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This spectral density could be arbitrarily small, providing the unbound measurement strength. However, there is a significant obstacle on the way towards back-action free measurement: the optimal correlation should be frequency dependent in the right and rather peculiar way. Another drawback of such back-action evasion via noise cancellation resides in the dissipation that is always present in real measurement setups and, according to Fluctuation-Dissipation 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
back-action 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 cross-correlations between the measurement and back-action 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
)):
Similar to Eq. (98), renormalize this output signal as the equivalent test object displacement:
Now we can perform the transition to the continuous measurement limit as we did in Section 4.1.2:
which transform inequality (190 In the particular case of the light pulses in a coherent quantum state (120), the measurement
error (188
), the momentum perturbation (103
), and the cross-correlation term (189
) are equal to:
The cross-correlation between the measurement and back-action fluctuations is equivalent to
the virtual rigidity as one can conclude looking at Eqs. (141
).
Indeed,
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The sum noise does not change under the above transformation and can be written as:
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where the new effective dynamics that correspond to the new noise are governed by the following differential operator
The above explains why we refer toTo see how virtual rigidity created by cross-correlation 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 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 back-action 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
Another evident flaw of the virtual rigidity, which it shares with the real one, is the narrow-band
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.
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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