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 test-mass position at the moment of reflection:
The reflected pulses are detected by a phase-sensitive 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:
Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to
The energy and the phase of each pulse are canonically conjugate observables and thus obey the following 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 back-action 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 kilogram-scale test masses of GW interferometers). In this case, many pulses should be used to accumulate the measurement precisions; at the same time, the test-mass 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:
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 test-mass position as a result. We need more adequate parameters to characterize its ‘strength’ than and . For continuous measurement we introduce the following parameters instead: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 (double-sided) spectral density of this noise, and is a power double-sided 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:
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)):
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 fundamentally-looking sensitivity limitations, the SQL. We will call linear quantum meters, which obey these limitations (that is, with mutually non-correlated and stationary noises and ), Simple Quantum Meters (SQM).
Living Rev. Relativity 15, (2012), 5
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