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 26. The main difference of this scheme from the position meters considered above (see Figures 20, 25) 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 . An outgoing pulse acquires a phase shift proportional to the difference of the test-object positions at time moments separated by , which is proportional to the test-mass average velocity in this time interval ( indicates the time moment after the second reflection):

where

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:

In spectral representation these equations yield:
where
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:

and in spectral form as:

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

and allows one to expand the exponents in Eqs. (208, 209) into a Taylor series:
where
Spectral densities of theses noises are equal to
where
Note also that

The apparent difference of the spectral densities presented in Eq. (215) from the ones describing the ‘ordinary’ position meter (see Eqs. (191)) 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, , allows the reduction of the sum noise spectral density to arbitrarily small values. One can easily see it after the substitution of Eq. (215) into Eq. (144) with a free mass in mind:

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

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 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 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:
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 (218) conditioned on uncertainty relation (217), one needs to set
which allows one to overcome the SQL simply by choosing the right fixed homodyne angle:
Then the sum noise is equal to
and, in principle, can be made arbitrarily small, if a sufficient value of 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. (122), hence the sum noise power (double-sided) spectral density for the speed meter takes the following form:

where for our scheme is defined in Eq. (157). 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 of a free mass is directly proportional to its momentum . And the momentum in turn is, as an integral of motion, a QND-observable, i.e., it satisfies the simultaneous measurability condition (131):

But this connection between and 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 twice, with opposite signs of the coupling factor. Therefore, the Lagrangian of this scheme can be written as:

where
is the test-mass velocity, stands for the meter’s observable, which provides coupling to the test mass, is the self-Lagrangian of the meter, and is the coupling factor, which has the form of two short pulses with the opposite signs, separated by the time , see Figure 27. We suppose for simplicity that the evolution of the meter observable 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 from consideration.

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 with the operator of measured observable does not vanish [28]. However, it can be shown that a more general condition is satisfied:

where is the evolution operator of probe-meter dynamics from the initial moment when the measurement starts till the final moment when it ends. Basically, the latter condition guarantees that the value of 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 , the Lagrangian (225) can be converted to the form, satisfying the simple condition of [28]:

where
see Figure 27. 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:

Note that the antisymmetric shape of the function guarantees that the coupling factor becomes equal to zero when the measurement ends. The canonical momentum of the mass is equal to
and the Hamiltonian of the system reads

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

(if is proportional to the number of quanta in the light pulse then is proportional to its phase), which represents the output signal of the meter. The Heisenberg equations of motion for these observables are the following:
These equations show clearly that (i) a canonical momentum is preserved by this measurement scheme while (ii) the test-mass velocity , as well as its kinematic momentum , are perturbed during the measurement (that is, while ), yet restore their initial values after the measurement, and (iii) the output signal of the meter carries the information about this perturbed value of the velocity.

Specify to be of a simple rectangular shape:

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

with

the test-mass velocity during the measurement [compare with Eqs. (203, 204)].

Therefore, by detecting the variable , the perturbed value of velocity is measured with an imprecision

where is the initial uncertainty of . The test-mass position perturbation after this measurement is proportional to the initial uncertainty of :
It follows from the last two equations that
The sum error of the initial velocity estimate yielding from this measurement is thus equal to:
As we see, it is limited by a value of the velocity measurement SQL:

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

instead of , which gives a sum measurement uncertainty for the initial velocity , proportional to the initial uncertainty of only:
and hence not limited by the SQL.