Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble

TL;DR

This paper derives expressions for reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, analyzing their correlation functions and showing convergence to grand canonical results in the thermodynamic limit.

Contribution

It provides a new derivation of reduced density matrices in the canonical ensemble and clarifies the relation to grand canonical ensemble results for ideal quantum gases.

Findings

Fermions exhibit antibunching with $g^{(2)}(0)=0$ due to Pauli exclusion.

Bosons show bunching with $g^{(2)}(0) o 2$, demonstrating the Hanbury Brown--Twiss effect.

Below the Bose--Einstein condensation temperature, off-diagonal long-range order develops in bosons.

Abstract

We derive an expression for the reduced density matrices of ideal Bose and Fermi gases in the canonical ensemble, which corresponds to the Bloch--De Dominicis (or Wick's) theorem in the grand canonical ensemble for normal-ordered products of operators. Using this expression, we study one- and two-body correlations of homogeneous ideal gases with $N$ particles. The pair distribution function $g^{(2)}(r)$ of fermions clearly exhibits antibunching with $g^{(2)}(0)=0$ due to the Pauli exclusion principle at all temperatures, whereas that of normal bosons shows bunching with $g^{(2)}(0)\approx 2$, corresponding to the Hanbury Brown--Twiss effect. For bosons below the Bose--Einstein condensation temperature $T_0$, an off-diagonal long-range order develops in the one-particle density matrix to reach $g^{(1)}(r)=1$ at $T=0$, and the pair correlation starts to decrease towards $g^{(2)}(r)\approx 1$ at $T=0$. The results for $N\rightarrow \infty$ are seen to converge to those of the grand canonical ensemble obtained by assuming the average $\langle\hat\psi({\bf r})\rangle$ of the field operator $\hat\psi({\bf r})$ below $T_0$. This fact justifies the introduction of the "anomalous" average $\langle\hat\psi({\bf r})\rangle\neq 0$ below $T_0$ in the grand canonical ensemble as a mathematical means of removing unphysical particle-number fluctuations to reproduce the canonical results in the thermodynamic limit.