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THE ANNUAL REPORT OF THE MSU GROUP (Jan.-Dec. 2001)
Contributors: V.B.Braginsky (P.I.), I.A.Bilenko, M.L.Gorodetsky,F.Ya.Khalili, V.P.Mitrofanov, K.V.Tokmakov, S.P.Vyatchanin The researches were supported by NSF grant #PHY98-00097 "Suspensions and suspension noise for LIGO test masses" (July 1998 { June 2001) and by NSF grant"Low noise suspensions and readout systems for LIGO" (July 2001 { June 2004)

Contents

I

Summary

3

A The improvement of violin modes Q-factors in fused silica suspension of the test mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 B Measurement of electric charges on the fused silica test masses . . . . . . 3 C The development and the improvement of methods of registration of small excess noise in the violin modes of all fused silica suspension . . . . . . . . 4 D The development of the method of measurement of thermorefractive noise in fused silica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 E The analysis of table-top quantum measurement with macroscopic masses 6 F Frequency-dependent rigidity in large-scale interferometric gravitationalwave detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 G Parametric Oscillatory Instabilityin Fabry-Perot Resonator . . . . . . . . 8 H Collaboration of MSU group with group of K. S. Thorne . . . . . . . . . . 9 1 Conversion of conventional gravitational-wave interferometers into QND interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Reducing of Thermoelastic Noise by Reshaping the Light Beams and Test Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1


I

Collaboration of MSU group with LIGO team . . . . . . . . . . . . . . . . 10

II

APPENDIXES 11 Appendix D The development of the method of measurement of thermorefractive noise in fused silica . . . . . . . . . . Appendix E The analysis of table-top quantum scopic masses . . . . . . . . . . . . . . . . . . Appendix F Frequency-dependent rigidity in gravitational-wave detectors . . . . . . . . . . Appendix G Parametric Oscillatory Instabilityin ometer . . . . . . . . . . . . . . . . . . . . . ................ measurement with macro................ large-scale interferometric ................ Fabry-Perot (FP) Interfer................ 11 14 30 44

2


I. SUMMARY A. The improvement of violin modes Q-factors in fused silica suspension of the test mass
According to the list of tasks formulated in the program of this grant V.P.Mitrofanovand K.V.Tokmakovhave carried out direct measurements of the Q-factors of violin modes in a suspension of the test mass model. Before this new test the highest obtained value of the Q was about 1 108 (see V.B.Braginsky et al., Physics-Doklady, 40 (1995) 11, S.Rowan et al, Proceeding of the Third Edoardo Amaldi Conference, AIP , Melville, NY, 2000). In these new tests special precautions were made to reduce the possible additional losses caused by sedimentation of fused silica vapour on the surface of the pin and the ber during the welding (hydro uoric acid was used to remove visible sedimentation). Another procedure was used in the new tests whichallowed to reduce the second possible source of additional losses: the adsorbed and absorbed water. The ber was baked during 6 hours at the temperature 260C inside the vacuum chamber by a special oven that was installed around the ber. The net result of these tests was the rise of Q of the lowest violin mode up to Q =2:45 108 ( 10%). The expected value of Q in this test (based on the value of the intrinsic loss and on the dilution factor) would be Q =(5 10) 108 . It is necessary to note that in a similar test P.Willems at Caltech has reached recently the value Q =5 108. The gap between these twovalues probably is due to not perfect (in MSU experiment) system of reduction of the recoil losses in the supporting elements of the installation and more perfect system of the ber fabrication developed at Caltech.

B. Measurement of electric charges on the fused silica test masses
The second task formulated in the program of the grant is the measurementof dc and ac components of the electric charge on the mirror. These two components may mimic the action of gravitational wave on the mirror if an electrostatic actuator consists of a at 3


fused silica plate that is covered with two sets of metal stripes. If the surface of the mirror without coating has electric potential U = 30V and the electric charge on it changes by q =8 10;16C ' 5 103 electrons then the "jump" of the Coulomb force acting on the mirror will be Fc ' 3 10;8 dyn for the actuator with the square area 10cm2 and actuatormirror gap d =0:2cm. This value Fc is in the order of the amplitude of force corresponding to the action of gravitational wave in LIGO II (h =10;22, m =104 g). Two experiments were realized by V. Mitrofanov, K.Tokmakov and postgraduate student I.Elkin. In the rst one the vibrating capacitor probe technique was used to measure the value of the electric charge on a fused silica plate situated in vacuum. In this experiment the level of the obtained sensitivity corresponds to variation of the charge density ' 4 10;17 C/cm2 ' 3 102 electrons/cm2 (see details in: I.A.Elkin, V.P.Mitrofanov, Method of measurement of electrostatic charges on the surface of dielectric specimens, MoscowUniv. Phys. Bull., N5, (2001) 47). In the second experiment the electric charge on 0.5kg model of the fused silica mirror suspended by two silica bers in the high vacuum chamber was measured. In this one the rise of the electric charge with a rate of about 2 10;12 C/cm2 ' 107 electrons/cm2 per twoweeks was observed. At present these experiments continue.

C. The developmentand theimprovement of methods of registration of small excess noise in the violin modes of all fused silica suspension
The task of measurements of small excess noise in the violin modes of fused silica bers turn out to be a much more di cult one in comparison with the measurements of the same class of noise in steel and tungsten wires. During the last year a new readout system (after two preceding ones which failed to guarantee the necessary sensitivity) was tested. In this system a small diamond-shaped at plate of fused silica covered with high re ective coating was welded into the middle of a fused silica ber. After the welding direct tests have shown that the welding to the sharp corners of the plate does not damage the coating and the mechanical quality factor of the violin mode (the value of the obtained nesse is higher than 4


50). This value is su cient to reach the sensitivityat level 5 10;13cm/ Hz with few milliwatts power from a He-Ne laser). Apart from this problem special precautions were used for not to ruin the high quality factor of the violin modes in the support structure and in the load. This will allow to decrease substantially the recoil losses and thus to savehigh Q factors of violin modes. At present the process of assembling of all elements of the test installation is in progress.

p

D. The development of the method of measurement of thermorefractive noise in fused silica
M.L.Gorodetsky and his student I.S.Grudinin are nishing the assembling of new installation aimed to observe thermorefractive noise in fused silica microspheres and to test in this way predictions of this noise in LIGO antenna. The installation is based on small chamber for the resonator whichmay either be evacuated or lled with pure nitrogen. In this chamber having optical windows, fused silica cantilever is installed for ne positioning of microspheres relative to coupling prism. Nd:YAG diode pumped laser (Lightwave Electronics, 1:064 m) with narrow linewidth analogous to the lasers in LIGO antennas will be used for the measurements. In preliminary tests of microspheres (diameter 600 ; 200 m)in the chamber with this laser, optical modes with Q-factors of the order of (109)were observed with pronounced thermal nonlinearity. The main di culty in measurements should be stabilization of frequency of the laser on the slope of the resonantcurve in presence of strong thermal nonlinearity. Computer based stabilization scheme is now in preparation. Numerical simulation of thermal nonlinearity in microspheres is also in progress now aiming to nd optimal methods for locking the laser on the mode with strong thermal nonlinearity. The modeling revealed regimes of oscillatory instability in microspheres which were experimentally observed but not explained several years ago in MSU group. The theoretical calculations of spectral densities of this noise in microspheres were given 5


in the previous annual report and presented at international symposium Photonics West'2001 in San-Jose and published in Proc. of SPIE 4270 (2001). See in appendix D the theoretical analysis of the connection between thermorefractive noise and nonlinear response of the microresonator which may simplify the experimental veri cation.

E. The analysis of table-top quantum measurement with macroscopic masses
At the stage II of the LIGO pro ject (years 2006-8) the antennae sensitivity is expected to be close to the Standard Quantum Limit, and in the stage III the sensitivitywill haveto be better. The members of the MSU group share the point of view that before the implementation of any version of QND measurement in the LIGO pro ject, apart from scrupulous analysis it will be fruitful to realize at table-top scale a real experiment with small test masses. This type of test may permit to discover undesirable e ects whichmight be missed in the modeling. F.Ya.Khalili and his studentP.S.Volikov carried out analysis of suchascheme. The scheme of experiment is based on two concepts: the di erence between the free test mass and the oscillator SQL sensitivity, and the use of mechanical rigidity produced by an optical pumping eld in Fabry-Perot resonator to convert the free test mass into the mechanical oscillator having very low intrinsic noises. The analysis shows that proposed scheme allows to circumvent the free test mass Standard Quantum Limit by the factor ' 0:1, using good, but not unique mirrors with 1 ; R ' 10;5 (R is the re ectivity factor), and moderate pumping power W ' 50mWt. The main goal of the proposed experimentwill be to show that various noises of non quantum origin does not preventto achieve sensitivity better than the SQL at room temperatures. Evidently there are others, more sophisticated schemes of measurements whichmay provide better sensitivity. At present there are two evident \candidates". In the rst one (which is a simpler one) it is possible to use variational measurementby periodic modulation 6


of the phase of the reference beam. The second one is the stroboscopic measurement which is in essence a Quantum Non-Demolition measurement of the probe oscillator quadrature amplitude (see details in the Appendix E).

F. Frequency-dependent rigidity in large-scale interferometric gravitational-wave detectors
Several methods to overcome the SQL has been proposed but most of them encounter serious technological limitation and/or has some other disadvantages whichmake implementation of these methods hard in the near future. On the other hand, the SQL itself is not an absolute limit but, in particular, it depends on the dynamic properties of the test ob ject which is used in the experiment. The well known example is the harmonic oscillator, which allows to obtain sensitivity better than the SQL for the free mass when the signal frequency is close to the eigen frequency of the oscillator m . F.Ya.Khalili has shown, that electromagnetic rigidity which exists in large-scale optical resonators if pumping frequency is detuned from the eigen frequency of resonator have sophisticated spectral dependence which allows to obtain sensitivity better than the SQLs both for the free test mass and the harmonic oscillator. Depending on the detuning, the bandwidth of the resonator and the pumping power, several regimes of this frequencydependent rigidity are possible. In particular, the second-order-pole regime allows to \dive" deep below the SQL in the narrow spectral band which is, however, much wider than if the usual frequency independent rigidity is used. It is important that the pumping energy in this regime does not depend on the sensitivity and remains approximately equal to the energy which is necessary to achieve the SQL in the traditional scheme of the interferometric position meter. The third-order-pole regime provides sensitivity a few times better than the SQL in relatively wide spectral band and extremely lowlevel of the measurement noise in this band. However its backaction noise is rather large. This regime looks as a good candidate for use 7


in advanced topologies of the gravitational-waveantennae, based on use of separate optical modes for measurement and for creating rigidity, or/and which eliminate back-action noise by using variational measurement (see details in the Appendix F).

G. Parametric Oscillatory Instability in Fabry-Perot Resonator
S. P.Vyatchanin and his student S. E. Strigin have analyzed undesirable e ect of parametric instabilitywhich (being ignored) may cause very substantial decrease of the antennae sensitivity and even may make the antenna unable to work properly. This e ect is originated from the fact that in LIGO-II high value of optical energy E0 is planned to be stored in the Fabry-Perot resonator optical mode: E0 > 30 J (it corresponds to the circulating power W of the order 0:8 megawatt). This e ect takes place if mode of elastic oscillations in FP resonator mirrors with frequency !m and Stokes optical mode with frequency !1 are coupled with optical main mode with frequency !0 (which is pumped) by parametric condition !0 ' !1 + !m . The origin of parametric instability can be described qualitatively in the following way: small mechanical oscillations with the resonance frequency !m modulate the distance L between mirrors that causes the excitation of optical elds with frequencies !0 !m . Therefore, the Stokes mode amplitude will rise linearly in time if time interval is shorter than relaxation time of Stokes mode. The presence of two optical elds with frequencies !0 and !1 will produce the component of ponderomotive force (which is proportional to square of sum eld) with di erence frequency !0 ; !1. Thus this force will increase the initially small amplitude of mechanical oscillations. It is obvious that at some threshold level of initial energy E0 in main mode this "feedback" prevails the damping in Stokes and elastic mode | it maygive birth to the parametric oscillatory instability. The calculations for simpli ed model shows that in resonance case parametric instability may take place when value of optical energy is about 300 times smaller that planned in LIGO-II. Several recommendations to obtain a guarantee to evade this non-desirable e ect 8


and details of calculations are presented in Appendix G.

H. Collaboration of MSU group with group of K. S. Thorne
1. Conversion of conventional gravitational-wave interferometers into QND interferometers

This part of researches was done byS. P.Vyatchanin in collaboration with H.J.Kimble, Yuri Levin, Andrey B.Matsko and K.S.Thorne. The possible designs for interferometers that can beat the SQL (LIGO-III) was analyzed. These designs are identical to a conventional broad-band interferometer (without signal recycling) except for new input and/or output optics. Three variants of design were analyzed: 1. A squeezed-input interferometer in which squeezed vacuum with frequency-dependent (FD) squeeze angle is injected into the interferometer's dark port. 2. A variational-output interferometer, in which homodyne detection with FD homodyne phase is performed on the output light 3. A squeezed-variational interferometer with squeezed input and FD-homodyne output. It is shown that the FD squeezed-input light can be produced by sending ordinary squeezed light through two successive Fabry-Perot lter cavities before injection into the interferometer, and FD-homodyne detection can be achieved by sending the output light through two lter cavities before ordinary homodyne detection. With anticipated technology (power squeeze factor e;2R =0:1 for input squeezed vacuum and net fractional loss of signal power in arm cavities and output optical train = 0:01) and using an input laser power Io in units of that required to reach the SQL (the planned LIGO-II power, ISQL), the three types of interferometer could beat the amplitude SQL at 100 Hz by the following pS =qS SQL and with the following corresponding increase V =1= 3 in the amounts h h volume of the universe that can be searched for a given non-cosmological source: p Squeezed-Input | ' e;2R ' 0:3and V' 1=0:33 ' 30 using Io=ISQL =1. 9


Variational-Output | ' 1=4 ' 0:3 and V ' 30 but only if times larger power: Io=ISQL ' 1=p =10. Squeezed-Variational | = 1:3(e;2R )1=4 ' 0:24 and V ' p ' (e;2R )1=4 ' 0:18 and V' 180 using Io=ISQL = e;2R= The details see in LANL Archive (qr-qc/0008026). The pape D.

the optics can handle a ten 80 using Io=ISQL = 1 and ' 3:2. r is submitted to Phys. Rev.

2. Reducing of Thermoelastic Noise by Reshaping the Light Beams and Test Masses

This part of researches are being done byS.P.Vyatchanin and his student S. E. Strigin in collaboration with ErikaD'Ambrosio, Richard O'Shaughnessy and Kip S.Thorne. Thermoelastic uctuations of mirror's surface decrease considerably if beam with larger radius is used. This source of noise is very signi cant in sapphire test masses (as it was shown in Physics Letters A A264 (1999) 1). In order to decrease the thermoelastic uctuations of mirrors surface it was proposed to use so called "Mexican-hat" light beam in the LIGO interferometers. These beams has larger radius of beam than Gaussian beams at the same difractional losses and hence thermoelastic noise for such beams is smaller. It was also proposed to use conic shape of test masses (instead of cylindric one). The preliminary results show that both these modi cations allow to decrease thermoelastic noise by factor 4 8. The article is in preparation (see also LIGO documents G010151-00 and G010297-00).

I. Collaboration of MSU group with LIGO team
In 2001 the fruitful collaboration between LIGO team and MSU group continued. V.Mitrofanov has visited Caltech and Phil Willems has visited MSU. Now P.Willems and V.Mitrofanov together with other colleagues prepare paper "Investigation of the Dynamics and Mechanical Dissipation of a Fused Silica Suspension" which describes the highest violin modes Q's yet measured in fused silica. 10


II. APPENDIXES Appendix D. The development of the method of measurement of thermorefractive noise in fused silica
Thermodynamical uctuations of temperature in the mirrors of LIGO are essential for the estimates of ultimate sensitivityof the antenna. These uctuations lead to thermoelastic and thermorefractive noises due to e ects of thermal expansion and thermal dependence of index of refraction. Thermorefractive noise may be analysed in laboratory conditions using microspheres with whispering gallery modes. These microresonators combine very small e ectiv volume (the diameter is of the order of 100 m, and e ectvevolume is of the order of 10;8 cm3) with very high quality factor of optical modes ( 109). For correct interpretation of measurements of thermorefractive noise spectral density in microspheres is complicated by the di culties with precise identi cation of the excited mode and hence the uncertainty of the distribution of the optical eld. One may show, however, that this spectral densitymaybe veri ed and compared with the results of another independent and simpler measurement - nonlinear responce of resonance frequency on the weak modulation of input optical power. The modulation of intensity in the mode leads to the modulation of the temperature of dn the mode volume and hence through the change of refraction index n = dT u to the change of resonance frequency. This e ect is known as thermal nonlinearity and may be analysed using equation of temperature conductivity:

@u ; a2 u = !W @t Qa C 2 jE j2 where W = n 8 ~ is internal energy density in the mode and Qa =2 n= ( is absorption). ~ ~ ~r jE j2 = I (t)jE0j2, where R jE0j2d~ = 1, C is speci c heat capacity, and is density of the material of the microresonator.
11


In spectral form we obtain:
r G( ~ )ei~~ d ~ u( ~)= I ( ) a2 2 + i (2 )3 r
Z

where

= 8 nc C Z ~ G( ~ )= jE0j2e

;i ~~ r

d~ r

is the spacial spectrum of energy distribution of the mode. Averaging temperature over the mode volume we obtain the spectrum of temperature modulation: Z ZZZ ~ rr jE0(~)j2jE0(~0)j2ei~(~;~0 ) d~ ~0 d ~ r ~r ~ 0j2d~ = I ( ) u( ) = ujE r rdr (2 )3 a2 2 + i Z j (~ ) 2 d~ = I ( ) a2G2 + ji (2 )3 "Z # Z 2jG( ~ )j2 d ~ jG( ~ )j2 d ~ = I ( ) a2 a4 4 + 2 (2 )3 + i (1) a4 4 + 2 (2 )3 This is the responce of the temperature on intensity modulation that may be measured 1 dn by measuring the dependence of resonance frequency modulation !! ( ) = n dT u( ) from the modulation of input power. From the other hand the spectral density of thermodynamical uctuations of temperature. averaged over the mode volume mayt be found see the papers as:

u = u(~ t)jE0(~)j2d~ r rr
where

Z

F ( ~ ) ei t+i~ a2 2 + i 2 kT 2 = 2a C 2 ( (see our previous report or Physics Letters A, A264 Hence u(~ t)= r
Z

~ r

d d~ (2 )4

; 0) ( ; 0 )
(1999) 1)

12


2 Su

2 = 2a kT C

2Z Z Z

a

44

r 2 r0 2 2 jE0(~ )j jE0 (~ )j e +
2 = 2kT a2 C
Z

2

i ~(~;~0 ) rr

dr 2jG( ~ a4 4 +

d~ dr0 (2 ) )j2 d ~ 2 (2 )

3 3

(2)

Now comparing this result with (1) we nd

kT 2 2 Su = 16 nc
13


Appendix E. The analysis of table-top quantum measurement with macroscopic masses
Introduction

There is an evident steady progress in improvement of the sensitivityin manytypes of physical measurements. Particularly, in the previous century late 80-s several groups of experimentalists successfully demonstrated the resolution better than the Standard Quantum Limit (SQL) in optical domain using QND methods (e.g. see review article 1]). At the end of the 90-s even more impressive experimentwas realized in the microwave domain. In this experiment single microwave quanta were counted without absorption 2]. At the same time experiments with resolution better than SQL using mechanical test ob jects are not yet realized. There is at least one area in the experimental physics where the necessity to circumvent the SQL of sensitivity using mechanical test masses is crucially important. This is the terrestrial gravitational waveantennae creation. At the stage II of the LIGO pro ject (years 2006-8) the antennae sensitivity is expected to be close to the SQL, and in the stage III the sensitivity will have to be better 3]. There were several articles with di erent schemes of measuring devices aimed to \beat" the SQL for mechanical ob jects (see, for example, articles 4], 5], 6], 7]. In the ma jority of these articles only concepts of new methods were presented and only one of them 7] has provided rather detailed analysis of the measurementscheme using mechanical ob ject (mirror of the gravitational-waveantenna) based on QND principle. It was shown that proposed scheme can be implemented provided that very sophisticated cryogenic technique is used. In this article we present the analysis key parts for a simple experimentscheme using relatively small test masses that is able to provide the sensitivity better than the free test mass SQL and can be realized in relatively modest laboratory conditions. The rst initial principle of the scheme is based on the di erence between the sensitivity 14


SQLs for the force F acting on the free mass m and mechanical oscillator having the same mass and eigenfrequency m:

F

free mass SQL oscillator SQL

'

r

F

p~m '

~

m

2 F

(3) (4)

F

where F is the mean frequency of the force and is its duration. These equations are valid if & 1= F and j F ; mj . 1= . Comparing equations (3) and (4) one may conclude that it is possible to \beat" the free mass SQL by the factor
oscillator FSQL = free mass ' FSQL

1
F

1=2

(5)

using test mass with su ciently low noise rigidity m 2 attached to it. It can be shown (see m Appendix II) that the exact form of this condition is =
1=2

F

(6)

where ' 1= is the bandwidth of the force. The second initial principle of the scheme is based on the possibility to create mechanical rigidityprovided by the dependence of light pressure on the detuned Fabry-Perot resonator mirrors position. Wehave already analyzed the possibility to create very low noise rigidity using the pumping frequency !p detuned much far from the resonator eigenfrequency !o 8]. It was supposed in article 8] that separate resonator must be used as a measuring device. In this article we propose another simpler scheme where the same resonator serves as the meter and the rigidity source. In this case optimal detuning = !p ; !o should be close to the resonator semi-bandwidth = !o =2QFP,where QFP is the quality factor of the resonator. Weshow that this scheme allows to reach the limiting value see formula (6)]. 15


Wehave to note that the potential possibilities to use the mechanical rigidity of optical origin in di erent LIGO readout meters were already discussed in 9], 10]. These possibilities are indicating that SQL can be circumvented in narrow bandwidth. In the presented below analysis wewere trying to give answers to most important practical issues whichmay appear in the implementation of such an experiment.
The main elements of the design and the experiment scheme

The test mass suspension The most important elements of the experimental setup are the test mass suspension, optical rigidity, and optical readout scheme. Simpli ed sketchof experimental scheme is presented on Fig.1.

The necessary condition for such kind of experiments is a su ciently low dissipation in the suspension. The quality factor of the mechanical test oscillator has to exceed value

T 104 s;1 (7) 300K F F where is the Boltzmann constant, and T is the heatbath temperature. In fact, this inequality represents the condition that the uctuating force originated from mechanical losses in the suspension (according to FDT) has to be ;1 times smaller than the force free FSQLmass . The estimate (7) show that an ordinary mechanical spring can't be used here. It is necessary to use \arti cial" rigidity with very lowintrinsic noises, and optical ponderomotive rigidity does look promising. In this case the suspension can be similar to the Galileo pendulum with eigenfrequency pend F . The relaxation time of pend ' 2 108 s has been already obtained for all fused silica suspension of the LIGO mirror model. This value of pend allows in principle to obtain the quality factor of Qm ' m pend ' 2 1012 (even if the viscous model of friction is valid) and thus allows to reach ' 0:1. In the Appendix c more rigorous analysis of the suspension noises is presented, whichshows that these noises do not prevent from obtaining the sensitivityof . 0:1.
10 Qm & ~2 T 2 ' 102

16


It is evident that the platform the suspending ber has to be welded to must be a compact one: the mechanical eigenmodes of the platform have to be substantially higher than the chosen value of F . If the value m ' 2 10;2 g (a few millimeters in dimension cylinder covered with high-re ectivitymultilayer coating) then for F ' 104s;1 the meter has to register the oscillations of the mass with the amplitude of

x = m~ ' 2 10;15cm : F
We omit here calculations which show that if the platform has sizes of a few centimeters and is manufactured from fused silica then quality factor of the eigenmodes that is higher than 105 will be su cient to register this value of x. It seems appropriate to use the rst mirror of the Fabry-Perot resonator as the test mass and the second mirror must be attached rigidly to the platform (see Fig.1). The optical rigidity It is a relatively easy task to \convert" the mass m ' 2 10;2 ginto a mechanical oscillator with eigenfrequency m ' 104s;1 . If the mass is the Fabry-Perot resonator mirror (see Fig. 1) and the laser is tuned on the one of the resonator resonance curves slope then the rigidity will be equal to
M1 M2

r

m

2 m

2 = = 16!o2W2F c 1+ ( = )

22

' 2 106 dyn=cm

W 50mW

10

F

2

3

= : 1+( = )2 2 (8)

where !o = 2 1015s;1 is the optical pumping frequency, F is the nesse of the optical resonator, W is the pumping power. Fig.2 illustrates the dependence of the laser ponderomotive force on the distance between the mirrors. Dashed line corresponds to the pendulum rigidity m 2 nd. There is a relatively pe big number n of static equilibrium points (which correspond to the crossings of the right slopes of the resonant curves with the horizontal axis these points are marked by Xsonthe Fig.2): 17


!o W n = 24mc3 F ' 500 2 pend

W 50mW

10

F

3

10s;1
pend

2

:

(9)

Thus bychoosing one of these points experimentalists maychange the rigidity. Another way for changing it is to apply a d.c. force onto the mirror (using for example the light pressure from another laser). This rigidity has one disadvantage: it is associated with negative friction which corresponds to the characteristic time (\negative relaxation time") equal to =2 1+ ( = )2 : (10)

instab

2 m

If the nesse of the resonator is F ' 103 and its length is L . 1cm then & 5 107 s;1, and the value of instab can be large enough to provide su cient time for the measurement, 4 m instab & 10 . The optical readout scheme The optical readout scheme is presented on the left part of the Fig.1 The pumping laser beam is splitted into two beams (the signal beam and the reference one). The signal beam enters the Fabry-Perot resonator and passes back carrying information about the displacement of the mirror relativeto in its phase. The re ected beam is separated from the input one by polarization beam splitter and Faraday isolator , and then is combined with the reference beam on the beamsplitter . This beamsplitter together with photodetectors and form standard balanced homodyne detector. It is assumed that an arbitrary phase shift LO can be added to the reference beam. The di erence between the photocurrents in such a scheme depends on the phase shift of the signal beam relative to the reference one,
M1 M2 PBS FI BS2 D1 D2

WW I1 ; I2 = 2e ~! ref cos o

p

1

+ 2 !o 2 x ; 2+ L

2

;

LO

18


and thus provides information about x. Here Wref is the reference beams power, 1, 2 are the initial phases of the signal and reference beams. It can be shown (see Appendix II) that if the Fabry-Perot resonator bandwidth is de ned by the nonzero transmittance of the mirror only, if there are no losses in all optical elements which the signal beam passed through, if quantum e ciency of the photodetectors are equal to unity and quantum state of the pumping beam is pure coherent one then the value (6) can be achieved in this ideal case. In more realistic case when the above conditions are not ful lled the total achievable value of 2 is a sum of two terms: the \ideal" value described by formula (6) and additional 2 value optics which depends on the parameters of the optical scheme. General expression 2 for optics is very cumbersome. In asymptotic case when = F 1, losses are small and 2 su ciently large value of the pumping power can be provided, the value of optics can be presented as
M1

2 optics

2 p1 1+=( )=3=2) + A = P(

(11)

where P is a dimensionless parameter proportional to the pumping power:

! P = mc2 64(1o W ) 2 ;R2 F

2

1:5 10

4

W 50mW

20mg m

5 10;5 1 ;R2

2

104 s

;1 2

F

(12)

R2 is the mirror

M2

re ectivity, (1 ;A0)(1 ;A1) 1 ; + A0 + A (13)

A =1 ;

PD

PD

1

is total \external losses", PD is quantum e ciency of the photodetectors, A0 is total absorption factor of the optical elements between the Fabry-Perot resonator and the photodetectors, A1 is the mirror absorption factor. Expression (11) is valid if P 1 and A 1.
M1

19


It is evident that the sensitivity depends on the detuning . Optimal value of depends p on whether rst or second term prevails in the expression (11). If A 1= P then the p optimal value is = 3 and
2 optics

3 3P In the opposite case the detuning must be large, = =(4A2P )
2 optics

=

p

p

4

1:75 : p P

(14)
1=3

1 and in this case (15)

=3 A 2P

1=3

:

These estimates show that even in the case of moderate conditions for the optical elements parameters losses in them don't prevent from obtaining the value of optics ' 0:1 when the pumping power is su ciently large, e.g. W & 50mW. If such a value of pumping power can't be provided then, nevertheless, the sensitivity slightly better than the SQL can be obtained, i.e. optics ' 0:3 0:5. Sensitivity for this case is calculated numerically. The results for the case when = =0:01 are presented on the Fig.3 (solid line). Dashed line is the asymptote (32).
Conclusion

The scheme of measurement presented abovemay be regarded only as the rst step along the route of \divine quantum" measurements with macroscopic quantum ob jects. The main goal of the proposed experiment will be to showthatvarious noises of nonquantum origin do not preventto achieve sensitivity better than the SQL even at room temperatures. Evidently there are others, more sophisticated schemes of measurements which may provide better sensitivity. At present there are two evident \candidates". In the rst one (which is the simplier one) it is possible to use variational measurement 6] by periodic modulation of the phase of the reference beam LO. The second one is the stroboscopic measurement 5] which 20


is in essence a Quantum Non-Demolition measurement of the probe oscillator quadrature amplitude. In the second case it will be necessary to use two pumping lasers: the rst one has to be permanently on and it will provide the rigidity, and the second one has to be turned on periodically during the time interval much shorter than ;1 . m The authors of this article have no doubts that the sensitivity better than the SQL may be obtained at the present level of experimental \culture" in the measurement with mechanical ob ject.
Acknow ledgments

This paper was supported in part by the California Institute of Technology, US National Science Foundation, by the Russian Foundation for Fundamental Research grants #9602-16319a, #97-02-0421g and #99-02-18366-q, and the Russian Ministry of Science and Technology.

21


FIGURES

µ
€°

¸­

µ µ
¸¬

«¸



µ µ

‰

º µ €µ Š­ Š¬

­

¬

ïìµ ¯

FIG. 1. Sketch of the experimental scheme

X

X

X

X

X

FIG. 2. Dependence of the ponderomotive force on the distance between the mirrors

22


0.9 0.8 0.7 0.6

0.5

ª©¼ ½
0.4

0.3

10

100

«

FIG. 3. Sensitivity as a function of the pumping power

Appendix A. The Standard Quantum Limits for the free mass and the oscil lator

Total net noise of linear scheme for detection of classical force acting on the free test mass m is equal to 12]

F

free mass

d2 (t)= m dt2 x

uct

(t)+ F

uct

(t) :

(16)

where x uct(t) is the additive noise of the meter and F uct(t) is its back-action noise. For an ordinary position meter which sensitivity is limited by the SQL, these twonoises are non-correlated and have frequency-independent spectral densities Sx and SF , correspondingly, which satisfy the uncertainty relation

SxS

F

~2

4:

(17)

23


We will suppose that noises are as small as possible and exact equalitytakes place in this formula. In this case spectral density of the total noise (16) is equal to

S

free mass

( )=m
F 24 F

24

Sx + SF :

(18)

For any given observation frequency = ratio of the spectral densities SF =Sx = m

this value can be minimizedby adjusting the , giving (19)

S

SQL free mass

( 0)= ~m 2: 0

This is the spectral form of the SQL for a free test mass 7]. In the case of an oscillator total net noise is equal to

F

oscillator(

d2 t)= m dt2 +

2 0

x uct(t)+ F uct(t)

(20)

and its spectral density is equal to

S

oscillator(

) = m2( 2 ; 2)2Sx + SF : 0

(21)

By adjusting the ratio SF =Sx in order to provide minimum of the spectral density at the edges of the narrow vicinity of the eigenfrequency 0, = 0 =2, where 0, we obtain:

S

SQL oscillator

(

0

=2) = ~m

0

:

(22)

The ratio of the spectral densities (19) and (22) is equal to (6). 24


Appendix B. The sensitivity limitation due to optical losses

a. Spectral density of the total net noise We will suppose that the Fabry-Perot resonator bandwidth is much larger than the observation frequency. It can be shown (we omit lengthy but quite straightforward calculations) that in the case of the Fabry-Perot position meter spectral densities Sx and SF of the noises x uct and F uct introduced in the previous Appendix and their cross spectral density are equal to
2 ~ Sx = 2m 2 1+(2 = ) sin LO 2 SF = ~m 1+ (1 = ) 2

2

SxF = ~ cot 2

LO

(23)

and the electromagnetic rigidity is equal to
2 = K = m 2 = m2 1+ (= = )2 = 4!o 1(12;A1)W 0 3 L 1+ ( = )

22

(24)

where
2

4! = mLo2E

=

1

(1 ;A)

12

;R = 1 4L=c1

2

( 1+ 2= )

(25)

E is pumping energy in the resonator, R1 R2 are the mirrors

M1 M2

re ectivities, A is total

\external losses" see formula (13)]. Spectral density of the total noise of the meter in this case is equal to

m ( 2; S ( ) = ~2
where
2 F

2 )2 1+ ( F 2 sin2 LO

=)

2

2 + 1 ; cos ) 1+( =

LO 2

2

:

(26)

=

2 0

(1 1 2 sin 2 ; 2 1+( = )LO = 4!o 1mL;A1)W = ; sin 22 2 23 1+ ( = ) 25

LO 2

:

(27)


b. Large pumping power Now our goal is to minimize the expression

=2) = ( )2 1+ ( = )2 + 1 ; cos2 LO : (28) 2 m!F = ; sin 2 LO 2 2 sin2 LO It is evident that in order to obtain 1 it is necessary to have = F 1, 1 ; 1 and j LOj 1. Taking it into account one can show that the expression (28) is minimal if
2

S(

F ~

LO

=

opt LO

;

s

F

= 2

(29)

and the minimum is equal to + = =+ A F From this expression it is evident that the larger other hand, the larger is this ratio the larger is the 1and 2 F 0 , see formula (24). If A1
2 2
s p

2 opt : 1+ =LO

(30)

is the ratio = the smaller is . On the pumping power required to provide given 1 then from (27) it follows that
LO 1; = : opt

= 4!o W 2 1+( = ) mL2 F

2

(31)

Substitution of this expression into formula (30) gives that
opt opt 2 1 + p 1+( )=3=2) 1+ 3 =LO + A 1+ 2 =LO (32) = P (= F Omitting here small terms proportional to opt we obtain formula (11). LO c. Small pumping power If P ' 1 then it is possible to neglect the second term in the formula (11) because any good optical components can provide the value A . 0:1. In this case it is necessary to minimize expression (28) with respect to LO , 1 and with given values of the W and 2 and with additional condition (27). This minimization was performed numerically. Results are presented on the Fig.3. 2

26


Appendix C. The suspension noises

We will base our consideration on the formula (11) in the article 13]. If the observation frequency F satis es condition pend F v , where v is the eigenfrequency of the suspension ber fundamental violin mode then this formula can be rewritten as:
2 2 I I 4T susp (33) Sx = I 2 6 l2 m ; Rh top + m ; (R + l)h bot F where l is the length of the suspension ber, I is the test-mass moment of inertia for rotation about the center of masses, R is the radius of the mirror face, m is the mass of the test mirror, h is the displacement of the laser beam spot from the center of the mirror, top bot are values characterizing dissipation at the top and the bottom of the ber. Following authors of the article 13] we suppose that

(34) 2 where Y is the Young modulus of the ber material and J = S 2=4 is the ber geometrical momentum of inertia. If h is chosen optimally:
top bot

=

==

F

pYJ mg

2R +1 I h = R2 +(R + l)2 M then (we suppose that R

I Ml

(35)

l) S
susp x

T 4 = m2 6 R42 +(R + l)2] m2 T6 l2 (36) F F This value of Sx corresponds to the spectral density of the uctuating force acting on the test mass S
susp

=m

2 4 S susp Fx

= 4 2T 2 = 2 T Fl 27

pYJ mg : l2
F

(37)


So the value of

2

limited by the suspension noise is equal to
2 susp

2 = ~Ssusp2 = ~ T l 3 mF F

r

2

YJ g = T g r 2~v 2 3=2 m o

2 v 3 F

(38)

where vo = Y= is the speed of sound in the ber material, stress factor of the ber. For the room temperature and fused silica it will be
2 susp

p

mg = YS is dimensionless

4 10

;4

r ;8 dyn=cm 10

104 s

;1

F

v F

2

10;

3 3=2

(39)

Taking into accountthatvalues r . 10;8 dyn=cm has been already obtained experimentally 14,15] it is possible to conclude that suspension noises don't prevent from obtaining the sensitivity . 0:1.

28


REFERENCES

1] P.Grangier, J.A.Levenson and J.-P.Poizat, Nature 396 (537) 1998 2] G.Nogues et al, Nature 400 (239) 1999 3] Proceedings of Third Edoardo Amaldi Conference, ed. by Sydney Meshkov, 1999 4] C.M.Caves et al, Review of Modern Physics 52 341 (1980) 5] V.B.Braginsky,Yu.I.Vorontsov, F.Ya.Khalili, Sov. Phys. JETP Lett. 27 (1978) 276 6] S.P.Vyatchanin, Physycs Letters A239 (1998) 201. 7] V.B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili and K.S.Thorne, Physical Review D61 (2000) 044002 8] V.B.Braginsky,F.Ya.Khalili, Physics Letters A 257 (1999) 241 9] F.Ya.Khalili, \Quantum experiments with macroscopic mechanical ob jects", proceedings of ICQO, Minsk, 2000 (in press) 10] A.Buonanno, Y.Chen, \Quantum noise in second generation, signal-recycled laser interferometric gravitational-wave detectors", Phys.Rev.D (in press) 11] V.B.Braginsky,V.P.Mitrofanov, K.V.Tokmakov, Physics Letters A218 (1996) 164 12] V.B.Braginsky, F.Ya.Khalili, Quantum Measurement, ed. by K.S.Thorne, Cambridge Univ. Press, 1992. 13] V.B.Braginsky,Yu.Levin, S.P.Vyatchanin, Meas. Sci. Technol 10 (1999) 598 14] V.P.Mitrofanov, O.I.Ponomareva, Vestnik Moskovskogo Universiteta, series 3, #5 (1987) 28 15] S.D.Penn, G.M.Harry, A.M.Gretarsson, S.E.Kittelberger, P.R.Saulson, J.J.Schiller, J.R.Smith, S.O.Swords, Syracuse Univ. Gravitational Physics Preprint 2000/8-11 29


Appendix F. Frequency-dependent rigidity in large-scale interferometric gravitational-wave detectors
Introduction

The standard quantum limit (SQL) 1] is one of the most fundamental factors which prevent the gravitational-wave antennae 2] sensitivity from increasing. The basis for this limit is the uncertainty relation for two kind of noises inherent in position meters: the measurement noise and the back action noise. In the interferometric position meters, these noises are proportional to phase uctuations of the pumping beam and radiation pressure noise, correspondingly. Several methods to overcome the SQL has been proposed (see, for example, articles 3], 4], 5], 6], 7]) but most of them encounters serious technological limitation and/or has some other disadvantages which do not permit implementation of these methods in the near future. On the other hand, the SQL itself is not an absolute limit but, in particular, it depends on the dynamic properties of the test ob ject which is used in the experiment. The well known example is the harmonic oscillator. Its response to a resonant force is relatively strong and it allows to use less sensitive meter (with larger measurement noise and therefore with smaller back action noise). Due to this property the harmonic oscillator allows to obtain sensitivity better than the SQL for the free mass when the signal frequency is close to the eigen frequency of the oscillator m 8], 9], 10], 11]. In the articles 8], 10] it was shown that it is possible to create very low noise mechanical rigidity using Fabry-Perot resonators with detuned pumping. In the article 11] it was also shown that such a rigidity exists in the signal-recycled topology of the gravitational-wave antennae and it permits to overcome the SQL for a free mass in narrow band. One can imagine the following frequency dependent rigidity:

K( ) = m
30

2

(40)


where m is the mass it is attached to, and is an arbitrary observation frequency. In principle, such a rigiditywould allow to obtain arbitrarily high sensitivity throughout the spectral range where the formula (40) is valid. On the other hand, it is well known that the mechanical rigidity created by parametrical opto-mechanical systems can be frequency dependent. In this article we show that the mechanical rigidity frequency dependence in large-scale interferometric meters with bandwidth comparable to or smaller than the signal frequency can be close to the formula (40) in some spectral band.
The simple example: Second-order-pole regime

Consider the simpli ed interferometric detector scheme presented in Fig. 4. Here the signal force with the amplitude Fsignal which had to be detected, acts on the test mass m. This mass serves also as mirror and together with second mirror forms Fabry-Perot resonator. In this article we suppose that refraction of the mirror is equal to unityand there is no absorption in the mirror . The resonator is pumped at the frequency !pump which is detuned far from its eigen frequency !o:
M1 M2 M1 M2

=!

pump

;!

o

(41)

where is the half-bandwidth of the resonator. This detuned pumping creates a ponderomotive rigidity. One of the re ected beam quadrature amplitudes is measured, giving information about the mirrors relative position.
M2 M1

!

pump

= !o +

-

-

Fsignal -

L

FIG. 4. Simpli ed scheme of the interferometric detector

31


We will refer to this simple scheme in this article but it can be shown that all results obtained here are valid for the signal-recycled topology 12] planned for the second stage of the LIGO program. By solving this system equations of motion it is easy to show that if ! 0 then mechanical rigidity created by the optical pumping will be equal to

K( )

L

2!o 2( 2

E ; 2)

(42)

where is the observation frequency, E is the optical energy stored in the resonator, and L is the resonator length. 1

0.5

K=m ( =)
0 0 0.5 ( = )2

2 2

1

FIG. 5. Second-order pole ( ! 0)

Ourgoalis toset K ( ) as close to the ideal frequency dependence (40) as possible nearby some given value of , so we require that

K( ) = m
and 32

2

(43)


dK ( ) = d(m 2) : d d
It is easy to show that these conditions can be ful lled if (and only if ) = and
23 23 p E = mL = mL 2 : 8!o 2 2!o

(44)

2

p

2

(45)

(46)

In this case susceptibility of test ob ject which consists of mass m and such a rigidity ( ) = ;m 2 1 K ( ) + will have a second-order pole at the frequency
2

(47) then we will obtain (48)

, i.e if j ; 2j
2

;m 2 + K ( ) 4m( ; 2)

where subscript \2" means \second-order pole" (see Fig. 5, where second-order-pole point marked by\ "). It means if the signal force has the form of a sinusoidal train with duration F and the mean frequency F ' 2 then the amplitude of the mass m oscillations caused by this force will be proportional to1

x
1

signal

Fsignal m

2 F

(49)

Wewant to remind that the free mass has second-order pole at zero frequency, and the harmonic

oscillator has rst-order pole at resonance frequency.

33


(in this section we omit all numerical factors of the order of unity). On the other hand, it can be shown that the Standard Quantum Limit for such a second-order-pole system is equal to (50) m: Hence using this system and ordinary position meter, it is possible to detect the force

x

(2) SQL

r

~F

F
This value is
FF

signal

mx(2) L SQ
2 F

(2) = FSQL =

s

~

m: 3
F

(51)

times smaller than the SQL value corresponding to the free test mass
s

and F F times smaller than the SQL for the harmonic oscillator with ordinary frequencyindependent rigidity,
oscillator FSQL

p

F

f ree mass SQL

=

~m 2 F F

(52)

p~m F = :
F

(53)

It had to be noted that the energy (46) is close to the energy
23 E = mL! : 2

which is necessary to achieve the SQL using traditional scheme of the interferometric position meter.
Sensitivity for di erent regimes of the frequency-dependent rigidity

o

(54)

In this section we will use spectral approach based on the total net noise of the meter (see article 13]). This noise is normalized in such a way that the signal-to-noise ratio is equal to 34


where Fsignal this noise. In the case of the interferometric detector (see Fig. 4) spectral density of the net noise is equal to
ef Stotal( ) = SF f ( ) +

s = Z 1 jFsignal( )j2 d (55) n ;1 Stotal( ) 2 ( ) is the spectrum of the signal force, and Stotal( ) is the spectral densityof

;2 ef f

( )Sx ( )

(56)

where
ef SF f ( ) = 4S~( ) (57) x is the residual back-action noise of the meter (i.e. part of the back-action noise SF which does not correlate with the measurement noise),
2

~2 Sx( ) = 8!LE o is the measurement noise,

4

+2 2( 2 ; 2)+( 2 + 2)2 2+ 2

(58)

ef f

( ) = ;m 2 +1K ( ) ef f

(59)

is the e ective susceptibility of the system, and

K

ef f

= 2!o2E L

4

3 2+ 2; 2 +2 2( 2 ; 2)+( 2 + 2)

2

(60)

is the e ective rigidity which is the sum of two terms: (i) the real physical rigidity which exists in the system due to the dependence of the optical energy in the resonator on the 35


mirrors position, and (ii) the \virtual" rigidity introduced by the cross-correlation of the measurement noise and back-action noise (see article 14]). It should be noted that in our case the real rigidity is much larger than the \virtual" one. However, it is the \virtual" rigidity that compensates the imaginary part of the real physical rigidity which describes dynamical instability of the system. In other words, the instability does exist and must be compensated by some feed-backscheme but the meter does not \see" it. Expressions (58,60) are obtained for the case where the phase quadrature amplitude of the output optical wave is measured. Our results does not depend essentially on which quadrature amplitude is measured but choosing of the phase quadrature amplitude provides slightly better results and also allows to simplify the formulae. The behavior of the Kef f ( ) is rather sophisticated and allows a several di erent regimes, depending on the pumping energy, resonator bandwidth and detuning: with three rstorder poles Fig. c(a)] with one second-order and one rst-order poles Fig. c(b,c)] and with one third-order pole Fig. c(d)]. Which one should be chosen depends on the signal form. Detailed analysis of all of them exceeds the frames of this short article. One of these regimes was considered in details in the article 11]. Here we consider two other regimes which for our opinion are the most interesting ones from both theoretical and \consumer" points of view.

36


1
(a)

1
(b)

0.5

0.5

0 1

0

0.5 ( =)

2

1

0 1

0

0.5 ( =)

2

1

(c)

(d)

0.5

0.5

K=m ( =)
0 0 0.5 ( = )2 1 0 0 0.5 ( =)
2

2 2

1

FIG. 6. Di erent regimes of the frequency-dependent rigidity: (a) | three rst-order poles (b,c) | one second-order and one rst-order poles (d) | one third-order pole

The second-order-pole regime If the bandwidth of the Fabry-Perot resonator is small, , and the pumping energy is equal to2

mL2 3 1+ 6 E 8! o then in the narrow vicinity of the frequency
2

2 2

(61)

The exact expressions are too cumbersome so we present here only second-order Taylor expan=

sions with respect to small parameter

37


r

2

2

+11 2

2

(62)

the formula (56) can be presented as

S

total

( ) = ~m

2

4 + 2 ( ; 4 2) : 2 2

(63)

The value of = can be adjusted to provide minimum of this spectral density at the edges of some given spectral band 2 =2: 1 =2 In this case there will be
2 2

:

(64)

S

total

(

2

=2) = ~m(

)2 :

(65)

This is the spectral equivalent of the formula (51). In Fig. 7 spectral density of the total noise (56) is presented for several values of and for the pumping energy (61) corresponding to the second-order pole (dashed line is the SQL level).

38


1 0.8 0.6 0.4 0.2 0 0.3
r r

= =0:1 Stotal ~m 2 ~m 2 ~m 2
0.5 0.6

= =0:01
0.7

0.4

=

= =0:001 0.8 0.9

1

FIG. 7. Sensitivity for the second-order-pole regime

It is useful to compare sensitivity of this regime with the sensitivity provided by the usual probe oscillator with frequency independent rigidity and with eigen frequency m. In the latter case (see article 10])

Stotal( m =2) = (66) SSQL( m) m where SSQL( ) = ~m 2 is the spectral density corresponding to the SQL for the free test mass. In the case of second-order pole regime will be
2 2 (2 =2) = =2 : (67) SSQL( 2) 2 The third-order-pole regime Parameters of the scheme can be tuned also to create the third order pole of the e ective susceptibility(59) bysetting 2

S

total

E = 9 177 ; 113 49

p

mL2 !o
39

3

0:14mL !o

23

(68)


and = 280 ;721 177 The pole frequency is equal to = 2 177 ; 22 7
s p

p

0:11 :

(69)

p

3

0:81 :

(70)

In Fig. 8 spectral density of the total noise (56) is presented for the case of the third-orderpole regime. 1 0.8 0.6 0.4 0.2 0 0.3
r r

Stotal ~m 2 ~m 2 ~m 2
0.5 0.6

0.4

=

0.7

0.8

0.9

1

FIG. 8. Sensitivity for the third-order-pole regime

It is necessary to note that this third-order-pole regime is \overpumped": the second term in the formula (56), that is proportional to the measurement noise, is several orders of magnitude smaller than the rst one (back-action noise) in the frequency area of interest. It is evident from Fig. 9, where these two terms are plotted separately. Usually, in such a 40


situation the reduction of the total noise is possible by increasing the measurement noise and proportional decreasing the back-action noise due to using, for example, smaller value of the pumping energy. Unfortunately, within the framework of our simple scheme it is impossible because the same optical pumping is used both for measurement and for creating the rigidity. There is no additional \degree of freedom" here: values of all parameters are xed by the equations (68,69). It is probable, however, that more sophisticated topologies based on use of separate optical modes for measurement and for creating rigidity, or/and which eliminate back-action noise by using variational measurement 5,6,15], will allow to create \well-balanced" third-order-pole regime with very low total noise. 1 0.8 0.6 0.4 0.2 0 0.3
r r

;2

m e SF ~m
~

e

S

x 2

2

0.4

0.5

0.6

=

0.7

0.8

0.9

1

FIG. 9. Measurement noise and back-action noise for the third-order-pole regime

Conclusion

It is evidently impossible to consider thoroughly in one short article all the possible variants of the use of the frequency-dependent rigidity which exists in large-scale optical resonators. Such a consideration has to be based on a priori information about the signal 41


spectrum provided, for example, by astrophysical predictions in the similar as it had been done in the article 11]. It is evident, however, that: The second-order-pole regime allows to \dive" deep below the SQL in the narrow spectral band which is, however, much wider than if the usual frequency independent rigidity is used compare formulae (66) and (67)]. The recent achievements in fabrication of high-re ectivity mirrors 16] allows to expect that it will be possible to obtain relaxation time of the large-scale interferometers ;1 & 1 s and thus to reach the sensitivity at the level of 2 . 10;3 , if 103 s;1 . It is important that the pumping energy in this regime does not depend on the sensitivity and remains approximately equal to the energy (54) which is necessary to achieve the SQL in the traditional scheme of the interferometric position meter. The third-order-pole regime provides sensitivity a few times better than the Standard Quantum Limit in relatively wide spectral band and at extremely low level of the measurement noise in this band. This regime looks as a good candidate for use in advanced topologies of the gravitational-waveantennae.

42


REFERENCES

1] V.B.Braginsky, Sov. Phys. JETP 26, 831 (1968). 2] K.S.Thorne, ThreeHundred Years of Gravitation,Cambridge University Press, 1987. 3] V.B.Braginsky, M.L.Gorodetsky,F.Ya.Khalili, Physics Letters A 232, 340 (1997). 4] V.B.Braginsky, M.L.Gorodetsky,F.Ya.Khalili, Physics Letters A 246, 485 (1998). 5] A.B.Matsko, S.P.Vyatchanin, JETP 77, 218 (1993). 6] S.P.Vyatchanin, Physics Letters A 239, 201 (1998). 7] H.J.Kimble,Yu.Levin, A.B.Matsko, K.S.Thorne and S.P.Vyatchanin, Physical Review D , in press (2001). 8] V.B.Braginsky,F.Ya.Khalili, Physics Letters A 257, 241 (1999). 9] F.Ya.Khalili, Optics and Spectroscopy 91, 550 (2001). 10] V.B.Braginsky,F.Ya.Khalili, S.P.Volikov, Physics Letters A A 287, 31 (2001). 11] A.Buonanno, Y.Chen, Physical Review D , in press (2001). 12] B.Meers, Physical Review D 38, 2217 (1988). 13] V.B.Braginsky, M.L.Gorodetsky,F.Ya.Khalili and K.S.Thorne, Physical Review D 61, 044002 (2000). 14] A.V.Syrtsev, F.Ya.Khalili, JETP 79, 409 (1994). 15] S.L.Danilishin, F.Ya.Khalili, S.P.Vyatchanin, Physics Letters A 278, 123 (2000). 16] G.Rempe, R.Tompson, H.J.Kimble, Optics Letters 17, 363 (1992).

43


Appendix G. Parametric Oscillatory Instability in Fabry-Perot (FP) Interferometer
Introduction

The full scale terrestrial gravitational wave antennae are in process of assembling and tuning at present. One of these antennae (LIGO-I pro ject) sensitivity expressed in terms of the metric perturbation amplitude is pro jected to achievesoon the level of h ' 1 10;21 1,2]. In 2008 the pro jected level of sensitivity has to be not less than h ' 1 10;22 3]. This value is scheduled to achieveby substantial improvement of the test masses (mirrors in the big FP resonator) isolation from di erent sources of noises and by increasing the optical readout system sensitivity. This increase is expected to be obtained by rising the value of optical energy E0 stored in the FP resonator optical mode: E0 > 30 J (it corresponds to the circulating power W bigger than 1 megaWatt). So high values of E0 and W maybe a source of the nonlinear e ects whichwillprevent from reaching the pro jected sensitivityof h 1 10;22 . Authors of this article already described two such e ects: photo-thermal shot noise 4] (the random absorption of optical photons in the surface layer of the mirror causes the uctuating of mirror surface due to nonzero coe cient of thermal expansion) and photo-refractive shot noise 5] (the same random absorption of optical photons causes the uctuations of the re ected wave phase due to the dependence of refraction index on temperature). In this paper we analyze undesirable e ect of parametric instability | another "trap" of pure dynamical nonlinear origin which (being ignored) may cause very substantial decrease of the antennae sensitivityand even maymake the antenna unable to work properly. It is appropriate to remind that nonlinear coupling of elastic and lightwaves in continuous media produces Mandelstam-Brillouin scattering. It is a classical parametric e ect, however, it is often explained in terms of quantum physics: one quantum ~!0 of main optical wave transforms into two, i. e. ~!1 in the additional optical wave (Stokes wave: !1 < !0) and ~!m in the elastic wavesothat !0 = !1 + !m (it is Manley-Rowe condition for parametric process). The irradiation into the anti-Stokes wave is also possible (!1 = !0 + !m), however, 44


in this case the part of energy is taken from the elastic wave. The physical "mechanism" of this coupling is the dependence of refractive index on densitywhich is modulated by elastic waves. If the main wavepower is large enough the stimulated scattering will take place, the amplitudes of elastic and Stokes waves will increase substantially. The physical description is the following: the ux of energy into these waves is so large that before being irradiated from the volume of interaction, the oscillations with frequencies !1 and !m stimulate each other substantially increasing the power taken from the main wave. Note that stimulated scattering causes irradiation only into Stokes wave because the additional energy pump into elastic wavemust take place for radiation into anti-Stokes wave. In gravitational waveantennae elastic oscillations in FP resonator mirrors will interact with optical ones being coupled parametrically due to the boundary conditions on one hand, and due to the ponderomotive force on the other hand. Two optical modes may play roles of the main and Stokes waves. High quality factors of these modes and of the elastic one will increase the e ectiveness of the interaction between them and may give birth to the parametric oscillatory instabilitywhich is similar to stimulated Mandelstam-Brillouin e ect 6]. This instabilitymay create a speci c upper limit for the value of energy E0. It is worth to note that this e ect of parametric instability is a particular case of the more general phenomenon related to the dynamical back action of parametric displacement meter on mechanical oscillator or free mass. This dynamical back action was analyzed and observed more than 30 years ago 7,8]. Usually parametric meter consists of e.m. resonator with high quality factor Q (radiofrequency, microwave or optical ones) and high frequency stability pumping self-sustained oscillator. The displacement of the resonator movable element modulates its eigenfrequency which in its turn produces the modulation of the e.m. oscillations amplitude (it can produce also phase or output power modulations). If the experimentalist attaches a probe mass to the movable element of the meter he is inevitably confronted with the e ect of dynamical back action, the ponderomotive force produces a rigidity and due to nite e.m. relaxation time | a mechanical friction. Both these values may be positive and negative ones. In the case when the negative friction is su ciently 45


high the behavior of mechanical oscillator and meter becomes oscillatory unstable. This e ect was observed and explained for the case when value of mechanical frequency !m was substantially smaller than the bandwidth of e.m. resonator 7,8]. In this article we analyze the parametric oscillatory instabilityin two optical modes of FP resonator and elastic mode of the mirror. In this case the value of !m is much larger than the bandwidth of the optical modes. In section II we present the analysis of this e ect for simpli ed one-dimensional model which permits to obtain approximate estimates for the instability conditions. In section III we present considerations and preliminary estimates for nonsimpli ed three-dimensional model, and in section c | the program of necessary mode numerical analysis.
Simpli ed one-dimensional model

a)

-

-

-

A

m

A A

A

b)
-

-

x
2
1

A

; ; ; ; ; ; ; ; ; ; ; ;

2
m

m

-

!

!1

!

0

-

FIG. 10. Scheme of FP resonator with movable mirror (a) and frequency diagram (b).

For approximate estimates we present in this section the simpli ed model analysis where we assume that: The mechanical oscillator (model of mirror) is a lumped one with single mechanical degree of freedom (eigenfrequency !m and quality factor Qm = !m=2 m ). 46


This oscillator mass m is the FP resonator right mirror (see g. 10) having ideal re ectivity and the value of m is of the order of the total mirror's mass. The left mirror (through which FP resonator is pumped) has an in nite mass, no optical losses and nite transmittance T =2 L=( 0 Qopt)( 0 is the optical wavelength, Qopt is the quality factor, L is the distance between the mirrors). We take into account only the main mode with frequency !0 and relaxation rate 0 = !0 =2Q0 and Stokes mode with !1 and 1 = !0 =2Q1 correspondingly (Q0 and Q1 are the quality factors), !0 ; !1 ' !m . Laser is pumping only the main mode which stored energy E0 is assumed to be a constant one (approximation of constant eld). It is possible to calculate at what level of energy E0 the Stokes mode and mechanical oscillator becomes unstable. The origin of this instability can be described qualitatively in the following way: small mechanical oscillations with the resonance frequency !m modulate the distance L that causes the exitation of optical elds with frequencies !0 !m. Therefore, ; the Stokes mode amplitude will rise linearly in time if time interval is shorter than 1 1. The presence of two optical elds with frequencies !0 and !1 will produce the componentof ponderomotive force (which is proportional to square of sum eld) on di erence frequency !0 ; !1. Thus this force will increase the initially small amplitude of mechanical oscillations. In other words, wehave to use two equations for Stokes mode and mechanical oscillator and nd the conditions when this "feedback" prevails the damping which exists due to the nite values of Qm and Q1. Belowwe present only the scheme of calculations (see details in Appendix c). We write down the eld components of optical modes and the displacement x of mechanical oscillator in rotating wave approximation:

47


E0 = A0 D0e E1 = A1 D1e x = Xe

;i!0 t ;i!1 t

+ D0 ei!0t] + D1 ei!1t]
i!m t

;i!m t

+X e

where D0 and D1 are the slowly changing complex amplitudes of the main and Stokes modes correspondingly and X is the slowly changing complex amplitude of mechanical displacement. Normalizing constants A0 A1 are chosen so that energies E0 1 stored in each 2 mode are equal to E0 1 = !0 1jD0 1j2=2. Then it is easy to obtain the equations for slowly changing amplitudes:

D @tD1 + 1D1 = iX L 0 !0 e;i 0D @tX + mX = iDm!1 !0!1 e L
m

!t

(71) (72)

;i !t

where ! = !0 ; !1 ; !m is the possible detuning. Remind that we assume D0 as a constant. One can nd the solutions of (71, 72) in the following form D1(t) = D1e( ; !=2)t, X (t)= X e( + !=2)t and write down the characteristic equation. The parametric oscillatory instability will appear if real part of one of the characteristic equation roots is positive. In LIGO design the values of 0 and 1 are of the order of the bandwidth the gravitational burst spectrum is expected to lie in, i.e. ' 2 100 s;1. On the other hand many e orts were made to reduce the value of m to the lowest possible level and thus to decrease the threshold of sensitivity caused by Brownian noise. In existing today fused silica mirrors Qm ' 106 ; 2 107 and even for !m = 107 s;1 the value of m 10 s;1. Thus we can assume that m 1 and obtain the instability condition in simple form: 1+

R0

!2
2 1

>1

(73) (74)

E E 0Q R0 = 2mL02!2 !1!m = 2mL21!Qm : 2 m 1m m
For estimates we assume parameters corresponding to LIGO-II to be: 48


!m = 2 105 sec;1 m = 5 10;3 sec;1 1 = 6 102 sec;1 !1 ' 2 1015 sec;1 (75) 8 erg 5 cm E0 ' 3 10 L = 4 10 m = 104 g The mechanical frequency !m is about the frequency of the lowest mirror elastic (longitudinal or drum) mode and it has the same order as the intermodal interval c=L ' 2 105 sec;1 between optical modes of FP resonator. The mechanical relaxation rate ' 5 10;8 (quality factor Qm ' 2 107 ) for m corresponds to the loss angle fused silica. The value of energy E0 corresponds to the value of circulating power about W ' cE0=2L ' 1013 erg/s = 106 Watt. For these parameters wehave obtained the estimate of coe cient R0 for the resonance case (j !j 1):

R0 ' 300 1
It means that the critical value of stored energy E0 for the instability initiation will be 300 times smaller than the planned value ' 3 108 erg = 30 J 3. For nonresonance case and planned value of E0 the "borders" of detuning !crit within p the system is unstable, are relatively large: !crit = 1 R0 ' 1:7 104 sec;1.
Considerations of Three-Dimensional Modes Analysis

The numerical estimates for the values of factor R0 and detuning ! obtained in the preceding section have to be regarded as some kind of warning about the reality of the
It is worth to note that if sapphire is chosen then due to the larger value of Qm the factor R0 will be even bigger: R0 ' 5 103: It is another argument whichisnot in favor of this material for
3

mirrors.

49


undesirable parametric instability e ect. In the simpli ed analysis we have ignored the nonuniform distribution of optical elds and of mechanical displacements over the mirror's surface. It is evident that more accurate analysis has to be done. Belowwe presentseveral considerations about further necessary analysis.
The frequency range of "dangerous" optical and elastic modes

The values of the mirror's radius R and thickness H for LIGO-II are not yet nally de ned. Due to the necessity to decrease the level of thermoelastic and thermorefractive noises 4,5,9,10] the size of the light spot on the mirror's surface is likely to be substantially larger than in LIGO-I and the light density distribution in the spot is not likely to be a gaussian one (to evade substantial di ractional losses) 10]. Thus the presented below estimates for gaussian optical modes may be regarded only as the rst approximation in which the use of analytical calculations is still possible. The resonance conditions j!0 ; !1 ; !m j < 1 may be obtained with a relatively high probability for many optical Stokes and mirror elastic modes combinations. If we assume the main optical mode to be gaussian one with waist radius w0 of the caustic (the optical eld amplitude distribution in the middle between the mirrors is e;r2=w02 )), and if we assume also that the Stokes mode may be described by generalized Laguerre functions (GaussLaguerre beams) then the set of frequency distances !opt between the main and Stokes modes is determined by three integer numbers: (76) !optic ' Lc K + 2(2N + M ) arctan 2L w02 0 where 0 is the wave length, K =0 1 2 ::: is the longitudinal index, N =0 1 2 ::: , and M =0 1 2 ::: are the radial and angular indices. For w0 ' 5:9 cm the beam radius on the mirror's surface is equal to w ' 6 cm, corresponding to the level of di ractional losses about 20 ppm for mirror radius of R =14 cm. In this case the equation (76) has the following form:

!optic ' (2:4 K +0:56 N +0:28 M ) 105 s;1:
50

(77)


We see that the distance between optical modes is not so large, i.e. ' 3 104 s;1 . In units of optical modes bandwidth 2 1 ' 103 s;1 it is about 3 104 =2 1 ' 30. Thus assuming that the value of elastic mode frequency can be an arbitrary one we can roughly estimate the probability that the resonance condition is ful lled as 1=30. The order of the distance !m0 between the frequencies for the rst several elastic modes is about !m0 ' vs=d ' 2 105 s;1 (d is the dimension of the mirror and vs is the sound velocity). It is about one order larger than the distance between the optical modes. However, for higher frequencies !m these intervals become smaller and can be estimated by formula

!3 !m ' 2!2m m

0

Even for !m ' 6 105 s;1 the intervals between the elastic and optical modes become equal to each other and has value about ' 3 104 s;1. And for !m 107 s;1 the distances between elastic modes become of the order of optical bandwidth 2 1. Therefore, the resonance condition for these frequencies is practically always ful lled. On the other hand according to (74) the factor R0 decreases for higher elastic frequencies !m . In addition the loss angle in fused silica usually slightly increases for higher frequencies 11,12]. Assuming that the upper value !m ' 2 106 s;1 , Qm ' 3 106 and other parameters correspond to (75) we obtain R0 ' 1. Therefore, the elastic modes which "deserve" accurate calculations lie within the range between several tens and several hundreds kiloHerz. The total number of these modes is about several hundreds.
The Matching between the Mechanical Displacements and Light Density Distributions

The simpli ed model described in section c is approximately valid for the uniform over all the mirror's surface distribution of optical eld density and pure longitudinal elastic mode. The equations for this model can be extended for any distributions of mechanical displacements in the chosen elastic mode and for any distribution of the light eld density in chosen optical modes. This extension can be done by adding a dimensionless factor in (73): 51


1+ !12 2 ;R V f0(~R ) f1(~?) uR d~? 2 r r r = R jf j2d~ ? jf j2d~ z j~ j2dV : 0 r? 1 r? u

R

0

>1

(78) (79)

Here f0 and f1 are the functions of the distributions over the mirror's surface of the optical elds in the main and Stokes optical modes correspondingly, vector ~ is the spatial vector u of displacements in elastic mode, uz is the componentof ~ , normal to the mirror's surface, u R R d~? corresponds to the integration over the mirror's surface and dV |over the mirror's r volume V . It is necessary to know the functions f0 f1 ~ in order to calculate the factors for u di erent mode combinations. But to the best of our knowledge there is no analytical form for ~ in the case of cylinder with free boundary conditions 11]. Our approximate estimates u show that there is substantial number of modes combinations for which factor is large enough to satisfy the condition (78) for parameters (75). We do not present the details of these estimates here because they are rather rough. It is evident that a complete numerical analysis which includes nongaussian distributions of optical elds is necessary to do.
Conclusion

The simpli ed model analysis of parametric oscillatory instability and considerations about the real model presented abovemay be regarded only as the rst step along the route to obtain a guarantee to evade this nondesirable e ect. Summing up we may formulate several recommendations for the next steps: 1. Due to the nite size of the mirror and to the use of the nongaussian distribution of light densitywe think that the accurate numerical analysis of di erent optical and elastic mode combinations (candidates for the parametric instability) is inevitably necessary. This problem (numerical calculations for the elastic modes) has been already solved partialy 13]. 52


2. In the same time the numerical analysis maynot give an absolute guarantee because the fused silica pins and bers will be attached to the mirror. This attachmentwill change the elastic modes frequency values (and may be also the distribution). In addition the unknown Young modulus and fused silica density inhomogeneity will limit the numerical analysis accuracy. Thus the direct measurements for several hundreds of probe mass elastic modes eigenfrequencies values and quality factors are also necessary. 3. When more "dangerous" candidates of elastic and Stokes modes will be known, their undesirable in uence can be depressed. For example it can be done by small the change of mirror's shape. 4. It is also reasonable to perform direct tests of the optical eld behavior with smooth increase of the input optical power: it will be possible to register the appearence of the photons at the Stokes modes and the rise of the Qm in the corresponding elastic mode until the power W in the main optical mode is below the critical value. 5. Apart from above presented case of the oscillatory instability it is likely that there are similar instability in which other mechanical modes are involved (especialy violin ones which also have eigen frequencies several tens kHz and higher). There are also additional instability for the pendelum mode in the mirror's suspension (in the case of small detuning of pumping optical frequency out of resonance). These potencial "dangers" also deserve accurate analysis. We think that the parametric oscillatory instability e ect can be excluded in the laser gravitational antennae after this detailed investigation.
Acknow ledgements

Authors are very grateful to H. J. Kimble, S. Witcomb and especially to F. Ya. Khalili for help, stimulating discussions and advises. This work was supported in part by NSF and 53


Caltech grants and by Russian Ministry of Industry and Science and Russian Foundation of Basic Researches.
Appendix A: Lagrangian Approach

Let us denote q0(t) and q1(t) as generalized coordinates for the FP resonator optical modes with frequencies !0 and !1 correspondingly, sothattheir vector potentials (A0 A1), electrical (E0 E1) and magnetic (H0 H1) elds are the following:

c2 ; Ai(t)= 2S L fieikiz ; fi e;iki z qi(t) i r 2; Ei(t)= ; S L fieiki z ; fi e;iki z @tqi(t) i r 2; Hi(t)= S L fieikiz + fi e;ikiz !iqi(t)
i

s

fi = fi(~? z) Si = r

Z

~ jfij2dr?:

Let also denote x(t) as generalized coordinate of the considered elastic oscillations mode with displacement spatial distribution described bythe vector ~ (~). Nowwe can write down ur the lagrangian:

22 2 Lm = M (@tx) ; M!2m x Z2 M= j~ (~)j2 dV ur V Z L = ; xuz hH0 + H1i

L = L0 + L1 + Lm + Lint Z 2 2 2 2 ~ L0 = L(hE0i 8;hH0 i ) dr? = @t2q0 ; !02q 2 22 L1 = @t2q1 ; !02q1

2 0

int

d~? = r 8 z=0 x = ;2!0!1 q0q1 B L R r r r B = qRf0(~?)f1(~R?)uz d~? jf0j2d~? jf1j2d~? r r
54

2


We consider only one mechanical mode below. Now we can write down the equations of motion (adding losses in each degree of freedom): x 2 @t2q0 +2 0@tq0 + !0 q0 = ;B 2L !0!1q1 2 @t2q1 +2 1@tq1 + !1 q1 = ;B 2x !1!0 q0 L 2x +2 @ x + ! 2 x = ;B 2!0 !1 q q : @t mt m ML 0 1 Introducing slowly varying amplitudes we can rewrite these equations as:

q0(t)= D0(t) e q1(t)= D1(t) e x(t)= X (t) e

;i!0 t ;i!1 t

+ D0 (t) e + X (t) e

i!0 t i!1 t

+ D1 (t) e

;i!m t

i!m t

! = !0 ; !1 ; !m D @tD0 + 0D0 = iB XL 1 !1 ei !t @tD1 + 1D1 = iB X LD0!0 e;i !t ! @tX + mX = iB D0D1L0!1 e;i M!
m

(80)
!t

(81)

We can see that this system (81, 80) coincides with (72, 71) if B =1, and M = m. For the simplest resonance case ! = 0 it is easy to substitute (81) into (80) and to obtain the condition of parametric instability (in the frequency domain): ;iB D ! D1( 1 ; i ) = iB D0!0 M! L(0 D1+0i!1) L mm Condition of instability: 2 j2 2 1 < B jD0L2!0 !1 M!m 1 m Nowwe can express the energy E0 in mode "0" in terms of jD0j2: 2 22 E0 = @t2q0 + !02q0 = = 1 (;i!0)2 D0 e;i!0t ; D0 ei!0t 2 + 2
2 + !0 D0 e

;i!0 t

+ D0 ei!

0t 2

=

2 =2!0 jD0j2:

55


Nowwe can write down the condition of parametric instability in the following form: !1 !m > 1 E0 B (82) 2 L2 2m!m 1m ;R r r zr B 2m = R V f0(~? )f1(~? )uRd~? 2 R =M (83) 2d~? jf1 j2d~? j~ j2 dV jf0j r ru which accurately coincides with (78, 79). Let us deduce the instability condition for nonresonance case. We are looking for the solution of (71, 72) in the following form:

D1(t)= D1 e ;t X (t)= X e +t i! i! += + ;= ; 2 2 and writing down the characteristic equation as: 22 ( + + 1)( ; + m) ; A =0 D0 !0 !12 = A: m!mL The solutions of characteristic equation are: 1+ m p Det 12 = ; 2 2 1; m i ! Det = 2 ; 2 + A:
The condition of instability is the following: Using a convenient formula: Det = a + ib p qp p < Det = 22 a2 + b2 + a: we can rewrite the condition (84) as: 1 pa2 + b2 + a > 1 + m 2 2 2 2 !2 a2 + b2 = A2 + ( 1 ; m) + 4 4 ( 1 ; m)2 ; !2 +2A 4 4 56

p < Det > 1 + m : 2

(84)

(85)
2

+ (86)


Note that for the resonance case ( ! = 0) the solution of (84 or 85) is known: A> 1 m. 2 For our case m 1 it means that A 1 . Therefore for small detuning ! 1 wecan expand a2 + b2 in series in terms of A and rewrite condition (85) as: 1 pa2 + b2 + a ' A 2 2 + ( 1 ; m)2 > A ( ; )2 + !2 1 1 m Or A> 1 m (
2 + ( 1 ; m) + 4 A ( 1 ; m)2 ; ! 2 ( 1 ; m)2 + ! 2 2

m
1

; m)2 + !2 : ( 1 ; m)2

(87)

Let us underline that condition (87) is obtained for small detuning ! 1. However, considering situation more attentively one can conclude that expansion in series (and consequently the formula (87) ) is valid for the condition:

A

( 1 ; m)2 + ! 4 4

2

(88)

We see that this condition is ful lled for the solution (87). Therefore we conclude that solution (87) is approximately valid for any detunings !.

57


REFERENCES

1] A. Abramovici et al., Science 256(1992)325. 2] A. Abramovici et al., Physics Letters A218, 157 (1996). 3] Advanced LIGO System Design (LIGO-T010075-00-D), Advanced LIGO System requirements (LIGO-G010242-00), available in .
http://www.ligo.caltech.edu

4] V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, Physics Letters A264, 1 (1999) cond-mat/9912139 5] V. B. Braginsky, M. L. Gorodetsky, and S. P. Vyatchanin, Physics Letters, A 271, 303-307 (2000). 6] M.Pinard, Z. Hadjar and A. Heidman, http//:xxx.lanl.gov/quant-ph/9901057 (1999). 7] V. B. Braginsky and I. I. Minakova, Bull. of MSU, ser. III, no.1, 83 (1964). 8] V. B. Braginsky A. B. Manukin, and M. Yu. Tikhonov, Sov. Phys. JETP 58, 1550 (1970). 9] Yu. T. Liu and K. S. Thorne, submited to Phys. Rev. D. 10] V. B. Braginsky,E.d'Ambrosio, R. O'Shaughnessy, S. E. Strigin, K. Thorne, and S. P. Vyatchanin, report on LSC Meeting, Baton Rouge, LA, 16 March 2001, LIGO document G010151-00-R (http//:www.ligo.caltech.edu/). 11] "Physical acoustics. Principles and Methods." Edited byW. P.Mason,Vol. I, Methods and devices,Part A, Academic press , New York and London (1964). 12] Kenji Numata, \Intrinsic losses of various kind of fused silica", proceedings of Forth Edoardo Amaldi Conference, Pert , 2001. (Trancparencies are temporary available in ftp://t-munu.phys.s.u-tokyo.ac.jp/pub/numata/Transparencies/Amaldi/Amaldi4.pdf ) 13] A. Gillespi and F. Raab, Phys. Rev. D 52, 577 (1995). 58