The abstract scheme of such a device is drawn in Figure 21. It consists of a probe that is exposed to the action of a classical force , and the meter. The action of this force on the probe causes its displacement that is monitored by the meter (e.g., light, circulating in the interferometer). The output observable of the meter is monitored by some arbitrary classical device that makes a measurement record . The quantum nature of the probe–meter interaction is reflected by the backaction force that randomly kicks the probe on the part of the meter (e.g., radiation pressure fluctuations). At the same time, the meter itself is the source of additional quantum noise in the readout signal. Quantum mechanically, this system can be described by the following Hamiltonian:
where and are the Hamiltonians describing the free evolution of the probe and the meter, respectively, i.e., when there is no coupling between these systems, and is the interaction Hamiltonian. The evolution of this system can be found, in general, by solving Heisenberg equations for all of the system observables. However, it is convenient to rewrite it first in the interaction picture factoring out the free evolution of the probe and the meter (see Appendix 3.7 of [43] for detailed derivation): where , and and are the Heisenberg operators of the probe’s displacement and the meter backaction force, respectively, in the case of no coupling between these systems, i.e., the solution to the following system of independent Heisenberg equations:

and is the arbitrary initial moment of time that can be set to without loss of generality.
The following statement can be proven (see [94], Section VI of [22], and Theorems 3 and 4 in Appendix 3.7 of [43] for proof): For a linear system with Hamiltonian (124), for any linear observable of the probe and for any linear observable of the meter, their full Heisenberg evolutions are given by:
where and stand for the free Heisenberg evolutions in the case of no coupling, and the functions and are called (timedomain) susceptibilities and defined as: The second clauses in these equations maintain the causality principle.For time independent Hamiltonian and operator (in the Schrödinger picture), the susceptibilities are invariant to time shifts, i.e., , therefore they depend only on the difference of times: . In this case, one can rewrite Eqs. (126) in frequency domain as:
where the Fourier transforms of all of the observables are defined in accordance with Eq. (6).Let us now use these theorems to find the full set of equations of motion for the system of linear observables , and that fully characterize our linear measurement process in the scheme featured in Figure 21:
where timedomain susceptibilities are defined asThe meaning of the above equations is worth discussing. The first of Eqs. (129) describes how the readout observable of the meter, say the particular quadrature of the outgoing light field measured by the homodyne detector (cf. Eq. (39)), depends on the actual displacement of the probe, and the corresponding susceptibility is the transfer function for the meter from to . The term stands for the free evolution of the readout observable, provided that there was no coupling between the probe and the meter. In the case of the GW detector, this is just a pure quantum noise of the outgoing light that would have come out were all of the interferometer test masses fixed. It was shown explicitly in [90] and we will demonstrate below that this noise is fully equivalent to that of the input light except for the insignificant phase shift acquired by the light in the course of propagation through the interferometer.
The following important remark should be made concerning the meter’s output observable . As we have mentioned already, the output observable in the linear measurement process should be precisely measurable at any instance of time. This implies a simultaneous measurability condition [30, 41, 147, 22, 43, 33] on the observable requiring that it should commute with itself at any moment of time:
Initially, this condition was introduced as the definition of the quantum nondemolition (QND) observables by Braginsky et al. [27, 28]. In our case it means that the measurement of at some moment of time shall not disturb the measurement result at any other moments of time and therefore the sample data can be stored directly as bits of classical data in a classical storage medium, and any noise from subsequent processing of the signal can be made arbitrarily small. It means that all noise sources of quantum origin are already included in the quantum fluctuations of [43, 33]. And the fact that due to (131) this susceptibility turns out to be zero reflects the fact that should be a classical observable.The second equation in (129) describes how the backaction force exerted by the meter on the probe system evolves in time and how it depends on the probe’s displacement. The first term, , meaning is rather obvious. In GW interferometer, it is the radiation pressure force that the light exerts on the mirrors while reflecting off them. It depends only on the mean value and quantum fluctuations of the amplitude of the incident light and does not depend on the mirror motion. The second term here stands for a dynamical backaction of the meter and since, by construction, it is the part of the backaction force that depends, in a linear way, from the probe’s displacement, the meaning of the susceptibility becomes apparent: it is the generalized rigidity that the meter introduces, effectively modifying the dynamics of the probe. We will see later how this effective rigidity can be used to improve the sensitivity of the GW interferometers without introducing additional noise and thus enhancing the SNR of the GW detection process.
The third equation of (129) concerns the evolution of the probe’s displacement in time. Three distinct parts comprise this evolution. Let us start with the second and the third ones:
Here is the probe’s response on the signal force and is, actually, the part we are mostly interested in. This expression also unravels the role of susceptibility : it is just the Green’s function of the equation of motion governing the probe’s bare dynamics (also known as impulse response function) that can be shown to be a solution of the following initial value problem:

The second value, , is the displacement of the probe due to the backaction force exerted by the meter on the probe. Since it enters the probe’s response in the very same way the signal does, it is the most problematic part of the quantum noise that, as we demonstrate later, imposes the SQL [16, 22].
And finally, simply features a free evolution of the probe in accordance with its equations of motion and thus depends on the initial values of the probe’s displacement , momentum , and, possibly, on higher order time derivatives of taken at , as per the structure of the operator governing the probe’s dynamics. It is this part of the actual displacement that bears quantum uncertainties imposed by the initial quantum state of the probe. One could argue that these uncertainties might become a source of additional quantum noise obstructing the detection of GWs, augmenting the noise of the meter. This is not the case as was shown explicitly in [19], since our primary interest is in the detection of a classical force rather than the probe’s displacement. Therefore, performing over the measured data record the linear transformation corresponding to first applying the operator on the readout quantity that results in expressing in terms of the probe’s displacement:

with standing for the meter’s own quantum noise (measurement uncertainty), and then applying a probe dynamics operator that yields a force signal equivalent to the readout quantity :

The term vanishes since is the solution of a freeevolution equation of motion. Thus, we see that the result of measurement contains two noise sources, and , which comprise the sum noise masking the signal force .
Since we can remove initial quantum uncertainties associated with the state of the probe, it would be beneficial to turn to the Fourier domain and rewrite Eqs. (129) in the spectral form:
where spectral susceptibilities are defined as: with , and we omit the term in the last equation for the reasons discussed above. The solution of these equations is straightforward to get and reads:It is common to normalize the output quantity of the meter to unit signal. In GW interferometers, two such normalizations are popular. The first one tends to consider the tidal force as a signal and thus set to 1 the coefficient in front of in Eq. (135). The other one takes GW spectral amplitude as a signal and sets the corresponding coefficient in to unity. Basically, these normalizations are equivalent by virtue of Eq. (12) as:
In both cases, the renormalized output quantities can be considered as a sum of the noise and signal constituents: And it is the noise term in both cases that we are seeking to calculate to determine the sensitivity of the GW detector. Let us rewrite in force normalization: where we introduce two new linear observables and of the meter defined as: that have the following meaning:These two new observables that embody the two types of noise inherent in any linear measurement satisfy the following commutation relations:
that can be interpreted in the way that and can be seen at each instance of time as the canonical momentum and the coordinate of different effective linear measuring devices (meter+probe), thus defining an infinite set of subsequent measurements similar to the successive independent monitors of von Neumann’s model [157]. In this case, however, the monitors described by and are not, generally speaking, independent. In GW detectors, these monitors appear correlated due to the internal dynamics of the detector, i.e., the noise processes they describe are nonMarkovian.In particular, this can be seen when one calculates the power (doublesided) spectral density of the sum noise :
where spectral densities: are not necessarily constant and, thus, describe nonMarkovian random processes. It can also be shown that since and satisfy commutation relations (142), their spectral densities shall satisfy the Schrödinger–Robertson uncertainty relation: that is the generalization of a Heisenberg uncertainty relation in the case of correlated observables.The general structure of quantum noise in the linear measurement process, comprising two types of noise sources whose spectral densities are bound by the uncertainty relation (148), gives a clue to several rather important corollaries. One of the most important is the emergence of the SQL, which we consider in detail below.
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Living Rev. Relativity 15, (2012), 5
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