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Progress of Theoretical Physics

Bose-Einstein Condensation and the Canonical Conjugates to the Density Fluctuations

Shuichiro Yamasaki, Masahiko HIROOKA

Published Oct 1, 1980• 2 citations
Publisher

Abstract

Concrete expressions are given to the operators φk forming a set of canonically conjugate variables to the density fluctuations ρl in a Bose-Einstein condensed many-boson system. The Fourier transform of φk to the coordinate space admits in classical terms an interpretation as the phase of the boson field ψ(x). Various objections against the existence of canonically conjugate variables to the density are critically examined. They do not apply to the proposed operator φk which is non local and canonically conjugate only to the density fluctuation.

DOI
10.1143/ptp.64.1248

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Concrete expressions are given to the operators φk forming a set of canonically conjugate variables to the density fluctuations ρl in a Bose-Einstein condensed many-boson system. The Fourier transform of φk to the coordinate space admits in classical terms an interpretation as the phase of the boson field ψ(x). Various objections against the existence of canonically conjugate variables to the density are critically examined. They do not apply to the proposed operator φk which is non local and canonically conjugate only to the density fluctuation.
The existence of a Bose-Einstein condensation in an interacting many-boson system at T = 0°K is proved under certain conditions on the particle density and the interparticle potential. Starting with the tentative assumption that the condensation exists, we study the fluctuation in the occupation number of the condensate with due regard to its interactions (1) with particles outside the condensate as well as (2) with the fluctuation itself. If the condensate fluctuation has a normalizable ground state, then the assumed existence of the condensation is tenable. For the case of the pair-Hamiltonian model satisfying the conditions for condensation, the interactions of the second category are of no importance. In the limit of infinite volume, this Hamiltonian can be diagonalized in an irreducible representation of a Bose-field operatorφ(x), whereφ(x) has nonvanishing ground state expectation value, in accordance with the usual c-number replacement of creation and destruction operators for the condensate particles. The full Hamiltonian for a system of pairwise interacting bosons is studied only in a low-density limit. Bose-Einstein condensation exists when the over-all space integral of the interparticle potential is positive. In this case the interactions of the second category play an important role in ensuring a normalizable ground state for the condensate fluctuation. There is an indication that in the limit of infinite volume the Hamiltonian cannot be diagonalized in any irreducible representation of the field operatorφ. Yet the c-number replacement of the condensate operators is legitimate as far as states of particles outside the condensate are concerned. Some speculations are made as to what may happen for systems of moderate density.
The velocity operator vk studied in a previous paper admits further reduction in a Bose-Einstein condensed system. The reduced expression manifests the longitudinal nature of the operator vk(k≠0), and is proportional to the operator φk canonically conjugate to the density fluctuation, ρk. The value of v0 is the momentum to mass ratio of a condensed particle. As a dynamical variable describing the Hamiltonian, v0 plays peculiar role reminiscent of the super-velocity introduced in phenomenological theories. The collective formalism of condensed boson systems is found to provide the microscopic foundation of the quantum hydrodynamics incorporating the super-velocity.
Bose-Einstein condensation in the grand canonical ensemble admits a formulation in terms of a phase-density decomposition of the condensate mode operator ψ^0\hatψ_{\bf 0}ψ^​0​. In the presence of macroscopic condensate number fluctuations this representation presents nontrivial implications. In particular, we show that, for the ideal gas, under the assumption of a well-defined phase and a fluctuating condensate density, the square modulus of the anomalous average ⟨ψ^0⟩\langle \hatψ_{\bf 0}\rangle⟨ψ^​0​⟩ can provide only a fraction of the whole condensate density ρ0ρ_{\bf 0}ρ0​ and for the grand canonical statistics of the ideal Bose gas one obtains the value ∣⟨ψ^0⟩∣2=(π/4)ρ0|\langle {\hat ψ}_{\bf 0}\rangle|^2 =(π/4) ρ_{\bf 0}∣⟨ψ^​0​⟩∣2=(π/4)ρ0​. The remaining part is supplemented by the (macroscopic) fluctuations of ψ^0\hatψ_{\bf 0}ψ^​0​, which become a distinctive feature of the BEC in this setting. This provides a transparent physical picture of a condensate of photons with a well-defined phase but large number fluctuations, as observed in dye-filled microcavity photon experiments. We also propose a way to access the square modulus of the anomalous average to test theoretical predictions.
For the phenomenon of Bose-Einstein condensation we construct the canonical pair of field operators of the Goldstone Bosons explicitly as fluctuation operators in the ground state. We consider the imperfect Bose gas as well as the weakly interacting Bose gas. We prove that a canonical pair of fluctuation operators is always related to the order parameter and the generator of the broken symmetry fluctuations. We find that although the first one has an anomalous behaviour, the second one is squeezed by the same inverse rate. Furthermore, we prove that this canonical pair separates from the other variables of the system and that it behaves dynamically as oscillator variables. Finally the long wavelength behaviour of the spectrum determines the lifetime of this pair. 1 Aspirant van het Fonds voor Wetenschappelijk Onderzoek- Vlaanderen