A quantum speed meter epitomizes the approach to the broadband SQL beating, in some sense, opposite to
the one based on the quantum noises cross-correlation tailoring with filter cavities, considered above. Here,
instead of fitting the quantum noise spectral dependence to the Fabry–Pérot–Michelson interferometer
optomechanical coupling factor (389), the interferometer topology is modified in such a way as to mold the
new optomechanical coupling factor
so that it turns out frequency-independent in the
low- and medium-frequency range, thus making the frequency-dependent cross-correlation not
necessary.
The general approach to speed measurement is to use pairs of position measurements separated by a
time delay , where
is the characteristic signal frequency (cf. the simplified consideration in
Section 4.5). Ideally, the successive measurements should be coherent, i.e., they should be performed by the
same photons. In effect, the velocity
of the test mass is measured in this way, which gives the necessary
frequency dependence of the
.
In Section 4.5, we have considered the simplest toy scheme that implements this principle and which
was first proposed by Braginsky and Khalili in [21]. Also in this paper, a modified version of this scheme,
called the sloshing-cavity speed meter, was proposed. This version uses two coupled resonators (e.g.,
microwave ones), as shown in Figure 40
(left), one of which (2), the sloshing cavity, is pumped on resonance
through the input waveguide, so that another one (1) becomes excited at its eigenfrequency
. The eigenfrequency of resonator 1 is modulated by the position
of the test mass
and puts a voltage signal proportional to position
into resonator 2, and a voltage signal
proportional to velocity
into resonator 1. The velocity signal flows from resonator 1 into
an output waveguide, from which it is monitored. One can understand the production of this
velocity signal as follows. The coupling between the resonators causes voltage signals to slosh
periodically from one resonator to the other at frequency
. After each cycle of sloshing, the sign of
the signal is reversed, so the net signal in resonator 1 is proportional to the difference of the
position at times
and
, thus implementing the same principle of the double position
measurement.
Later, the optical version of the sloshing-cavity speed-meter scheme suitable for large-scale laser GW
detectors was developed [20, 126, 127]. The most elaborated variant proposed in [127
] is shown in
Figure 40
(right). Here, the differential mode of a Michelson interferometer serves as the resonator 1 of the
initial scheme of [21], and an additional kilometer-scale Fabry–Pérot cavity – as the resonator 2, thus
making a practical interferometer configuration.
In parallel, it was realized by Chen and Khalili [42, 85
] that the zero area Sagnac
interferometer [140, 11, 145] actually implements the initial double-measurement variant of the quantum
speed meter, shown in Figure 26
. Further analysis with account for optical losses was performed in [50
] and
with detuned signal-recycling in [113
]. Suggested configurations are pictured in Figure 41
. The core idea is
that light from the laser gets split by the beamsplitter (BS) and directed to Fabry–Pérot cavities in the
arms, exactly as in conventional Fabry–Pérot–Michelson interferometers. However, after it leaves the
cavity, it does not go back to the beamsplitter, but rather enters the cavity in the other arm, and
only afterwards returns to the beamsplitter, and finally to the photo detector at the dark port.
The scheme of [42
] uses ring Fabry–Pérot cavities in the arms to spatially separate ingoing
and outgoing light beams to redirect the light leaving the first arm to the second one evading
the output beamsplitter. The variant analyzed in [85, 50
] uses polarized optics for the same
purposes: light beams after ordinary beamsplitter, having linear (e.g., vertical) polarization, pass
through the polarized beamsplitter (PBS), then meet the
plates that transform their linear
polarization into a circular one, and then enter the Fabry–Pérot cavity. After reflection from the
Fabry–Pérot cavity, light passes through a
-plate again, changing its polarization again to
linear, but orthogonal to the initial one. As a result, the PBS reflects it and redirects to another
arm of the interferometer where it passes through the same stages, restoring finally the initial
polarization and comes out of the interferometer. With the exception of the implementation
method for this round-robin pass of the light through the interferometer, both schemes have
the same performance, and the same appellation Sagnac speed meter will be used for them
below.
Visiting both arms, counter propagating light beams acquire phase shifts proportional to a sum of end
mirrors displacements of both cavities taken with time delay equal to average single cavity storage time
:
Both versions of the optical speed meter, the sloshing cavity and the Sagnac ones, promise about the same sensitivity, and the choice between them depends mostly on the relative implementation cost of these schemes. Below we consider in more detail the Sagnac speed meter, which does not require the additional long sloshing cavity.
We will not present here the full analysis of the Sagnac topology similar to the one we have provided
for the Fabry–Pérot–Michelson one. The reader can find it in [42, 50
]. We limit ourselves
by the particular case of the resonance tuned interferometer (that is, no signal recycling and
resonance tuned arm cavities). It seems that the detuned Sagnac interferometer can provide a
quite interesting regime, in particular, the negative inertia one [113]. However, for now (2011)
the exhaustive analysis of these regimes is yet to be done. We assume that the squeezed light
can be injected into the interferometer dark port, but consider only the particular case of the
classical optimization,
, which gives the best broadband sensitivity for a given optical
power.
In order to reveal the main properties of the quantum speed meter, start with the simplified case of
the lossless interferometer and the ideal photodetector. In this case, the sum quantum noise
power (double-sided) spectral density of the speed meter can be written in a form similar to the
one for the Fabry–Pérot–Michelson interferometer [see, e.g., Eqs. (386), (387
) and (388
)]:
The key advantage of speed meters over position meters is that at low frequencies, ,
is
approximately constant and reaches the maximum there:
This spectral density is plotted in Figure 42 (left). For comparison, spectral densities for the lossless
ordinary Fabry–Pérot–Michelson interferometer without and with squeezing, as well as for the ideal
post-filtering configuration [see Eq. (408
)] are also given. One might conclude from these plots that
the Fabry–Pérot–Michelson interferometer with the additional filter cavities is clearly better
than the speed meter. However, below we demonstrate that optical losses change this picture
significantly.
In speed meters, optical losses in the arm cavities could noticeably affect the sum noise at low frequencies, even if
because the radiation pressure noise component created by the arm cavity losses has a frequency dependence similar to the one for position meters (remember that This spectral density is plotted in Figure 42 (right), together with the lossy variants of the same
configurations as in Figure 42
(left), for the same moderately optimistic value of
, the losses part
of the bandwidth and for
[which corresponds to the losses
per bounce in
the 4 km length arms, see Eq. (322
)]. These plots demonstrate that the speed meter in more robust with
respect to optical losses than the filter cavities based configuration and is able to provide better sensitivity
at very low frequencies.
It should also be noted that we have not taken into account here optical losses in the filter cavity.
Comparison of Figure 42 with Figure 39
, where the noise spectral density for the more realistic
lossy–filter-cavity cases are plotted, shows that the speed meter has advantage over, at least, the short and
medium length (tens or hundred of meters) filter cavities. In the choice between very long (and hence
expensive) kilometer scale filter cavities and the speed meter, the decision depends, probably, on the
implementation costs of both configurations.
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Living Rev. Relativity 15, (2012), 5
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