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:
The two output measured quantities will then be given by the two homodyne photocurrents:
Now we can write down the equation of motion for the mirror assuming it is pulled by a GW tidal force :10. 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
and the term represents the dynamical back action with
We can reduce both our readout quantities to the units of the signal force according to Eq. (140):
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:
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.
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:
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.
Living Rev. Relativity 15, (2012), 5
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