4.5 Quantum speed meter

4.5.1 The idea of the quantum speed meter

The toy scheme that demonstrates a bare idea of the quantum speed meter is shown in Figure 26View Image. The main difference of this scheme from the position meters considered above (see Figures 20View Image, 25View Image) is that each light pulse reflects from the test mass twice: first from the front and then from the rear face after passing the delay line with delay time τ. An outgoing pulse acquires a phase shift proportional to the difference of the test-object positions at time moments separated by τ, which is proportional to the test-mass average velocity ˆ¯v(tj) in this time interval (tj indicates the time moment after the second reflection):

ϕˆrefl(t ) = ϕˆ(t) + 2гk τˆ¯v(t ), (203 ) j j p j
ˆ¯v(t) = xˆ(tj)-−-ˆx(tj −-τ-). (204 ) j τ
View Image

Figure 26: Toy example of the quantum speed-meter scheme.

We omit here mathematical details of the transition to the continuous measurement limit as they are essentially the same as in the position measurement case, see Section 4.1.2, and start directly with the continuous time equations. The output signal of the homodyne detector in the speed-meter case is described by the following equations:

ˆ Oˆ(t) = − ˆϕrefl(t) sin ϕ + ℐ(t) −-ℐ0-cosϕ , (205 ) LO 2ℐ0 LO ˆrefl ˆ ϕ (t) = ϕ (t) + 2гkp [ˆx(t) − ˆx(t − τ)]. (206 )
In spectral representation these equations yield:
ˆO(Ω ) x&tidle;(Ω) ≡ − -------------= ˆx(Ω) + ˆxfl(Ω ), (207 ) 2гkp sin ϕLO
[ ] 1 ˆℐ(Ω ) − ℐ ˆxfl(Ω ) = --------------- ϕˆ(Ω ) − ---------0cot ϕLO (208 ) 2гkp (1 − eiΩτ) 2ℐ0
is the equivalent displacement measurement noise.

The back-action force with account for the two subsequent light reflections off the faces of the probe, can be written as:

ˆ 2-г ˆ ˆ Fb.a.(t) = c [ℐ (t + τ) − ℐ(t)] (209 )
and in spectral form as:
2г −iΩ τ Fˆb.a.(Ω ) = -c-ˆℐ(Ω )(e − 1). (210 )

Then one can make a reasonable assumption that the time between the two reflections τ is much smaller than the signal force variation characteristic time (∼ Ω −1) that spills over into the following condition:

Ωτ ≪ 1, (211 )
and allows one to expand the exponents in Eqs. (208View Equation, 209View Equation) into a Taylor series:
ˆvfl(Ω) ˆxfl(Ω ) = ------, Fˆb.a.(t) = − iΩ ˆpb.a.(Ω), (212 ) − iΩ
[ ˆ ] ˆvfl(Ω) = ---1--- ϕˆ(Ω) − ℐ-(Ω-) −-ℐ cotϕLO , (213 ) 2гkp τ 2ℐ ˆp (Ω) = Fˆb.a.(Ω-) = 2г-τℐˆ(Ω). (214 ) b.a. − iΩ c
Spectral densities of theses noises are equal to
S Sx (Ω) = --v2, SF(Ω ) = Ω2Sp (Ω ), SxF (Ω) = − Svp(Ω ), (215 ) Ω
( ) 1 Sℐ 2 Sv = ---2-2--2 Sϕ + ---2 cot ϕLO , 4 г kpτ 4ℐ 4-г2τ2S-ℐ Sp = c2 , S Svp = − ---ℐ- cotϕLO (216 ) 2ωp ℐ
Note also that
2 Sx (Ω )SF (Ω ) − S2xF(Ω ) = SvSp − S2vp = SϕSℐ-≥ ℏ-. (217 ) ω2p 4

The apparent difference of the spectral densities presented in Eq. (215View Equation) from the ones describing the ‘ordinary’ position meter (see Eqs. (191View Equation)) is that they now have rather special frequency dependence. It is this frequency dependence that together with the cross-correlation of the measurement and back-action fluctuations, SxF ⁄= 0, allows the reduction of the sum noise spectral density to arbitrarily small values. One can easily see it after the substitution of Eq. (215View Equation) into Eq. (144View Equation) with a free mass χxx (Ω ) = − 1∕(M Ω2 ) in mind:

SF = M 2Ω4S (Ω ) + S (Ω) − 2M Ω2S (Ω) = Ω2 (M 2S + 2M S + S ). (218 ) x F xF v vp p

If there was no correlation between the back-action and measurement fluctuations, i.e., Svp = 0, then by virtue of the uncertainty relation, the sum sensitivity appeared limited by the SQL (161View Equation):

( ℏ2M 2 ) SF = Ω2 ------+ Sp ≥ ℏM Ω2. (219 ) 4Sp
One might wonder, what is the reason to implement such a complicated measurement strategy to find ourselves at the same point as in the case of a simple coordinate measurement? However, recall that in the position measurement case, a constant cross-correlation SxF ∝ cotϕLO allows one to get only a narrow-band sub-SQL sensitivity akin to that of a harmonic oscillator. This effect we called virtual rigidity, and showed that for position measurement this rigidity Kvirt = SxF ∕Sx is constant. In the speed-meter case, the situation is totally different; it is clearly seen if one calculates virtual rigidity for a speed meter:
KSM = -SxF-- = − Ω2 Svp. (220 ) virt Sx (Ω ) Sv
It turns out to be frequency dependent exactly in the way that is necessary to compensate the free mass dynamical response to the back-action fluctuations. Indeed, in order to minimize the sum noise spectral density (218View Equation) conditioned on uncertainty relation (217View Equation), one needs to set
SxF = − Svp = -SF-- = Sp- = const., (221 ) M Ω2 M
which allows one to overcome the SQL simply by choosing the right fixed homodyne angle:
8г2τ-2ωp-ℐ0- cot ϕLO = M c2 . (222 )
Then the sum noise is equal to
S S M 2Ω2 M 2Ω2 SF (Ω ) = -ϕ2-ℐ------ = ---2-2--2Sϕ. (223 ) ωp Sp 4 г kpτ
and, in principle, can be made arbitrarily small, if a sufficient value of S p is provided; that is, ifthe optomechanical coupling is sufficiently strong.

Simple case: light in a coherent state.
Let us consider how the spectral density of a speed meter will appear if the light field is in a coherent state. The spectral densities of phase and power fluctuations are given in Eqs. (122View Equation), hence the sum noise power (double-sided) spectral density for the speed meter takes the following form:

( )2 F F -ℏM-2c2Ω2--- ℏM-- Ω-- S-SQLf.m. S (Ω) = 16ωpℐ0г2 τ2 = 2τ2 Ωq = 2Ω2 τ2 , (224 ) q
where Ωq for our scheme is defined in Eq. (157View Equation). This formula indicates the ability of a speed meter to have a sub-SQL sensitivity in all frequencies provided high enough optical power and no optical loss.

4.5.2 QND measurement of a free mass velocity

The initial motivation to consider speed measurement rested on the assumption that a velocity ˆv of a free mass is directly proportional to its momentum ˆp = M ˆv. And the momentum in turn is, as an integral of motion, a QND-observable, i.e., it satisfies the simultaneous measurability condition (131View Equation):

′ ′ [ˆp(t), ˆp(t )] = 0 ∀t,t .

But this connection between ˆp and ˆv holds only if one considers an isolated free mass not coupled to a meter. As the measurement starts, the velocity value gets perturbed by the meter and it is not proportional to the momentum anymore. Let us illustrate this statement by our simple velocity measurement scheme. The distinctive feature of this example is that the meter probes the test position ˆx twice, with opposite signs of the coupling factor. Therefore, the Lagrangian of this scheme can be written as:

ˆ M--ˆv2 ˆ ˆ ℒ = 2 + β(t)ˆx𝒩 + ℒmeter, (225 )
dˆx- vˆ= dt (226 )
is the test-mass velocity, 𝒩 stands for the meter’s observable, which provides coupling to the test mass, ℒmeter is the self-Lagrangian of the meter, and β (t) is the coupling factor, which has the form of two short pulses with the opposite signs, separated by the time τ, see Figure 27View Image. We suppose for simplicity that the evolution of the meter observable 𝒩 can be neglected during the measurement (this is a reasonable assumption, for in real schemes of the speed meter and in the gedanken experiment considered above, this observable is proportional to the number of optical quanta, which does not change during the measurement). This assumption allows one to omit the term ℒmeter from consideration.
View Image

Figure 27: Real (β(t)) and effective (α(t)) coupling constants in the speed-meter scheme.

This Lagrangian does not satisfy the most well-known sufficient (but not necessary!) condition of the QND measurement, namely the commutator of the interaction Hamiltonian ℋˆint = − β(t)ˆx𝒩ˆ with the operator of measured observable ˆx does not vanish [28Jump To The Next Citation Point]. However, it can be shown that a more general condition is satisfied:

ˆ [U (0,τ), ˆx] = 0

where ˆU(0,τ ) is the evolution operator of probe-meter dynamics from the initial moment tstart = 0 when the measurement starts till the final moment tend = τ when it ends. Basically, the latter condition guarantees that the value of xˆ before the measurement will be equal to that after the measurement, but does not say what it should be in between (see Section 4.4 of [22] for details).

Moreover, using the following nice trick 9, the Lagrangian (225View Equation) can be converted to the form, satisfying the simple condition of [28]:

ˆ′ ˆ dα-(t)ˆx-ˆ ℒ = ℒ − dt 𝒩 , (227 )
∫ t α (t) = β(t′)dt′, (228 ) − ∞
see Figure 27View Image. It is known that two Lagrangians are equivalent if they differ only by a full time derivative of and arbitrary function of the generalized coordinates. Lagrangian equations of motion for the coordinates of the system are invariant to such a transformation.

The new Lagrangian has the required form with the interaction term proportional to the test-mass velocity:

M ˆv2 ˆℒ = -----− α (t)ˆv𝒩ˆ . (229 ) 2
Note that the antisymmetric shape of the function β (t) guarantees that the coupling factor α(t) becomes equal to zero when the measurement ends. The canonical momentum of the mass M is equal to
∂ ℒ p = -∂v = M v − α(t)𝒩 , (230 )
and the Hamiltonian of the system reads
ˆ [p +-α-(t)𝒩-]2- ℋ = pv − ℒ = 2M . (231 )

The complete set of observables describing our system includes in addition apart to ˆx, ˆp, and ˆ𝒩, the observable ˆΦ canonically conjugated to 𝒩ˆ:

[ˆΦ, ˆ𝒩 ] = iℏ (232 )
(if ˆ 𝒩 is proportional to the number of quanta in the light pulse then ˆ Φ is proportional to its phase), which represents the output signal ˆO of the meter. The Heisenberg equations of motion for these observables are the following:
dˆx (t) pˆ+ α (t)𝒩ˆ dpˆ(t) ----- = ˆv (t) = -----------, ----- = 0, dt M dt (233 ) dΦˆ(t) α(t)[ˆp + α (t)𝒩 ˆ ] d𝒩ˆ (t) ------= ----------------= α(t)ˆv(t), ------ = 0. dt M dt
These equations show clearly that (i) a canonical momentum ˆp is preserved by this measurement scheme while (ii) the test-mass velocity ˆv, as well as its kinematic momentum M ˆv, are perturbed during the measurement (that is, while α ⁄= 0), yet restore their initial values after the measurement, and (iii) the output signal of the meter ˆ Φ carries the information about this perturbed value of the velocity.

Specify α (t) to be of a simple rectangular shape:

{ 1, if 0 ≤ t < τ α(t) = 0, if t < 0 or t ≥ τ.

This assumption does not affect our main results, but simplifies the calculation. In this case, the solution of Eqs. (233View Equation) reads:

ˆx(τ) = ˆx(0) + ˆ¯vτ, (234a ) ˆΦ(τ) = ˆΦ (0) + ˆ¯v τ, (234b )


ˆ ˆp-+-𝒩ˆ- ¯v = M (235 )
the test-mass velocity during the measurement [compare with Eqs. (203View Equation, 204View Equation)].

Therefore, by detecting the variable Φ (τ), the perturbed value of velocity ¯v is measured with an imprecision

Δv = ΔΦ-, (236 ) meas τ
where Δ Φ is the initial uncertainty of Φ. The test-mass position perturbation after this measurement is proportional to the initial uncertainty of 𝒩:
Δ-𝒩--τ Δxb.a. = Δvb.a.τ = M . (237 )
It follows from the last two equations that
Δ Φ Δ𝒩 ℏ Δvmeas Δxb.a. = ---M---- ≥ 2M--. (238 )
The sum error of the initial velocity estimate vinit = p ∕m yielding from this measurement is thus equal to:
∘ (----)-----(-----)-- ∘ ----- Δ-Φ- 2 Δ𝒩--- 2 -ℏ-- Δvsum = τ + M ≥ M τ . (239 )
As we see, it is limited by a value of the velocity measurement SQL:
∘ ----- ℏ ΔvSQL = ----. (240 ) M τ

To overcome this SQL one has to use cross-correlation between the measurement error and back-action. Then it becomes possible to measure vinit with arbitrarily high precision. Such a cross-correlation can be achieved by measuring the following combination of the meter observables

ˆΦ(τ) − 𝒩ˆτ∕m = Φˆ(0) + ˆpτ- (241 ) M
instead of Φˆ(τ), which gives a sum measurement uncertainty for the initial velocity ˆp∕m, proportional to the initial uncertainty of ˆΦ only:
Δ-Φ- Δvsum = τ , (242 )
and hence not limited by the SQL.

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