The interplay between Matsubara terms and the contribution from the bath modes to the low-temperature zero-phonon line shapes for weakly damped modes

A new form of spectral density is presented for a system of weakly damped Brownian oscillators. This spectral density is shown to lead to an energy gap correlation function which neglects the Matsubara terms. It is found that, in the low-temperature limit, the solution is approximately valid for γ/ω ≤ 0.2, where γ and ω are the damping constant and frequency of the Brownian oscillators. This finding provides a rough idea of how weakly damped should the oscillator be when applying this solution. The upper limit increases with increasing temperature. The corresponding linear absorption spectrum expression is derived. This will cause consequent maximum contribution from the bath modes to the zero-phonon line (ZPL) profile. It is further concluded that there is interplay between the predominance of Matsubara terms and the contribution from the bath modes to the ZPL region, in the sense that as one is depressed the other is elevated. Model calculations are presented.