Practical speed meter designs for quantum nondemolition gravitational-wave interferometers
Patricia Purdue and Yanbei Chen
Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125
共Received 27 July 2002; published 27 December 2002兲
In the quest to develop viable designs for third-generation optical interferometric gravitational-wave detec-
tors 共e.g., LIGO-III and EURO兲, one strategy is to monitor the relative momentum or speed of the test-mass
mirrors, rather than monitoring their relative position. A previous paper analyzed a straightforward but imprac-
tical design for a speed-meter interferometer that accomplishes this. This paper describes some practical
variants of speed-meter interferometers. Like the original interferometric speed meter, these designs in prin-
ciple can beat the gravitational-wave standard quantum limit 共SQL兲 by an arbitrarily large amount, over an
arbitrarily wide range of frequencies. These variants essentially consist of a Michelson interferometer plus an
extra ‘‘sloshing’’ cavity that sends the signal back into the interferometer with opposite phase shift, thereby
cancelling the position information and leaving a net phase shift proportional to the relative velocity. In
practice, the sensitivity of these variants will be limited by the maximum light power W
circ
circulating in the
arm cavities that the mirrors can support and by the leakage of vacuum into the optical train at dissipation
points. In the absence of dissipation and with squeezed vacuum 共power squeeze factor e
⫺ 2R
⯝0.1) inserted into
the output port so as to keep the circulating power down, the SQL can be beat by h/h
SQL
⬃
冑
W
circ
SQL
e
⫺ 2R
/W
circ
at all frequencies below some chosen f
opt
⯝100 Hz. Here W
circ
SQL
⯝800 kW(f
opt
/100 Hz)
3
is the power required to reach the SQL in the absence of squeezing. 共However, as the
power increases in this expression, the speed meter becomes more narrow band; additional power and reopti-
mization of some parameters are required to maintain the wide band. See Sec. III B.兲 Estimates are given of the
amount by which vacuum leakage at dissipation points will debilitate this sensitivity 共see Fig. 12兲; these losses
are 10% or less over most of the frequency range of interest (f ⲏ 10 Hz). The sensitivity can be improved,
particularly at high freqencies, by using frequency-dependent homodyne detection, which unfortunately re-
quires two 4-km-long filter cavities 共see Fig. 4兲.
DOI: 10.1103/PhysRevD.66.122004 PACS number共s兲: 04.80.Nn, 03.67.⫺a, 42.50.Dv, 95.55.Ym
I. INTRODUCTION
This paper is part of the effort to explore theoretically
various ideas for a third-generation interferometric
gravitational-wave detector. The goal of such detectors is to
beat, by a factor of 5 or more, the standard quantum limit
共SQL兲—a limit that constrains interferometers 关1兴 such as
the first generation of the Laser Interferometric Gravitational
Wave Observatory 共LIGO-I兲 which have conventional opti-
cal topology 关2,3兴, but does not constrain more sophisticated
‘‘quantum nondemolition’’ 共QND兲 interferometers 关4,5兴.
The concepts currently being explored for third-
generation detectors fall into two categories: external read-
out and intracavity readout. In interferometer designs with
external readout topologies, light exiting the interferometer
is monitored for phase shifts, which indicate the motion of
the test masses. Examples include conventional interferom-
eters and their variants 共such as LIGO-I 关2,3兴, LIGO-II 关6兴,
and those discussed in Ref. 关7兴兲, as well as the speed-meter
interferometers discussed here and in a previous paper 关8兴.In
intracavity readout topologies, the gravitational-wave force
is fed via light pressure onto a tiny internal mass, whose
displacement is monitored with a local position transducer.
Examples include the optical bar, symphotonic state, and op-
tical lever schemes discussed by Braginsky, Khalili, and
Gorodetsky 关9–11兴. These intracavity readout interferom-
eters may be able to function at much lower light powers
than external readout interferometers of comparable sensitiv-
ity because the QND readout is performed via the local po-
sition transducer 共perhaps microwave-technology based兲, in-
stead of via the interferometer’s light; however, the designs
are not yet fully developed.
At present, the most complete analysis of candidate de-
signs for third-generation external-readout detectors has been
carried out by Kimble, Levin, Matsko, Thorne, and Vyatcha-
nin 关7兴共KLMTV兲. They examined three potential designs for
interferometers that could beat the SQL: a squeezed-input
interferometer, which makes use of squeezed vacuum being
injected into the dark port; a variational-output scheme in
which frequency-dependent homodyne detection was used;
and a squeezed-variational interferometer that combines the
features of both. 共Because the KLMTV designs measure the
relative positions of the test masses, we shall refer to them as
position meters, particularly when we want to distinguish
them from the speed meters that, for example, use
variational-output techniques.兲 Although at least some of the
KLMTV position-meter designs have remarkable perfor-
mance in the lossless limit, all of them are highly susceptible
to losses.
In addition, we note that the KLMTV position meters
each require four kilometer-scale cavities 共two arm cavities
⫹ two filter cavities兲. The speed meters described in this pa-
per require at least three kilometer-scale cavities 关two arm
cavities⫹one ‘‘sloshing’’ cavity 共described below兲兴.Ifwe
use a variational-output technique, as KLMTV did, the re-
sulting interferometer will have five kilometer-scale cavities
关two arm cavities⫹ one sloshing cavity⫹two filter cavities
共again, see below兲兴. The speed meter described in
PHYSICAL REVIEW D 66, 122004 共2002兲
0556-2821/2002/66共12兲/122004共24兲/$20.00 ©2002 The American Physical Society66 122004-1
this paper can achieve a performance significantly better than
a conventional position meter, as shown in Fig. 1. 共By ‘‘con-
ventional,’’ we mean ‘‘without any QND techniques.’’ An
example is LIGO-I.兲 The squeezed-input speed meter 共SISM兲
noise curve shown in Fig. 1 beats the SQL by a factor of
冑
10
in amplitude and has fixed-angle squeezed vacuum injected
into the dark port 关this allows the interferometer to operate at
a lower circulating power than would otherwise be necessary
to achieve that level of sensitivity, as described by Eq. 共3兲
below兴. The squeezed-variational position meter 共SVPM兲,
which requires squeezed vacuum and frequency-dependent
homodyne detection, is more sensitive than the squeezed-
input speed meter over much of the frequency range of in-
terest, but the speed meter has the advantage at low frequen-
cies. It should also be noted that the squeezed-variational
position meter requires four kilometer-scale cavities 共as de-
scribed in the previous paragraph兲, whereas the squeezed-
input speed meter requires three.
If frequency-dependent homodyne detection is added to
the squeezed-input speed meter, the resulting squeezed-
variational speed meter 共SVSM兲 can be optimized to beat the
squeezed-variational position meter over the entire frequency
range. Figure 1 contains two squeezed-variational speed
meter curves; one is optimized to match the squeezed-input
speed meter curve at low frequencies, and the other is opti-
mized for comparison with the squeezed-variational postion-
meter curve 共resulting in less sensitivity at high frequencies兲.
The original idea for a speed meter, as a device for mea-
suring the momentum of a single test mass, was conceived
by Braginsky and Khalili 关12兴 and was further developed by
Braginsky, Gorodetsky, Khalili, and Thorne 共BGKT兲关13兴.In
their appendix, BGKT sketched a design for an interferomet-
ric gravity wave speed meter and speculated that it would be
able to beat the SQL. This was verified in Ref. 关8兴共Paper I兲,
where it was demonstrated that such a device could in prin-
ciple beat the SQL by an arbitrary amount over a wide range
of frequencies. However, the design presented in that paper,
which we shall call the two-cavity speed-meter design, had
three significant problems: it required 共i兲 a high circulating
power (⬃8 MW to beat the SQL by a factor of 10 in noise
power at 100 Hz and below兲, 共ii兲 a large amount of power
coming out of the interferometer with the signal 共⬃0.5 MW兲,
and 共iii兲 an exorbitantly high input laser power
(ⲏ300 MW). The latter two problems are effectively elimi-
nated by the alternate class of speed meters presented here—
designs that are based on the same QND mechanism de-
scribed in Refs. 关8,12,13兴 but implemented by different
optical configurations. In addition, techniques for reducing
the needed circulating power are also discussed. These im-
provements bring interferometric speed meters into the realm
of practicality.
A simple version of the three-cavity speed-meter design to
be discussed in this paper is shown in Fig. 2. In 共an idealized
theorist’s version of兲 this speed meter, the input laser light
关with electric field denoted I(
) in Fig. 2兴 passes through a
power-recycling mirror into a standard Michelson interfer-
ometer. The relative phase shifts of the two arms are adjusted
so that all of the input light returns to the input port, leaving
the other port dark 关i.e., the interferometer is operating in the
symmetric mode so D(
)⫽0 in Fig. 2兴. In effect, we have a
resonant cavity shaped like ⬜. When the end mirrors move,
they will put a phase shift on the light, causing some light to
enter the antisymmetric mode 共shaped like ) and come out
the dark port. So far, this is the same as conventional inter-
FIG. 1. Comparison of noise curves 共with losses兲 of several
interferometer configurations. Each of these curves has been opti-
mized in a way that is meant to illustrate their relative advantages
and disadvantages. The conventional position meter 共CPM兲关7兴 has
W
circ
⫽ 820 kW and bandwidth
␥
⫽ cT/4L⫽ 2
⫻ 100 Hz. The
squeezed-input speed meter 共SISM兲—optimized to agree with the
conventional position meter at high frequencies—has power
squeeze factor e
⫺ 2R
⫽ 0.1, optimal frequency
opt
⫽ 2
⫻ 105 Hz,
extraction rate
␦
⫽ 2
opt
, and sloshing frequency ⍀⫽
冑
3
opt
. The
squeezed-variational position meter 共SVPM兲关7兴 has the same pa-
rameters as the conventional position meter, with power squeeze
factor e
⫺ 2R
⫽ 0.1. There are two squeezed-variational speed-meter
curves 共SVSM兲.One共black dashes兲 uses the same parameters as
the squeezed-input speed meter. The other 共solid curve兲 has been
optimized to compare more directly with the squeezed-variational
position meter; it has ⍀⫽ 2
⫻ 95 Hz and
␦
⫽ 2
⫻ 100 Hz 共note
that our
␦
is equivalent to the bandwidth
␥
used to describe the
interferometers in Ref. 关7兴兲.
FIG. 2. Simple version of three-cavity design for speed-meter
interferometer. The main laser input port is denoted by I(
), where
⫽ t⫺ z/c. The signal is extracted at the bottom mirror 关denoted
Q(
), where
⫽ t⫹ z/c]. The difference between the one- and
two-port versions is the mirror shown in gray.
P. PURDUE AND Y. CHEN PHYSICAL REVIEW D 66, 122004 共2002兲
122004-2
ferometer designs 共but without the optical cavities in the two
interferometer arms兲.
Next, we feed the light coming out of the dark port
关
D(
)
兴
into a sloshing cavity 关labeled K(
) and L(
)in
Fig. 2兴. The light carrying the position information sloshes
back into the ‘‘antisymmetric cavity’’ with a phase shift of
180°, cancelling the position information in that cavity and
leaving only a phase shift proportional to the relative veloc-
ity of the test masses.
1
The sloshing frequency is
⍀⫽
c
冑
T
s
2L
, 共1兲
where T
s
is the power transmissivity of the sloshing mirror, L
is the common length of all three cavities, and c is the speed
of light. We read the velocity signal
关
Q(
)
兴
out at an extrac-
tion mirror 共with transmissivity T
o
), which gives a signal-
light extraction rate of
␦
⫽
cT
o
L
. 共2兲
We have used the extraction mirror to put the sloshing cavity
parallel to one of the arms of the Michelson part of the in-
terferometer, allowing this interferometer to fit into the ex-
isting LIGO facilities. The presence of the extraction mirror
essentially opens two ports to our system. We can use both
outputs, or we can add an additional mirror to close one port
共the gray mirror in Fig. 2兲. We will focus on the latter case in
this paper.
The sensitivity h of this interferometer, compared to the
SQL, can be expressed as
2
h
h
SQL
⬃
冑
W
circ
SQL
e
2R
W
circ
⯝
冑
800 kW
e
2R
W
circ
, 共3兲
where W
circ
is the power circulating in the arms, W
circ
SQL
⯝800 kW(f
opt
/100 Hz)
3
is the power required to reach the
SQL in the absence of squeezing 共for the arms of length L
⫽ 4 km and test masses with mass m⫽ 40 kg), and e
2R
is the
power squeeze factor.
3
With no squeezed vacuum, the
1
The net signal is proportional to the relative velocities of the test
masses, assuming that the frequencies
of the test masses’ motion
are
Ⰶ ⍀⫽ (sloshing frequency). However, the optimal regime of
operation for the speed meter is
⬃⍀. As a result, the output
signal contains a sum over odd time derivatives of position 共see the
discussion in Sec. III A兲. Therefore, the speed meter monitors not
just the relative speed of the test masses, but a mixture of all odd
time derivatives of the relative positions of the test masses.
2
It should be noted that, as the power increases in Eq. 共3兲, the
speed-meter performance becomes more narrow band. Additional
power and a re-optimization of some of the speed meter’s param-
eters are required to maintain the same bandwidth at higher sensi-
tivities. See Sec. III B for details.
3
For an explanation of squeezed vacuum and squeeze factors, see,
for example, KLMTV and references cited therein. In particular,
their work was based on that of Caves 关14兴 and Unruh 关4兴. Also,
KLMTV state that a likely achievable value for the squeeze factor
共in the LIGO-III time frame兲 is e
2R
⯝10, so we use that value in our
discussion.
FIG. 3. Schematic diagram showing the practical version of the
three-cavity speed-meter design, which reduces the power flowing
through the beam splitter. Three additional mirrors, with transmis-
sivity T
i
, are placed around the beam splitter. The ‘‘⫹ ’’ and ‘‘⫺ ’’
signs near the mirrors indicate the sign of the reflectivities in the
junction conditions for each location. The mirror shown in gray
closes the second port of the interferometer.
FIG. 4. Schematic diagram showing the practical three-cavity
speed-meter design with squeezed vacuum injected at the dark port
and two filter cavities on the output. Note that the circulator is a
four-port optical device that separates the injected 共squeezed兲 input
and the interferometer’s output.
PRACTICAL SPEED METER DESIGNS FOR QUANTUM . . . PHYSICAL REVIEW D 66, 122004 共2002兲
122004-3
squeeze factor is e
2R
⫽ 1, so the circulating power W
circ
must
be 8 MW in order to beat the SQL at f
opt
⯝100 Hz by a
factor of
冑
10 in sensitivity. With a squeeze factor of e
2R
⫽ 10, we can achieve the same performance with W
circ
⯝800 kW, which is the same as LIGO-II is expected to be.
This performance 共in the lossless limit兲 is the same as that
of the two-cavity 共Paper I兲 speed meter for the same circu-
lating power, but the three-cavity design has an overwhelm-
ing advantage in terms of required input power. However,
there is one significant problem with this design that we must
address: the uncomfortably large amount of laser power,
equal to W
circ
, flowing through the beam splitter. Even with
the use of squeezed vacuum, this power will be too high.
This type of problem was addressed by Weiss and Drever,
who showed, respectively, that inserting optical delay lines
关15兴 or Fabry-Pe
´
rot 共FP兲 cavities 关16兴 into the arms can
achieve a high circulating power with relatively low input
power at the beam splitter. In particular, using FP cavities in
the arms is now the standard design for most conventional
interferometers, such as LIGO-I. However, applying these
techniques alone will alter the propagation of the
gravitational-wave sidebands inside the interferometer and
jeopardize the performance of our speed meter. Fortunately,
there is a technique, based on the work of Mizuno 关17兴 that
allows us to use FP cavities in the arms without affecting the
propagation of the sidebands. This method requires an addi-
tional mirror between the beam splitter and the extraction
mirror, placed such that light with the carrier frequency reso-
nates in the subcavity formed by this mirror and the arms’
internal mirrors. We shall call this design the practical three-
cavity speed meter; the three new mirrors are labeled T
i
in
Fig. 3.
As claimed by Mizuno 关17兴 and tested experimentally by
Freise et al. 关18兴 and Mason 关19兴, when the transmissivity of
the third mirror decreases from 1, the storage time of side-
band fields in the arm cavity due to the presence of the in-
ternal mirrors will decrease. This phenomenon is called reso-
nant sideband extraction 共RSE兲; consequently, the third
mirror is called the RSE mirror. One special case, which is of
great interest to us, occurs when the RSE mirror has the same
transmissivity as the internal mirrors. In this case, the effect
of the internal mirrors on the gravitational-wave sidebands
should be exactly cancelled out by the RSE mirror. The three
new mirrors then have just one effect: they reduce the carrier
power passing through the beam splitter—and they can do so
by a large factor.
Indeed, we have confirmed that this is true for our speed
meter, as long as the distances between the three additional
mirrors 共the length of the ‘‘RSE cavity’’兲 are small 共a few
meters兲, so that the phase shifts added to the slightly off-
resonance sidebands by the RSE cavity are negligible. We
can then adjust the transmissivities of the power-recycling
mirror and of the three internal mirrors to reduce the amount
of carrier power passing through the beam splitter to a more
reasonable level.
With this design, the high circulating power is confined to
the Fabry-Pe
´
rot arm cavities, as in conventional LIGO de-
signs. There is some question as to the level of power that
mirrors will be able to tolerate in the LIGO-III time frame.
Assuming that several megawatts is not acceptable, we shall
show that the circulating power can be reduced by injecting
fixed-angle squeezed vacuum into the dark port, as indicated
by Eq. 共3兲.
Going a step farther, we shall show that if, in addition to
injected squeezed vacuum, we also use frequency-dependent
共FD兲 homodyne detection, the sensitivity of the speed meter
is dramatically improved at high frequencies 共above f
opt
⯝100 Hz); this is shown in Fig. 1. The disadvantage of this
is that FD homodyne detection requires two filter cavities of
the same length as the arm cavities 共4 km for LIGO兲,as
shown in Fig. 4.
Our analysis of the losses in these scenarios indicates that
our speed meters with squeezed vacuum and/or variational-
output are much less sensitive to losses than a position meter
using those techniques 共as analyzed by KLMTV兲. Losses for
the various speed meters we discuss here are generally quite
low and are due primarily to the losses in the optical ele-
ments 共as opposed to mode-mismatching effects兲. Without
squeezed vacuum, the losses in sensitivity are less than 10%
in the range 50–105 Hz, lower at higher frequencies, but
higher at low frequencies. Injecting fixed-angle squeezed
vacuum into the dark port allows this speed meter to operate
at a lower power 关see Eq. 共3兲兴, thereby reducing the domi-
nant losses 共which are dependent on the circulating power
because they come from vacuum fluctations contributing to
the back action兲. In this case, the losses are less than 4% in
the range 25–150 Hz. As before, they are lower at high fre-
quencies, but they increase at low frequencies. Using FD
homodyne detection does not change the losses significantly.
This paper is organized as follows: In Sec. II we give a
brief description of the mathematical method that we use to
analyze the interferometer. In Sec. III A, we present the re-
sults in the lossless case, followed in Sec. III B by a discus-
sion of optimization methods. In Sec. III C, we discuss some
of the advantages and disadvantages of this design, including
the reasons it requires a large circulating power. Then in Sec.
IV, we show how the circulating power can be reduced by
injecting squeezed vacuum through the dark port of the in-
terferometer and how the use of frequency-dependent homo-
dyne detection can improve the performance at high frequen-
cies. In Sec. V, we discuss the effect of losses on our speed
meter with the various modifications made in Sec. IV, and we
compare our interferometer configurations with those of
KLMTV. Finally, we summarize our results in Sec. VI.
II. MATHEMATICAL DESCRIPTION
OF THE INTERFEROMETER
The interferometers in this paper are analyzed using the
techniques described in Paper I 共Sec. II兲. These methods are
based on the formalism developed by Caves and Schumaker
关20,21兴 and used by KLMTV to examine more conventional
interferometer designs. For completeness, we will summa-
rize the main points here.
The electric field propagating in each direction down each
segment of the interferometer is expressed in the form
P. PURDUE AND Y. CHEN PHYSICAL REVIEW D 66, 122004 共2002兲
122004-4
E
field
共
兲
⫽
冑
4
ប
0
Sc
A
共
兲
. 共4兲
Here A(
) is the amplitude 关which is denoted by other
letters—B(
), P(
), etc.—in other parts of the interferom-
eter; see Fig. 2兴,
⫽ t⫺ z/c,
0
is the carrier frequency, ប is
the reduced Planck’s constant, and S is the effective cross-
sectional area of the light beam; see Eq. 共8兲 of KLMTV. For
light propagating in the negative z direction,
⫽ t⫺ z/c is
replaced by
⫽ t⫹ z/c. We decompose the amplitude into
cosine and sine quadratures,
A
共
兲
⫽ A
1
共
兲
cos
0
⫹ A
2
共
兲
sin
0
, 共5兲
where the subscript 1 always refers to the cosine quadrature,
and 2 to sine. Both arms and the sloshing cavity have length
L⫽4 km, whereas all of the other lengths z
i
are short com-
pared to L. We choose the cavity lengths to be exact half
multiples of the carrier wavelength so e
i2
0
L/c
⫽ 1 and
e
i2
0
z
i
/c
⫽ 1. There will be phase shifts put onto the sideband
light in all of these cavities, but only the phase shifts due to
the long cavities are significant.
The aforementioned sidebands are put onto the carrier by
the mirror motions and by vacuum fluctuations. We express
the quadrature amplitudes for the carrier plus the sidebands
in the form
A
j
共
兲
⫽ A
j
共
兲
⫹
冕
0
⬁
关
a
˜
j
共
兲
e
⫺ i
⫹ a
˜
j
†
共
兲
e
i
兴
d
2
. 共6兲
Here A
j
(
) is the carrier amplitude, a
˜
j
(
) is the field ampli-
tude 共a quantum mechanical operator兲 for the sideband at
sideband frequency
共absolute frequency
0
⫾
) in the j
quadrature, and a
˜
j
†
(
) is the Hermitian adjoint of a
˜
j
(
); cf.
Eqs. 共6兲–共8兲 of KLMTV, where commutation relations and
the connection to creation and annihilation operators are dis-
cussed. In other portions of the interferometer 共Fig. 2兲, A
j
(
)
is replaced by, e.g., C
j
(
); A
j
(
), by C
j
(
); a
˜
j
(
), by
c
˜
j
(
), etc.
Since each mirror has a power transmissivity and comple-
mentary reflectivity satisfying the equation T⫹R⫽ 1, we can
write out the junction conditions for each mirror in the sys-
tem, for both the carrier quadratures and the sidebands 关see
particularly Eqs. 共5兲 and 共12兲–共14兲 in Paper I兴. We shall de-
note the power transmissivities for the sloshing mirror as T
s
,
for the extraction 共output兲 mirror as T
o
, the power-recycling
mirror as T
p
, for the beam-splitter as T
b
⫽ 0.5, for the inter-
nal mirrors as T
i
, and for the end mirrors as T
e
; see Figs. 2
and 3.
The resulting equations can be solved simultaneously to
get expressions for the carrier and sidebands in each segment
of the interferometer. Since those expressions may be quite
complicated, we use the following assumptions to simplify
our results. First, we assume that only the cosine quadrature
is being driven 共so that the carrier sine quadrature terms are
all zero兲. Second, we assume that the transmissivities obey
1ⰇT
o
Ⰷ T
s
Ⰷ T
e
and 1Ⰷ
兵
T
p
,T
i
其
Ⰷ T
e
. 共7兲
The motivations for these assumptions are that 共i兲 they lead
to speed-meter behavior; 共ii兲 as with any interferometer, the
best performance is achieved by making the end-mirror
transmissivities T
e
as small as possible; and 共iii兲 good per-
formance requires a light extraction rate comparable to the
sloshing rate,
␦
⬃⍀ 关cf. the first paragraph of Sec. III B in
Paper I兴, which with Eqs. 共1兲 and 共2兲 implies T
o
⬃
冑
T
s
so
T
o
Ⰷ T
s
. Throughout the paper, we will be using these as-
sumptions, together with
L/cⰆ 1, to simplify our expres-
sions.
III. SPEED METER IN THE LOSSLESS LIMIT
For simplicity, in this section we will set T
e
⫽ 0 共end mir-
rors perfectly reflecting兲. We will also neglect the 共vacuum-
fluctuation兲 noise coming in the main laser port (i
˜
1,2
) since
that noise largely exits back toward the laser and produces
negligible noise on the signal light exiting the output port. As
a result of these assumptions, the only 共vacuum-fluctuation兲
noise that remains is that which comes in through the output
port (p
˜
1,2
). An interferometer in which this is the case and in
which light absorption and scattering are unimportant (R
⫹ T⫽ 1 for all mirrors, as we have assumed兲 is said to be
‘‘lossless.’’ In Sec. V, we shall relax these assumptions; i.e.,
we shall consider lossy interferometers.
It should be noted that the results and discussion in this
section and in Sec. IV apply to both the simple and practical
versions of the three-cavity speed meter 共Figs. 2 and 3兲. The
two versions are completely equivalent 共in the lossless limit兲.
A. Mathematical analysis
The lossless interferometer output for the speed meters in
Fig. 2 and 3, as derived by the analysis sketched in the pre-
vious section, is then
q
˜
1
⫽⫺
L
*
共
兲
L
共
兲
p
˜
1
, 共8a兲
q
˜
2
⫽
2i
冑
0
␦
W
circ
冑
បcLL
共
兲
x
˜
⫺
L
*
共
兲
L
共
兲
p
˜
2
. 共8b兲
Here p
˜
j
(
) is the side-band field operator 关analogue of
a
˜
j
(
) in Eq. 共6兲兴 associated with the dark-port input P(
),
and q
˜
j
(
) associated with the output Q(
); see Fig. 2. Also,
in Eqs. 共8兲, L(
)isac number given by
L
共
兲
⫽ ⍀
2
⫺
2
⫺ i
␦
共9兲
关recalling that ⍀⫽ c
冑
T
s
/2L is the sloshing frequency,
␦
⫽ cT
o
/L the extraction rate兴, the asterisk in L
*
(
) denotes
the complex conjugate, x
˜
(
) is the Fourier transform of the
relative displacement of the four test masses—i.e., the Fou-
rier transform of the difference in lengths of the interferom-
eter’s two arm cavities—and W
circ
is circulating power in the
each of the interferometer’s two arms. Note that the circulat-
PRACTICAL SPEED METER DESIGNS FOR QUANTUM . . . PHYSICAL REVIEW D 66, 122004 共2002兲
122004-5
