4.1 Quantum measurement of a classical force

4.1.1 Discrete position measurement

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 20View Image, a sequence of very short light pulses are used to monitor the displacement of a probe body M. The position x of M is probed periodically with time interval πœ—. In order to make our model more realistic, we suppose that each pulse reflects from the test mass Π³ > 1 times, thus increasing the optomechanical coupling and thereby the information of the measured quantity contained in each reflected pulse. We also assume mass M large enough to neglect the displacement inflicted by the pulses radiation pressure in the course of the measurement process.

View Image

Figure 20: Toy example of a linear optical position measurement.

Then each j-th pulse, when reflected, carries a phase shift proportional to the value of the test-mass position x(tj) at the moment of reflection:

refl ˆΟ•j = ˆΟ•j − 2Π³kpxˆ(tj), (96 )
where k = ω βˆ•c p p, ω p is the light frequency, j = ...,− 1,0,1,... is the pulse number and ˆΟ• j is the initial (random) phase of the j-th pulse. We assume that the mean value of all these phases is equal to zero, ⟨ˆΟ•j⟩ = 0, and their root mean square (RMS) uncertainty ⟨(Ο•ˆ2⟩ − ⟨ˆΟ•βŸ©2)1βˆ•2 is equal to Δ Ο•.

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 Ο•ˆrejfl measurement error introduced by the detector is much smaller than the initial uncertainty of the phases Δ Ο•. In this case, the initial uncertainty will be the only source of the position measurement error:

Δxmeas = -Δ-Ο•-. (97 ) 2Π³kp
For convenience, we renormalize Eq. (96View Equation) as the equivalent test-mass displacement:
ˆrefl &tidle;x ≡ − -Ο•j-- = ˆx(t) + ˆx (t ), (98 ) j 2 Π³kp j fl j
where
-ˆΟ•j-- ˆxfl(tj) = − 2Π³k (99 ) p
are the independent random values with the RMS uncertainties given by Eq. (97View Equation).

Upon reflection, each light pulse kicks the test mass, transferring to it a back-action momentum equal to

2Π³ ˆpajfter− ˆpbjefore= ˆpbj.a.= ---𝒲ˆj, (100 ) c
where ˆpbjefore and ˆpafjter are the test-mass momentum values just before and just after the light pulse reflection, and 𝒲j is the energy of the j-th pulse. The major part of this perturbation is contributed by classical radiation pressure:
⟨ˆpb.a.⟩ = 2Π³-𝒲, (101 ) j c
with 𝒲 the mean energy of the pulses. Therefore, one could neglect its effect, for it could be either subtracted from the measurement result or compensated by an actuator. The random part, which cannot be compensated, is proportional to the deviation of the pulse energy:
b.a. b.a. b.a. 2Π³ ( ) ˆp (tj) = ˆpj − ⟨ˆpj ⟩ = --- 𝒲ˆj − 𝒲 , (102 ) c
and its RMS uncertainly is equal to
2Π³ Δ π’² Δpb.a. = --------, (103 ) c
with Δ π’² the RMS uncertainty of the pulse energy.

The energy and the phase of each pulse are canonically conjugate observables and thus obey the following uncertainty relation:

ℏωp Δ π’² Δ Ο• ≥ ----. (104 ) 2
Therefore, it follows from Eqs. (97View Equation and 103View Equation) that the position measurement error Δxmeas and the momentum perturbation Δpb.a. due to back action also satisfy the uncertainty relation:
ℏ Δxmeas Δpb.a.≥ -. (105 ) 2

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 (98View Equation) and (100View Equation), provided that both the measurement uncertainty and the object back-action perturbation (xˆfl(tj) and b.a. pˆ (tj) 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).

4.1.2 From discrete to continuous measurement

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 T, which, on the one hand, is long enough to comprise a large number of individual pulses:

T N = --≫ 1, (106 ) πœ—
and, on the other hand, is sufficiently short for the test-mass position x not to change considerably during this time due to the test-mass self-evolution. Then one can use all the N measurement results to refine the precision of the test-mass position x estimate, thus getting √ --- N times smaller uncertainty
Δx ∘ πœ—-- ΔxT = -√-meas = Δxmeas --. (107 ) N T
At the same time, the accumulated random kicks the object received from each of the pulses random kicks, see Eq. (102View Equation), result in random change of the object’s momentum similar to that of Brownian motion, and thus increasing in the same diffusive manner:
∘ --- Δp = Δp √N--= Δp T-. (108 ) T b.a. b.a. πœ—

If we now assume the interval between the measurements to be infinitesimally small (πœ— → 0), keeping at the same time each single measurement strength infinitesimally weak:

Δx → ∞ ⇔ Δp → 0, meas b.a.

then we get a continuous measurement of the test-mass position ˆx(t) as a result. We need more adequate parameters to characterize its ‘strength’ than Δxmeas and Δpb.a.. For continuous measurement we introduce the following parameters instead:

S (Δp )2 4Π³2S Sx = lim (Δxmeas )2πœ— = ---Ο•2-2, SF = lim ----b.a.-- = ---2-ℐ-, (109 ) πœ—→0 4Π³ kp πœ—→0 πœ— c
with
2 (Δ π’² )2 SΟ• = lπœ—im→0(Δ Ο• )πœ—, Sℐ = liπœ—m→0 -------. (110 ) πœ—
This allows us to rewrite Eqs. (107View Equation) and (108View Equation) in a form that does not contain the time interval πœ—:
∘ --- ----- Δx = Sx, Δp = ∘ S T . (111 ) T T T F
To clarify the physical meaning of the quantities Sx and SΟ• let us rewrite Eq. (98View Equation) in the continuous limit:
x&tidle;(t) = ˆx (t) + ˆxfl(t), (112 )
where
ˆΟ•(t) ˆxfl(t) = − ----- (113 ) 2Π³kp
stands for measurement noise, proportional to the phase ˆΟ•(t) 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 Sx is a power (double-sided) spectral density of this noise, and SΟ• is a power double-sided spectral density of ˆΟ•(t).

If we turn to Eq. (100View Equation), which describes the meter back action, and rewrite it in a continuous limit we will get the following differential equation for the object momentum:

dˆp(t)= ˆF (t) + ... (114 ) dt fl
where ˆF (t) fl is a continuous Markovian random force, defined as a limiting case of the following discrete Markov process:
ˆpb.a.(tj) 2Π³ 𝒲ˆj − 𝒲 2Π³ Fˆfl(tj) = lim --------= ---lim ---------= ---[ˆβ„(tj) − ℐ0], (115 ) πœ—→0 πœ— c πœ—→0 πœ— c
with ℐˆ(t) the optical power, ℐ0 its mean value, and ‘...’ here meaning all forces (if any), acting on the object but having nothing to do with the meter (light, in our case). Double-sided power spectral density of Fˆb.a. is equal to SF, and double-sided power spectral density of ℐˆ is S ℐ.

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 (112View Equation) and also for back action (114View Equation). The precision of this measurement and the object back action in this case are described by the spectral densities Sx and SF 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. (109View Equation)):

S Ο•Sℐ ℏ2 SxSF = ---2- ≥ --. (116 ) ωp 4
This relation (as well as its more general version to be discussed later) for continuous linear measurements plays the same role as the uncertainty relation (105View Equation) for discrete measurements, establishing a universal connection between the accuracy of the monitoring and the perturbation of the monitored object.

Simple case: light in a coherent state.
Recall now that scheme of representing the quantized light wave as a sequence of short statistically-independent pulses with duration πœ€ ≡ πœ— we referred to in Section 3.2. It is the very concept we used here, and thus we can use it to calculate the spectral densities of the measurement and back-action noise sources for our simple device featured in Figure 20View Image assuming the light to be in a coherent state with classical amplitude ∘ ---------- Ac = 2ℐ0βˆ•(ℏωp ) (we chose As = 0 thus making the mean phase of light ⟨ˆΟ•βŸ© = 0). To do so we need to express phase ˆΟ• and energy 𝒲ˆ in the pulse in terms of the quadrature amplitudes ˆac,s(t). This can be done if we refer to Eq. (61View Equation) and make use of the following definition of the mean electromagnetic energy of the light wave contained in the volume vπœ— ≡ π’œc πœ— (here, π’œ is the effective cross-sectional area of the light beam):

vπœ— ------ vπœ— ∫ πœ—βˆ•2 𝒲ˆ = ---ˆE2(t) = ---- dτEˆ2 (τ) = 𝒲 + δ𝒲ˆ, (117 ) 4π 4π πœ— −πœ—βˆ•2
where 𝒲 = vπœ—π’ž2 A2βˆ•(8π ) = ℐ0 πœ— 0 c is the mean pulse energy, and
π’œc π’ž2 ∫ πœ—βˆ•2 ∘ --------- ∘ -------- δ𝒲ˆ ≃ ----02Ac dτˆac(τ) = 2 ℏωpℐ0πœ— ˆXπœ—(t) = 2ℏ ωp𝒲 Xˆπœ—(t) (118 ) 4π − πœ—βˆ•2
is a fluctuating part of the pulse energy8. We used here the definition of the mean pulse quadrature amplitude operators introduced in Eqs. (67View Equation). In the same manner, one can define a phase for each pulse using Eqs. (14View Equation) and with the assumption of small phase fluctuations (ΔΟ• β‰ͺ 1) one can get:
∫ ∘ ------ ∘ ----- ˆ -1-- πœ—βˆ•2 -ℏωp-ˆ ℏωp-ˆ Ο• ≃ Ac πœ— dτ ˆas(τ ) = 2ℐ0πœ— Yπœ— = 2𝒲 Yπœ—. (119 ) −πœ—βˆ•2
Thus, since in a coherent state Δ Xˆ2 = Δ ˆY 2= 1βˆ•2 πœ— πœ— the phase and energy uncertainties are equal to
∘ ---- 1 ℏω ∘ ------ Δ Ο• = -- ---p, Δ π’² = ℏωp 𝒲, (120 ) 2 𝒲
and hence
∘ ------ -c- --ℏ-- 2Π³-∘ ------ Δxmeas = 4 Π³ ωp𝒲 , Δpb.a.= c ℏωp𝒲. (121 )
Substituting these expressions into Eqs. (110View Equation, 109View Equation), we get the following expressions for the power (double-sided) spectral densities of the measurement and back-action noise sources:
S = ℏωp-, S = ℏω ℐ , (122 ) Ο• 4ℐ0 ℐ p 0
and
ℏc2 4ℏ ωpℐ0Π³2 Sx = ---------2, SF = -----2----. (123 ) 16ωp ℐ0Π³ c

We should emphasize that this simple measurement model and the corresponding uncertainty relation (116View Equation) are by no means general. We have made several rather strong assumptions in the course of derivation, i.e., we assumed:

  1. energy and phase fluctuations in each of the light pulses uncorrelated: βŸ¨π’²ˆ(tj)ˆΟ•(tj)⟩ = 0;
  2. all pulses to have the same energy and phase uncertainties Δ π’² and ΔΟ•, respectively;
  3. the pulses statistically independent from each other, particularly taking βŸ¨π’²ˆ(ti)𝒲 ˆ (tj)⟩ = ⟨ˆΟ•(ti)ˆΟ•(tj)⟩ = βŸ¨π’²ˆ(ti)ˆΟ•(tj)⟩ = 0 with ti ⁄= tj.

These assumptions can be mapped to the following features of the fluctuations ˆxfl(t) and Fˆb.a.(t) in the continuous case:

  1. these noise sources are mutually not correlated;
  2. they are stationary (invariant to the time shift) and, therefore, can be described by spectral densities Sx and SF;
  3. they are Markovian (white) with constant (frequency-independent) spectral densities.

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 ˆxfl and ˆ Fb.a.), Simple Quantum Meters (SQM).


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