4.2 General linear measurement

In this section, we generalize the concept of linear quantum measurement discussed above and give a comprehensive overview of the formalism introduced in [22Jump To The Next Citation Point] and further elaborated in [33, 43Jump To The Next Citation Point]. This formalism can be applied to any system that performs a transformation from some unknown classical observable (e.g., GW tidal force in GW interferometers) into another classical observable of a measurement device that can be measured with (ideally) arbitrarily high precision (e.g., in GW detectors, the readout photocurrent serves such an observable) and its value depends on the value of unknown observable linearly. For definiteness, let us keep closer to GW detectors and assume the continuous measurement of a classical force.
View Image

Figure 21: General scheme of the continuous linear measurement with G standing for measured classical force, ˆð’³ the measurement noise, ˆâ„± the back-action noise, Oˆ the meter readout observable, ˆx the actual probe’s displacement.

The abstract scheme of such a device is drawn in Figure 21View Image. It consists of a probe 𝒫 that is exposed to the action of a classical force G (t), and the meter. The action of this force on the probe causes its displacement ˆx that is monitored by the meter (e.g., light, circulating in the interferometer). The output observable of the meter ˆO is monitored by some arbitrary classical device that makes a measurement record o(t). The quantum nature of the probe–meter interaction is reflected by the back-action force ˆ F that randomly kicks the probe on the part of the meter (e.g., radiation pressure fluctuations). At the same time, the meter itself is the source of additional quantum noise ˆO fl(t) in the readout signal. Quantum mechanically, this system can be described by the following Hamiltonian:

ˆ ˆ(0) ˆ (0) ˆ ℋ = ℋ probe + ℋ meter + V(t), (124 )
where ˆâ„‹ (0) probe and ℋˆ(0) meter are the Hamiltonians describing the free evolution of the probe and the meter, respectively, i.e., when there is no coupling between these systems, and ˆV(t) = − ˆx(G (t) + ˆF ) is the interaction Hamiltonian. The evolution of this system can be found, in general, by solving Heisenberg equations for all of the system observables. However, it is convenient to rewrite it first in the interaction picture factoring out the free evolution of the probe and the meter (see Appendix 3.7 of [43Jump To The Next Citation Point] for detailed derivation):
{ } { } ˆâ„‹I (t) = exp iℋˆ(0)(t − t0) ˆV(t)exp − iℋˆ(0)(t − t0) = − ˆx(0)(t)(G (t) + ˆF (0)(t)), (125 ) ℏ ℏ
where ˆ (0) ˆ(0) ˆ (0) ℋ = ℋ probe + ℋ meter, and (0) ˆx (t) and ˆ(0) F (t) are the Heisenberg operators of the probe’s displacement and the meter back-action force, respectively, in the case of no coupling between these systems, i.e., the solution to the following system of independent Heisenberg equations:
dˆx(0)(t)- -i[ ˆ(0) (0) ] d ˆF-(0)(t) i-[ ˆ (0) ˆ(0) ] dt = ℏ ℋ probe,xˆ (t) , dt = ℏ ℋ meter,F (t) ,

and t0 is the arbitrary initial moment of time that can be set to − ∞ without loss of generality.

The following statement can be proven (see [94], Section VI of [22Jump To The Next Citation Point], and Theorems 3 and 4 in Appendix 3.7 of [43Jump To The Next Citation Point] for proof): For a linear system with Hamiltonian (124View Equation), for any linear observable ˆA of the probe and for any linear observable Bˆ of the meter, their full Heisenberg evolutions are given by:

∫ t Aˆ(t) = Aˆ(0)(t) + dt′χAx (t,t′)[F ˆ(t′) + G (t′)], ∫t0 (0) t ′ ′ ′ ˆB (t) = Bˆ (t) + dtχBF (t,t)xˆ(t), (126 ) t0
where ˆA(0)(t) and Bˆ(0)(t) stand for the free Heisenberg evolutions in the case of no coupling, and the functions χAx (t,t′) and χBF (t,t′) are called (time-domain) susceptibilities and defined as:
({ i [ ] ′ -- ˆA(0)(t), ˆx(0)(t′) , t ≥ t′ χAx(t,t) ≡ ( ℏ ′ 0, t < t ({ i [ ] ′ -- ˆB (0)(t),Fˆ(0)(t′) , t ≥ t′ χBF (t,t) ≡ ( ℏ ′ . (127 ) 0, t < t
The second clauses in these equations maintain the causality principle.

For time independent Hamiltonian ˆ(0) ℋ and operator ˆ F (in the Schrödinger picture), the susceptibilities are invariant to time shifts, i.e., ′ ′ χ(t,t) = χ(t + τ,t + τ), therefore they depend only on the difference of times: χ (t,t′) → χ(t − t′). In this case, one can rewrite Eqs. (126View Equation) in frequency domain as:

ˆ ˆ(0) ˆ ˆ ˆ (0) A (Ω) = A (Ω) + χAx (Ω)[F(Ω ) + G (Ω )], B (Ω ) = B (Ω) + χBF (Ω )xˆ(Ω), (128 )
where the Fourier transforms of all of the observables are defined in accordance with Eq. (6View Equation).

Let us now use these theorems to find the full set of equations of motion for the system of linear observables ˆx, ˆF and Oˆ that fully characterize our linear measurement process in the scheme featured in Figure 21View Image:

(0) ∫ t ′ ′ ′ ˆO(t) = ˆO (t) + dt χOF (t − t )ˆx(t), ∫t0 ˆ ˆ (0) t ′ ′ ′ F(t) = F (t) + dt χF F(t − t )ˆx(t), ∫t0t [ ] (0) ′ ′ ′ ˆ ′ ˆx(t) = ˆx (t) + t dtχxx (t − t ) G (t) + F (t) , (129 ) 0
where time-domain susceptibilities are defined as
′ i [ (0) (0) ′] χOF (t − t ) =-- ˆO (t),Fˆ (t) , ℏ [ ] χFF (t − t′) = i ˆF(0)(t),Fˆ(0)(t′) , ℏ ′ i-[ (0) (0) ′] χxx (t − t ) = ℏ xˆ (t),xˆ (t) . (130 )

The meaning of the above equations is worth discussing. The first of Eqs. (129View Equation) describes how the readout observable Oˆ(t) of the meter, say the particular quadrature of the outgoing light field measured by the homodyne detector (cf. Eq. (39View Equation)), depends on the actual displacement ˆx(t) of the probe, and the corresponding susceptibility χ (t − t′) OF is the transfer function for the meter from ˆx to ˆ O. The term ˆ(0) O (t) stands for the free evolution of the readout observable, provided that there was no coupling between the probe and the meter. In the case of the GW detector, this is just a pure quantum noise of the outgoing light that would have come out were all of the interferometer test masses fixed. It was shown explicitly in [90Jump To The Next Citation Point] and we will demonstrate below that this noise is fully equivalent to that of the input light except for the insignificant phase shift acquired by the light in the course of propagation through the interferometer.

The following important remark should be made concerning the meter’s output observable ˆO (t). As we have mentioned already, the output observable in the linear measurement process should be precisely measurable at any instance of time. This implies a simultaneous measurability condition [30, 41, 147, 22Jump To The Next Citation Point, 43Jump To The Next Citation Point, 33] on the observable ˆO (t) requiring that it should commute with itself at any moment of time:

[ ] ˆO (t),Oˆ(t′) = 0, ∀t,t′. (131 )
Initially, this condition was introduced as the definition of the quantum non-demolition (QND) observables by Braginsky et al. [27, 28Jump To The Next Citation Point]. In our case it means that the measurement of Oˆ(t ) 1 at some moment of time t1 shall not disturb the measurement result at any other moments of time and therefore the sample data { } Oˆ(t1), ˆO(t1),..., ˆO (tn) can be stored directly as bits of classical data in a classical storage medium, and any noise from subsequent processing of the signal can be made arbitrarily small. It means that all noise sources of quantum origin are already included in the quantum fluctuations of ˆO (t) [43Jump To The Next Citation Point, 33]. And the fact that due to (131View Equation) this susceptibility turns out to be zero reflects the fact that Oˆ(t) should be a classical observable.

The second equation in (129View Equation) describes how the back-action force exerted by the meter on the probe system evolves in time and how it depends on the probe’s displacement. The first term, ˆ (0) F (t), meaning is rather obvious. In GW interferometer, it is the radiation pressure force that the light exerts on the mirrors while reflecting off them. It depends only on the mean value and quantum fluctuations of the amplitude of the incident light and does not depend on the mirror motion. The second term here stands for a dynamical back-action of the meter and since, by construction, it is the part of the back-action force that depends, in a linear way, from the probe’s displacement, the meaning of the susceptibility χF F(t − t′) becomes apparent: it is the generalized rigidity that the meter introduces, effectively modifying the dynamics of the probe. We will see later how this effective rigidity can be used to improve the sensitivity of the GW interferometers without introducing additional noise and thus enhancing the SNR of the GW detection process.

The third equation of (129View Equation) concerns the evolution of the probe’s displacement in time. Three distinct parts comprise this evolution. Let us start with the second and the third ones:

∫ t ∫ t xs(t) = dt′χxx (t − t′)G (t′), and ˆxb.a.(t) = dt′χxx(t − t′) ˆF(t′). (132 ) t0 t0
Here xs(t) is the probe’s response on the signal force G(t) and is, actually, the part we are mostly interested in. This expression also unravels the role of susceptibility ′ χxx(t − t): it is just the Green’s function of the equation of motion governing the probe’s bare dynamics (also known as impulse response function) that can be shown to be a solution of the following initial value problem:
′ ′ ′ D χxx(t − t ) = δ(t − t ), χxx(t − t )|t→t′ = 0, ... ∂n−2χxx (t − t′)| ∂n −1χxx(t − t′)| 1 ... -------n−2-----|t→t′+0 = 0, -------n−1-----|t→t ′+0 = --, ∂t ∂t an
where ∑ k D = nk=0 ak ddtk is the linear differential operator that is governed by the dynamics of the probe, e.g., it is equal to D = M d2- f.m. dt2 for a free mass M and to D = M -d2+ M Ω2 osc dt2 m for a harmonic oscillator with eigenfrequency Ωm. Apparently, operator D is an inverse of the integral operator χχχxx whose kernel is ′ χxx(t − t ):
∫ t xs(t) = D −1G (t) = χχχxxG (t) = dt′χxx(t − t′)G(t′). t0

The second value, ˆxb.a.(t), is the displacement of the probe due to the back-action force exerted by the meter on the probe. Since it enters the probe’s response in the very same way the signal does, it is the most problematic part of the quantum noise that, as we demonstrate later, imposes the SQL [16, 22Jump To The Next Citation Point].

And finally, (0) ˆx (t) simply features a free evolution of the probe in accordance with its equations of motion and thus depends on the initial values of the probe’s displacement (0) ˆx (t0), momentum (0) ˆp (t0), and, possibly, on higher order time derivatives of ˆx(0)(t) taken at t0, as per the structure of the operator D governing the probe’s dynamics. It is this part of the actual displacement that bears quantum uncertainties imposed by the initial quantum state of the probe. One could argue that these uncertainties might become a source of additional quantum noise obstructing the detection of GWs, augmenting the noise of the meter. This is not the case as was shown explicitly in [19], since our primary interest is in the detection of a classical force rather than the probe’s displacement. Therefore, performing over the measured data record o(t) the linear transformation corresponding to first applying the operator χχχ −1 OF on the readout quantity that results in expressing o(t) in terms of the probe’s displacement:

−1 (0) &tidle;x(t) = χχχOF O (t) = xfl(t) + x (t) + xs (t) + xb.a.(t),

with −1 x fl(t) = χχχ OFO (0)(t) standing for the meter’s own quantum noise (measurement uncertainty), and then applying a probe dynamics operator D that yields a force signal equivalent to the readout quantity o(t):

&tidle; F(t) = D &tidle;x(t) = Dx fl(t) + Fb.a.(t) + G (t).

The term Dx (0)(t) vanishes since x(0)(t) is the solution of a free-evolution equation of motion. Thus, we see that the result of measurement contains two noise sources, ˆxfl(t) and ˆFb.a.(t), which comprise the sum noise masking the signal force G (t).

Since we can remove initial quantum uncertainties associated with the state of the probe, it would be beneficial to turn to the Fourier domain and rewrite Eqs. (129View Equation) in the spectral form:

(0) Oˆ(Ω) = Oˆ (Ω ) + χOF (Ω )ˆx(Ω), Fˆ(Ω) = Fˆ(0)(Ω ) + χF F(Ω )ˆx(Ω), ˆ ˆx(Ω) = χxx (Ω )[F (Ω ) + G (Ω )], (133 )
where spectral susceptibilities are defined as:
∫ ∞ χAB (Ω ) = d τχAB (τ)eiΩτ, (134 ) 0
with (A, B ) ⇒ (O, F,x ), and we omit the term ˆx(0)(Ω ) in the last equation for the reasons discussed above. The solution of these equations is straightforward to get and reads:
χ (Ω )χ (Ω) [ ] ˆO (Ω ) = ˆO (0)(Ω) + ----xx-----OF------ G (Ω ) + ˆF (0)(Ω) , (135 ) 1 − χxx(Ω )χFF (Ω) ˆ ---------1-------- [ ˆ (0) ] F (Ω ) = 1 − χ (Ω )χ (Ω) F (Ω) + χF F(Ω)χxx (Ω )G (Ω) , (136 ) xx FF [ ] ˆx (Ω ) = ------χxx(Ω-)----- G (Ω ) + ˆF(0)(Ω ) . (137 ) 1 − χxx(Ω )χFF (Ω)

It is common to normalize the output quantity of the meter Oˆ(Ω ) to unit signal. In GW interferometers, two such normalizations are popular. The first one tends to consider the tidal force G as a signal and thus set to 1 the coefficient in front of G (Ω) in Eq. (135View Equation). The other one takes GW spectral amplitude h(Ω ) as a signal and sets the corresponding coefficient in ˆ O (Ω) to unity. Basically, these normalizations are equivalent by virtue of Eq. (12View Equation) as:

2 2 Lh(Ω-)- 2 − M Ω xh(Ω ) ≡ − M Ω 2 = G (Ω) ⇒ h(Ω ) = − 2G(Ω )∕(M L Ω ). (138 )
In both cases, the renormalized output quantities can be considered as a sum of the noise and signal constituents:
ˆ F ˆF ˆh ˆh O = 𝒩 + G or O = 𝒩 + h(Ω ). (139 )
And it is the noise term in both cases that we are seeking to calculate to determine the sensitivity of the GW detector. Let us rewrite ˆO (Ω ) in force normalization:
( ) F 1 − χF F(Ω )χxx(Ω ) Oˆ(0)(Ω ) (0) χF F(Ω ) (0) ˆO (Ω) = ------------------ˆO(Ω ) = ---------------+ ˆF (Ω) − --------ˆO (Ω ) + G (Ω ) χOF (Ω )χxx(Ω ) χOF (Ω )χxx (Ω) χOF (Ω ) 𝒳ˆ(Ω ) ≡ χ--(Ω)-+ ℱˆ(Ω ) + G (Ω), (140 ) xx
where we introduce two new linear observables 𝒳ˆ and ℱˆ of the meter defined as:
(0) 𝒳ˆ(Ω) ≡ ˆO---(Ω-), ˆâ„±(Ω ) ≡ ˆF (0)(Ω) − χF-F(Ω-)ˆO (0)(Ω), (141 ) χOF (Ω ) χOF (Ω )
that have the following meaning:

These two new observables that embody the two types of noise inherent in any linear measurement satisfy the following commutation relations:

[ ] [ ] [ ] [ ] 𝒳ˆ(Ω ), ˆð’³ †(Ω ′) = ℱˆ(Ω ), ˆâ„± †(Ω ′) = 0, ⇐ ⇒ 𝒳ˆ(t), ˆð’³ (t′) = ˆâ„± (t),ℱˆ(t′) = 0, (142 ) [ ] [ ] 𝒳ˆ(Ω),ℱˆ†(Ω′) = − 2πiℏδ(Ω − Ω ′), ⇐ ⇒ 𝒳ˆ(t), ˆâ„± †(t′) = − iℏδ(t − t′), (143 )
that can be interpreted in the way that 𝒳ˆ(t) and ℱˆ(t) can be seen at each instance of time as the canonical momentum and the coordinate of different effective linear measuring devices (meter+probe), thus defining an infinite set of subsequent measurements similar to the successive independent monitors of von Neumann’s model [157]. In this case, however, the monitors described by 𝒳ˆ(t) and ˆâ„± (t) are not, generally speaking, independent. In GW detectors, these monitors appear correlated due to the internal dynamics of the detector, i.e., the noise processes they describe are non-Markovian.

In particular, this can be seen when one calculates the power (double-sided) spectral density of the sum noise 𝒩ˆF(t):

∫ ∞ [ ] SF (Ω ) = dt⟨𝒩ˆ F(t) ∘ ˆð’© F(t′)⟩eiΩ(t−t′) =-S𝒳-𝒳(Ω-) + S ℱℱ(Ω ) + 2Re S𝒳-ℱ(Ω-) , (144 ) −∞ |χxx(Ω )|2 χxx (Ω)
where spectral densities:
∫ ∞ ′ iΩ(t−t′) S 𝒳𝒳 (Ω ) = dt⟨𝒳ˆ(t) ∘ 𝒳ˆ(t)⟩e , (145 ) ∫−∞∞ ˆ ˆ ′ iΩ (t−t′) S ℱℱ (Ω ) = −∞ dt⟨ℱ (t) ∘ ℱ (t)⟩e , (146 ) ∫ ∞ S 𝒳ℱ (Ω ) = dt⟨𝒳ˆ(t) ∘ ℱˆ(t′)⟩eiΩ (t−t′), (147 ) −∞
are not necessarily constant and, thus, describe non-Markovian random processes. It can also be shown that since ˆ 𝒳 (t) and ˆ ℱ(t) satisfy commutation relations (142View Equation), their spectral densities shall satisfy the Schrödinger–Robertson uncertainty relation:
2 S 𝒳𝒳 (Ω)Sℱ ℱ(Ω ) − |S𝒳 ℱ(Ω )|2 ≥ ℏ- (148 ) 4
that is the generalization of a Heisenberg uncertainty relation in the case of correlated observables.

The general structure of quantum noise in the linear measurement process, comprising two types of noise sources whose spectral densities are bound by the uncertainty relation (148View Equation), gives a clue to several rather important corollaries. One of the most important is the emergence of the SQL, which we consider in detail below.


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