz solution and the Tonks-Girardeau limit. 14. Introduce scale invariant quantum gases. 15. Introduce the Josephson effect and the self-trapping as two different transport phenomena driven by phase coherence. 16. Discuss the conjugate relation between the relative phase fluctuation and the relative particle number fluctuation, and show strong enough phase fluctuation renders the system into a Fock state. 17. Discuss an example of quantum measurement that can project two independent condensates to a coherent superposition state. 18. Discuss interference effects in density and density-density correlations. 19. Discuss the relation between the instability of the Sch\"odinger Cat state and the stability of symmetry breaking.