This scheme works similar to the ordinary Michelson interferometer considered briefly in Section 2.1.2. The beamsplitter BS distributes the pump power from the laser evenly between the arms. The beams, reflected off the Fabry–Pérot cavities are recombined on the beamsplitter in such a way that, in the ideal case of perfect symmetry of the arms, all the light goes back to the laser, i.e., keeping the signal (‘south’) port dark. Any imbalance of the interferometer arms, caused by signal forces acting on the end test masses (ETMs) makes part of the pumping light leak into the dark port where it is monitored by a photodetector.
The Fabry–Pérot cavities in the arms, formed by the input test masses (ITMs) and the end test masses, provide the increase of the optomechanical coupling, thus making photons bounce many times in the cavity and therefore carry away a proportionally-amplified mirror displacement signal in their phase (cf. with the factor in the toy systems considered in Section 4). The two auxiliary recycling mirrors: the PRM and the signal recycling (SRM) allow one to increase the power, circulating inside the Fabry–Pérot cavities, for a given laser power, and provide the means for fine-tuning of the quantum noise spectral density [103, 155], respectively.
It was shown in  that quantum noise of this dual (power and signal) recycled interferometer is equivalent to that of a single Fabry–Pérot cavity with some effective parameters (the analysis in that paper was based on earlier works [112, 128], where the classical regime had been considered). Here we reproduce this scaling law theorem, extending it in two aspects: (i) we factor in optical losses in the arm cavities by virtue of modeling it by the finite transmissivity of the ETMs, and (ii) we do not assume the arm cavities tuned in resonance (the detuned arm cavities could be used, in particular, to create optical rigidity in non-signal-recycled configurations).
We start with Eqs. ((279)) and ((280)) for the arm cavities. The notation for the field amplitudes is shown in Figure 30. The fields referring to the interferometer arms are marked with the subscripts (‘northern’) and (‘eastern’) following the convention of labeling the GW interferometer parts in accordance with the cardinal directions they are located at with respect to the drawing (up-direction coincides with north). In order to avoid subscripts, we rename some of the field amplitudes as follows:
Rewrite those Eqs. (279), (280), (281) that are relevant to our consideration, in these notations:
Assume then that the beamsplitter is described by the matrix (32), with . Let be the power recycling cavity length (the optical distance between the power recycling mirror and the input test masses) and – power recycling cavity length (the optical distance between the signal recycling mirror and the input test masses). In this case, the light propagation between the recycling mirrors and the input test masses is described by the following equations for the classical field amplitudes:
The last group of equations that completes our equations set is for the coupling of the light fields at the recycling mirrors:
The striking symmetry of the above equations suggests that the convenient way to describe this system is to decompose all the optical fields in the interferometer arms into the superposition of the symmetric (common) and antisymmetric (differential) modes, which we shall denote by the subscripts and , respectively:
It is easy to see that the classical field amplitudes of the antisymmetric mode are equal to zero. For the common mode, combining Eqs. (320), (327), (330), (331), it is easy to obtain the following set of equation:
In the differential mode, the first non-vanishing terms are the first-order quantum-field amplitudes. In this case, using Eqs. (321), (329), (330), (331), and taking into account that
Eqs. (332) and (333), on the one hand, and (335) and (339), on the other, describe two almost independent optical configurations each consisting of the two coupled Fabry–Pérot cavities as featured in Figure 31. ‘Almost independent’ means that they do not couple in an ordinary linear way (and, thus, indeed represent two optical modes). However, any variation of the differential mechanical coordinate makes part of the pumping carrier energy stored in the common mode pour into the differential mode, which means a non-linear parametric coupling between these modes.
The mechanical elongation modes of the two Fabry–Pérot cavities are described by the following equations of motion [see Eq. (305)]:
Equations (339) and (341) together form a complete set of equations describing the differential optomechanical mode of the interferometer featured in Figure 31(b). Eq. (341) implies that the effective mass of the differential mechanical degree of freedom coincides with the single mirror mass:
Return to Eqs. (333) for the common mode. Introduce the following notations:
Taking into account that the main goal of power recycling is the increase of the power circulating in the arm cavities, for a given laser power , the optimal tuning of the power recycling cavity corresponds to the critical coupling of the common mode with the laser:
Consider now the differential mode quantum field amplitudes as given in Eqs. (339). Note a factor that describes a frequency-dependent phase shift the sideband fields acquire on their pass through the signal recycling cavity. It is due to this frequency-dependent phase shift that the differential mode cannot be reduced, strictly speaking, to a single effective cavity mode, and a more complicated two-cavity model of Figure 31 should be used instead. The reduction to a single mode is nevertheless possible in the special case of a short signal-recycling cavity, i.e., such that:[156, 66]. In this case, the phase shift can be approximated by the frequency-independent value:
The mechanical equations of motion for the effective cavity are absolutely the same as for an ordinary Fabry–Pérot cavity considered in Section 5.2 except for the new values of the effective mirrors’ mass and effective circulating power . Bearing this in mind, we can procede to the quantum noise spectral density calculation for this interferometer.
The scaling law we have derived above enables us to calculate spectral densities of quantum noise for a dual-recycled Fabry–Pérot–Michelson featured in Figure 30 as if it were a bare Fabry–Pérot cavity with movable mirrors pumped from one side, similar to that shown in Figure 32.
We remove some of the subscripts in our notations, for the sake of notational brevity:
With this in mind, we rewrite I/O-relations (360) and (361) in the two-photon quadratures notation:
Suppose that the output beam is registered by the homodyne detector; see Section 2.3.1. Combining Eqs. (366) and (272), we obtain for the homodyne detector a readout expressed in units of signal force:
The dynamics of the interferometer is described by the effective susceptibility that is the same as the one given by Eq. (318) where
Suppose then that the input field of the interferometer is in the squeezed quantum state that is equivalent to the following transformation of the input fields:
Using the rules of spectral densities computation given in Eqs. (89) and (92), taking into account unitarity conditions (300), one can get the following expressions for the power (double-sided) spectral densities of the dual-recycled Fabry–Pérot–Michelson interferometer measurement and back-action noise sources as well as their cross-correlation spectral density:rodinger–Robertson uncertainty relation:
In order to compute the sum quantum noise spectral density one has to first calculate , and using Eqs. (376), (377), and (378) and then insert them into the general formula (144).
However, there is another option that is more convenient from the computational point of view. One can compute the full quantum noise transfer matrix of the Fabry–Pérot–Michelson interferometer in the same manner as for a single mirror in Section 5.1.4. The procedure is rather straightforward. Write down the readout observable of the homodyne detector in units of signal force:
The power (double-sided) spectral density of the sum quantum noise then reads:
In conclusion, we should say that the quantum noise of the Fabry–Pérot–Michelson interferometer has been calculated in many papers, starting from the seminal work by Kimble et al.  where a resonance-tuned case with was analyzed, and then by Buonanno and Chen in [32, 34], who considered a more general detuned case. Thus, treading their steps, we have shown that the quantum noise of the Fabry–Pérot–Michelson interferometer (as well as the single cavity Fabry–Pérot one) has the following distinctive features:
All of these features can be used to decrease the quantum noise of the interferometer and reach a sensitivity below the SQL in a decent range of frequencies as we show in Section 6.
Living Rev. Relativity 15, (2012), 5
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