4.4 Beating the SQL by means of noise cancellation

The SQL is not a fundamental limitation as we have mentioned already, and the clue to how to overcome it can be devised from the expression for the general linear measurement sum noise spectral density (144View Equation). One can see that a properly constructed cross-correlation between the measurement noise ˆ ˆ(0) 𝒳 = O ∕χOF (Ω ) and back-action noise ˆ(0) F, i.e., the right choice of χF F(Ω ) that should be at any frequency equal to:
χFF (Ω) = S 𝒳F(Ω )∕S𝒳𝒳 (Ω) (184 )
can compensate the back-action term and leave only the measurement noise-related contribution to the final sum quantum noise:
SF (Ω) = S (Ω )∕|χ (Ω)|2. 𝒳𝒳 xx

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 [148Jump To The Next Citation Point, 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 [162Jump To The Next Citation Point, 159Jump To The Next Citation Point, 158Jump To The Next Citation Point, 160Jump To The Next Citation Point, 161Jump To The Next Citation Point, 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.

View Image

Figure 25: Toy example of a linear optical position measurement.

The advanced version of that example is shown in Figure 25View Image. The only difference between this scheme and the initial one (see Figure 20View Image) 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 ϕLO (cf. Eq. (39View Equation)):

𝒲ˆ − 𝒲 ˆO(tj) = − ˆϕrjefl sin ϕLO + ---j-----cos ϕLO, (185 ) 2𝒲
(we subtracted the regular term proportional to the mean energy 𝒲 from the output signal Oˆ(tj)).

Similar to Eq. (98View Equation), renormalize this output signal as the equivalent test object displacement:

-----Oˆj------ &tidle;x (tj) ≡ 2гkp sin ϕLO = ˆx(tj) + ˆxfl(tj), (186 )
where the noise term has in this case the following form:
( ˆ ) ˆx fl(tj) = --1-- − ˆϕ(tj) + 𝒲j-−-𝒲--cot ϕLO . (187 ) 2гkp 2𝒲
RMS uncertainty of this value (the measurement error) is equal to
∘ ------------------------- --1-- 2 (Δ-𝒲-)2 2 Δxmeas = 2гk (Δ ϕ) + 4𝒲2 cot ϕLO. (188 ) p
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. (103View Equation). However, the additional term in Eq. (185View Equation) 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 cross-correlation of the measurement error and the back action:
(Δ 𝒲 )2 Δ (xp) = ⟨ˆxfl(tj) ∘ ˆpb.a.(tj)⟩ =-------cot ϕLO. (189 ) 2 ωp𝒲
It is easy to see that the uncertainties (103View Equation, 188View Equation, 189View Equation) satisfy the following Schrödinger–Robertson uncertainty relation:
2 2 2 (Δ ϕ)2(Δ 𝒲 )2 ℏ2 (Δxmeas ) (Δpb.a.) − [Δ(xp )] = ------ω2----- ≥ 4-. (190 ) p

Now we can perform the transition to the continuous measurement limit as we did in Section 4.1.2:

( ) 2 ---1-- S-ℐ- 2 Sx = l𝜗im→0(Δxmeas ) 𝜗 = 4г2k2 Sϕ + 4ℐ2 cot ϕLO , 2 2 p S = lim (Δpb.a.)--= 4г--Sℐ, F 𝜗→0 𝜗 c2 Sℐ SxF = l𝜗im→0 Δ (xp ) = -----cot ϕLO., (191 ) 2ωpℐ
which transform inequality (190View Equation) to the Schödinger–Robertson uncertainty relation for continuous measurements:
2 SϕS ℐ ℏ2 SxSF − SxF = --2--≥ --. (192 ) ωp 4

In the particular case of the light pulses in a coherent quantum state (120View Equation), the measurement error (188View Equation), the momentum perturbation (103View Equation), and the cross-correlation term (189View Equation) are equal to:

∘ -------------- c ℏ 2г ∘ ------ ℏ Δxmeas = --- -------2-----, Δpb.a.= --- ℏ ωp𝒲, Δ (xp) = --cotϕLO (193 ) 4г ωp𝒲 sin ϕLO c 2
(the momentum perturbation Δp b.a. 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:
2 (Δx )2(Δp )2 − [Δ (xp )]2 = ℏ--. (194 ) meas b.a. 4
Correspondingly, substituting the coherent quantum state power (double-sided) spectral densities (122View Equation) into Eqs. (191View Equation), we obtain:
ℏc2 4 ℏωpℐ0г2 ℏ Sx = ---------2---2----, SF = -----2---- ,SxF = --cot ϕLO, (195 ) 16 ωpℐ0г sin ϕLO c 2
2 ℏ2 SxSF − S xF = --- (196 ) 4
[compare with Eqs. (123View Equation)].

The cross-correlation between the measurement and back-action fluctuations is equivalent to the virtual rigidity χviFrFt(Ω ) ≡ − Kvirt = const. as one can conclude looking at Eqs. (141View Equation). Indeed,

ˆxfl(t) = 𝒳ˆ(t), and ˆF(0) = ℱˆ(t) + Kvirtˆxfl(t).

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

ˆF (t) = D ˆx (t) + Fˆ(0)(t) = D ˆx (t) + ℱˆ(t), sum fl fl eff fl

where the new effective dynamics that correspond to the new noise are governed by the following differential operator

Deff = D + Kvirt. (197 )
The above explains why we refer to Kvirt as ‘virtual rigidity’.

To 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:

-d2 De ff = M dt2 + Kvirt, SxF = KvirtSx ⁄= 0, (198 )
correspond to a harmonic oscillator with eigenfrequency 2 Ω0 = Kvirt∕M, and as we have demonstrated in Eq. (171View Equation) provide a narrow-band sensitivity gain versus a free mass SQL near the resonance frequency Ω0 .

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 S F, which is a good measure of measurement strength according to Eqs. (156View Equation), for the virtual rigidity against the real one. For the latter, to overcome the free mass SQL by a factor

ξ2 = Δ-Ω- (199 ) Ω0
(see Eq. (171View Equation)) at a given frequency Ω 0, the back-action noise spectral density has to be reduced by this factor:
ℏM Ω2 SF = ------q= ξ2SoFptf.m., (200 ) 2
see Eqs. (156View Equation, 160View Equation, 167View Equation). Here, SoFptf.m.= ℏM Ω0 ∕2 is the back-action noise spectral density, which allows one to reach the free mass SQL (161View Equation) at frequency Ω0. Such a sensitivity gain is achieved at the expense of proportionally reduced bandwidth:
2 Δ Ω = ξ Ω0. (201 )
For the virtual rigidity, the optimal value of SF results from Eq. (167View Equation):
2 2 ( 4) ( ) ℏ-∕4-+-S-xF- ℏM-- 2 Ω0- optf.m. 2 1- SF = Sx = 2 Ω q + Ω2q = S F ξ + ξ2 . (202 )
Hence, the better the sensitivity (the smaller ξ2), the larger SF must be and, therefore, measurement strength.

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 Ω0 and that this band shrinks as the sensitivity gain rises (cf. Eq. (199View Equation)). In order to provide a broadband enhancement in sensitivity, either the real rigidity K = M Ω20, or the virtual one Kvirt = SxF ∕SF should depend on frequency in such a way as to be proportional (if only approximately) to Ω2 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 [21Jump To The Next Citation Point] and the filter cavities [90Jump To The Next Citation Point] 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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