In GW detectors, one deals normally with a close to monochromatic laser light with carrier frequency , and a pair of modulation sidebands created by a GW signal around its frequency in the course of parametric modulation of the interferometer arm lengths. The light field coming out of the interferometer cannot be considered as the continuum of independent modes anymore. The very fact that sidebands appear in pairs implies the twophoton nature of the processes taking place in the GW interferometers, which means the modes of light at frequencies have correlated complex amplitudes and thus the new twophoton operators and related formalism is necessary to describe quantum light field transformations in GW interferometers. This was realized in the early 1980s by Caves and Schumaker who developed the twophoton formalism [39, 40], which is widely used in GW detectors as well as in quantum optics and optomechanics.
One starts by defining modulation sideband amplitudes as

and use the fact that enables us to expand the limits of integrals to . The operator expressions in front of in the foregoing Eqs. (51) are quantum analogues to the complex amplitude and its complex conjugate defined in Eqs. (14):

Again using Eqs. (14), we can define twophoton quadrature amplitudes as:
Note that sointroduced operators of twophoton quadrature amplitudes are Hermitian and thus their frequency domain counterparts satisfy the relations for the spectra of Hermitian operator:

Now we are able to write down commutation relations for the quadrature operators, which can be derived from Eq. (50):
The commutation relations represented by Eqs. (53) indicate that quadrature amplitudes do not commute at different times, i.e., , which imply they could not be considered for proper output observables of the detector, for a nonzero commutator, as we would see later, means an additional quantum noise inevitably contributes to the final measurement result. The detailed explanation of why it is so can be found in many works devoted to continuous linear quantum measurement theory, in particular, in Chapter 6 of [22], Appendix 2.7 of [43] or in [19]. Where GW detection is concerned, all the authors are agreed on the point that the values of GW frequencies , being much smaller than optical frequencies , allow one to neglect such weak commutators as those of Eqs. (53) in all calculations related to GW detectors output quantum noise. This statement has gotten an additional ground in the calculation conducted in Appendix 2.7 of [43] where the value of the additional quantum noise arising due to the nonzero value of commutators (53) has been derived and its extreme minuteness compared to other quantum noise sources has been proven. Braginsky et al. argued in [19] that the twophoton quadrature amplitudes defined by Eqs. (52) are not the real measured observables at the output of the interferometer, since the photodetectors actually measure not the energy flux
but rather the photon number flux: The former does not commute with itself: , while the latter apparently does and therefore is the right observable for a selfconsistent quantum description of the GW interferometer output signal.In the course of our review, we shall adhere to the approximate quadrature amplitude operators that can be obtained from the exact ones given by Eqs. (52) by setting , i.e.,
The new approximate twophoton quadrature operators satisfy the following commutation relations in the frequency domain:
and in the time domain:Then the electric field strength operator (48) can be rewritten in terms of the twophoton quadrature operators as:
Hereafter, we will omit the spatial mode factor since it does not influence the final result for quantum noise spectral densities. Moreover, in order to comply with the already introduced division of the optical field into classical carrier field and to the 1st order corrections to it comprising of laser noise and signal induced sidebands, we adopt the same division for the quantum fields, i.e., we detach the mean values of the corresponding quadrature operators via the following redefinition with . Here, by we denote an ensemble average over the quantum state of the light wave: . Thus, the electric field strength operator for a plain electromagnetic wave will have the following form: Further, we combine the twophoton quadratures into vectors in the same manner as we used to do for classical fields:Now, when we have defined a quantum Heisenberg operator of the electric field of a light wave, and introduced quantum operators of twophoton quadratures, the last obstacle on our way towards the description of quantum noise in GW interferometers is that we do not know the quantum state the light field finds itself in. Since it is the quantum state that defines the magnitude and mutual correlations of the amplitude and phase fluctuations of the outgoing light, and through it the total level of quantum noise setting the limit on the future GW detectors’ sensitivity. In what follows, we shall consider vacuum and coherent states of the light, and also squeezed states, for they comprise the vast majority of possible states one could encounter in GW interferometers.
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Living Rev. Relativity 15, (2012), 5
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