List of Footnotes

1 Here, we adopt the system of labeling parts of the interferometer by the cardinal directions, they are located with respect to the interferometer central station, e.g., Mn and Me in Figure 2View Image stand for ‘northern’ and ‘eastern’ end mirrors, respectively.
2 Here and below we keep to a definition of the reflectivity coefficient of the mirrors that implies that the reflected wave acquires a phase shift equal to π with respect to the incident wave if the latter impinged the reflective surface from the less optically dense medium (air or vacuum). In the opposite case, when the incident wave encounters reflective surface from inside the mirror, i.e., goes from the optically more dense medium (glass), it is assumed to acquire no phase shift upon reflection.
3 In fact, the argument of x should be written as t∗, that is the moment when the actual reflection takes place and is the solution to the equation: c(t− t∗) = x (t∗), but since the mechanical motion is much slower than that of light one has δx∕c ≪ 1. This fact implies t ≃ t∗.
4 In the resonance-tuned case, the phase modulation of the input carrier field creates equal magnitude sideband fields as discussed in Section 2.2.2, and these sideband fields are transmitted to the output port thanks to Schnupp asymmetry in the same state, i.e., they remain equal in magnitude and reside in the phase quadrature. In detuned configurations of GW interferometers, the upper and lower RF-sideband fields are transformed differently, which influences both their amplitudes and phases at the readout port.
5 Insofar as the light beams in the interferometer can be well approximated as paraxial beams, and the polarization of the light wave does not matter in most of the considered interferometers, we will omit the vector nature of the electric field and treat it as a scalar field with strength defined by a scalar operator-valued function ˆE(x,y,z,t).
6 Herein, we make use of a double-sided power spectral density defined on a whole range of frequencies, both negative and positive, that yields the following connection to the variance of an arbitrary observable ˆo(t):
∫ ∞ Var[oˆ(t)] ≡ ⟨ˆo2(t)⟩ − ⟨ˆo(t)⟩2 = dΩSo(Ω ), − ∞ 2π
It is worth noting that in the GW community, the sensitivity of GW detectors as well as the individual noise sousces are usually characterized by a single-sided power spectral density S+ (Ω ) o, that is simply defined on positive frequencies Ω ≥ 0. The connection between these two is straightforward: S+(Ω) = 2So (Ω ) o for Ω ≥ 0 and 0 otherwise.
7 Hereafter we will omit, for the sake of brevity, the factor 2πδ(Ω− Ω ′) in equations that define the power (double-sided) spectral densities of relevant quantum observables, as well as assume Ω = Ω′, though keeping in mind that a mathematically rigorous definition should be written in the form of Eq. (87View Equation).
8 Here, we omitted the terms of δ𝒲ˆ proportional to ˆa2c,s(t) since their contribution to the integral is of the second order of smallness in ˆac,s∕Ac compared to the one for the first order term.
9 Personal communication with Yanbei Chen.
10 Note the second term proportional to √ ℐ1ℐ2-cos Φ0, which owes its existence to the interference of the two traveling waves running in opposite directions. An interesting consequence of this is that the radiation pressure does not vanish even if the two waves have equal powers, i.e., ℐ1 = ℐ2, that is, in order to compensate for the radiation pressure force of one field on the semi-transparent mirror, the other one should not only have the right intensity but also the right phase with respect to the former one:
∘ -- cosΦ0 = 1 R-ℐ1√-− ℐ2-. 2 T ℐ1ℐ2