A Hamiltonian for cavity decay

Summary form only given. The conventional way to treat cavity dissipation in quantum optics is by applying the system-reservoir approach. The small system consists of the quantized discrete modes of a fictitious perfect cavity. The reservoir is the external field described by continuous annihilation and creation operators for the modes of free-space. This conventional approach has been used since the early days of the laser. The Hamiltonian that describes this approach became known as the Gardiner-Collett Hamiltonian. Despite being useful in describing several phenomena in cavities with a high Q factor, it is well known that this approach breaks down when the Q decreases. We have derived the Gardiner-Collett Hamiltonian from a rigorous modes of the universe expansion, as an approximation for high-Q, and obtained the coupling strength between the discrete cavity annihilation and creation operators (small system) and the continuous annihilation and creation operators of free-space (reservoir). We have found that the coupling strength depends on the frequency even when the reflectivity is frequency independent. The physical consequence of this frequency dependence is that for short times compared to one cavity round-trip time, the reservoir (free-space) is no longer Markovian. Our explicit expression for this frequency dependent coupling strength unlocks this fast time non-Markovian regime that had been previously inaccessible within the Gardiner-Collett Hamiltonian. Finally, we also show how to generalize the Gardiner-Collett Hamiltonian to arbitrary quality factors. This generalized Hamiltonian can be applied to develop a full quantum theory of excess noise in lasers.