Exact solutions of the Caldeira-Leggett master equation: a factorization theorem for decoherence

Abstract

Exact solutions of the Caldeira-Leggett Master equation for the reduced density matrix <x'|ρ(t)|x> for a free particle and for a harmonic oscillator system coupled to a heat bath of oscillators are obtained for arbitrary initial conditions. The solutions prove that the Fourier transform of ρ(t) with respect to (x+x')/2 factorizes exactly into a part depending linearly on ρ(0) and a part independent of it. The theorem yields the exact initial state dependence of ρ(t) and its eventual diagonalization in the energy basis.