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 [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 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:
where
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:
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 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:
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:
(compare Figures 32 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:
where the squeezing matrix is defined by Eq. (74 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:
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:
where matrices
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which yields:
In the GW community, it is more common to normalize the signal of the interferometer in units of GW amplitude spectrumThe 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. [90] 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.
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Living Rev. Relativity 15, (2012), 5
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