
This spectral density could be arbitrarily small, providing the unbound measurement strength. However, there is a significant obstacle on the way towards backaction free measurement: the optimal correlation should be frequency dependent in the right and rather peculiar way. Another drawback of such backaction evasion via noise cancellation resides in the dissipation that is always present in real measurement setups and, according to FluctuationDissipation Theorem [37, 95] is a source of additional noise that undermines any quantum correlations that might be built in the ideal system.
The simplest way is to make the relation (4.4) hold at some fixed frequency, which can always be done either (i) by preparing the meter in some special initial quantum state that has measurement and backaction fluctuations correlated (Unruh [148, 147] proposed to prepare input light in a squeezed state to achieve such correlations), or (ii) by monitoring a linear combination of the probe’s displacement and momentum [162, 159, 158, 160, 161, 51, 53] that can be accomplished, e.g., via homodyne detection, as we demonstrate below.
We consider the basic principles of the schemes, utilizing the noise cancellation via building crosscorrelations between the measurement and backaction noise. We start from the very toy example discussed in Section 4.1.1.
The advanced version of that example is shown in Figure 25. The only difference between this scheme and the initial one (see Figure 20) is that here the detector measures not the phase of light pulses, but linear combination of the phase and energy, parametrized by the homodyne angle (cf. Eq. (39)):
(we subtracted the regular term proportional to the mean energy from the output signal ).Similar to Eq. (98), renormalize this output signal as the equivalent test object displacement:
where the noise term has in this case the following form: RMS uncertainty of this value (the measurement error) is equal to At first glance, it seems like we just obtained the increased measurement error, for the same value of the test object perturbation, which is still described by Eq. (103). However, the additional term in Eq. (185) can be viewed not only as the additional noise, but as the source of information about the test object perturbation. It can be used to subtract, at least in part, the terms induced by this perturbation from the subsequent measurement results. Quantitatively, this information is characterized by the crosscorrelation of the measurement error and the back action: It is easy to see that the uncertainties (103, 188, 189) satisfy the following Schrödinger–Robertson uncertainty relation:Now we can perform the transition to the continuous measurement limit as we did in Section 4.1.2:
which transform inequality (190) to the Schödinger–Robertson uncertainty relation for continuous measurements:In the particular case of the light pulses in a coherent quantum state (120), the measurement error (188), the momentum perturbation (103), and the crosscorrelation term (189) are equal to:
(the momentum perturbation evidently remains the same as in the uncorrelated case, and we provided its value here only for convenience), which gives the exact equality in the Schrödinger–Robertson uncertainty relation: Correspondingly, substituting the coherent quantum state power (doublesided) spectral densities (122) into Eqs. (191), we obtain: with [compare with Eqs. (123)].The crosscorrelation between the measurement and backaction fluctuations is equivalent to the virtual rigidity as one can conclude looking at Eqs. (141). Indeed,

The sum noise does not change under the above transformation and can be written as:

where the new effective dynamics that correspond to the new noise are governed by the following differential operator
The above explains why we refer to as ‘virtual rigidity’.To see how virtual rigidity created by crosscorrelation of noise sources can help beat the SQL consider a free mass as a probe body in the above considered toy example. The modified dynamics:
correspond to a harmonic oscillator with eigenfrequency , and as we have demonstrated in Eq. (171) provide a narrowband sensitivity gain versus a free mass SQL near the resonance frequency .However, there is a drawback of virtual rigidity compared to the real one: it requires higher measurement strength, and therefore higher power, to reach the same gain in sensitivity as provided by a harmonic oscillator. This becomes evident if one weighs the backaction spectral density , which is a good measure of measurement strength according to Eqs. (156), for the virtual rigidity against the real one. For the latter, to overcome the free mass SQL by a factor
(see Eq. (171)) at a given frequency , the backaction noise spectral density has to be reduced by this factor: see Eqs. (156, 160, 167). Here, is the backaction noise spectral density, which allows one to reach the free mass SQL (161) at frequency . Such a sensitivity gain is achieved at the expense of proportionally reduced bandwidth: For the virtual rigidity, the optimal value of results from Eq. (167): Hence, the better the sensitivity (the smaller ), the larger must be and, therefore, measurement strength.Another evident flaw of the virtual rigidity, which it shares with the real one, is the narrowband character of the sensitivity gain it provides around and that this band shrinks as the sensitivity gain rises (cf. Eq. (199)). In order to provide a broadband enhancement in sensitivity, either the real rigidity , or the virtual one should depend on frequency in such a way as to be proportional (if only approximately) to in the frequency band of interest. Of all the proposed solutions providing frequency dependent virtual rigidity, the most well known are the quantum speed meter [21] and the filter cavities [90] schemes. Section 4.5, we consider the basic principles of the former scheme. Then, in Section 6 we provide a detailed treatment of both of them.
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Living Rev. Relativity 15, (2012), 5
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