### A.1 Input/Output relations derivation for a Fabry–Pérot cavity

Here we consider the derivation of the I/O-relations for a Fabry–Pérot cavity given by Eqs. (274) in
Section 5.2. We start by writing down the input fields in terms of reduced complex amplitudes for
the classical part of the field and annihilation operators for quantum corrections, respectively, as
prescribed by Eqs. (48) and (51):
with c.c. standing for “complex conjugate” and h.c. for “Hermitian conjugate”. It would be convenient for
us to make the preliminary calculations in terms of complex amplitudes before going for two-photon
quadrature amplitudes.
Start with the equations that connect the classical field amplitudes of the fields shown in Figure 29. At
each mirror, the corresponding fields are related according to Eqs. (27) with mirror transfer matrix :

and two equations for the running waves inside the cavity:
describes the free propagation of light between the mirrors. The solution to these equations is the following:
The equations set for the quantum fields has the same structure, but with more sophisticated boundary
conditions, which include mirrors’ motion as described in Section 2.2.5:

where
and
The solution to these equations reads: