Generalized Wick's theorem at finite temperature for a quadratic Hamiltonian

TL;DR

This paper extends Gaudin's finite-temperature Wick's theorem to quadratic Hamiltonians and demonstrates its application by analyzing particle number fluctuations in a weakly interacting Bose gas below the condensation temperature.

Contribution

It generalizes the finite-temperature Wick's theorem for quadratic Hamiltonians and applies it to evaluate particle number fluctuations in Bose gases.

Findings

Sub-Poissonian particle number distribution at zero temperature

Validation of quadratic Hamiltonian approximation below Bose-Einstein condensation temperature

Extended theorem facilitates calculations in finite-temperature quantum many-body systems

Abstract

In Gaudin (1960), Michel Gaudin showed the Wick's theorem at finite temperature using a diagonal Hamiltonian. We extend the Gaudin's prove for a statistical density operator which depend on a quadratic Hamiltonian. To illustrate the utility of the theorem, we evaluate the ratio $\sqrt{[<\hat{N}^2>-<\hat{N}>^2]/<\hat{N}>}$ of a homogeneous weakly interacting Bose gas at temperature below the Bose-Einstein condensation temperature. At this condition, the quadratic Hamiltonian approximation is valued and, in this evaluation, we show the sub-Poissonian behaviour of the fundamental state distribution at zero temperature.