Quantum Field Theoretical Description of Unstable Behavior of Trapped Bose-Einstein Condensates with Complex Eigenvalues of Bogoliubov-de Gennes Equations

TL;DR

This paper develops a quantum field theoretical framework to analyze the unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues in the Bogoliubov-de Gennes equations, revealing new insights into condensate metastability and density dynamics.

Contribution

It introduces a complete set of modes including complex eigenvalues, formulates physical state conditions, and studies the instability via linear response theory, extending the understanding of BEC dynamics.

Findings

Complex eigenmodes lead to blow-up and damping in density distributions.

The quantum state space is indefinite metric, and the Hamiltonian is not diagonalizable.

Results are consistent with time-dependent Gross-Pitaevskii analyses.

Abstract

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo's linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.