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THE SEMIANNUAL REPORT OF THE MSU GROUP (Jul.-Dec. 1998)
Contributors: V.B.Braginsky (P.I.), I.A.Bilenko, M.L.Gorodetsky, F.Ya.Khalili, V.P.Mitrofanov, K.V.Tokmakov, S.P.Vyatchanin, Collaboration with the theoretical group of prof. K.S.Thorne

I. SUMMARY A. Quantum limits of the sensitivity in the gravitational waveantennae on free masses and intracavity readout meters (M.Gorodetsky, F.Khalili)
The universal formula for the minimal required energy in optical resonators of the antennae attaining resolution at the level of the standard quantum limit is obtained (see Appendix A). The complete analysis of a new topology of interferometricgravitational waveantenna is presented. This new scheme is based on princilples of quantum intracavity measurements of crossquadrature observable and utilizes features of speci c quantum state { symphotonic state. It is shown that this scheme can provide the same sensitivity as traditional topologies but at signi cantly lower levels of circulating power. If characteristic frequency of the gravitational signal is equal to 10 =2 H z then the optical energy stored in the resonators has to be 10 erg for the parameters of LIGO, that is 3 orders of magnitude lower than the energy required in traditional schemes if the sensitivity has to be close to the SQL. In optical bars scheme, intracavityquantum measurementallows to obtain better resolution (better than SQL) with moderate requirements to circulating power. The key element of the suggested intracavityscheme of gravitational waveantenna is mechanical quantum limited sensor. Currently we are analysing in collaboration with K.Thorne and his group a practical scheme of mechanical QND speed meter.
3 6

1


B. The e ect of individual microdust particles in LIGO antenna tubes (M.Gorodetsky)
Microdust particles inside the tubes of LIGO interferometers may simulate gravitational signal (see Appendix B). Scattering on these particles leads to the shift of resonance frequencies and hence additional phaseshifts between the arms of interferometers. It is shown that even one particle falling from the ceiling of the tube through the beam of interferometer may produce 30ms pulse in output signal with amplitude in dimensionless units 3 10; for a particle with radius 0:3 m and 3 10; for a particle with radius 3 m.
21 19

C. The improvement of the Q -factors of the suspensions' modes and the searchof the damping e ect due to the electric eld (V.Mitrofanov, N.Styazhkina, K.Tokmakov)
The new vacuum chamber (where the tests of the pendulum and violin Qs are performed) was isolated from the rest of the room by a special box which permited to reduce the level of the contamination of the ber surface by dust approximately by one order. This box (which is supplied with the special dust free ventilation) also allows to fabricate the bers for the suspension and to make the welding in the dust free enviroment as well as to install the pendulum in the chamber within few hours after the fabrication of the bers. Two pendulums were fabricated during this half of year. Testing of the second one is now in progress. The preliminary result for this pendulum which is a 2-kg fused silica cylinder suspended by two welded fused silica bers is the following: the quality factor of the torsional-pendulum mode is about 1:4 10 . The investigation of the weak dissipation mechanisms at this level of losses is now the main goal of the current researches. The group continued the study of the electric eld damping of the test mass pendulum mode. They have found that the usage of electrodes covered by a gold lm in order to applied the electric eld creating the control force allows to decrease the eld induced losses
8

2


more than one order in comparison with alumimium cover. This result has to be regarded as a preliminary one (see details in Appendix C).

D. Excess noise and thermal noise in the elements of the antenna (I.A.Bilenko, S.P.Vyatchanin)
During last six month the method of excess noise measurement in the fused silica mirror suspension was under development. The presence of excess noise in the well stressed metal wire suspension has been proved recently (see the previous annual report). However, it is necessary to obtain a displacement sensitivity approximately 100 times better in order to resolve a thermal uctuation during the short time intervals as compared to the relaxation time on the fused silica threads, because the quality factor of these threads is about 10 against 10 for the metal ones. The necessity of preparation and keeping of high quality factor during the measurement is an additional problem. In order to obey the requirements the transducer head placed right on the test mass prototype is designed. The numerical analyses and experimental testing of various transducers are in progress now. For example, the optical displacement sensor based on twin balls whispering-gallery microresonators allows one to p reach the su cient sensitivityof 10; cm= Hz, but its application to the measurementof thread oscillation amplitude meets a number of technique and principal di culties. At the same period the distribution of equilibrium thermal uctuations inside the antenna mirrors was analysed. This analyses shows a principal possibility to extract a contribution of these uctuations from the antenna output signal. We are planning to nish this analyses soon and obtain the gain parameter and correspondingly the reduction of the requirements to the quality factors of mirrors internal modes.
8 4 13

3


II. APPENDIX A. Quantum limits and symphotonic states in free-mass gravitational-wave antennae (B.Braginsky, M.L.Gorodetsky, F.Ya.Khalili)
1. Introduction

In 1,2], we presented an analysis of two qualitatively new schemes for the extraction of information from free-mass gravitational-wave antennas 3]. Common features of these schemes are the use of nonclassical quantum states of the optical eld inside the resonators and of QND methods for intracavity measurements of the variations of these states. This becomes possible only with the realization of optical eld relaxation times o much longer than the measurementtime meas ' 10; 10; s. One signi cantadvantage of intracavity measurements is that they require lower levels of circulating power than traditional schemes with an antenna with a coherent pump. In 1] and in the subsequent article by Levin 4], the optical cubic nonlinearity of thin plates inserted in an antenna was exploited. The idea of our second scheme 2], which, in our opinion, can be implemented relatively easily,was to place an additional partially transparent mirror{probe mass at the intersection of the two arms of a gravitational antenna. This results in the formation of two coupled Fabry-Perot resonators. Displacement of the end mirrors under the action of a gravitational wave leads to a redistribution of the energies in the arms, which pushes the central mass. The absolute displacement under optimal conditions is simply equal to the relative displacements of the end mirrors (hL=2, where L is the arm length and h is the amplitude of the variation of the metric), and the light in the system behaves like a rigid bar. The displacement associated with an independent mass that does not interact with the optical eld can be registered without consuming a large amountof power. A rigorous general relativistic justi cation of the schemes in 1,2] can be found in 5]. The merits of this intracavity measurement are the following: a) in the resonators, the required nonclassical quantum state (close to a Fock state) is formed automatically b) direct
2 3 (3)

4


measurement of a displacement hL=2 consumes relatively little power c) precision higher than the standard quantum limit can be obtained. In 2], we did not make an analysis of the minimal energy of the optical eld E in the system required to preserve the sensitivity. Another important unanalysed problem is the connection between the achievable resolution and a chosen procedure for displacement measurement. It is important to note here that the provision of substantial values of E is a key problem for large-scale gravitational waveantennas, and that this problem has not a technical but a fundamental nature. Indeed, the proposed sensitivitylevels of such antennas will be close to the standard quantum limit for the displacement of the masses M of the end mirrors:

x

SQL

(M )= LhSQL ' M!h gr
2

v u u t

gr

(1)

where !gr is the frequency of the gravitational signal and gr is its duration (we omit in our estimates numerical terms of the order of unity that depend on the form of the signal). According to the Heisenberg uncertainty relation, the momentum should be perturbed bya value of the order of:
q h p = 2 x ' hM ! SQL
2 gr gr

(2)

This perturbation must be provided by the uncertainty in the energy E in the interferometer, which, thus, cannot be less than
q E = !gr L p = L hM ! s

gr

gr
5 3 1

3

(3) 10 cm, !gr = 10 s; , and (4)

This value is not especially large for example, for L = 4 M =10 g (the parameters of the LIGO antenna),
4

E' 4 10; erg
2

and in the case of nonclassical states of the optical eld, in which E E , the necessary resolution can be obtained at very low energies. However, for coherent states in which 5


E = h!o E
where !o is the optical frequency, the requirements are very strict: E' ML !gr : !o For the same parameters as before and !o =2 10 s; ,
2 3 15 1

q

(5)

(6)

E
and if !gr =10 s; ,then
4 1

SQL

' 10 erg
9

(7)

E

SQL

' 10 erg:
12

(8)

In this paper, we analyze a new intracavityscheme that is, in some sense, complementary to the \optical bars" scheme. In this scheme, the optical eld forms in a quantum state that is close to states with squeezed phase this is known to allow, in principle, a dramatic decrease in the optical quanta because ' 1=N . (Non-QND measurement of a similar observable was proposed in 6]).
2. A crossquadrature quantum observable and a scheme for its measurement

The basic idea of the new scheme for an intracavity readout system is the use of two modes excited in the Fabry-Perot resonators of the antenna's orthogonal arms. If the modes are not linearly coupled (this is critical in this scheme), they can be tuned as close to each other as (! ; ! ) meas 1. As a result, the frequency variation in one (or both) resonators produced by a gravitational wave will lead to the appearance of a phase di erence with the oscillation amplitude ' ' h!o (9) !gr which we propose to register. Since no meter has been invented thus far to directly register the phase di erence between twoquantum electromagnetic oscillators, another variable proportional to ' is required.
1 2

6


We propose to measure the averaged product of the two quadrature components of two di erent oscillators, which, in the limit of large numbers of quanta, is very close to a phase measurement. One possible scheme for the realization of the proposed crossquadrature observable is depicted in Fig.1. This scheme is based on the use of ponderomotive nonlinearity in a way similar to that in 2]. Mirrors A0 and B 0 direct the optical beams re ected from the end mirrors A and B and transmitted by the 50% beamsplitter C on opposite sides of the double highly re ecting (zero transmission) mirror D (to eliminate linear coupling). In the engineering realization of this scheme, A0 and B 0 can be rigidly connected to the beamsplitter, and can be focusing re ectors, making it possible for the mass m of D to be smaller. It is easy to see that, due to the beamsplitter, the optical beams from arms A{C and B{C interfere in the shorter arms such that one of them has amplitude proportional to a + ia and the other has amplitude proportional to a + ia (a are the complex eld amplitudes in the longer arms). This is valid if the geometrical conditions in Fig.1 are satis ed. As a result, the ponderomotive force Fpond acting on mirror D will be proportional to:
1 2 2 1 12

F

pond

/ja + ia j ;ja + ia j ' 4ja jja j ':
1 2 2 2 1 2 1 2

(10)

Provided that the initial optical energy E =2in the two arms is nearly the same (E = h!o N = h!oa , a = ja j = ja j), in a quasistatic approximation, this force will be
2 1 2

E Fpond ' L '
(3)

(11)

Note here that there is no direct linear coupling between modes in this scheme. In other words, modes in the resonator are coupled via the nonlinearity resulting from the ponderomotive e ect. Linear coupling is due only to the movement of the mirror D. The shift of D changes the lengths of the shorter arms, changing the interference conditions on the beamsplitter, which consequently leads to a redistribution of the optical photons between the two modes. This scheme realizes indirect QND measurement of the operator 7


^ X = = i(^ ^ ; a a ) aa ^^
2 + 12 + 2 1 + 12 12

(12)

where a and a are the creation and annihilation operators for two di erent oscillators ^ ^ ^ with the same frequencies !. The operator X = presents a special case of the family of operators
2

^ a^ X =^ a ei +^ a e a^
+ 12 + 21

;i

(13)

which we propose to name crossquadrature operators. These operators commute with the Hamiltonian of the two modes: ^ X h!(^ a +^ ^ )] = 0 a^ aa
+ 11 + 2 2

(14)

i.e., they are, indeed, QND variables. The eigenstates of the crossquadrature operators have the form (^ +^ e;i )n(^ +^ ei )N ;nj0i aa aa (15) jN ni = q N 1 2 n!(N ; n)! where N is the sum of quanta in the system and n is an integer in the range from 0 to N . In this state, each of the N quanta has equal probability to reside in either arm of the interferometer. However, the amplitudes of these probabilities for n quanta are orthogonal to those of the other N ; n quanta. Due to this peculiar entanglementbetween the modes, we shall call eigenstates of the crossquadrature operator symphotonic quantum states. ^ The eigenvalues of the operator X are n ; (N ; n) = 2n ; N , i.e., measuring the crossquadrature variable, the observer determines the di erence between the two kinds of quanta. Symphotonic states (15) are very sensitive to the change of the phase di erence in the two oscillators. As weshow in Appendix A, a phase shift leads to a transition between states with di erent n (preserving the total number of quanta), that can be detected by ^ measuring X = . The probability of this transition is equal in the case ' 1to p = ' (N +2n(N ; n)) (16) 4 p and when n ' N=2, p tends to unity when ' ' 8=N , thus allowing, in principle, the registration of these small phase shifts.
+ 1 + 2 1 2 2 2

8


3. Limitations on the sensitivity

It is not di cult to show that the nite masses of the mirrors A0 and B 0,aswell as the mass of the beamsplitter C , do not in uence the behavior of the system if these masses are substantially greater than the mass m. We will use a standard linear approximation, in which the optical eld can be represented as the sum of the large classical dimensionless amplitude A and the quantum annihilation operator ^, neglecting terms of the order of a a ^ and higher. We suppose also that o and relaxation time m of the mass m is very large in comparision with other characteristic times. In this case, the equations of motion will have the form: s ! da (t) = ! A ix (t) ; x(t) + ih(t) + Z1 o ^ (!)e;i ! !o td! ^ ^ ^ b o dt L 2 s ! ^ ^ da (t) = ! A ix (t)+ x(t) ; ih(t) + Z1 o ^ (!)e;i ! !o td! ^ b o dt L 2 x ^ ^ ^ m d dt(t) = ih!oA a (t) ; ^ (t)+^ (t) ; ^ (t) + F meter (t)+ F mech(t) a a a L^ x ^ M d dt (t) = h!o A ^ (t)+^ (t) a La x ^ M d dt (t) = h!o A ^ (t)+^ (t) a (17) La where x are the displacements of the mirrors A and B , x is the displacementof D, o = 1=2 o is the decrement of the optical losses in the resonators, ^ (!) are the corresponding b annihilation operators for the heatbath modes, which satisfy the commutational relations
2 1 1 1 (+ ) 0 2 2 2 (+ ) 0 2 2 + 1 1 2 + 2 2 1 2 1 + 1 + 2 2 2 2 2 12 12

^ (!) ^ (!0)] = (! ; !0) b b
12 + 12

(18)

F meter is the uctuational reaction of the coordinate meter on the mirror D with mass m, h(t)=2 is the relative change of the optical lengths of the resonators (in the case of a gravitational antenna, this is the dimensionless metric variation), and F mech is the Nyquist uctuational force acting on the mass m. The characteristic equation of this system is: p+
6 6

=0 9

(19)


where 2!o = mME L
2 2 4

!1=

6

:

(20)

It has roots with positive real parts of the order of . Thus, there exists in the system a dynamic instabilitywith a characteristic time ; . To suppress this with a feedbackloop, it is necessary to have
1

< !gr :
The signal-to-noise ratio is equal to (see Appendix B):

(21)

s = !o E nL
2 2

2

1 Z ;1

! jh! (!)j d! m ( ; ! ) Sx +2m! ( ; ! )SxF + ! (SF + Sm + So) 2
6 2 2 6 62 4 6 6 8

(22)

where h! (!) is the signal spectrum,

Sm = 2 T m
m

(23)

is the spectral densityof F

mech

( is the Boltzmann constantand T is the temperature),

E So = Lh!o ! o
2

2

(24)

is the spectral Sx and SF are and SxF is the must obey the

density of the uctuational force due to dissipation in the optical resonators, the spectral densities of the additive noise xmeter of the meter and of F meter , cross spectral densityof xmeter and F meter . The values of SF Sx and SxF (!) Heisenberg inequality 7]:

Sx(!)SF (!) ; SxF (!) h : 4
2 2

(25)

The condition for the detection of a signal can be represented in the form:

h
where

q

h

2

meter

+ hmech + h
2

2

opt

(26) 2T m gr M

h

mech

= L!gr 2 T m =2 ! E !o m gr 10

s

gr
3

s

(27)


is the limit due to the thermal noise of the mass m,

hopt = ! 1 N o o gr
2

s

(28)

is the limit due to the optical losses ( gr is the duration of the signal), and hmeter is the limit due to the quantum noise of the meter. It is important to note that the limitation (28) is also valid for the previous scheme 2], based on a di erent principle for intracavity measurement (this follows from formula (10) of 2]). The value of hmeter is determined by the magnitudes of the spectral densities Sx(!) SF (!), and SxF (!) and their frequency dependence. In the case of a plain coordinate meter:

Sx(!) = const SF (!) = const SxF (!)=0:
be:

(29)

With regard to limitation (21), values corresponding to optimal tuning of the meter will

SF =

hm!gr 2
2 4

h Sx = 2m! : gr
2

(30)

The ultimate sensitivity of the meter is determined, in this case, by the formula

h

meter

hm!gr p ! = !LE =2 o gr

v u u t

gr

3

h

SQL

(M ):

(31)

Thus, because of (21), it is impossible in this case to reach a sensitivity corresponding to hSQL(M ). To preserve a sensitivity at the level of hSQL (M ) and lower the requirements on the energy, one can use an advanced meter providing higher precision for monitoring the mass m. A speed meter 9] can be used for this monitoring. This can be realized in the form of an ordinary parametric electromagnetic displacement transducer (operating at microwave wavelengths) with an additional bu ering cavity, coupled with the main (working) cavity 9]. We show in Appendix C that, in this case, 11


e Sx(!)= 4! !hd sin e We
2 4 2

2

SF (!)= h!de We ! e
2 4

2

SxF (!)= ; h cot 2

(32)

where !e is the microwave frequency, We is the microwave pump power, d is an equivalent parameter with the dimensions of length, whichcharacterizes the tunability of the transducer 10]: 1 d; = ! @!e e @x
1

(33) bu the the eter ering resonators, whichmust satisfy relaxation time due to the coupling local oscillator used for detection of parameters, when (34)

is the beat frequency between the working and the conditions e !gr and e= e !gr ( e is with the transmission line), and is the phase of the microwave signal. For optimal tuning of the m
e
2 2 4

! We = md! e gr 2e
6 6

and cot = ; the limiting sensitivity will be

!

gr
6

6

(35)

hmeter = 2h
15 1

p

SQL
5

(M ):
3 1

(36)

If, for example, !o =2 10 s; , L =4 10 cm, !gr =10 s; (these values correspond to the values for the LIGO antenna 3]), m =1g,and E =10 erg, then ' 5 10 s; ,and condition (21) is satis ed. If in addition d = 1cm (the value achieved in high-Q sapphire disk resonators 11]) and e =3 10 s; , then the required microwavepump power will be We =3 10 erg=s. Thus, the analyzed scheme makes it possible to dramatically decrease the requirements for the optical circulating energy by using a microwave transducer with a reasonable set of parameters. Under these conditions, however, the requirements for the level of dissipation in the probe mass m increase as the signal that must be registered decreases:
6 2 1 3 1 4

12


vm ' ! v gr
3

3

SQL

(37)

where

v

SQL

= mh

v u u t

gr

(38)

If the above parameters are chosen, vm ' 1=8 vSQL . In order for the dissipation not to deteriorate the sensitivity, it is necessary that hmech 6 4 8

(39)

For example, for T =4K , m > 3 10 s. Thus, the requirements for the dissipation in the mass m are severe, but achievable 12]. Losses in the optical resonator will not in uence the sensitivityif hopt o

> ESQL E !gr
o

(40)

or, for the parameter values introduced above, detector with optical mirrors available today.

> 1s. This is quite possible in a LIGO-type

4. Comparison with the \optical bar" scheme

In 2], not all regimes for the \optical bar" scheme were analyzed in detail. Moreover, there was unfortunately an error in the formula following formula (12) (term \1" under the root should be omitted). Here, we shall limit our treatment to \wideband" regimes, when the range of the signal frequencies is far from the resonant frequencies in the signal-to-noise integral these regimes are the most useful from the practical point of view. The \narrowband" regime, in whichit is possible to attain sensitivity better than the SQL, has already been considered in detail in 2]. In our analysis, we shall assume that !gr < ( is the beat frequency in the system of two coupled optical resonators the case of !gr > is di cult to realize in practice, and 13


does not provide anyinteresting new results) and m M . The behavior of the system is determined by the parameter with the dimensions of frequency
o = 2!LE
2

1+ 1 m 2M

14

=

(41)

This frequency describes the in uence of the ponderomotive force on the dynamics of the system, and plays a role analogous to (see formula (20)). It is possible to distinguish three cases, depending on the level of the circulating energy (the value of ).
5. Weak pump power,
2


If a plain coordinate meter is used, the calculations give the following result:

h
where

meter

=!

gr
2

2

hSQL(m) >h
v

SQL

(m)

(42)

h

SQL

u 1u h (m)= L t m! gr
2

gr

(43)

If a speed meter is used, the sensitivity can be higher:

h

meter

=!

gr
2

h

SQL

(m) >h

SQL

(m)

(44)

but is still lower than even hSQL(m).
Intermediate case, !
gr

<

2


gr

q

2M=m

For a plain coordinate sensor, the best sensitivity in this case is

h

meter

=!

gr
2

h

SQL

(m)
2

(45)

i.e., hmeter is smaller than hSQL(m), and hmeter ! hSQL (M ) if required optical energy in this case is 14

! !gr

q

2M=m. The


E = ! ESQL > ESQL: gr

(46)

The use of a speed meter in this regime does not give a gain in sensitivity, however an increase in sensitivity is possible if an advanced coordinate detector with correlated noises is used (SxF 6= 0) 8]. In this case h = !gr h (m) (47)
2

meter

2

SQL

if

2


gr

q
4

2M=m,and

h

meter

=h

SQL

(M )
SQL
2 2

(48) , but with respect (49)

otherwise. The required energy in the latter case can be lower than E to a possible dynamical instability,which appears when =4: s E > 8m ESQL M
8

Strong pump power,

2

!

gr

q

2M=m

This is connected sensitivity makes it p

the \optical bar" regime, when the masses M and m move together, and are by electromagnetic rigidity. In this case, a plain coordinate meter provides a corresponding to the standard quantum limit hSQL(M ), and use of a speed meter ossible to overcome this limit, but with higher energy: s !gr h (m)= ESQL h (M ) hmeter = (50) SQL E !gr SQL Note that, in this case, also, the total mass 2M is present in the expression for the thermal limit. This result is quite understandable, since, in this regime, thermal uctuations of the small mass m act on the large compound mass 2M + m.
2

6. Conclusion

Quantum mechanics sets severe limits on the sensitivity and the required circulating energy in traditional free-mass gravitational-waveantennas. One possible way to beat these 15


limits is to use intracavity QND measurements. In this paper, wehave analyzed a new QND observable and its corresponding symphotonic quantum states, which possess a number of features that make it promising for experiments requiring registration of small phase variations: 1) Unlike other known QND observables, this one is a joint integral of motion for two quantum oscillators with equal frequencies. 2) The crossquadrature observable is very sensitive to the phase di erence of the oscillators. Phase di erences of the order of 1=N (the theoretical limit for phase measurements) can be detected, where N is number of quanta in the system. 3) Well-known methods for the QND measurement of electromagnetic energy can be used to measure this new observable. Wehave considered a practical optical scheme in which the new observable can be used for the detection of gravitational waves. Our estimates show that, in combination with advanced coordinate meters, this scheme provides a sensitivity of the same order as that for planned antennas at signi cantly lower energies. Summarizing the results of this article and of 1,2,4], we conclude that intracavity measurements with automatically organizing nonclassical optical quantum states make it possible, in principle, to lower the required power levels and in several cases to achieve sensitivity better than the standard quantum limit. We note also that the schemes wehave analyzed do not cover all possible geometries for intracavity measurements with ponderomotive nonlinearity. Better realizations with higher responses are probably possible.
7. appendix

a. The evolution of a symphotonic state

^ U(

1

2

) = exp 16

1

n+ n ^ ^ ih
1 2

!

2

(51)


where n are the operators for the number of quanta in the modes. Hence ^
12

^ U( ^ U( and

1

2

)^ U ( a^
+ 1 +

1

2

)=^ e; a
+ 1

i i

1

(52) (53)

1

2

)^ U ( a^
+ 2 +

1

2

)=^ e; a
+ 2

2

^ U(

1

2

)j0i = j0i:
i
( 1+ 2 ) 2

(54)
N

, we can Taking into account formula (15) and omitting the unimportant factor e; obtain: ^ ^ ^ U( )jN ni = q N 1 (A cos + iB sin )n(B cos + iA sin )N ;nj0i 2 n!(N ; n)! (55)
1 2 + + + +

where ^ A =a +a e
+ + 1 + 2

;i

^ B =a ;a e
+ + 1 + 2

;i

(56)

and = If 1 then ^ U(
1 2

;:
1

(57)
!

i
2

)jN ni' 1 ; 8 (N +2n(N ; n)) jN ni + q q n(N ; n +1)jN n ; 1i + (n +1)(N ; n)jN n +