Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://hbar.phys.msu.ru/articles/04a2Da.pdf Äàòà èçìåíåíèÿ: Fri Oct 28 00:00:00 2005 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:07:38 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: narrow band filter

PHYSICAL REVIEW D 69, 102003 2004

Sensitivity limitations in optical speed meter topology of gravitational-wave antennas
S. L. Danilishin*
Department of Physics, Moscow State University, Moscow 119992, Russia Received 8 December 2003; published 27 May 2004 The possible design of a quantum nondemolition gravitational-wave detector based on the speed meter principle is considered with respect to optical losses. A detailed analysis of a speed meter interferometer is performed and the ultimate sensitivity that can be achieved is calculated. It is shown that unlike with the position meter signal recycling can hardly be implemented in speed meter topology to replace the arm cavities, as is done in signal-recycled detectors such as GEO 600. It is also shown that a speed meter can beat the standard quantum limit by a factor of 3 in a relatively wide frequency band, and by a factor of 10 in a narrow band. For wideband detection the speed meter requires a quite reasonable amount of circulating power 1 MW. The advantage of the considered scheme is that it can be implemented with minimal changes in the current optical layout of the LIGO interferometer. DOI: 10.1103/PhysRevD.69.102003 PACS number s : 04.80.Cc, 04.30.Db

I. INTRODUCTION

It is a well known fact that in order to detect gravitational waves very high precision devices are needed. The current technological progress in this area of knowledge suggests that a detector will achieve the level of sensitivity where its quantum behavior will play the main role. The sensitivity of the third-generation interferometric gravitational wave detectors is planned to be higher than the standard quantum limit SQL 1,2 . Therefore, to overcome the SQL one needs to monitor not the coordinate of the detector probe body, as in contemporary detectors, but an observable that is not perturbed by the measurement. Such an observable has been called a quantum nondemolition QND observable 3­ 6 . One needs to monitor this kind of observable because, if there is no force acting upon the probe object, the observable value will remain unperturbed after the measurement. This means that there is no back action and, therefore, no SQL, limiting the sensitivity. In order to detect external action on the object it is reasonable to measure its integral of motion, which is a QND observable at the same time. For a free mass, its momentum can be that QND observable, but the realization of momentum measurement is not an easy task. In the article 7 it was proposed to measure the velocity of a free mass instead of momentum. Although it is not a QND observable and is perturbed during the measurement, it returns to its initial value after the measurement, and therefore can be used to beat the SQL. A device that measures the object velocity is called a speed meter. The first realization of a gravitational-wave detector based on speed meter principle was proposed in 7 . In this article the authors suggested measuring the velocity of a free mass placed in a gravitational wave field. Analysis of the scheme has shown that measuring velocity instead of body coordinates allows the back action noise to be canceled and, therefore, significantly increases the scheme sensitivity. The measuring system was presented as two coupled microwave cavities whose coupling constant had to be chosen so that

*Email address: stefan@hbar.phys.msu.ru
0556-2821/2004/69 10 /102003 17 /$22.50

phase shifts proportional to the coordinate were fully compensated, while the output signal contained information about the probe mass velocity only. Later, in 8 the possibility of applying the speed measurement technique to interferometric gravitational-wave detectors was analyzed. It was suggested to attach to the probe ´ bodies of the detector small rigid Fabry-Perot cavities. These cavities, fully transparent for light at a certain frequency 0 when immobile, introduce a phase shift in the output light due to the Doppler effect. By measuring this phase shift it is possible to measure the probe body velocity. It was shown in this article that the speed meter can potentially beat the SQL if the pumping power is larger than one for a SQL limited position meter: W W SQL . The realization of a speed meter based on two microwave coupled resonators, suggested in 7 , was considered in 9 . This speed meter was proposed to be attached to the detector probe body to measure its velocity. It was demonstrated that it is feasible, with current technology, to construct such a speed meter that beats the SQL in a wide frequency band by a factor of 2. A possible design of a speed meter for an optical frequency band was also proposed. This design demands construction of four large scale cavities instead of two as in the traditional LIGO detector. The disadvantage of this scheme, common for all speed meters, is the extremely high power circulating in the cavities. A further comprehensive analysis of the scheme proposed in 9 was carried in 10 . It was shown that in principle the interferometric speed meter can beat the gravitational-wave standard quantum limit by an arbitrarily large amount, over an arbitrarily wide range of frequencies, and can do so without the use of a squeezed vacuum or any auxiliary filter cavities at the interferometer 's input or output. However, in practice, to reach or beat the SQL, this specific speed meter requires exorbitantly high input light power. The influence of losses on the speed meter sensitivity was also analyzed, and it was shown that optical losses in the considered scheme influence the sensitivity at low frequencies. In 11,12 more practical schemes for large scale interferometric speed meters based on the Sagnac effect 13 were analyzed and proposed for use in possible design of
©2004 The American Physical Society

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third-generation LIGO detectors. It should be noted that use of Sagnac interferometers in gravitational-wave detection was suggested earlier in 14 ­ 17 . In 11,12 it was shown that to lower the value of circulating power one should use the squeezed vacuum input with a squeezing factor of 0.1, and variational output detection 18 . Then it is possible to 10. The schemes analyzed beat the SQL by a factor of ´ are based on using either three large scale Fabry-Perot cavities 11 or ring cavities and optical delay lines 12 . A comprehensive analysis of the above mentioned schemes including optical losses was carried out. Another variant of a Sagnac based speed meter was proposed in 19 . This design requires little change in the initial LIGO equipment and seems to be a good candidate for implementation. The scheme of the interferometer proposed in 19 is considered in this paper. We analyze here how this meter will behave when there are optical losses in the interferometer mirrors and obtain the optimal value of the circulating power and how it depends on cavity parameters. This article is organized as follows. In Sec. II we consider the simple scheme of a speed meter with a single lossy element in order to study how optical losses influence the scheme sensitivity. In Sec. III we compare the sensitivities of a signal-recycled position meter and a speed meter, and demonstrate that the later has worse sensitivity at the same level of pumping power if signal recycling is applied. In Sec. IV we evaluate the sensitivity of the more realistic speed meter scheme proposed in 19 and obtain the optimal values of the parameters that minimize the influence of noise sources. In Sec. V we summarize our results.

FIG. 1. Simplified scheme of speed meter.

II. SIMPLE SAGNAC SPEED METER SCHEME WITH OPTICAL LOSSES A. Input-output relations for simple speed meter scheme

gravitational-wave detectors this approximation is satisfied quite well. After the three above mentioned reflections the light beams return to the beam splitter where they are mixed up. We suppose that the arm lengths are chosen so that all the light power returns to the laser. Then the photodetector located in the ``southern'' part of the figure will register only vacuum oscillations whose phase is modulated by the end mirror motion. The input light can be presented as a sum of a classical pumping wave with frequency 0 and sideband quantum oscillations. The electric field strength of the incident wave can be written as ^ Et ^ a
0

In this section we will analyze the action of a simple speed meter scheme based on the Sagnac effect to estimate how optical losses affect its sensitivity. This scheme is a model that is quite easy for analysis and yet contains all features specific for a speed meter. It differs from real, complicated schemes only in the fact that there is no bending up of the noise curve at high frequencies that arises due to the ´ finite bandwidth of Fabry-Perot cavities being used in real speed meter interferometers. Therefore, it seems convenient to investigate this simple scheme before analyzing complicated speed meters designed for LIGO. The scheme we want to consider is presented in Fig. 1. Its action can be described as follows: the light beam from a laser enters the scheme from the ``western'' side of the figure and is divided by the beam splitter into two beams propagating in the ``northern'' and ``eastern'' directions. Each beam is reflected sequentially from two movable end mirrors. Between these reflections the beams are reflected from the central mirror. The end mirrors we suppose to be ideally reflective, while the central mirror has the amplitude reflectivity r and transmittance t. We suppose that all optical losses are concentrated in the central mirror and therefore can be expressed by a single parameter t. It can be shown that for

Ae

i 0t

A *e

i 0t 0

^ a

e

it



e

it

d , 2

1

^ where A is the classical wave quadrature amplitude, a ( ) is the quantum fluctuation sideband operator, () 2 / Ac is a normalization factor, and A is the beam cross section. Using this formalism we can write down inputoutput relations for the scheme. The incident wave quadra^ ture amplitude and fluctuations we will denote as A and a w as they come from the ``western'' side of the figure. Zero oscillations entering the scheme from the ``southern'' side we ^ will denote as a s . After the beam splitter we will have two light beams propagating in the ``northern'' and ``eastern'' directions. These beam values we will mark by indices ``n '' and ``e , '' respectively. Moreover, there is a transparent cen^ tral mirror that is the source of additional noise denoted as g n ^ and g e see Fig. 1 .

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SENSITIVITY LIMITATIONS IN OPTICAL SPEED . . . TABLE I. Expressions for values used in Sec. II. x( ) x
e

PHYSICAL REVIEW D 69, 102003 2004

S

S

F

m

2

4

S

x

2m

2

RS

xF

,

3

x 2

n

0

^ g

^ ig
s

e

^ g 2

n

()
in put loss

0

/c

where is the observation frequency, and S x ( ), S F ( ), S xF ( ) are the spectral densities of fluctuational mirror displacement x f luct , fluctuational radiation pressure force F f luct , and their cross correlation, respectively. In our specific case these spectral densities are equal to see Appendix A 8 S
e
i( 0 )

ire

4i 2i

0

W

ite 2 ( ) A ( ire

F

c

2

· 1 r cos 2

, 4a

Ksim P

ple

i( 0 3 )

)

16
sim ple

W mc 2
0

2

S

c
x

2 2

16

0

W sin

·

1 1r
2

2 r cos 2

, 4b

Let us denote the beam that travels in the interferometer arm for the first time as the ``primary'' beam, while the same beam that has left its first arm and entered the second one we will denote as the ``secondary'' beam. Due to absorption, the powers of the primary and secondary beams relate to each 2 other as W secondar y / W primar y r 2 1 loss , where loss is the interferometer absorption coefficient. This coefficient is quite small and in practice can be neglected for the values of the classical powers but, in principle, there is an opportunity to compensate these losses by introducing some additional power into arms. This opportunity can be taken into account 2 , where is some coby assuming W secondar y / W primar y 2 efficient that is equal to 1 loss without additional pumping, and can be larger with it. We will assume that 1 as optical losses in the considered schemes are supposed to be very small ( loss 10 5 ) . Then the output wave can be written as see Appendix A ^ b
s in put a s

S

xF

2

cot

.

4c

Here W is the pumping power at the end mirrors, and is the homodyne angle that allows us to minimize the value of Eq. 3 significantly. This angle is chosen so that one measures not the amplitude or phase quadrature component but their mixture. This principle provides the basis for the variational measurement technique 18,20 . Here is one of the optimization parameters that allows one to overcome the SQL when chosen in the proper way. Suppose that the signal varies slowly compared to the scheme characteristic time , and, therefore, 1, 2 , 5a 5b

^

loss g s

^

Ksim
ple

ple

x.

2

Expressions for in marized in Table I.

put

,

loss

, Ksim

^ , x , and g s are sum-

L / c , and L is the distance between the central and where end mirrors. These assumptions we will call the narrowband approximation. Later we will see that for real detectors this approximation works pretty well. Using this approximation we can rewrite Eqs. 4 as 8 S
F 0

B. Quantum noise spectral density

W

In order to evaluate the scheme sensitivity we need to calculate the spectral density of total quantum noise. Suppose the mirror dynamics can be described by a free body equation of motion, i.e., ¨ mx F . Here F F GW F f luct , where F GW is the gravitational-wave force measured, F f luct is the fluctuational force that arises due to radiation pressure of the incident light, and x x GW x f luct is the mirror displacement that consists of displacement due to gravitational-wave action and a noisy part that arises due to incident light phase fluctuations. The total noise spectral density then can be written as

c

2

· 1 r 2r

2

, 6a

c S
x

2

16

0

W sin2 .

·

1 1r
2

4r

2

,

6b

S

xF

2

cot

6c

If we substitute the expressions obtained in Eq. 3 and divide by the value of the radiation pressure noise SQL specm 2 , we will obtain tral density for the free mass S SQL the factor by which the speed meter beats the SQL:

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PHYSICAL REVIEW D 69, 102003 2004

2

1P 2

sim ple 2 2

· 1 r 2r 2 1 cot2 1r
2

2

P

sim ple

4r

2

2 cot

.

7

The expression for P sim ple is presented in Table I. The main goal of optimization is to beat the SQL in a frequency band that is as wide as possible. First of all we should find the optimal value of cot and P sim ple . These parameters should not depend on frequency; therefore we . This will optimize them at high frequencies, i.e., optimization will result in the following values: cot 2 rP
sim ple

,

8

FIG. 3. Signal-recycled position meter.

and being substituted in Eq. 7 will produce
2 HF

1 4 rP sim

.
ple

9

Obviously, the rise of 2 at low frequencies is determined by the radiation pressure spectral density as all other items in Eq. 3 are proportional to 2 and 4 and, therefore, cannot influence at low frequencies. Hence we see that optical losses lead to a decrease of speed meter sensitivity when the observation frequency is small. It is useful to calculate the value of the parameter P sim ple that provides the minimum of 2 at the defined frequency * . It can be shown that for 1 r 1 this value is equal to P
op t sim ple

where r is the central mirror amplitude reflectivity coefficient that characterizes the scheme optical losses. Here we suppose t 1. Comparison of the expressions 9 and 11 shows that one needs to increase the optical power ( P sim ple ) in order to obtain high sensitivity, while to have a wide frequency band and increase the sensitivity at low frequencies it is necessary to have a low value of the pumping power to decrease radiation pressure noise. In Fig. 2 several curves are presented that demonstrate how 2 depends on the frequency at different values of P sim ple . The dot-dashed lines are for the ideal case where losses are equal to zero ( r 1).

*
. 21 r 10

III. SIGNAL-RECYCLED SPEED METER VS SIGNAL-RECYCLED POSITION METER

One can also readily obtain the minimal frequency where the speed meter sensitivity is equal to the SQL, i.e., where 2 1 is satisfied, is defined by the expression P
min sim ple

21 r ,
sim ple

11

In this section we will consider semiqualitatively the influence of a signal-recycling mirror on the sensitivity of the traditional position meter and speed meter. Let us consider the two schemes presented in Figs. 3 and 4. Suppose T FP 1/N FP and T SR M 1/N SR M are the trans-

4P

1

FIG. 2. Typical curves for

2

S/S

SQL

at different P

sim ple

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FIG. 4. Signal-recycled speed meter.


SENSITIVITY LIMITATIONS IN OPTICAL SPEED . . .

PHYSICAL REVIEW D 69, 102003 2004

mittances of the arm cavity input mirrors ITMs and signalrecycling mirrors SRMs , respectively. Here N FP shows how many times the light beam is reflected from the end mirrors until it leaves the cavity, and N SR M is the number of light beam reflections from the SRM before leaving the whole scheme. The first number characterizes the Fabry´ Perot half bandwidth, while the second one represents the signal-recycling cavity half bandwidth. Let be the time the light needs to travel from one cavity mirror to another. Suppose also the mirror motion is uniform and can be expressed during the light storage time ( * N FP N SR M GW / GW , max ) by the formula x
i

S S A

a 2

.

The expression for S a can be readily obtained and is equal to S
a

1 , 4

and A 2 can be represented in terms of the power circulating in the arms as A
2 0

W N
FP

x

i ,0

v it ,

N

.
SR M

where the subscript i 1,2 denotes the interferometer arm, x i ,0 are the end mirror initial positions, and v i are their velocities. In the case of the position meter the light path is the following. After leaving the beam splitter light enters both cavities, it passes from one arm cavity mirror to another 2 N FP times, then it is reflected from the SRM and returns to the cavity. This cycle repeats N SR M times until the light leaves the scheme and goes to the homodyne detector. The output beam phase shift in our simple case can be written as 2
signal 0

Then for S one will have the following expression: S
0

N FP N 4W

SR M

.

14

On the other hand, for the position meter this spectral density can be expressed in terms of the mirror displacement spectral density caused by radiation shot noise corresponding to the / m 2 , as SQL, S SQL x 0 S
PM

4

2 0

N

2 FP

N

2 SR M

c

N

FP

N SR M¯ , x

12

c

2

S

SQL x

4

2 0

N

2 FP

N
2 0

2 SR M

mc

2

,

15

¯ 1 ¯ 2 is the mean end mirror relative displacewhere ¯ x xx ment during the storage time, and c is the speed of light. As for the speed meter, the light path in this scheme differs from the previous one. Light that leaves, for example, the first cavity goes not to the beam splitter and then to the SRM, but to the second cavity, where it experiences N FP reflections from the end mirror in addition to those in the first cavity. Then it is reflected from the SRM and so on for N SR M times until it leaves the scheme. It can be shown that in this specific case the output beam phase shift will be written as 2
signal 0

where 0 is some fixed observation ror mass, and for the speed meter velocity spectral density caused by /m responding to the SQL S SQL v S
SM

frequency, m is the mirin terms of the mirror radiation shot noise coras
22 0

4

22 0

N c

4 FP 2

N

2 SR M

S

SQL v

4

N

4 FP 2

N

2 SR M

. 16

mc

Substituting Eq. 14 into Eqs. 15 and 16 we will obtain the following values of the circulating power for both schemes considered: mc W
PM 2 2 0 SR M 0

c

N

SR M

N

2 FP

¯ v

,

13

mc , W
SM

2 2

16N

FP

N

16N

3 FP

. 17

N

SR M

0

¯ 1 ¯ 2 is the relative velocity of the end mirrors. where ¯ v vv We can readily see that the position meter and speed meter output signals depend on the arm cavity ( N FP ) and signal-recycling cavity ( N SR M ) parameters in different ways. It can be shown that this difference significantly influences the sensitivity of these schemes. Let us calculate the amount of circulating optical power necessary to achieve the level of SQL in each of the above mentioned schemes. The output signal field quadrature component a signal for both schemes is proportional to the product a
signal

If we divide W SM by W PM we will obtain the factor of by which the speed meter circulating power necessary to achieve the SQL is larger than that for the position meter: W W
SM PM

1 N
FP 0 2

N

2 SR M

,

18

A

signal

,

where A is the input radiation field quadrature amplitude. Therefore the spectral density of the phase fluctuations can be expressed in terms of the spectral density of a signal as

where we have taken into account that the light storage time should be much smaller than the period of the gravitational / 0 . Obviously, the above wave, i.e., * N FP N SR M estimates show that a speed meter with signal recycling can hardly be considered as the best variant for implementation as a QND meter, as it requires much more circulating power than the signal-recycled position meter with the same param-

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FIG. 5. Speed meter interferometer.

eters. More precise calculations that prove the above simple considerations are performed in Appendix B.
IV. OPTICAL LOSSES IN SPEED METER ´ INTERFEROMETERS WITH FABRY-PEROT CAVITIES IN ARMS A. Speed meter interferometer action and output signal

Let us consider the scheme of a gravitational-wave detector based on the speed meter principle presented in Fig. 5. This interferometer differs from the traditional LIGO interferometers by an additional polarization beam splitter PBS and two quarter-wave plates ( /4) . Quarter-wave plates are needed to transform the input light polarization from linear ( 0 ° and 90° ) to circular ( and ) during each pass. After ´ reflecting from one of the 4 km long Fabry-Perot FP cavities the light polarization varies over 90° , and the light beam is reflected from the PBS to the second FP cavity. Hence, the light beam passes not only one FP cavity, but both cavities in sequence. As there are two beams that propagate in opposite directions in clockwise and counterclockwise directions, respectively the scheme output does not contain any information about the symmetric mechanical mode of the mirrors, i.e., about the sum of the end mirror displacements. This mode is not coupled with a gravitational-wave signal, and its presence can significantly lower the sensitivity and prevent it from beating the SQL. The output signal of this scheme can be found in the same manner as was done in 19 . The only problem is to include additional types of noise, arising due to internal losses in the optical elements, in the calculations. To do so, suppose each mirror to be a source of two independent additional types of noise, that stand for the interaction of electromagnetic radiation with the mirror medium excitations. This approach is based on the Huttner-Barnett scheme

21 of electromagnetic field quantization in linear lossy dielectrics. Using this approach one can obtain input-output relations for any optical device with losses see 22 ­ 24 . It should also be noted that we do not consider the beam splitters, polarization PBS and ordinary BS , and quarterwave plates as separate sources of additional noise that arise due to their internal losses. The reason is that the laser beam interacts with these optical elements N ef f 2/ ln r1r2 2/1 ´ r 1 r 2 102 times more seldom than with the Fabry-Perot cavity mirrors here N ef f is the effective number of laser beam reflections from the FP cavity mirrors . Therefore, the contribution of these elements to the total optical loss coefficient will be N ef f times smaller than that for the FP mirrors, and can be neglected. Let us use the same notations as in Sec. II for the scheme parameters. In order to distinguish values that correspond to inner and end mirrors we will denote the first ones by the subscript 1 and the second ones by the subscript 2. Each ´ Fabry-Perot cavity has the following parameters: the mirror reflectivity, transmittance, and absorption coefficients are equal to r 1 , r 2 , it 1 ,0, and ia 1 , ia 2 , respectively, and the cavity length is L. The end mirror transmittance we suppose to be equal to zero because there is no difference whether the light exits or is absorbed in the end mirror. Suppose that the classical pumping wave amplitude is A ^ and the sideband fluctuation operator is a w . Suppose also that zero oscillations that enter the scheme from the ``south^ ern'' side are described by the operator a s . We will also need ^ ^ ^ ^ the operators g s 11, g s 12, g s 21, and g s 22 to describe noises due to internal losses in the FP mirrors the second numerical subscript stands for the number of noise type that arise in the mirror due to internal losses . The superscripts I and II denote which types of noise arise during the first and the second reflection, respectively. The parameter we assume to be equal to unity, where has the same meaning as in the simple scheme and is equal to the ratio of the classical amplitudes inside the cavity during the second and the first reflection, respectively see Appendix C . Finally, we are able to write down the output sideband ^ fluctuation operator c s that contains information about the mirror movements, and therefore about the gravitational wave force: ^ c 1
s

L

2

iB
2i

2 1

^ a

I s i

e

^ r 2t 1a 1g e
2i

I s 11

e
II w 11

^ t 1a 2g e
i

I s 21

iL KSM x

^ r 2t 1a 1g
0

^ t 1a 2g

II w 21

.

19

The notation used in this expression is presented in Table II.
B. Speed meter spectral densities and sensitivity

Here we will obtain the expressions for radiation pressure noise and shot noise and their cross-correlation spectral den-

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PHYSICAL REVIEW D 69, 102003 2004

TABLE II. Expressions for values used in Sec. IV. Here E is the classical complex amplitude of the pumping field near the movable mirror after the first reflection. Exact form
1

Narrowband approximation t 2 /4 1 a
1 2 2 1

-- --

a

2 2

4 1 r1r2 2

-- W L( ) B1 ( ) r
1 0

1

E 2 /4 1
2 1

0

E 2 /4 i)

r 1r 2e e
2i

2i

2( t)
2 1

r 2( r

2( e
i

2
1

1

i)
1 2

KSM

0

r2t1 B1 L
2

L

2 E

0

/

i E

i

sities. We will suppose that the same additional pumping procedure as in Sec. II is taking place. Here we will use the same approximation that is defined by Eq. 5 of Sec. II. It is useful to evaluate if this approximation is valid for LIGO. The parameters for the LIGO interferometer are the following:
0

1.77 1015 s

1

,

1.33 10

5

s,

mirror mass. The multiplier 1/4 appears when we suppose not only the end mirrors to be movable, but also the inner ones. Now we can write down the expression for 2 S SM / S SQL , where S SM is the total quantum noise of the speed meter that is calculated in accordance with Eq. 3 , and m 2 is the SQL spectral density for the fluctuaS SQL tional force:
2

m 40 kg,

L 4 103 m.

8

0 2

W
2 2 0 2

2

2 2

1 22 2

2

2

If we suppose that the gravitational signal upper frequency is about 103 s 1 , then there remain no doubts that for LIGO the condition of the narrowband approximation appli1.33 10 2 1. cability is satisfied as One is able to obtain the following expressions for the spectral densities see derivation in Appendix C : S L
x 2 2 22 1 2 2 2 2 2

mL

mL 32

22 2 2

W2

1

sin2

cot

. 21

1

32 8

0

W2 W

1

sin2
2 1

,

20a

This value characterizes the scheme sensitivity. Our goal is to have this value as small as possible. In the next subsection we will perform two possible optimizations of 2 , in narrow and wide frequency bands.
C. Optimization of
2

S

0

2

F

L S

2

22

,

20b

1. Narrowband optimization
xF

2

cot

,

20c

where is the homodyne angle, and W is the pumping power at the end mirrors. The expressions for 1 and are given in Table II. One can obtain the result that the minimum force that can be measured by the speed meter presented in Fig. 5 depends on the total measurement noise spectral density. This spectral density can be expressed by the formula 3 with one exception: m in this formula is equal to one-fourth of the real end

Let optimize Eq. 21 at some fixed frequency. A possible situation when such optimization can be useful is the detection of gravitational radiation emitted by quasimonochromatic sources. Compact quickly rotating neutron stars, i.e., pulsars, may be an example of such sources. Gravitational radiation from pulsars is quasimonochromatic and relatively weak so it is crucial to have high sensitivity in a narrow frequency band to detect these sources. So it is convenient to find the minimum of 2 at the source main frequency 0 . If we suppose that optical losses are small enough, i.e., will be equal to 0 , then the minimal value of

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PHYSICAL REVIEW D 69, 102003 2004

FIG. 6. Plot of S SM / S SQL , optimized at some fixed frequency 0 . For given 0 103 s 1 and fixed optical loss contribution to the total half bandwidth 1 s 1 see Table II , one should take the following parameters to obtain the best sensitivity at given frequency: total half bandwidth 0 , circulating power 23 W ( mL 0 /8 0 ) 27 MW, and cot 0/ 0/ 3 10 2 . The minimal in that case will be equal to 4 / 0 0.18.
4

FIG. 7. Plots of for circulating optical power W 1 MW and several fixed homodyne angles thin curves; the curve with rightmost minimum corresponds to the least angle . Bold black curve corresponds to frequency-dependent homodyne angle.

4 min

,
0

22

and is reached at op t 0 , optical circulating power W op t ( mL 2 3 /8 0 ) / , and homodyne angle defined by 0 0 see Appendix D for dethe formula cot opt 0/ 1 s 1 it will be equal to 0.18. For the tails . For LIGO interferometer the optical power necessary to reach the above value of is equal to W op t 27 MW. Unfortunately, at frequencies different from 0 the value of is much worse. The behavior of the function at different frequencies is presented in Fig. 6. We can see that significant gain in sensitivity compared to the SQL can be achieved in a very narrow frequency band of about several tens of hertz. Thus, this regime of the speed meter operation, which we prefer to call the ``narrowband'' regime, can be used to beat the SQL by a significant amount only in a narrowband near some arbitrarily chosen frequency for the purposes of weak quasimonochromatic sources detection.
2. Wideband optimization

tion that the speed meter is capable of. Therefore we will suppose that the optical power W circulating in the arms of the interferometer and the resonator halfbandwidth are fixed. Then by varying the homodyne angle it is possible to obtain a variety of different sensitivity curves. These curves for three different values of the circulating optical power W 1, 3 , and 10 MW are presented by thin curves in Figs. 7, 8, and 9, respectively. By varying the homodyne angle it is possible to reach a sensitivity even three times better than SQL in a relatively wide frequency band. It is also possible to choose the frequency where the best sensitivity is reached by changing the homodyne angle, i.e., an increase of leads to a sensitivity curve offset into the lower frequency domain. It should also be noted that the increase of circulating power in this regime leads to a reduc-

Contrary to the narrowband case considered above, the vast majority of gravitational-wave sources either radiate in a relatively wide frequency band, or their main frequency is unknown. In both cases it is necessary to perform a wideband detection procedure. This problem cannot be solved as easily as the previous one, because there are no criteria for what should be the frequency bandwidth value and what sensitivity should be reached within the ranges of this band. As there is no clarity on this question it seems convenient to represent the variety of different possible regimes of opera-

FIG. 8. Plots of for circulating optical power W 3 MW and several fixed homodyne angles thin curves; the curve with rightmost minimum corresponds to the least angle . Bold black curve corresponds to frequency-dependent homodyne angle.

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SENSITIVITY LIMITATIONS IN OPTICAL SPEED . . .

PHYSICAL REVIEW D 69, 102003 2004

FIG. 9. Plots of for circulating optical power W 10 MW and several fixed homodyne angles thin curves; the curve with rightmost minimum corresponds to the least angle . Bold black curve corresponds to frequency-dependent homodyne angle.

mirrors close to each other. It might be difficult to find an additional place for one more mirror in existing interferometer facilities. On the other hand, the RC interferometer has an important advantage. It does not require such devices as a polarization beam splitter and quarter-wave plates to be used, and can be constructed using only traditional optical elements, whose properties are well studied. As for delay lines, they will provide practically the same sensitivity as FP cavities even taking optical losses into account. Nevertheless, in large scale interferometers it is quite difficult, in our opinion, to implement delay lines, as it requires larger mirrors compared to FP cavities in order to provide a necessary number of light beam round trips. Moreover, the delay line, we suppose, should be very sensitive to mirror tilts, which is quite crucial for kilometer-long cavities. Therefore, it seems impractical to use delay lines in large scale interferometers, while it can be a good candidate for shorter QND interferometers.
V. CONCLUSION

tion of the frequency band where is less than 1; therefore it is reasonable to use moderate values of the optical powers. Using the technique of frequency-dependent variational readout suggested in 24 , it is possible to increase the sensitivity of the speed meter significantly in a wide frequency band. The achievable sensitivities in this case are represented in the same figures by bold black curves. Here is the dependence of ( ) that allows us to obtain the above mentioned results: mL 2 · 32 0 W
2 1 2 2 22 2

arctan

.

23

´ D. Delay line and ring cavity vs Fabry-Perot in speed meter topology

It seems resonable to discuss some issues concerning the performance of the analyzed scheme and the schemes of speed meter interferometers proposed in 12 , that are based on using delay lines DLs and ring cavities RCs instead of FP cavities. In the lossless limit all the above mentioned schemes provide practically the same sensitivity and there are no significant differences in their performance. When taking internal optical losses in cavity elements into account, one can readily realize that the RC interferometer will have approximately a 1.5 times higher loss coefficient compared to the FP interferometer, as the ring cavity requires three mirrors instead of two. A simple evaluation can show that in the narrowband regime the RC speed meter will require 1.5 1.22 times more optical power to achieve the best sensitivity, which is 4 1.5 1.11 22 times worse than the FP speed meter. The wideband regime is less susceptible to an increase of , but it can slightly decrease the frequency band where the speed meter sensitivity is better than the SQL. Another point that, in our opinion, should be taken into account is that ring cavity design requires placing two large

In this section we will sum up the results of our consideration. The following conclusions can be made. The attempts to increase speed meter sensitivity by introducing a signal-recycling mirror or replacing arm cavities by this mirror will not be successful, because unlike the position meter, the speed meter sensitivity depends nonsymmetrically upon the transmittances of the arm cavity input mirrors and SRM. The influence of the ITM is considerably greater than the SRM influence because in the speed meter the light beam passes consequently through both cavities before it is reflected from the SRM. Therefore, it is convenient to use arm cavities in gravitational-wave detectors based on the speed meter principle. The speed meter topology of gravitational-wave antennas allows one to achieve a sensitivity about three times better than the SQL in a relatively wide frequency band, preserving the optical power circulating in the arms at a reasonable level of 1 MW. Moreover, its sensitivity can be improved and its frequency band can be significantly increased if one applies a variational readout technique. The ultimate sensitivity a speed meter is capable of is defined by two factors, the optical losses and bandwidths of arm cavities and, therefore, the circulating power. This sensitivity can be expressed in terms of these factors as h h
SQL 4

,

where h is the metric variation which can be measured by the speed meter, h SQL is the standard quantum limit for h, and is the optical loss contribution to the total interferometer halfbandwidth . However, a high sensitivity small ) requires a large amount of circulating optical power about tens of megawatts and can be achieved in a relatively narrow frequency band. In our opinion, the best operation mode for the speed meter is the wideband regime with frequencydependent readout.

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S. L. DANILISHIN ACKNOWLEDGMENTS

PHYSICAL REVIEW D 69, 102003 2004 2. Radiation pressure noise spectral density S F ,, ...

The author would like to express his sincere gratitude to Professor F. Ya. Khalili for the interesting subject of investigation and help and advice. The author also would like to thank V. B. Braginsky, S. P. Vyatchanin, and S. E. Strigin for helpful discussions and encouragement. Special thanks go to K. Danzmann, R. Schnabel, and all members of the GEO600 team for fruitful talks, valuable conversations, and hospitality. This paper is supported in part by NSF and Caltech Grant No. PHY0098715, by the Russian Foundation for Basic Research, and by the Russian Ministry of Industry and Science.
APPENDIX A: DERIVATION OF SPECTRAL DENSITIES FOR SIMPLE CASE 1. Input-output relation derivation

The radiation pressure force corresponding to system mode x is equal to ^^ FF 2
0 e

^ F

n

A*

ie

i(

0)

e

i

re

i (3

0)

^ a

s

2 H.c.,

e

i

e

i(

0)

^ g

n

^ ig 2

e

e

i( 0

)t

d 2 A5

Here we will obtain the formulas for the quantum noise spectral densities, presented in Sec. II by the expressions 4 . The input light can be described by the formula 1 . The input amplitudes are the following the subscripts w,e,s,n are for light beams corresponding to that part of the scheme; for example, A w means the classical amplitude of a light beam propagating in the western direction : A
w

^ ^ where F e and F n are the radiation pressure fluctuational forces acting upon the eastern and northern movable mirrors, respectively. In order to calculate the radiation pressure spectral density one should calculate the symmetric correlation function BF t t 1 ^ ^ 0 FtFt 2 ^ ^ Ft Ft 0, A6

A,

A

s

0,

A1

where 0 is the radiation field ground state. If we calculate this value we obtain BF t t 1 2 F
0 2 i( 0 2 i( 0 e )( t t )

and the corresponding sideband operators are ^ aw , ^ as . A2 d 2 d 2 , A7

As we mentioned above we are able to introduce additional pumping through the central mirror. This additional pumping increases the light power in the scheme and does not create additional noise; therefore, it increases the scheme sensitivity. To describe it we will introduce the complex parameter ei 2 C / A which is equal to the ratio of the light amplitude at the end mirrors with additional pumping ( C A / 2 A add ) and the light amplitude after the beam splitter ( A / 2) which is equal to the amplitude at the end mirror provided that the central mirror is ideally reflecting, i.e., in the lossless case . Taking all the above into account one can easily obtain that the classical output of the scheme is equal to B
s

F
0

e

)( t t )

where F
2

4

22

A
0)

2

e

i(

i e i re

i (3

0)

2

2

2

.

In order to obtain this result one should take into account that ^ ^ 0 a ( ) a ( ) 0 2 ( ) . The radiation pressure spectral density can be defined as S BF t e 16 c
2 0 it

1 dt
2

0,

A3

F

2 /2

S r sin 1
2

S cos 2 . A8

and the corresponding sideband output is equal to ^ g i
n

W 1/2 ·

^ b

s

^ ira

s

e

4i

^ ig 2

e

e
)

2i

2
in put a s

A ire ^

i( 0 3 )

e
ple

i( 0

x

0

In order to provide the speed meter mode of operation we /2; then one will obtain that need to set A4 S
F

loss g s

^

Ksim

x;

8 c

0 2

see Eq. 2 for notation.
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W1 ·

2

2 r cos 2 1
2

.

A9


SENSITIVITY LIMITATIONS IN OPTICAL SPEED . . . 3. Shot noise spectral density S x ,, ...

PHYSICAL REVIEW D 69, 102003 2004

The output signal of the scheme is mixed up with the local oscillator wave in order to detect the phase shift due to the end mirror displacement in the optimal way. This mixed radiation enters the homodyne detector. The photocurrent of the detector is proportional to the following time-averaged value: ^ I
p.d.

t

2E

bs

t cos ^ b

0

t

LO

FIG. 10. Power- and signal-recycled speed meter interferometer.

0

s

e

i( 0

) t i LO

d 2 H.c. d 2 , A10

K

^ x

f luct

x

e

it

^ where LO is the local oscillator phase, r ( ) is the noise operator whose spectral density is equal to unity, and K ( ) is equal to K 4
3/2 0

recycling mirrors installed see Fig. 10 . Here we will confirm by exact calculations the results of the qualitative consideration presented in Sec. III. Let consider the influence of additional optical elements on the input-output relations. Figure 11 represents the situation common for PRM and SRM. Beams a and b represent light entering and leaving the beam splitter, and beams a and b stand for light entering and leaving the PRM/SRM. The length L of the recycling cavity should be chosen in the way that provides the maximal values of circulating power in the case of the PRM, and signal sidebands in the case of the SRM.
a. Power recycling mirror

A

c

e

3i

r sin

e

2i

cos

, A11

where LO arg A . The fluctuations of the coordinate are described by the operator ^ x Finally, assuming density is equal to S c
x 2

f luct

^ r K

.

A12

Power recycling influences the value of the circulating power only, as quantum fluctuations from the laser that are influenced by the PRM return back to the laser due to dark port tuning of the interferometer, and do not contribute to the output signal. The interferometer circulating power depends upon the quadrature amplitude A of the beam a as W
0

A 2.

/2, the coordinate noise spectral
2

32

0

W sin2

·

1 r
2 2

Then we need to express A in terms of A , the amplitude of the beam a entering the PRM. One can write down the following equations: . A13

2 r cos 2

4. Cross-correlation spectral density S

xF

,,

...

Now we know everything needed to calculate the crosscorrelation spectral density S xF ( ) . In order to do this one should find the cross-correlation function of Eq. A5 and ^ x f luct ( ) . Using the same algorithm as in previous subsections one obtains that S
xF

2

cot

.

A14

APPENDIX B: POWER- AND SIGNAL-RECYCLED SPEED METER INTERFEROMETER SENSITIVITY: PRECISE ANALYSIS 1. Input-output relations for power- and signal-recycled speed meter interferometer

In this appendix we will analyze the scheme of a speed meter interferometer with power PRM and signal SRM
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FIG. 11. Power recycling/signal recycling mirrors.


S. L. DANILISHIN

PHYSICAL REVIEW D 69, 102003 2004

A B

it it

PR PR

A B

r r

PR PR

B, A,

A B

Ae Be

i PR i PR

,

B1a B1b B1c

,

B i1

loss

A,

i.e., KSR max. Therefore, the signal-recycling cavity /4. should be tuned in a way to provide SR Now we are able to calculate the spectral densities of radiation pressure and shot noise and their cross-correlation spectral density.
2. Radiation pressure noise spectral density S F ,, ...

where t PR and r PR are the PRM transmittance and reflectivity, PR 0 L / c , and loss represents losses in the entire scheme we will neglect these losses in this calculation as loss is sufficiently small . Solving these equations leads to the following expression: it A
PR

e

i PR

1 ir

PR

e

2 i PR

A.

B2

In order to calculate the radiation pressure noise spectral density for a power- and signal-recycled speed meter inter^ ferometer one needs to replace operator a s in formula A5 ^ of the previous appendix by the operator a s that can be ex^ s and g s as ^ pressed in terms of a ^ a it SR e
s i SR

^ a

In order to maximize the circulating power one should tune /4; then we obtain that the PR cavity so that PR t W
PR 2 PR

s

r

SR

loss

e

2 i SR

^ g

s

r SR e
2 i SR

2 i SR

Ksim

ple

x . B6

1r

SR

in put

e

W

0 2

4W t
0 2

0

,

B3

1r

PR

PR

where W

0

0

A , and W

PR

A 2.

After the same operations as in Appendix one obtains that the radiation pressure noise spectral density in the case of a power- and signal-recycled speed meter interferometer is equal to 16 S
F 0

b. Signal recycling mirror

W

PR

In the case of the signal-recycling cavity only the quantum fluctuation transformation is worth examining. As the interferometer is tuned so that the output port is kept ``dark,'' the classical amplitude of the beam leaving the BS is equal to zero. Now a and b stand for the fluctuations entering and leaving the BS, while a and b stand for fluctuations entering and leaving the SRM. Therefore we write down the following equations: ^ b it SR e
i SR

c

2

·

SR SR 2

SR

2 2

.

B7

4
...

3. Shot noise spectral density S x ,,

^ ^ b r SR a ,

^ ae

i SR

^ it SR a

r SR e

i SR

^ b, B4

To obtain the expression for the shot noise spectral density in the case of a signal-recycled interferometer we should use the same procedure as in Appendix but for one exception. We should substitute the value of Ksim ple by KSR . To account for power recycling we should also write down W PR instead of W. Finally we obtain the following formula: S c
x 2

where t SR and r SR are the SRM transmittance and reflectivity, and SR 0 L / c . In these equations we should suppose ^^ ^ ^ a a s , b b s , and the equation that gives the relation between these operators is represented by the expression 2 . The solution of these equations gives us the following expression for the PR and SR speed meter output signal: ^ b r
2 t SR e SR 2 i SR in put

64

0

W

PR

sin2

·

SR SR 2

2

4
2

2

.

B8

4. Cross-correlation spectral density S

xF

,,

...

The cross-correlation spectral density in this case is the same as in the previous section and is common for all interferometric schemes considered with homodyne detection: S
xF

s

1r
i SR

SR

in put

e

2 i SR

^ a

s

2

cot

.

B9

it SR e 1r where KSR
SR

loss

in put

e

2 i SR

^ g

s

KSR x ,

B5
5. Power- and signal-recycled speed meter sensitivity

it SR e 1r
SR

i SR

loss

in put

e

2 i SR

Ksim

ple

,

and in put , loss , and Ksim ple are defined in Eq. 2 . The value of SR is chosen so that the output signal is maximal,

The total noise of the PR and SR speed meter is described by the same formula as Eq. 3 . The spectral densities of noises in this particular case can be obtained from the formulas B7 , B8 , and B9 . Being substituted in Eq. 3 and divided by S SQL m 2 , these formulas will give us the expression for the factor 2 by which one can beat the SQL using the PR and SR speed meter:

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SENSITIVITY LIMITATIONS IN OPTICAL SPEED . . .
SR SR 2 SR 2 SR 2 2

PHYSICAL REVIEW D 69, 102003 2004

2

P

4

2 2 2

1 cot2 4 P SR

·

SR SR 2

4

2

cot

, B10

(1 r )/2 is the interferwhere P SR 16 0 W PR / mc 2 , ometer half bandwidth part due to optical losses, and SR (1 r SR )/2 is the part of the half bandwidth due to the signal-recycling mirror. We can now optimize this expression at some fixed frequency 0 with respect to the homodyne angle and circulating power W PR . The optimal circulating power W op t and homodyne angle op t for the considered scheme are equal to mc W
op t 2

´ FIG. 12. Fabry-Perot cavity with movable mirror.

into account it is reasonable to conclude that the scheme with ´ Fabry-Perot cavities in both arms without a SRM looks like the best candidate for implementing a speed meter interferometer.
APPENDIX C: DERIVATION OF SPECTRAL DENSITIES FOR SPEED METER INTERFEROMETER ´ WITH FABRY-PEROT CAVITIES IN ARMS ´ 1. Fabry-Perot input-output relations

32

14
0

2 0 2 SR

2 0 SR

,

op t

arctan

SR 2 0

,

B11

and the minimal value of 0 is
op t

that can be achieved at frequency
4

SR 2 0

.

B12

We can see that to beat the SQL considerably using a signalrecycled speed meter one needs to decrease internal losses. Decreasing SR or increasing r SR ) will also increase the scheme sensitivity at a given frequency but at the sacrifice of sensitivity at other frequencies. Let us estimate the optimal circulating power that is nec0.1. Let us substitute the following paessary to obtain rameters:
0 0

To obtain input-output IO relations for the scheme presented in Fig. 5 one needs to know the IO relations for a ´ Fabry-Perot cavity with movable end mirror. The case of two movable mirrors will give the same result but when investigating cavity dynamics one should replace the end mirror displacement x by the difference x 1 x 2 of each mirror displacement, and the mirror masses should be taken at one-half of the real value to account for changes in radiation pressure forces. Consider the FP cavity presented in Fig. 12. We suppose the end mirror to have zero transmittance for the same reason as in Sec. IV A. Suppose we know the classical amplitude of the internal ^ field C 1 and input quantum fluctuations a 1 of the input light. To obtain expressions for the internal and output fields one needs to solve the following equations: ^ b ^ d ^ d ^ c
1

103 s

1

, m 5 kg, L 600 m,
1

^ r 1a ^ it 1 a

1

^ it 1 c ^ r 1c

1

^ ia 1 g ^ ia 1 g

12

,

1.77 1015 s
5

, B13
1 1 1 11

1 r 10

,

,

that are typical for the GEO 600 interferometer. To obtain 1 (1 r SR 1.6 10 4 ) and op t 0.1 one needs SR 40 s circulating power of the order of W
op t

2

^ r 2c ^ d

2

^ ia 2 g
i

21

2i ^ c
i

r 2C 2x e .

0

,

1011 W.

B14

2

1

e

,

^ d

2

1

C1

We can see that the considered scheme of a power- and signal-recycled speed meter can hardly be implemented in gravitational-wave detection, as it requires an enormous amount of optical power and its sensitivity is high only in a very narrow frequency band. This result confirms the statement we made in Sec. III, i.e., in order to achieve the same sensitivity as a position ´ meter with Fabry-Perot cavities in the arms, the speed meter requires a highly reflecting SRM to be used and therefore an enormous amount of pumping power is needed. Taking this
102003-13

The solution can be written as 1
1

^ b

L

B ^ g e
i

^ a

1

e e
i

2i

^ r 2t 1a 1g ^ t 1a 2g
0 21

11

ia 1 L 2

12

r 2t 1C 2x

,

C2a


S. L. DANILISHIN

PHYSICAL REVIEW D 69, 102003 2004

^ c

1
2

L ie
2i

ie

i

^ t 1a
21

1

ie 2i

i

^ a 1g e
2i

11

^ r 1a 2g

r 1r 2C 2x

0

, C2b

^ d

1
2

L 2i

ie

i

^ t 1r 2a r 2C 2x

1

ie
0

i

^ r 2a 1g

11

^ ia 2 g

21

,

C2c

input and output moments is sequentially reflected from two FP cavities. Therefore, the sideband operator that describes the output beam for the first cavity at the same time describes the input beam for the second one. Moreover, the beam that enters the cavity for the first time and the beam that enters the cavity after being reflected once do not interact as they have different polarizations ( and in our case . Then, the beams that leave the scheme falling to the beam splitter from ^ the north and east are characterized by sideband operators c n ^ e that can be expressed in terms of input beam operaand c tors as ^ c 1
n

where L r 1r 2e
2i

1,

B

r

1

e

2i

r2 r

2 1

t2 . 1 C3

L e
i

B ^ t 1a 2g

^ b

I n

e

2i

^ r 2t 1a 1g e
i

II e 11

The expressions C2 represent the FP cavity input-output relations.
2. Radiation pressure noise in FP cavity

II e 21

i2

r 2 t 1 Fx

e

0

, C7a

In this subsection we will obtain the expression for the radiation pressure force acting upon the movable mirror of the FP cavity. In accordance with the formula for the radiation pressure we can write down the following: ^ F ^ W
rp c2

^ c

1
e

L e
i

B ^ t 1a 2g

^ b

I e

e

2i

^ r 2t 1a 1g e
i

II n 11

II n 21

2

r 2 t 1 Fx

n

0

, C7b

^ W c

d2

^ E

2 c2

^ E

2 d2

4

A.

C4

Taking into account the expressions C2 one can obtain ^ Ft F
0

0 i( 0 )t

^ C *c 2 d 2

2

*^ D2 d

2

e F

H.c. ^ F a1a e
i( 0

0

0

1

F d 2

g 11g 11

^

where F is the amplitude inside the cavity during the second ^I ^I reflection, b 1 and b 1 are defined by formula C2a by ren e I I placing the index 1 by 1 and 1 the superscript I means the n e first reflection and the superscript II means the second reflection from the FP cavity , and C 2 by E n iE / 2 and E e E / 2 , where E is the amplitude inside the cavity during the first reflection. Here we also introduced additional pumping to compensate the energy losses due to absorption in the FP cavities. We suppose that this pumping should feed the main beam before it enters the second cavity; then the paF e / E e F n / E n . Now we are able to calculate rameter ^ ^ ^ the output beam sideband operator c s ( ic n c e )/ 2: ^ c 1
s

F where F

g 21g 21

^

)t

H.c.,

C5 L

2

iB e e

2

^ a

I s I s 11 II w 11 i

2 it 1 e
a1

i

* C2

L ia 2 e
2i

,

F

2 ia 1 e
g 11

i

* C2

iB , iL C6 B1

2i

^ r 2t 1a 1g ^ r 2t 1a 1g L E

e

^ t 1a 2g
i

I s 21 II w 21

L

2i

e e
i

^ t 1a 2g

F

1 C* 2

g 21

L

,

r 2t 1x

0

, C8

2 2 2 and F 0 0 C 2 (1 r 2 )/ c 2 0 C 2 / c is the constant classical radiation pressure force. Items that correspond to other noise operators are neglected as they are small compared to the above values.

where x ( x n x e ) /2, ^ s ( ^ n i ^ e )/ 2 , and ^ w ^ e i ^ n / 2) here ^ stands for any bosonic operator . (
4. Radiation pressure noise spectral density S F ,, ...

3. Input-output relation derivation

Now we can write down the IO relations for our speed meter scheme. One can note that the light beam between

In order to calculate the radiation pressure noise for the speed meter as a whole we need to use the expression for radiation pressure fluctuational force acting upon the end

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SENSITIVITY LIMITATIONS IN OPTICAL SPEED . . .

PHYSICAL REVIEW D 69, 102003 2004

´ mirror of the Fabry ­ Perot cavity, presented in Sec. C 2 by formula C5 . As in the simple scheme one can present the fluctuational force acting upon each of the FP cavity end mirrors as a sum of two independent items: ^ F
e

F

,I gs
21

ia 2 e

2i

1 L 2 a 1e i L

*E * t 1a 2e · L *E *

i

,

C13c

^ F

e

^ F

e

and

^ F

n

^ F

n

^ Fn ,

C9

F

, II gs
11

,

and the net force acting upon the scheme can be presented as ^^ FF where ^ F and ^ F ^ F
n

^ F,

C10

F

, II gs
21

ia 2 e

2i

1 L

*E *

.

C13d

^ F

n

^ F

e

The spectral density of the radiation pressure noise can be calculated using the same technique as in Appendix A, i.e., 1 S 2

^ Fe .

C11

S

F

F

0

S

F

0

,

C14

Let us write down the explicit form of these expressions: ^ F t F 2
0 ,I gs
21

0

^ Fa a
s

s

F d 2

,I ^ I g s g s 11 11

where S 4
22

F

^ g

I s 21

e

i( 0

)t

F

0

0

H.c.,

C12a

F F

as

F

2 as ,I 2 gs 21

F

,I gs 11

F

,I 2 gs 11

2 where F 0 0 E / c is the corresponding classical radiation pressure force,

,I gs 21

F

F

, II 2 gs 11

F

, II 2 gs 21

. C15

F

as

2 t 1e L

i

E*

,

C12b If one uses the formula 5 and the notation introduced in Table II, then it is easy to obtain that the radiation pressure noise spectral density in the narrowband approximation is defined by the following expression: 1 E* , C12c S
F 2 2 2 2 2 2 2

F

,I gs 11

2 a 1e L

i

E*

,

F

,I gs
21

ia 2 e

2i

L

8 L

0 2

W

1

2

1 22

, C16

1

and ^ F t F 2
0 ,I ^ I g s g s 21 21 , II ^ II g s g s 21
21

0

^ Fa a
s

s

F

,I ^ I g s g s 11 11

where W mirror.

0

E 2 (1

2

) /8 is the light power at the end

F F

F e

, II ^ II g s g s 11 11 )t

i( 0

d 2

5. Shot noise spectral density S x ,,

...

H.c.,

C13a

The shot noise spectral density can be obtained in the same manner as in Appendix A. Here K 2i L
3/2 0

2 E 2 / c is the corresponding classical rawhere F 0 0 diation pressure force,

E

e L
2

i 0

r 2t

1

c
0

B1 ,

0

sin C17

F

as

2 t 1e i L 2 a 1e i L

*E * B · L *E * t 1a 1e · L
2i

,

C13b

sin

F

,I gs 11

,

, arg , and the spectral where LO arg C , density is defined by the following expression:
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PHYSICAL REVIEW D 69, 102003 2004

S

0 x

c
2

2

1
0

2

L L

0 0

2

K

32

0

W r 2 t 2 B1 21

sin

sin

2

.

C18

Using the narrowband approximation defined by Eq. 5 and Table II, the following expression for S x can be written if one suppose 0: S
x

Optimizing Eq. D2 with respect to P and A, one can readily show that it reaches minimum at b P and ab 1 A ab 1 1 , D4

L 32
0

2

1 W
1

2

2

22 1 2

sin2

1

2

1

2

2

,

where a and b should be taken at frequency 0 . When substituted in Eq. D2 , these expressions will turn it into
2

C19 where L c .
6. Real scheme cross-correlation spectral density S
xF

ab 1.

D5

It can be shown that the cross-correlation spectral density for the real speed meter scheme with optical losses is the same as for the ideal one, i.e., S
xF

The second step is the minimization of the above expression with respect to . Obviously, to obtain the optimal value of it is necessary to solve the equation K 0, and

2

cot

.

C20

/ where K ab . Here we should remember that 0 / ; then the above equation is transformed to K K 0,

The above expression does not depend on frequency and is the same in the narrowband approximation.
APPENDIX D: NARROWBAND OPTIMIZATION OF
2

which after simplification will be written as
2

In this section we will perform the optimization of the speed meter interferometer sensitivity at some fixed given frequency 0 . In this connection it seems convenient to introduce new dimensionless variables ,
1

2

1

2 0 2

2

1 0.

D6

16 ,P mL

0

W , A cot D1

The positive solution of this equation is
op t 2 2 0 0

23

,

D7

to rewrite Eq. 21 as
2

1 Pa 2

1 A2 b P

A,

D2

where the last approximate equality corresponds to the case of small losses 0 . Substituting the obtained results in Eq. D5 with respect to the case of small losses, one will have
4 2 3 4 2 3 4

where
2

min

1

,
0

D8

2
22

2

1

22 2 2

a

2

,

b 21

.

D3

1

which is the same expression that is presented in formula 22 .

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