In this section, we consider the interferometer configurations that use the idea of the cross-correlation of the shot and the radiation pressure noise sources discussed in Section 4.4. This cross-correlation allows the measurement and the back-action noise to partially cancel each other out and thus effectively reduce the sum quantum noise to below the SQL.
As we noted above, Eq. (378) tells us that this cross-correlation can be created by tuning either the
homodyne angle
, the squeezing angle
, or the detuning
. In Section 4.4, the simplest particular
case of the frequency-independent correlation created by means of measurement of linear combination of the
phase and amplitude quadratures, that is, by using the homodyne angle
, has been considered.
We were able to obtain a narrow-band sensitivity gain at some given frequency that was similar to the one
achievable by introducing a constant rigidity to the system, therefore such correlation was called effective
rigidity.
However, the broadband gain requires a frequency-dependent correlation, as it was first demonstrated
for optical interferometric position meters [148], and then for general position measurement case [82].
Later, this idea was developed in different contexts by several authors [81, 118, 159, 90
, 70, 69, 149, 9].
In particular, in [90
], a practical method of creation of the frequency-dependent correlation was proposed,
based on the use of additional filter cavities, which were proposed to be placed either between the
squeeze light source and the main interferometer, creating the frequency-dependent squeezing
angle (called pre-filtering), or between the main interferometer and the homodyne detector,
creating the effective frequency-dependent squeezing angle (post-filtering). As we show below, in
principle, both pre- and post-filtering can be used together, providing some additional sensitivity
gain.
It is necessary to note also an interesting method of noise cancellation proposed by Tsang and Caves recently [146]. The idea was to use matched squeezing; that is, to place an additional cavity inside the main interferometer and couple the light inside this additional cavity with the differential mode of the interferometer by means of an optical parametric amplifier (OPA). The squeezed light created by the OPA should compensate for the ponderomotive squeezing created by back-action at all frequencies and thus decrease the quantum noise below the SQL at a very broad frequency band. However, the thorough analysis of the optical losses influence, that as we show later, are ruinous for the subtle quantum correlations this scheme is based on, was not performed.
Coming back to the filter-cavities–based interferometer topologies, we limit ourselves here by the case of
the resonance-tuned interferometer, . This assumption simplifies all the equations considerably, and
allows one to clearly separate the sensitivity gain provided by the quantum noise cancellation due
to cross-correlation from the one provided by the optical rigidity, which will be considered in
Section 6.3.
We also neglect optical losses inside the interferometer, assuming that . In broadband
interferometer configurations considered here, with typical values of
, the influence of these
losses is negligible compared to those of the photodetector inefficiency and the losses in the filter cavities.
Indeed, taking into account the fact that with modern high-reflectivity mirrors, the losses per bounce do not
exceed
, and the arms lengths of the large-scale GW detectors are equal to several
kilometers, the values of
, and, correspondingly,
, are feasible. At the same
time, the value of photodetector quantum inefficiency
(factoring in the
losses in the interferometer output optical elements as well) is considered quite optimistic. Note,
however, that in narrow-band regimes considered in Section 6.3, the bandwidth
can be much
smaller and influence of
could be significant; therefore, we take these losses into account in
Section 6.3.
Using these assumptions, the quantum noises power (double-sided) spectral densities (376), (377
) and
(378
) can be rewritten in the following explicit form:
Eq. (381) and (385
) for the sum quantum noise and its power (double-sided) spectral density in this
case can also be simplified significantly:
where
In Section 6.1.2 we consider the optimization of the spectral density (391), assuming that the arbitrary
frequency dependence of the homodyne and/or squeezing angles can be implemented. As we see below, this
case corresponds to the ideal lossless filter cavities. In Section 6.1.3, we consider two realistic schemes,
taking into account the losses in the filter cavities.
It is evident that this minimum is provided by
In this case, the sum quantum noise power (double-sided) spectral density is equal to It is easy to note the similarity of this spectral density with the ones of the toy position meter considered above, see Eq. (173 The spectral density (394) was first calculated in the pioneering work of [38], where the
existence of two kinds of quantum noise in optical interferometric devices, namely the measurement
(shot) noise and the back action (radiation pressure) noise, were identified for the first time, and
it was shown that the injection of squeezed light with
into the interferometer dark
port is equivalent to the increase of the optical pumping power. However, it should be noted
that in the presence of optical losses this equivalence holds unless squeezing is not too strong,
.
The noise spectral density curves for the resonance-tuned interferometer are drawn in Figure 34. The
default parameters for this and all subsequent similar plots are chosen to be close to those planned for the
Advanced LIGO interferometer: the value of
corresponds to the
circulating power of
,
, and
; the interferometer bandwidth
is close to the one providing the best sensitivity for Advanced LIGO in
the presence of technical noise [93
]; 10 dB squeezing (
), which corresponds to the
best squeezing available at the moment (2011) in the low-frequency band [102, 150, 151]);
can be considered a reasonably optimistic estimate for the real interferometer quantum
efficiency.
Noteworthy is the proximity of the plots for the interferometer with 10 dB input squeezing and the one with 10-fold increased optical power. The noticeable gap at higher frequencies is due to optical loss.
The sum quantum noise power (double-sided) spectral density (403) is plotted in Figure 35
for the ideal
lossless case and for
(dotted line). In both cases, the optical power and the squeezing factor are
equal to
and
, respectively.
Compare this spectral density with the one for the frequency-dependent squeezing angle (pre-filtering)
case, see Eq. (403). The shot noise components in both cases are exactly equal to each other. Concerning
the residual back-action noise, in the pre-filtering case it is limited by the available squeezing, while in the
post-filtering case – by the optical losses. In the latter case, were there no optical losses, the back-action
noise could be removed completely, as shown in Figure 35
(left). For the parameters of the noise curves
presented in Figure 35
(right), the post-filtering still has some advantage of about 40% in the back-action
noise amplitude
.
Note that the required frequency dependences (404) and (407
) in both cases are similar to
each other (and become exactly equal to each other in the lossless case
). Therefore,
similar setups can be used in both cases in order to create the necessary frequency dependences
with about the same implementation cost. From this simple consideration, it is possible to
conclude that pre-filtering is preferable if good squeezing is available, and the optical losses are
relatively large, and vice versa. In particular, post-filtering can be used even without squeezing,
.
Concerning the squeezing angle, we can reuse Eqs. (400) and (401
). The minimum of the spectral
density (401
) in
corresponds to
It is easy to see that in the ideal lossless case the double-filtering configuration reduces to a post-filtering
one. Really, if , the spectral density (410
) becomes exactly equal to that for the post-filtering
case (408
), and the frequency dependent squeezing angle (411
) degenerates into a constant value (405
).
However, if
, then the additional pre-filtering allows one to decrease more the residual back-action
term. For example, if
and
then the gain in the back-action noise amplitude
is
equal to about 25%.
We have plotted the sum quantum noise spectral density (410) in Figure 34
, right (dash-dots). This
plot demonstrates the best sensitivity gain of about 3 in signal amplitude, which can be provided employing
squeezing and filter cavities at the contemporary technological level.
Due to the presence of the residual back-action term in the spectral density (410), there exists an
optimal value of the coupling factor
(that is, the optical power) which provides the minimum to the
sum quantum noise spectral density at any given frequency
:
In our particular case, the fact that the additional noise associated with the photodetector
quantum inefficiency does not correlate with the quantum fluctuations of the light
in the interferometer gives rise to this limit. This effect is universal for any kind of optical
loss in the system, impairing the cross-correlation of the measurement and back-action noises
and thus limiting the performance of the quantum measurement schemes, which rely on this
cross-correlation.
Noteworthy is that Eq. (410) does not take into account optical losses in the filter cavities. As we
shall see below, the sensitivity degradation thereby depends on the ratio of the light absorption
per bounce to the filter cavities length,
. Therefore, this method calls for long filter
cavities. In particular, in the original paper [90
], filter cavities with the same length as the main
interferometer arm cavities (4 km), placed side by side with them in the same vacuum tubes, were
proposed. For such long and expensive filter cavities, the influence of their losses indeed can be
small. However, as we show below, in Section 6.1.3, for the more practical short (up to tens
of meters) filter cavities, optical losses thereof could be the main limiting factor in terms of
sensitivity.
This narrow-band gain could be more interesting not for the full-scale GW detectors (where broadband
optimization of the sensitivity is required in most cases) but for smaller devices like the 10-m Hannover
prototype interferometer [7], designed for the development of the measurement methods with sub-SQL
sensitivity. Due to shorter arm length, the bandwidth in those devices is typically much larger than the
mechanical frequencies
. If one takes, e.g., the power transmissivity value of
for the ITMs
and length of arms equal to
, then
, which is above the typical working frequencies
band of such devices. In the literature, this particular case is usually referred to as a bad cavity
approximation.
In this case, the coupling factor can be approximated as:
In Figure 36, the SQL beating factor
The minima of these plots form the common envelope, given by Eqs. (410) and (416
):
In both cases, the filter cavity can be described by the input/output relation, which can be easily
obtained from Eqs. (290) and (291
) by setting
(there is no classical pumping in the filter cavity
and, therefore, there is no displacement sensitivity) and by some changes in the notations:
In order to demonstrate how the filter cavity works, consider the particular case of the lossless cavity. In this case,
and the reflection matrix describes field amplitude rotations with the frequency-dependent rotation angle: where The phase factor Let us now analyze the influence of the filter cavities on the interferometer sensitivity in post- and
pre-filtering variational schemes. Start with the latter one. Suppose that the light, entering the
interferometer from the signal port is in the squeezed state with fixed squeezing angle and
squeezing factor
and thus can be described by the following two-photon quadrature vector
In a similar manner, we can consider the post-filtering schemes. Consider a homodyne detection scheme
with losses, described by Eq. (272). Suppose that prior to detection, the light described by the quadrature
vector
, reflects from the filter cavity. In this case, the photocurrent (in Fourier representation) is
proportional to
It is easy to see that the necessary frequency dependencies of the homodyne and squeezing
angles (404) or (407
) (with the second-order polynomials in
in the r.h.s. denominators) cannot
be implemented by the rotation angle (432
) (with its first order in
polynomial in the
r.h.s. denominator). As was shown in the paper [90], two filter cavities are required in both
these cases. In the double pre- and post-filtering case, the total number of the filter cavities
increases to four. Later it was also shown that, in principle, arbitrary frequency dependence of the
homodyne and/or squeezing angle can be implemented, providing a sufficient number of filter
cavities [35].
However, in most cases, a more simple setup consisting of a single filter cavity might suffice.
Really, the goal of the filter cavities is to compensate the back-action noise, which contributes
significantly in the sum quantum noise only at low frequencies . However, when
Following this reasoning, we consider below two schemes, each based on a single filter cavity that realize pre-filtering and post-filtering, respectively.
Following the prescriptions of Section 6.1.2, we suppose the homodyne angle defined by Eq. (402). The
optimal squeezing angle should then be equal to zero at higher frequencies, see (404
). Taking into account
that the phase shift introduced by the filter cavity goes to zero at high frequencies, we obtain that the
squeezing angle
of the input squeezed vacuum must be zero. Combining Eqs. (390
) and (423
) taking
these assumptions into account, we obtain the following equation for the sum quantum noise of the
pre-filtering scheme:
In the ideal lossless filter cavity case, taking into account Eq. (431), this spectral density can be
simplified as follows:
Along similar lines, the post-filtering scheme drawn in the right panel of Figure 37 can be considered.
Here, the squeezed-vacuum produced by the squeezor first passes through the interferometer and then,
coming out, gets reflected from the filter cavity, gaining a frequency-dependent phase shift, which is
equivalent to introducing a frequency dependence into the homodyne angle, and then goes to the fixed
angle homodyne detector. Taking into account that this equivalent homodyne angle at high
frequencies has to be
, and that the phase shift introduced by the filter cavity goes to zero
at high frequencies, we obtain that the real homodyne angle must also be
. Assuming
that the squeezing angle is defined by Eq. (405
) and again using Eqs. (390
) and (423
), we
obtain that the sum quantum noise and its power (double-sided) spectral density are equal to
It is easy to show that substitution of the conditions (441) and (447
) into Eqs. (440
) and (445
),
respectively, taking Condition (437
) into account, results in spectral densities for the ideal frequency
dependent squeezing and homodyne angle, see Eqs. (403
) and (408
).
In the general case of lossy filter cavities, the conditions (404) and (407
) cannot be satisfied exactly by a
single filter cavity at all frequencies. Therefore, the optimal filter cavity parameters should be
determined using some integral sensitivity criterion, which will be considered at the end of this
section.
However, it would be a reasonable assumption that the above consideration holds with good precision, if losses in the filter cavity are low compared to other optical losses in the system:
This inequality can be rewritten as the following condition for the filter cavity specific losses: In particular, for our standard parameters used for numerical estimates ( Another more crude limitation can be obtained from the condition that should be small compared
to the filter cavity bandwidth
:
In the low and medium frequency range, where back-action noise dominates, and wherein our interest is
focused, the most probable source of signal is the gravitational radiation of the inspiraling binary systems of
compact objects such as neutron stars and/or black holes [131, 124]. In this case, the SNR is equal to
(see [58])
Since our goal here is not the maximal value of the SNR itself, but rather the relative sensitivity gain
offered by the filter cavity, and the corresponding optimal parameters and
, providing this gain,
we choose to normalize the SNR by the value corresponding to the ordinary interferometer (without the
filter cavities):
We optimized numerically the ratio , with filter cavity half-bandwidth
and
detuning
as the optimization parameters, for the values of the specific loss factor
ranging from
(e.g., very long 10 km filter cavity with
) to
(e.g., 10 m
filter cavity with
). Concerning the main interferometer parameters, we used the
same values as in all our previous examples, namely,
,
, and
.
The results of the optimization are shown in Figure 38. In the left pane, the optimal values of the filter
cavity parameters
and
are plotted, and in the right one the corresponding optimized values of
the SNR are. It follows from these plots that the optimal values of
and
are virtually the same as
, while the specific loss factor
satisfies the condition (448
), and starts to deviate sensibly from
only when
approaches the limit (451
). Actually, for such high values of specific losses, the
filter cavities only degrade the sensitivity, and the optimization algorithm effectively turns them off,
switching to the ordinary frequency-independent squeezing regime (see the right-most part of the right
pane).
It also follows from these plots that post-filtering provides slightly better sensitivity, if the optical losses
in the filter cavity are low, while the pre-filtering has some advantage in the high-losses scenario. This
difference can be explained in the following way [87]. The post-filtration effectively rotates the homodyne
angle from (phase quadrature) at high frequencies to
(amplitude quadrature) at
low frequencies, in order to measure the back-action noise, which dominates the low frequencies. As a result,
the optomechanical transfer function reduces at low frequencies, emphasizing all noises introduced after the
interferometer [see the factor
in the denominator of Eq. (445
)]. In the pre-filtering case
there is no such effect, for the value of
, corresponding to the maximum of the
optomechanical transfer function, holds for all frequencies (the squeezing angle got rotated
instead).
The optimized sum quantum noise power (double-sided) spectral densities are plotted in Figure 39 for
several typical values of the specific loss factor, and for the same values of the rest of the parameters, as in
Figure 38
. For comparison, the spectral densities for the ideal frequency-dependent squeezing angle
Eqs. (403
) and homodyne angle (408
) are also shown. These plots clearly demonstrate that providing
sufficiently-low optical losses (say,
), the single filter cavity based schemes can provide
virtually the same result as the abstract ones with the ideal frequency dependence for squeezing or
homodyne angles.
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Living Rev. Relativity 15, (2012), 5
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