The abstract scheme of such a device is drawn in Figure 21. It consists of a probe
that is exposed
to the action of a classical force
, and the meter. The action of this force on the probe causes its
displacement
that is monitored by the meter (e.g., light, circulating in the interferometer). The output
observable of the meter
is monitored by some arbitrary classical device that makes a measurement
record
. The quantum nature of the probe–meter interaction is reflected by the back-action force
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
in the readout
signal. Quantum mechanically, this system can be described by the following Hamiltonian:
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and 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 [22], and Theorems 3 and 4 in
Appendix 3.7 of [43
] for proof): For a linear system with Hamiltonian (124
), for any linear observable
of the probe and for any linear observable
of the meter, their full Heisenberg evolutions are given by:
For time independent Hamiltonian and operator
(in the Schrödinger picture), the
susceptibilities are invariant to time shifts, i.e.,
, therefore they depend only on
the difference of times:
. In this case, one can rewrite Eqs. (126
) in frequency domain
as:
Let us now use these theorems to find the full set of equations of motion for the system of linear
observables ,
and
that fully characterize our linear measurement process in the scheme
featured in Figure 21
:
The meaning of the above equations is worth discussing. The first of Eqs. (129) describes how the
readout observable
of the meter, say the particular quadrature of the outgoing light field
measured by the homodyne detector (cf. Eq. (39
)), depends on the actual displacement
of the probe, and the corresponding susceptibility
is the transfer function for
the meter from
to
. The term
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 [90
] 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 . 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, 22
, 43
, 33] on the observable
requiring that it should commute with itself
at any moment of time:
The second equation in (129) 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,
,
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
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 (129) 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:
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The second value, , 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, 22
].
And finally, 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
, momentum
,
and, possibly, on higher order time derivatives of
taken at
, as per the structure of the operator
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
the linear transformation corresponding to first applying the
operator
on the readout quantity that results in expressing
in terms of the probe’s
displacement:
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with standing for the meter’s own quantum noise (measurement uncertainty), and
then applying a probe dynamics operator
that yields a force signal equivalent to the readout quantity
:
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The term vanishes since
is the solution of a free-evolution equation of motion. Thus, we
see that the result of measurement contains two noise sources,
and
, which comprise the
sum noise masking the signal force
.
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. (129) in the spectral form:
It is common to normalize the output quantity of the meter to unit signal. In GW
interferometers, two such normalizations are popular. The first one tends to consider the tidal force
as a signal and thus set to 1 the coefficient in front of
in Eq. (135
). The other
one takes GW spectral amplitude
as a signal and sets the corresponding coefficient
in
to unity. Basically, these normalizations are equivalent by virtue of Eq. (12
) as:
These two new observables that embody the two types of noise inherent in any linear measurement satisfy the following commutation relations:
that can be interpreted in the way that In particular, this can be seen when one calculates the power (double-sided) spectral density of the sum
noise :
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 (148), 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.
http://www.livingreviews.org/lrr-2012-5 |
Living Rev. Relativity 15, (2012), 5
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