Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 60 / 5 / e / Solution

Take unilateral Laplace transforms of the cavity equation, keeping the initial operator:The input-output relation givesA transfer function describes the forced zero-state part of the response. Thus the passive cavity input-output transfer function isThe omitted initial-state term is a decaying transient $\sqrt{\gamma }{e}^{-\gamma t/2}a(0)$, and is present for a general initial cavity state. This distinction avoids silently setting a physical annihilation operator equal to zero.