Let us consider a very simple measurement scheme, which, nevertheless, embodies all key features of a general position measurement. In the scheme shown in Figure 20, a sequence of very short light pulses are used to monitor the displacement of a probe body . The position of is probed periodically with time interval . In order to make our model more realistic, we suppose that each pulse reflects from the test mass times, thus increasing the optomechanical coupling and thereby the information of the measured quantity contained in each reflected pulse. We also assume mass large enough to neglect the displacement inflicted by the pulses radiation pressure in the course of the measurement process.
Then each th pulse, when reflected, carries a phase shift proportional to the value of the testmass position at the moment of reflection:
where , is the light frequency, is the pulse number and is the initial (random) phase of the th pulse. We assume that the mean value of all these phases is equal to zero, , and their root mean square (RMS) uncertainty is equal to .The reflected pulses are detected by a phasesensitive device (the phase detector). The implementation of an optical phase detector is considered in detail in Section 2.3.1. Here we suppose only that the phase measurement error introduced by the detector is much smaller than the initial uncertainty of the phases . In this case, the initial uncertainty will be the only source of the position measurement error:
For convenience, we renormalize Eq. (96) as the equivalent testmass displacement: where are the independent random values with the RMS uncertainties given by Eq. (97).Upon reflection, each light pulse kicks the test mass, transferring to it a backaction momentum equal to
where and are the testmass momentum values just before and just after the light pulse reflection, and is the energy of the th pulse. The major part of this perturbation is contributed by classical radiation pressure: with the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator. The random part, which cannot be compensated, is proportional to the deviation of the pulse energy: and its RMS uncertainly is equal to with the RMS uncertainty of the pulse energy.The energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:
Therefore, it follows from Eqs. (97 and 103) that the position measurement error and the momentum perturbation due to back action also satisfy the uncertainty relation:This example represents a simple particular case of a linear measurement. This class of measurement schemes can be fully described by two linear equations of the form (98) and (100), provided that both the measurement uncertainty and the object backaction perturbation ( and in this case) are statistically independent of the test object initial quantum state and satisfy the same uncertainty relation as the measured observable and its canonically conjugate counterpart (the object position and momentum in this case).
Suppose the test mass to be heavy enough for a single pulse to either perturb its momentum noticeably, or measure its position with the required precision (which is a perfectly realistic assumption for the kilogramscale test masses of GW interferometers). In this case, many pulses should be used to accumulate the measurement precisions; at the same time, the testmass momentum perturbation will be accumulated as well. Choose now such a time interval , which, on the one hand, is long enough to comprise a large number of individual pulses:
and, on the other hand, is sufficiently short for the testmass position not to change considerably during this time due to the testmass selfevolution. Then one can use all the measurement results to refine the precision of the testmass position estimate, thus getting times smaller uncertainty At the same time, the accumulated random kicks the object received from each of the pulses random kicks, see Eq. (102), result in random change of the object’s momentum similar to that of Brownian motion, and thus increasing in the same diffusive manner:If we now assume the interval between the measurements to be infinitesimally small (), keeping at the same time each single measurement strength infinitesimally weak:

then we get a continuous measurement of the testmass position as a result. We need more adequate parameters to characterize its ‘strength’ than and . For continuous measurement we introduce the following parameters instead:
with This allows us to rewrite Eqs. (107) and (108) in a form that does not contain the time interval : To clarify the physical meaning of the quantities and let us rewrite Eq. (98) in the continuous limit: where stands for measurement noise, proportional to the phase of the light beam (in the continuous limit the sequence of individual pulses transforms into a continuous beam). Then there is no difficulty in seeing that is a power (doublesided) spectral density of this noise, and is a power doublesided spectral density of .If we turn to Eq. (100), which describes the meter back action, and rewrite it in a continuous limit we will get the following differential equation for the object momentum:
where is a continuous Markovian random force, defined as a limiting case of the following discrete Markov process: with the optical power, its mean value, and ‘’ here meaning all forces (if any), acting on the object but having nothing to do with the meter (light, in our case). Doublesided power spectral density of is equal to , and doublesided power spectral density of is .We have just built a simple model of a continuous linear measurement, which nevertheless comprises the main features of a more general theory, i.e., it contains equations for the calculation of measurement noise (112) and also for back action (114). The precision of this measurement and the object back action in this case are described by the spectral densities and of the two meter noise sources, which are assumed to not be correlated in our simple model, and thus satisfy the following relation (cf. Eqs. (109)):
This relation (as well as its more general version to be discussed later) for continuous linear measurements plays the same role as the uncertainty relation (105) for discrete measurements, establishing a universal connection between the accuracy of the monitoring and the perturbation of the monitored object.
We should emphasize that this simple measurement model and the corresponding uncertainty relation (116) are by no means general. We have made several rather strong assumptions in the course of derivation, i.e., we assumed:
These assumptions can be mapped to the following features of the fluctuations and in the continuous case:
The features 1 and 2, in turn, lead to characteristic fundamentallylooking sensitivity limitations, the SQL. We will call linear quantum meters, which obey these limitations (that is, with mutually noncorrelated and stationary noises and ), Simple Quantum Meters (SQM).
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Living Rev. Relativity 15, (2012), 5
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