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. (274View Equation) in Section 5.2. We start by writing down the input fields in terms of reduced complex amplitudes A1,2 for the classical part of the field and annihilation operators ˆa1,2 for quantum corrections, respectively, as prescribed by Eqs. (48View Equation) and (51View Equation):
∫ ∞ ∘ ω-- dω Aˆ1 (t − x ∕c) = A1e− iωp(t−x∕c) + c.c. + ---ˆa1(ω )e−iω(t−x∕c)---+ h.c., (535 ) ∫0 ∘ ωp- 2π − iωp(t+x∕c) ∞ ω −iω(t+x∕c)dω Aˆ2 (t + x ∕c) = A2e + c.c. + ω--ˆa2(ω )e 2π-+ h.c., (536 ) 0 p
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 29View Image. At each mirror, the corresponding fields are related according to Eqs. (27View Equation) with mirror transfer matrix 𝕄 real:

∘ ---- ∘ ---- B1,2 = − --R1,2A1,2 + --T1,2E1,2, (537 ) F = ∘ T A + ∘ R E , (538 ) 1,2 1,2 1,2 1,2 1,2
and two equations for the running waves inside the cavity:
E1,2 = F2,1eiωpτ (539 )
describes the free propagation of light between the mirrors. The solution to these equations is the following:
∘ ---- ∘ ---- √ ----- [ R2,1e2iωpτ − R1,2]A1,2 + T1T2A2,1eiωpτ B1,2 = -----------------√-------2iω-τ-------------, ∘ ----∘ ----1 − R1R2e∘ --p- R2,1 T1,2A1,2e2iωpτ + T2,1A2,1eiωpτ E1,2 = ---------------√-------2iωpτ-----------, ∘ ---- 1∘− --R1R∘2e--- T1,2A1,2 + R1,2 T2,1A2,1eiωpτ F1,2 = ------------√-------2iωpτ--------. (540 ) 1 − R1R2e

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:

∘ ----[ ∘ ---- ] ∘ ---- ˆb1,2(ω) = − R1,2 ˆa1,2(ω) − 2i kkpA1,2xˆ1,2(Ω) + T1,2ˆe1,2(ω), (541 ) ˆf (ω) = ˆs (ω) + ∘R----ˆe (ω ), 1,2 1,2 1,2 1,2 ˆe1,2(ω) = ˆf2,1(ω)eiωτ, (542 )
where
∘ ---- ∘ ---∘ ---- ˆs1,2(ω ) = T1,2ˆa1,2(ω) + 2i kkp R1,2E1,2ˆx1,2(Ω ), (543 )
and
Ω = ω − ωp, k = ω∕c, kp = ωp∕c. (544 )
The solution to these equations reads:
∘ ---- ∘ ---- ∘ ---- ∘ ----√ ------ ˆ [--R2,1e2iωτ-−---R1,2]ˆa1,2(ω)-+---T1,2eiω-τ[2i--kkp---R1R2E1,2xˆ1,2(Ω)-+-ˆs2,1(ω)] b1,2(ω ) = 1 − √R--R--e2iωτ ∘ ---- 1 2 +2iκ R1,2A1,2x1,2(Ω ) (545 ) ∘R----ˆs (ω )e2iωτ + ˆs (ω)eiωτ ˆe1,2(ω ) = ----2,1-1,2--√---------2,1-------, 1 − R1R2e2i ωτ ˆs1,2(ω) + ∘R1,2-ˆs2,1(ω )eiωτ ˆf1,2(ω ) = ---------√---------------. (546 ) 1 − R1R2e2iωτ

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