5.3 Fabry–Pérot–Michelson interferometer

In real-life high-precision experiments with mechanical test objects, interferometer schemes that are much more sophisticated than the simple Fabry–Pérot cavity are used. In particular, the best sensitivity in mechanical displacement measurements is achieved by the laser GW detectors. The typical scheme of such a detector is shown in Figure 30View Image. It is this scheme which is planned to be used for the next generation Advanced LIGO [143, 64, 137], Advanced VIRGO [4], and LCGT [96] GW detectors, and its simplified versions are (or were) in use in the contemporary first generation detectors: Initial LIGO [3, 98], VIRGO [156Jump To The Next Citation Point], GEO 600 [168, 66Jump To The Next Citation Point], and TAMA [8, 142].
View Image

Figure 30: Power- and signal-recycled Fabry–Pérot–Michelson interferometer.

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 г 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 [34Jump To The Next Citation Point] 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).

5.3.1 Optical I/O-relations

We start with Eqs. ((279View Equation)) and ((280View Equation)) for the arm cavities. The notation for the field amplitudes is shown in Figure 30View Image. The fields referring to the interferometer arms are marked with the subscripts N (‘northern’) and E (‘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:

a → a , b → b , a → g ; (319 ) 1N,E N,E 1N,E N,E 2N,E N,E
see also Figure 30View Image. Note that the fields gN,E now describe the noise sources due to optical losses in the arm cavities.

Rewrite those Eqs. (279View Equation), (280View Equation), (281View Equation) that are relevant to our consideration, in these notations:

BN,E = ℛarm (0)AN,E, ∘ ------ EN,E = ---1--- γ1arm-AN,E, (320 ) ℓarm(0) τ
√----- ˆ ˆb (ω) = ℛ (Ω)ˆa (ω ) + 𝒯 (Ω )ˆg (ω ) + 2-γ1armXN,E-(Ω-), N,E arm N,E arm N,E ℓarm (Ω) √ ----- √ --- ˆ ˆeN,E (ω) = --γ1armˆaN,E(ω-) +--γ2ˆgN√,E-(ω-) +-XN,E-(Ω-), (321 ) ℓarm (Ω) τ
γ = Tarm-, γ = Aarm-, (322 ) 1arm 4 τ 2 4τ
Tarm is the input mirrors power transmittance, Aarm is the arm cavities power losses per bounce,
γarm = γ1arm + γ2 (323 )
is the arm cavities half-bandwidth,
ℓarm(Ω ) = γarm − i(δarm + Ω ), (324 )
δarm is the arm cavities detuning,
2 γ 2 √γ----γ- ℛarm (Ω ) = ---1arm--− 1, 𝒯arm (Ω) = -----1arm-2 (325 ) ℓarm (Ω) ℓarm (Ω )
ikpEN,E ˆxN,E(Ω ) ˆXN,E = ------√---------. (326 ) τ

Assume then that the beamsplitter is described by the matrix (32View Equation), with R = T = 1∕2. Let lW = cτW be the power recycling cavity length (the optical distance between the power recycling mirror and the input test masses) and lS = cτS – 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:

DW eiϕW ± DSei ϕS AN,E = -------√---------, 2 C = BN√±-BE--eiϕW,S, (327 ) W,S 2
ϕW,S = ωpτW,S, (328 )
are the phase shifts gained by the carrier light with frequency ωp passing through the power and signal recycling cavities, and the similar equations:
ˆdW (ω )eiωτW ± ˆdS (ω)eiωτS ˆaN,E (ω) = -----------√-------------, 2 ˆbN (ω) ± ˆbE (ω) iωτ ˆcW,S(ω) = ------√--------e W,S (329 ) 2
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:

√ ---- √ ---- √---- √ ---- ˆbW = − RW ˆaW + TW ˆcW , dˆW = TW ˆaW + RW ˆcW , ˆ √ --- √ --- ˆ √ --- √ ---- (330 ) bS = − RS ˆaS + TSˆcS, dS = TSˆaS + RW ˆcS,
where RW, TW and RS, TS 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 zeroth-order classical amplitudes and for the first-order quantum ones.

5.3.2 Common and differential optical modes

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:

ˆ ˆ ˆa = ˆaN-√±-ˆaE , ˆb = bN-±√--bE, ˆe = ˆeN-±√--ˆeE, ˆg = ˆgN√±-ˆgE-. (331 ) ± 2 ± 2 ± 2 ± 2
It follows from Eqs. (329View Equation) 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. (320View Equation), (327View Equation), (330View Equation), (331View Equation), it is easy to obtain the following set of equation:

B+ = ℛarm (0)∘A+,---- ---1--- γ1arm- E+ = ℓarm(0) τ A+, iϕ A+ = DW e W , CW = B+ei ϕW, ∘ ---- ∘ ---- BW = −∘ --RW AW +∘ --TW CW , D = T A + R C . (332 ) W W W W W
Its solution is equal to (only those amplitudes are provided that we shall need below):
ℛ (0)e2iϕW − √R---- BW = ---arm-------√-------W--AW , (333a ) 1 − ℛarm (0) RW e2iϕW 1 ∘ γ1arm-- √TW--eiϕW E+ = ------- -----------------√-----2iϕ-AW . (333b ) ℓarm(0) τ 1 − ℛarm (0) RW e W

In the differential mode, the first non-vanishing terms are the first-order quantum-field amplitudes. In this case, using Eqs. (321View Equation), (329View Equation), (330View Equation), (331View Equation), and taking into account that

E EN = EE = √-+, (334 ) 2
we obtain:
2√ γ1arm-Xˆ− (Ω ) ˆb− (ω ) = ℛ1arm (Ω )ˆa − (ω) + 𝒯arm(Ω)ˆg− (ω) +-------------, ℓarm(Ω ) √ γ1armˆa − (ω) + √ γ2ˆg− (ω ) + Xˆ− (Ω ) ˆe− (ω ) =-------------------√--------------, (335 ) ℓarm(Ω ) τ ˆa− (ω ) = ˆdS (ω)eiωτS , ˆ iωτS ˆcS(ω ) = b −∘ (ω)e ,∘ --- ˆbS(ω ) = − RS ˆaS + TSˆcS(ω ), ∘ --- ∘ --- ˆdS(ω ) = TSˆaS + RSˆcS (ω ), (336 )
ˆ ikpE+-ˆx− (Ω-) X − (Ω ) = √ τ- , (337 ) ˆx = ˆxN-−--ˆxE-. (338 ) − 2
The solution of this equation set is the following:
√ --- √ --- [ 2√ γ1arm-ˆX − (Ω )] [ℛarm (Ω )e2iωτS − RS ]ˆaS (Ω ) + TSeiωτS 𝒯arm (Ω)ˆg− (ω) + --------------- ˆb (ω ) = -------------------------------------√-----------------------ℓarm(Ω-)-----, S 1 − ℛarm (Ω ) RSe2i ωτS √ ---iωτS√ ----- √--- 2iωτS √ --- ˆ ˆe (ω ) = --TSe------γ1arm-ˆaS(Ω)√-+-[1 +--RSe---√--][--γ2ˆg− (ω-) +-X-−-(Ω-)]. (339 ) − ℓarm (Ω) τ[1 − ℛarm (Ω ) RSe2iωτS]

Eqs. (332View Equation) and (333), on the one hand, and (335View Equation) and (339View Equation), on the other, describe two almost independent optical configurations each consisting of the two coupled Fabry–Pérot cavities as featured in Figure 31View Image. ‘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 x− 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.

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Figure 31: Effective model of the dual-recycled Fabry–Pérot–Michelson interferometer, consisting of the common (a) and the differential (b) modes, coupled only through the mirrors displacements.

5.3.3 Interferometer dynamics: mechanical equations of motion, radiation pressure forces and ponderomotive rigidity

The mechanical elongation modes of the two Fabry–Pérot cavities are described by the following equations of motion [see Eq. (305View Equation)]:

− μ Ω2ˆxN,E (Ω ) = 2ℏkpET ˆeN,E (Ω) + GN,E-(Ω)-, (340 ) N,E 2
where μ = M ∕2 is the effective mass of these modes and GN,E are the external classical forces acting on the cavities end mirrors. The differential mechanical mode equation of motion (338View Equation) taking into account Eq. (334View Equation) reads:
G (Ω) − 2μ Ω2ˆx − (Ω) = Fˆr−.p.(Ω ) +--−----, (341 ) 2
r.p. T F− (Ω ) = 2ℏkpE +ˆe− (Ω) (342 )
is the differential radiation-pressure force and
G − = GN − GE (343 )
is the differential external force.

Equations (339View Equation) and (341View Equation) together form a complete set of equations describing the differential optomechanical mode of the interferometer featured in Figure 31View Image(b). Eq. (341View Equation) implies that the effective mass of the differential mechanical degree of freedom coincides with the single mirror mass:

2μ = M, (344 )
which prescribes the mirrors of the effective cavity to be twice as heavy as the real mirrors, i.e., 2M. For the same reason Eq. (334View Equation) implies for the effective optical power a value twice as high as the power of light, circulating in the arm cavities:
2 2 ℐc = ℏωpE + = 2ℐarm = 2 ℏωpE N,E. (345 )

5.3.4 Scaling law theorem

Return to Eqs. (333) for the common mode. Introduce the following notations:

√ ---- 1 − RW e2iϕW γ1armTW γ1W = γ1arm Re ----√-----2iϕW-= -----√-------------------, (346 ) 1 + √ RW-e 1 + 2 RW cos2ϕ√W--+-RW 1-−---RW-e2iϕW- ---2γ1arm--RW--sin2ϕW---- δW = δarm − γ1arm Im 1 + √RW--e2iϕW = δarm + 1 + 2√RW---cos 2ϕW + RW , (347 ) γW = γ1W + γ2, (348 ) ℓW (0) = γW − iδW . (349 )
In these notation, Eqs. (333) have the following form:
B = ℛ A e2iαW, W W√--W- E = ---γ1W√--A eiαW , (350 ) + ℓW (0) τ W
2γ1W ℛW = ℓ--(0-) − 1, (351 ) W -----eiϕW------ αW = arg1 + √RW---e2iϕW . (352 )
It is easy to see that these equations have the same form as Eqs. (279View Equation) for the single Fabry–Pérot cavity, with the evident replacement of γ1 and δ with the effective parameters γ1W and δW. The only difference is an additional phase shift αW. 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. (350View Equation), complemented by light propagation over length αW ∕kp. Note also that the phase shift αW can be absorbed into the field amplitudes simply by renaming
AW eiαW → AW , BW e−iαW → BW , (353 )
which yields:
BW = ℛW√ AW-, ----γ1W--- E+ = ℓW (0)√ τAW . (354 )
The corresponding equivalent model of the common mode is shown in Figure 32View Image(a).
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Figure 32: Common (top) and differential (bottom) modes of the dual-recycled Fabry–Pérot–Michelson interferometer, reduced to the single cavities using the scaling law model.

Taking into account that the main goal of power recycling is the increase of the power ℐc = ℏ ωp|E+ |2 circulating in the arm cavities, for a given laser power ℐ0 = ℏωp |AW |2, the optimal tuning of the power recycling cavity corresponds to the critical coupling of the common mode with the laser:

γ1W = γ2, δW = 0. (355 )
In this case,
-AW---- -ℐ0-- E+ = 2√ γ-τ-=⇒ ℐc = 2 ℐarm = 4γ τ . (356 ) 2 2
Note that this regime can be achieved even with the detuned arm cavities, δ ⁄= 0.

Consider now the differential mode quantum field amplitudes as given in Eqs. (339View Equation). Note a factor iωτS e 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 31View Image 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:

|Ω |τS ≪ 1. (357 )
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 ϕS can be approximated by the frequency-independent value:
ω τS ≈ ωpτS ≡ ϕS. (358 )
It allows one to introduce the following effective parameters:
1 − √R--e2iϕS γ T γ1 = γ1armRe ----√--S------= -----√---1arm--S--------, 1 + √RSe2i ϕS 1 + 2 RS cos 2ϕ√S-+-RS 1 − RSe2i ϕS 2γ1arm RS sin 2ϕS δ = δarm − γ1armIm ----√----2iϕS-= δarm + -----√-----------------, 1 + RSe 1 + 2 RS cos2 ϕS + RS γ = γ1 + γ2, ℓ(Ω) = γ − i(δ + Ω) (359 )
and to rewrite Eqs. (339View Equation) as follows:
[ ] 2√ γ-ˆX (ω) ˆbS(ω) = ℛ1 (Ω )ˆaS(ω)eiαS + 𝒯 (Ω )ˆg− (ω ) +----1-−----- eiαS, (360 ) ℓ(Ω ) √ --- iαS √ --- ˆ ˆe− (ω) =--γ1ˆaS-(ω)e---+---γ√2ˆg−-(ω)-+-X-− (ω), (361 ) ℓ(Ω ) τ
where the reflectivity and transmittance of the equivalent Fabry–Pérot cavity are still defined by the Eqs. (285View Equation), but with new effective parameters (359View Equation), and
iϕS αS = arg ----√e-------- (362 ) 1 + RSe2iϕS
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
ˆa eiαS → ˆa , ˆb e−iαS → ˆb . (363 ) S S S S
Eqs. (360View Equation) and (361View Equation) have exactly the same form as the corresponding equations for the Fabry–Pérot cavity (280View Equation). Thus, we have successfully built a single cavity model for the differential mode, see Figure 32View Image(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 2M and effective circulating power ℐc = 2ℐarm. Bearing this in mind, we can procede to the quantum noise spectral density calculation for this interferometer.

5.3.5 Spectral densities for the Fabry–Pérot–Michelson 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 30View Image as if it were a bare Fabry–Pérot cavity with movable mirrors pumped from one side, similar to that shown in Figure 32View Image.

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Figure 33: The differential mode of the dual-recycled Fabry–Pérot–Michelson interferometer in simplified notation (364View Equation).

We remove some of the subscripts in our notations, for the sake of notational brevity:

ˆaS → ˆa, ˆbS → ˆb, ˆe− → ˆe, E− → E, x → x, F r.p.→ F , G → G, F b.a.→ F , (364 ) − − r.p. − − b.a.
(compare Figures 32View Image and 33View Image). We also choose the phase of the classical field E amplitude inside the arm cavities to be zero:
√ -- [ ] Im E = 0 =⇒ E = 2E 1 (365 ) 0
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/O-relations (360View Equation) and (361View Equation) in the two-photon quadratures notation:

∘ -------- 2M J γ D (Ω )ˆx(Ω) ˆb(Ω ) = ℝ1 (Ω)ˆa (Ω) + 𝕋(Ω )ˆg(Ω ) + ------1----------, (366 ) ℏ ∘ --𝒟(Ω ) 1 { √ --- √ --- M J D (Ω )ˆx(Ω )} ˆe(Ω ) = √--- 𝕃(Ω )[ γ1ˆa (Ω) + γ2ˆg(Ω )] + ---- ---------- , (367 ) τ 2ℏ 𝒟 (Ω)
where the matrices 𝕃, ℝ1, 𝕋 are defined by Eqs. (294View Equation) and (296View Equation),
[ ] [ ] D (Ω ) = 𝒟 (Ω)𝕃 (Ω ) 0 = − δ (368 ) 1 γ − iΩ
4ℏk2pE2 4ωp ℐc 8ωp ℐarm J = -M--τ-- = M--cL- = -M--cL-- (369 )
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. (366View Equation) and (272View Equation), we obtain for the homodyne detector a readout expressed in units of signal force:

F,loss 𝒳ˆloss G (Ω ) Oˆ (Ω ) = -eff,FP----+ ℱˆ(Ω) + ------, (370 ) χxx (Ω) 2
∘ -------- ˆloss ˆO(0),loss(Ω-)- ---ℏ-------𝒟-(Ω-)---- T 𝒳 (Ω) = χOF (Ω ) = 2M J γ1HT [ϕLO ]D (Ω)H [ϕLO ][ℝ1 (Ω)ˆa (Ω) + 𝕋(Ω )ˆg(Ω ) + 𝜖dˆn (Ω)(]371 )
stands for the measurement noise, which is typically referred to as shot noise in optomechanical measurement with the interferometer opto-mechanical response function defined as
∘ -------- T χOF (Ω) = 2M-J-γ1H---[ϕLO-]D-(Ω) (372 ) ℏ 𝒟(Ω )
and the back-action noise caused by the radiation pressure fluctuations equal to
√ -------[ ]T ˆℱ (Ω ) ≡ ˆFb.a.(Ω ) = 2ℏM J 1 𝕃 (Ω )[√γ1-ˆa(Ω) + √ γ2ˆg(Ω )] (373 ) 0

The dynamics of the interferometer is described by the effective susceptibility eff,FP χ xx (Ω) that is the same as the one given by Eq. (318View Equation) where

M-J-δ- K (Ω ) = 𝒟(Ω ) (374 )
is the frequency-dependent optical rigidity that has absolutely the same form as that of a single Fabry–Pérot cavity given by Eq. (314View Equation).

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:

aˆ = 𝕊sqz[r,𝜃]ˆavac, (375 )
where the squeezing matrix is defined by Eq. (74View Equation), and the quadrature vector vac ˆa corresponds to the vacuum state.

Using the rules of spectral densities computation given in Eqs. (89View Equation) and (92View Equation), taking into account unitarity conditions (300View Equation), 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:

ℏ |𝒟(Ω )|2 S 𝒳𝒳(Ω ) = -----------T-----------2- 4M{ Jγ1 |H [ϕLO ]D (Ω)| } × HT (ϕLO )ℝ1 (Ω )[𝕊sqz(2r,𝜃) − 𝕀]ℝ †(Ω)H [ϕLO ] + 1 + 𝜖2 , (376 ) [ ]T 1 [ ] d 1 † 1 S ℱℱ(Ω ) = ℏM J 0 𝕃 (Ω)[γ1𝕊sqz(2r,𝜃 ) + γ2]𝕃 (Ω ) 0 , (377 ) ∘ ------ [ ] S (Ω ) = ℏ-----𝒟-(Ω)-----HT (ϕ )[ℝ (Ω )𝕊 (2r,𝜃) + γ ∕ γ 𝕋(Ω )]𝕃 †(Ω) 1 . (378 ) 𝒳ℱ 2 HT [ϕLO ]D (Ω ) LO 1 sqz 2 1 0
These spectral densities satisfy the Schrodinger–Robertson uncertainty relation:
2 S (Ω)S (Ω ) − |S (Ω )|2 ≥ ℏ-- (379 ) 𝒳𝒳 ℱ ℱ 𝒳 ℱ 4
of the same form as in the general linear measurement case considered in Section 4.2, see Eq. (148View Equation), with the exact equality in the ideal lossless case:
γ2 = 0, ηd = 1. (380 )
see Appendix A.2

5.3.6 Full transfer matrix approach to calculation of the Fabry–Pérot–Michelson interferometer quantum noise

In order to compute the sum quantum noise spectral density one has to first calculate S 𝒳𝒳 (Ω), S ℱℱ(Ω ) and S𝒳 ℱ(Ω ) using Eqs. (376View Equation), (377View Equation), and (378View Equation) and then insert them into the general formula (144View Equation).

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:

∘ -------- ˆ𝒳 loss(Ω ) G (Ω ) G (Ω) ℏ − M Ω2 OˆF (Ω ) = -eff,FP----+ ℱˆ(Ω ) +------= ------+ ----------T----------HT [ϕLO ] χxx (Ω) 2 2 2M J γ1H [ϕLO ]D (Ω) { [ Jδ ] } × ℂ1 (Ω)𝕊sqz[r,𝜃 ]ˆavac(Ω ) + ℂ2 (Ω)ˆg(Ω ) + 𝒟 (Ω ) − --2 𝜖dˆn (Ω) , (381 ) Ω
where matrices ℂ1,2(Ω ) can be computed using the fact that
[ ] [ ] 1 T 0 0 [δ + D (Ω) 0 ]𝕃(Ω ) = 1 0 ,

which yields:

[ 2 ] ℂ (Ω ) = 2γ1(γ − iΩ ) − 𝒟 (Ω) + J δ∕Ω − 2γ1δ , (382 ) 1 2 γ1δ − 2Jγ1∕Ω2 2γ1(γ − iΩ) − 𝒟 (Ω) + Jδ ∕Ω2 √-----[ ] ℂ2(Ω ) = 2 γ1 γ2 γ − iΩ 2 − δ . (383 ) δ − J∕Ω γ − iΩ
In the GW community, it is more common to normalize the signal of the interferometer in units of GW amplitude spectrum h(Ω). This can easily be done using the simple rule given in Eq. (138View Equation) and taking into account that in our case G (Ω) → G (Ω)∕2:
∘ -------- Oˆh (Ω ) = hGW (Ω) + ˆhn(Ω ) = hGW (Ω) + -2 --ℏ----------1-------HT [ϕLO ] L 2M J γ1HT [ϕLO ]D (Ω ) { [ ] } × ℂ1 (Ω)𝕊sqz[r,𝜃 ]ˆavac(Ω) + ℂ2 (Ω)ˆg(Ω ) + 𝒟 (Ω ) − Jδ- 𝜖dˆn (Ω) , (384 ) Ω2
where h (Ω ) GW is the spectrum of the GW signal and the second term ˆh (Ω) n stands for the sum quantum noise expressed in terms of metrics variation spectrum units, i.e., in − 1∕2 Hz.

The power (double-sided) spectral density of the sum quantum noise then reads:

F Sh (Ω ) = -4S--(Ω-)-= ---ℏ------------1--------- M 2L2 Ω4 M J γ1L2 |HT [ϕLO]D (Ω)|2 { [ ] || Jδ||2 } × HT [ϕLO] ℂ1(Ω )𝕊sqz[2r,𝜃]ℂ†1(Ω ) + ℂ2(Ω )ℂ†2(Ω )H [ϕLO ] + ||𝒟 (Ω ) − --|| 𝜖2d . (385 ) Ω2

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. [90Jump To The Next Citation Point] where a resonance-tuned case with δ = 0 was analyzed, and then by Buonanno and Chen in [32Jump To The Next Citation Point, 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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