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):
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:
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
):
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
):
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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:
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:
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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 9,
the Lagrangian (225) can be converted to the form, satisfying the simple condition of [28]:
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 The complete set of observables describing our system includes in addition apart to ,
, and
,
the observable
canonically conjugated to
:
Specify to be of a simple rectangular shape:
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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 Therefore, by detecting the variable , the perturbed value of velocity
is measured with an
imprecision
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
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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