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 proportionallyamplified 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 finetuning of the quantum noise spectral density [103, 155], respectively.
It was shown in [34] 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 nonsignalrecycled 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 (updirection coincides with north). In order to avoid subscripts, we rename some of the field amplitudes as follows:
see also Figure 30. Note that the fields now describe the noise sources due to optical losses in the arm cavities.Rewrite those Eqs. (279), (280), (281) that are relevant to our consideration, in these notations:
where is the input mirrors power transmittance, is the arm cavities power losses per bounce, is the arm cavities halfbandwidth, is the arm cavities detuning, andAssume 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:
where are the phase shifts gained by the carrier light with frequency passing through the power and signal recycling cavities, and the similar equations: appling to the quantum fields’ amplitudes.The last group of equations that completes our equations set is for the coupling of the light fields at the recycling mirrors:
where , and , are the reflectivities and transmissivities of the power and signal recycling mirrors, respectively. These equations, being linear and frequency independent, are valid both for the zerothorder classical amplitudes and for the firstorder quantum ones.
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 follows from Eqs. (329) that the symmetric mode is coupled solely to the ‘western’ (bright) port, while the antisymmetric one couples exclusively to the ‘southern’ (dark) port of the interferometer.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:
Its solution is equal to (only those amplitudes are provided that we shall need below):In the differential mode, the first nonvanishing terms are the firstorder quantumfield amplitudes. In this case, using Eqs. (321), (329), (330), (331), and taking into account that
we obtain: where The solution of this equation set is the following: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 nonlinear 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)]:
where is the effective mass of these modes and are the external classical forces acting on the cavities end mirrors. The differential mechanical mode equation of motion (338) taking into account Eq. (334) reads: where is the differential radiationpressure force and is the differential external force.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:
which prescribes the mirrors of the effective cavity to be twice as heavy as the real mirrors, i.e., . For the same reason Eq. (334) implies for the effective optical power a value twice as high as the power of light, circulating in the arm cavities:
Return to Eqs. (333) for the common mode. Introduce the following notations:
In these notation, Eqs. (333) have the following form: where It is easy to see that these equations have the same form as Eqs. (279) for the single Fabry–Pérot cavity, with the evident replacement of and with the effective parameters and . The only difference is an additional phase shift . Thus, we have shown that the power recycling cavity formed by the PRM and the ITMs can be treated as a single mirror with some effective parameters defined implicitly by Eqs. (350), complemented by light propagation over length . Note also that the phase shift can be absorbed into the field amplitudes simply by renaming which yields: The corresponding equivalent model of the common mode is shown in Figure 32(a).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:
In this case, Note that this regime can be achieved even with the detuned arm cavities, .Consider now the differential mode quantum field amplitudes as given in Eqs. (339). Note a factor that describes a frequencydependent phase shift the sideband fields acquire on their pass through the signal recycling cavity. It is due to this frequencydependent phase shift that the differential mode cannot be reduced, strictly speaking, to a single effective cavity mode, and a more complicated twocavity model of Figure 31 should be used instead. The reduction to a single mode is nevertheless possible in the special case of a short signalrecycling cavity, i.e., such that:
The above condition is satisfied in a vast majority of the proposed schemes of advanced GW interferometers and in all current interferometers that make use of the recycling techniques [156, 66]. In this case, the phase shift can be approximated by the frequencyindependent value: It allows one to introduce the following effective parameters: and to rewrite Eqs. (339) as follows: where the reflectivity and transmittance of the equivalent Fabry–Pérot cavity are still defined by the Eqs. (285), but with new effective parameters (359), and is the phase shift introduced by the signal recycled cavity. Along similar lines as in the common mode case, we make the following change of variables Eqs. (360) and (361) have exactly the same form as the corresponding equations for the Fabry–Pérot cavity (280). Thus, we have successfully built a single cavity model for the differential mode, see Figure 32(b).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 dualrecycled 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:
(compare Figures 32 and 33). We also choose the phase of the classical field amplitude inside the arm cavities to be zero: that obviously does not limit the generality of our consideration, yet sets the reference point for all the classical and quantum fields’ phases.With this in mind, we rewrite I/Orelations (360) and (361) in the twophoton quadratures notation:
where the matrices , , are defined by Eqs. (294) and (296), and is the normalized optical power, circulating in the interferometer arms.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:
where stands for the measurement noise, which is typically referred to as shot noise in optomechanical measurement with the interferometer optomechanical response function defined as and the backaction noise caused by the radiation pressure fluctuations equal toThe dynamics of the interferometer is described by the effective susceptibility that is the same as the one given by Eq. (318) where
is the frequencydependent optical rigidity that has absolutely the same form as that of a single Fabry–Pérot cavity given by Eq. (314).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:
where the squeezing matrix is defined by Eq. (74), and the quadrature vector corresponds to the vacuum state.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 (doublesided) spectral densities of the dualrecycled Fabry–Pérot–Michelson interferometer measurement and backaction noise sources as well as their crosscorrelation spectral density:
These spectral densities satisfy the Schrodinger–Robertson uncertainty relation: of the same form as in the general linear measurement case considered in Section 4.2, see Eq. (148), with the exact equality in the ideal lossless case: see Appendix A.2In 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:
where matrices can be computed using the fact that

which yields:
In the GW community, it is more common to normalize the signal of the interferometer in units of GW amplitude spectrum . This can easily be done using the simple rule given in Eq. (138) and taking into account that in our case : where is the spectrum of the GW signal and the second term stands for the sum quantum noise expressed in terms of metrics variation spectrum units, i.e., in .The power (doublesided) 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. [90] where a resonancetuned 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.
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Living Rev. Relativity 15, (2012), 5
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