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Although spontaneous emission is required to start the laser, later it becomes a sort of parasite that only contributes to noise In an ideal optical transmitter, spontaneous emission should be negligible This is roughly what happens in a laser operating far above threshold To neglect spontaneous emission while keeping a quantum description oflight, we will replace the amplifying atoms or molecules by classical dipoles One ingredient that we cannot fail to include, though, is cavity damping Without it, the energy pumped into the laser just keeps piling up, making the intensity of the light inside the cavity grow indefinitely, quite unlike any real device A perfect cavity is the analog of a lossless LC circuit: It does not need an amplifying element to become a self-sustained oscillator We deal with cavity damping in the following phenomenological way Consider the closed system formed by the cavity and its environment.

This environment is whatever continuum set of degrees of freedom the energy that escapes the cavity dissipates into (eg, the external electromagnetic modes in the case of damping exclusively through transmissivity losses, or phonons in the cavity walls) The important thing is that this environment can be regarded as a set of harmonic oscillators For concreteness, we assume that the losses are due only to the transmissivity of the cavity walls In the zeroth-order approximation (in the transmissivity), the cavity is perfect and the Hamiltonian for the cavity and the outside is just the sum of the single-mode cavity Hamiltonian, (889) and the Hamiltonian for the external continuum of modes, (890).

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which we model as a one-dimensional continuum and for mathematical convenience extend to negative frequencies. 17 Now as the damping increases, we would expect to have corrections to the cavity and external modes themselves, as both cavity and outside would no longer satisfy the perfect reflector's boundary conditions (see Appendix B). We will assume that at least up to the first correction to this zeroth-order approximation, the modes of the cavity and outside are still those that are obtained using the boundary conditions of a perfect reflector. This correction then accounts for cavity damping through a simple interaction term Hint, where the annihilation of a photon in the cavity mode leads to the creation of a photon in one of the external modes, and vice versa (as it must be Hermitian),

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