Eigenvector Expansion and Petermann Factor for Ohmically Damped Oscillators

Thermal correlation functions C(t) ∼ 〈φ(t)φ(0)〉 in ohmically damped systems such as optical resonators can be expressed as a sum over modes j. In contrast to the conservative case, each term is multiplied by the Petermann factor Cj , which can exceed unity and lead to “excess noise”. A time-independent perturbation ∼ǫ also leads to a response ∼ǫCj . Moreover, C −1 j is proportional to a diagonal bilinear map central to eigenvector expansions for damped systems. “Giant excess noise” (Cj → ∞) occurs in the limit of a critical point, where the divergent parts of J (>1) contributions to C(t) cancel and time-independent perturbations lead to non-analytic shifts ∼ǫ . A sum rule implies that (for the normal case of positive masses) the average value of |Cj | over all modes is ≥ 1, showing that “excess noise” is not exceptional.