The correlation function for a quantum oscillator in a low-temperature heat bath

The momentum autocorrelation function c(t) for a quantum oscillator coupled with harmonic forces to a heat bath of oscillators is calculated at low temperatures. It is found that c(t) contains two distinct terms: one, the zero-point contribution c0(t), is temperature independent, and the other, c1(t), does depend on temperature. We concentrate our attention on the low-temperature case. An expression for c1(t) is obtained, which is valid for arbitrary strenghts of the coupling and for arbitrary times. It is shown that c1(t) is governed by the low-frequency behaviour of F(λ) = A2(λ)ϱ(λ), whereϱ(λ) is the density of normal modes and A(λ) is the central-oscillator component of the λth normal mode; other details of the problem are irrelevant. It is found that c1(t) decays in time as an inverse-power law, with a relaxation time tq ≈ ħ/kT.