Quantum Correlations of Ideal Bose and Fermi Gases in the Canonical Ensemble (1610.03617v1)

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.