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 two-photon 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 two-photon 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 two-photon 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
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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
):
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Again using Eqs. (14), we can define two-photon quadrature amplitudes as:
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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
two-photon 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
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 two-photon 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 two-photon quadrature
operators as:
Now, when we have defined a quantum Heisenberg operator of the electric field of a light wave, and introduced quantum operators of two-photon 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.
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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