### 5.1 Movable mirror

The scheme of the mirror is drawn in Figure 28. It is illuminated from both sides by the two independent laser sources with frequency , and mean power values and . In terms of the general linear measurement theory of Section 4.2 we have two meters represented by these two incident light waves. The two arbitrary quadratures of the reflected waves are deemed as measured quantities and . Measurement can be performed, e.g., by means of two independent homodyne detectors. Let us analyze quantum noise in such a model keeping to the scheme given by Eqs. (133).

#### 5.1.1 Optical transfer matrix of the movable mirror

The first one of Eqs. (133) in our simple scheme is represented by the input-output coupling equations (30, 36) of light on a movable mirror derived in Section 2.2.4. We choose the transfer matrix of the mirror to be real:

according to Eq. (29). Then we can write down the coupling equations, substituting electric field strength amplitudes and by their dimensionless counterparts as introduced by Eq. (61) of Section 3.1:
where
Without any loss of generality one can choose the phase of the light incident from the left to be such that and . Then factoring in the constant phase difference between the left and the right beams equal to , one would obtain for the left light .

The two output measured quantities will then be given by the two homodyne photocurrents:

where vector was first introduced in Section 2.3.2 after Eq. (47) as:
Then, after substitution of (243) into (244), and then into (246), one gets
and

#### 5.1.2 Probe’s dynamics: radiation pressure force and ponderomotive rigidity

Now we can write down the equation of motion for the mirror assuming it is pulled by a GW tidal force :

that gives us the probe’s dynamics equation (the third one) in (133):
where the free mirror mechanical susceptibility . The term stands for the radiation pressure force imposed by the light that can be calculated as
where
is the regular part of the radiation pressure force. It is constant and thus can be compensated by applying a fixed restoring force of the same magnitude but with opposite direction, which can be done either by employing an actuator, or by turning the mirror into a low-frequency pendulum with by suspending it on thin fibers, as is the case for the GW interferometers, that provides a necessary gravity restoring force in a natural way. However, it does not change the quantum noise and thus can be omitted from further consideration. The latter term represents a quantum correction to the former one
where is the random part of the radiation pressure that depends on the input light quantum fluctuations described by quantum quadrature amplitudes vectors and with coefficients given by vectors:

and the term represents the dynamical back action with

being a ponderomotive rigidity that arises in the potential created by the two counter propagating light waves. Eq. (257) gives us the second of Eqs. (133). Here .

#### 5.1.3 Spectral densities

We can reduce both our readout quantities to the units of the signal force according to Eq. (140):

and define the two effective measurement noise sources as
and an effective force noise as
absorbing optical rigidity into the effective mechanical susceptibility:
One can then easily calculate their power (double-sided) spectral densities according to Eq. (144):
where, if both lights are in coherent states that implies
, one can get:
Comparison of these expressions with the Eqs. (195) shows that we have obtained results similar to that of the toy example in Section 4.4. If we switched one of the pumping carriers off, say the right one, the resulting spectral densities for would be exactly the same as in the simple case of Eqs. (195), except for the substitution of and .

#### 5.1.4 Full transfer matrix approach to the calculation of quantum noise spectral densities

It was easy to calculate the above spectral densities by parts, distinguishing the effective measurement and back-action noise sources and making separate calculations for them. Had we considered a bit more complicated situation with the incident fields in the squeezed states with arbitrary squeezing angles, the calculation of all six of the above individual spectral densities (264) and subsequent substitution to the sum spectral densities expressions (263) would be more difficult. Thus, it would be beneficial to have a tool to do all these operations at once numerically.

It is achievable if we build a full transfer matrix of our system. To do so, let us first consider the readout observables separately. We start with and rewrite it as follows:

where we omitted the frequency dependence of the constituents for the sake of brevity and introduced a full transfer matrix for the first readout observable defined as
with outer product of two arbitrary vectors and written in short notation as:
In a similar manner, the full transfer matrix for the second readout can be defined as:
Having accomplished this, one is prepared to calculate all the spectral densities (263) at once using the following matrix formulas:
where and is the -matrix of spectral densities for the two input fields:
with defined by Eq. (83).

Thus, we obtain the formula that can be (and, actually, is) used for the calculation of quantum noise spectral densities of any, however complicated, interferometer given the full transfer matrix of this interferometer.

#### 5.1.5 Losses in a readout train

Thus far we have assumed that there is no dissipation in the transition from the outgoing light to the readout photocurrent of the homodyne detector. This is, unfortunately, not the case since any real photodetector has the finite quantum efficiency that indicates how many photons absorbed by the detector give birth to photoelectrons, i.e., it is the measure of the probability for the photon to be transformed into the photoelectron. As with any other dissipation, this loss of photons gives rise to an additional noise according to the FDT that we should factor in. We have shown in Section 2.2.4 that this kind of loss can be taken into account by means of a virtual asymmetric beamsplitter with transmission coefficients and for the signal light and for the additional noise, respectively. This beamsplitter is set into the readout optical train as shown in Figure 8 and the -th readout quantity needs to be modified in the following way:

where
is the additional noise that is assumed to be in a vacuum state. Since the overall factor in front of the readout quantity does not matter for the final noise spectral density, one can redefine in the following way:

The influence of this loss on the final sum spectral densities (269) is straightforward to calculate if one assumes the additional noise sources in different readout trains to be uncorrelated. If it is so, then Eq. (269) modifies as follows:

Now, when we have considered all the stages of the quantum noise spectral densities calculation on a simple example of a single movable mirror, we are ready to consider more complicated systems. Our next target is a Fabry–Pérot cavity.