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

摘要

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) approximate to 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) approximate to 1 at T = 0. The results for N -> infinity are seen to converge to those of the grand canonical ensemble obtained by assuming the average <(psi) over cap (r)> of the field operator (psi) over cap (r) below T-0. This fact justifies the introduction of the "anomalous" average <(psi) over cap (r)> not equal 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.