Equilibrium Fluctuations and Stability of the Condensate of a Degenerate Boson Fluid

Abstract

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 $c$ and $c^{†}$, where $c=a_0-\alpha$. $a_0$ is the annihilation operator for the zero momentum state, and the order parameter $\alpha$ is the quasiaverage of $a_0$. 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.