Quantum theory of high-resolution length measurement with ...phys.lsu.edu/~jdowling/PHYS7353/lectures/Ley87.pdf · JOURNAL OF MODERN OPTICS ... Quantum theory of high-resolution length

JOURNAL OF MODERN OPTICS, 1987, VOL . 34, NO . 2, 227-255

Quantum theory of high-resolution length measurementwith a Fabry-Perot interferometer

M. LEY and R. LOUDONPhysics Department, Essex University,Colchester C04 3SQ, England

(Received 17 November 1986)

Abstract. The quantum limits on measurements of small changes in the lengthof a Fabry-Perot cavity are calculated . The cavity is modelled by a pair ofdissimilar mirrors oriented perpendicular to a one-dimensional axis of infiniteextent . The continuous spectrum of spatial modes of the system is derived, andthe electromagnetic field is quantized in terms of a continuous set of modecreation and destruction operators . Coherent state and squeezed vacuum-stateexcitations of the field are characterized by energy flow, or intensity, variables .The determination of small changes in the cavity length by observations of fringeintensity is considered for schemes in which the cavity is simultaneously excitedby coherent and squeezed vacuum-state inputs . The contributions to the limitingresolution from photocount and radiation-pressure length uncertainties areevaluated . These properties of the Fabry-Perot cavity are compared with thecorresponding results for the Michelson interferometer .

1 . IntroductionInterest in the limiting resolutions of interferometers for measurements of small

changes in length has been greatly stimulated in recent years by the development ofoptical methods for the detection of gravitational waves [1-3] . Most of the detailedtheoretical work on the limiting length resolution has been concerned with theMichelson interferometer [4-7], but practical systems that use the Fabry-Perotinterferometer are also being developed [8-10] . The main content of the presentpaper is a study of the quantum theory of the Fabry-Perot interferometer and itsapplication to the measurement of length. The interferometer is here treated inisolation, and we do not consider its incorporation into a gravitational-wavedetecting system .

The Fabry-Perot cavity is modelled by a pair of plane high-reflectivity mirrorsoriented perpendicular to a one-dimensional axis . No boundaries are placed on theaxis, and the spatial modes of the cavity system accordingly have a continuousdistribution of wave-vectors . The mirror reflectivities are in general allowed to bedifferent, and the mode structure derived here generalizes earlier work [11, 12] inwhich one of the mirrors was taken to be perfectly reflecting . The electromagneticfield is quantized by the association of creation and destruction operators with thesespatial modes. For a spatial axis of infinite extent, it is natural to work with the energyflow, or intensity, of the field rather than the photon-number variables often used inquantum optics theory . The flow variables also correspond more closely to what ismeasured in experimental determinations of fringe intensity, and we express theresults from a simple model of photodetection in terms of these variables .

228

M. Ley and R. Loudon

It is assumed throughout that the cavity is excited through one of its mirrors by abeam of coherent light with a narrow spread of frequencies . It has been pointed outby Caves [6] that the length resolution of a Michelson interferometer can in principlebe improved by the injection of squeezed vacuum-state light through the normallyunused input channel. We accordingly consider the effects of simultaneousexcitation of the Fabry-Perot cavity through its other mirror by a beam of squeezedvacuum-state light obtained from a degenerate parametric amplifier .

Small changes in the cavity length produce small changes in the Fabry-Perotfringe intensities . We treat length measurement schemes in which photodetectorsare placed on both sides of the cavity with intensity data taken from one, or the other,or from the difference of the two detector readings . The inaccuracy of the lengthdetermination is produced by two factors . The first of these is the uncertainty orfluctuation in the photocount rate that occurs for the coherent input light . Itsmagnitude can in principle be reduced without limit by increasing the intensity ofthe coherent input and by increasing the degree of squeezing of the auxiliary inputlight . However, both these increases have the counter-effect of increasing the secondcontribution to the length measurement inaccuracy, which is caused by fluctuationsin the cavity length associated with fluctuations in the radiation pressure . Anappropriate balance between the two contributions produces a minimum lengthuncertainty equal to a standard quantum limit that has the same value for a range oflength-measuring schemes .

The main results of the paper are summarized in its final section, where acomparison is made of the length-measuring capabilities of the Fabry-Perot andMichelson interferometers .

2. Cavity model and field modesThe optical system is treated as purely one-dimensional with plane-wave

propagation parallel to the z-axis. The Fabry-Perot interferometer consists of twopartially reflecting mirrors whose planes are at right angles to the z-axis . The detailsof the optical propagation within the mirrors are not important for the present study .These details can be suppressed by representing each mirror as a dielectric slab ofthickness e and real dielectric constant x, taken in the limit where (-+O and K--* 00 insuch a way that

µ=xe (1)remains finite . The appropriate limits of standard results for a dielectric slab thengive the complex amplitude reflection and transmission coefficients in the forms

r=ikp/(2-iky) and t=2/(2-ikit),

(2)where k is the optical wave-vector .

These coefficients satisfy the usual amplitude and phase requirements for asymmetrical mirror,

Ir12+It12=1

(3)and

rt*+r*t=0 or argr-argt=2n .

(4)They also have the additional properties

t-r=1, t+r=exp(2iargt)

(5)

and the optical phase changes on reflection and transmission are approximately

argr : n- j tj and argt x 217r-1tI .

(8)

The Fabry-Perot cavity, represented in figure 1, has different mirrors ofcharacteristic constants µl and µ 2 placed respectively at coordinates -2L and 22L .The cavity is conveniently specified by the position-dependent relative permittivity :

K(z)=1 +µ18(z+2L)+µ25(z-2L) .

(9)

The mirror reflection and transmission coefficients, denoted r 1 , t 1 and r2, t2 , aredefined by equations similar to (2) in terms of the mirror constants µl and µ 2 ,respectively .

For a fixed linear polarization, Maxwell's equations have solutions in which theelectric field has a time-dependence exp (- ickt) and a spatial variation described by amode function Uk(z) that satisfies the wave equation

(d 2Uk(z)/dz 2 ) + k2K(z) Uk (z) = 0 .

(10)

There are two solutions for each wave-vector magnitude k . We choose them so thatone mode, with function Uk(z), is purely outgoing on the right of the cavity atpositive z, while the other mode, with function Uk(z), is purely outgoing on the left ofthe cavity at negative z . These modes correspond respectively to illumination of thecavity from the left and from the right . The spatial dependences of the two kinds ofmode are taken to be

exp (ikz) + Rk exp (- ikz)

-oo<z<-4L

Uk(z)= Ik exp (ikz) + J, exp(-ikz) -ZL<z<1L

(11)Tk exp (ikz)

Uk i

IIII4I

Rk

1L2 <z 00

Figure 1 . Geometry of the Fabry-Perot cavity showing the two kinds of mode andnotation for the mode coefficients .

the

Ik

W i Tk

Jk

III-fLI

IIk

I I

J

Ik' Rk'I

Nz

Quantum theory of the Fabry-Perot interferometer 229

and

(6)sin (arg r) = Its,

sin (arg t) = Irk,

For the usual Fabry-Perot

cos (arg r) = - I rl, cos (arg t) = tl .

limit of highly reflecting mirrors where kµ>>1,

(7)Jri2 ~Z_'1 - (4/k2µ 2 ), JtJ 2 ~4/k2µ2 ,

230

M. Ley and R . Loudon

and

Tk exp(-ikz)

-oc<z<-zL

UU(z)= I'k exp(-ikz)+Jkexp(ikz) -2'L<z<zL

(12)

exp (- ikz) + Rkexp (ikz)

2L<z<oc,

where k is taken to be positive throughout .The four unknown coefficients for each mode are determined by the boundary

conditions at the mirrors . Thus continuity of tangential E imposes the condition

Uk( -2I' ) = Uk( 2I'+)=Uk(-2L) (13)

at the left-hand mirror . The tangential B field suffers a discontinuity at each mirrorbecause of the finite change in the time-derivative of the electrical displacementacross the infinitesimal mirror thickness . The resulting boundary condition at theleft-hand mirror is

(dUk( -!L )/dz) - (dUk( - iL+ )/dz)=k2u 1 Uk(-iL) .

(14)

Similar boundary conditions apply at the right-hand mirror and for the second kindof mode function . After some mildly tedious algebra, the mode function coefficientsare found to be

The coefficients Rk and Tk agree with the usual expressions for the amplitudereflection and transmission coefficients of a Fabry-Perot cavity . It is not difficult toverify that they satisfy the conditions

IRkI2+ITkI2=IRkI2+ITkI2=1

(21)

andRk Tk+RkTk =0,

(22)

or

argRk -arg Tk +argRk-arg Tk=x

(23)

as required of all lossless optical systems. The equality (17) of the transmissioncoefficients is a consequence of the time-reversal symmetry of the system, and itleads, in view of (21), to

IRkI = IRkI .

(24)

In addition, it can readily be shown that

1 - IRkI 2= ITkI2- IJkI 2= IIk12- IJkI2= ITkI 2 ,

(25)

Rk ={r l exp (-ikL) +r 2 exp (ikL+2iargt l )}/D k , (15)

Ik =tl /Dk, Jk -tl r 2 exp(ikL)/D k, (16)

Tk= T k = tlt2/Dk, ( 17 )

rk=t2/Dk, Jk=t2rlexp(1kL)/Dk, ( 18 )

Rk ={r2 exp (-ikL)+r 1 exp (ikL +2i arg t 2)}/Dk , (19)

where

Dk =1 -r lr 2 exp (2ikL) . (20)

Mid

Rk Tk = Jklk -Ik Jk = - Tk Rk .

(27)

The standard Sturm-Liouville form of orthogonality integral for the eigenvalueequation (10) is

J K(z)Uk(z)Uk(z) dz= k2 1 k,2 lim [Uk(z)dd z) - d dzz) Uk(z)]Z Z, (28 )

where the expression on the right is obtained by partial integration after substitutionfrom (10) for Uk(z) and Uk(z) respectively. The integral can thus be evaluated byinsertion of the explicit mode functions from (11) . With the use of (21) and a standardrepresentation of the delta function, retention only of the dominating term in thelimit of large Z gives the result

It can similarly be shown that

f 7K(Z)Uk(z) Ukf(z) dz=2nb(k-k'),

( 30)

and the mode normalization is therefore determined . The orthogonality condition

f '_O ooK(z)Uk(z) Uk*(z) dz = 0

(31)

follows with the use of (22) .The mode strength inside the cavity is determined by the coefficients Ik and Jk .

Consider the quantity

IIk1 2 =It11 2 /{(1 - Ir111r21) 2 +41r11Ir2Isin 2 (kL+Zargr1+ argr2)},

(32)

where (16) and (20) have been used . The strength is a maximum for wave-vectors k„that satisfy

k„L=ntt-Zargr l -2argr2i where n is an integer .

(33)

The value of (32) is then denoted

IIImax=It11 2/(1 -Iril Ir21)2 .

(34)

The same wave-vectors (33) give maximum transmission through the cavity, with

I Tk12 of order unity for mirrors whose constants y, and µ2 are not too different . Forwavevectors k close to k, (32) is approximately

IIk1 2 ~IM ,axT2/[(k-k„)2+h2],

( 35)

where

I'=(1-Ir1I Ir2I)/2L(Iril Ir2I) 1 ' 2 .

(36)

For a high-Q cavity with highly reflecting mirrors where (7) and (8) are valid, theresonant wave-vectors are

k„ : [(n-1)ir+il tl I +ijt2I ]/L

(37)

Quantum theory of the Fabry-Perot interferometer

231

f

Iklk _ J kJk = Tk (26)

K(Z)Uk(z)Uk(z) dz=2n8(k-k') .

( 29)oo

232

M. Lev and R . Loudon

and the linewidth from (36) is

2C',: (It1 1 2+It21 2)/2L .

(38)

The internal resonant modes correspond to standing waves of the cavity, and thelinewidth can be attributed to the rate of loss of energy by transmission through theend mirrors . The other mode coefficients also take Lorentzian forms similar to (35)with appropriate maximum values denoted

and

I ti e

IIlmax =41t 1 I 2 1(It J i 2 +It21 2 ) 2Ma. Z'

ITIm

-IRlm,nz-41t,1 2 1t21 2 /(It,l 2 +lt212 ) 2 .

(39)

(40)

The Lorentzian approximation (35) holds over the high-transmission ranges ofwave-vector for the high-Q cavity, and the sharpness of the transmission maxima isof course the feature responsible for the outstanding practical importance of theFabry-Perot interferometer . The maximum transmission mode strength (40) cannotexceed unity, but the mode strengths (39) inside the cavity take very high valueswhen the mirror reflectivities are close to 100 per cent .

Gardiner and Savage [13] have considered the relation between the input andoutput fields of a Fabry-Perot cavity . They use quantized fields with differentcreation and destruction operator pairs for the different spatial regions of the modesshown in figure 1 . Their results for an empty cavity are equivalent to (15) and (20) .Collett and Gardiner [14] have given an analogous treatment in which the cavitymodes are represented by a system of discrete internal standing waves coupledthrough the mirrors to continua of external modes . Each external mode is coupled inthis model to all of the internal standing waves, but the connections between inputand output fields remain the same .

The results of this section can also be compared to the expressions derived in [11]and [12] when the right-hand mirror is made perfectly reflecting, with

Ir21=1, argr2=9, It21=0, argt2=iir

(41)

in accordance with (8) . The mode function Uk(z) does not involve any excitation ofthe optical cavity in this case . The other mode function, Uk (z), exactly reproducesthe standing-wave spatial dependence derived by Baseia and Nussenzveig [11] whenaccount is taken of their different coordinate origin, different normalization, and adifference in overall phase amounting, in their notation, to an angle of zkl-2r-8k .

Finally, we point out that in the absence of any mirrors, when

The two modes are simple plane waves travelling in opposite directions, as expected,with mode functions

Uk(z) = exp (ikz) and Uk(z) = exp (- ikz) .

(44)

,=r2 =0 and t j =t 2 =1, (42)

the node coefficients (15)-(19) reduce to

J k =Jk=Rk =Rk=O and I k =Ik=Tk =Tk=1 . (43)


233

3 . Field quantizationThe electromagnetic field is quantized by the introduction of mode creation and

destruction operators . The operators for modes Uk(z) and Uk(z) are denoted &k, akand akt, ak respectively . With k taken to be a continuous variable, they satisfy thecommutation relations

Lak, 61] = Lak, akt] = 6(k - k' ),

Lak, ak ] = Lak, ak'] = 0 .(45)

With a unit quantization area in the xy-plane and a single linear polarization E, theusual procedure for quantization of the electromagnetic field [15] produces a vectorpotential operator of the form

When the Fabry-Perot mirrors are removed, and the mode functions have the plane-wave spatial dependences (44), the vector potential reduces to a one-dimensionalform of the usual free-space expression .

The electric field operator has two parts that satisfy relations similar to (46) and(48), with

'E + (z, t)=icJ

dk(hck/47cc0) 112 [OkUk(z)+O Uk(z)] exp (- ickt) .

(49)0

If the zero-point contributions are ignored, the Hamiltonian can be written

~_ 00

H=2J

dzCOK(z)t -( z, t) • E + ( z, t), (50)

and this reduces with the use of the orthogonality relations (29)-(3 1) to the expectedform

H=J0dc000'00 khk(k k+k) .

(51)

The photodetection rates considered in the following section depend upon time-dependent operators defined by

at(t)=(c/27t) 112 jdk akexp(ickt)

&(t)=(c/271)1/2 Jdk~k exP(_ickt),

(52)

and similar relations for the primed operators . The ranges of integration extend from0 to oc but it is a good approximation to take a range - oo to oo for narrow-bandwidth

whereA(z, t)=A + (z, t)+A - (z, t), (46)

A + (z, t)=E' dk(h/47tE Ock)112 [hkUk(z)+d' U' (z)] exp (-ickt) (47)

and

J 0

A (z, t)=[A + (z, t)]f . (48)

234


excitations . It can then be shown with the use of (45) that the above operators satisfythe commutation relations

[a(t), at(t')l =[a'(t), a't(t')]=8(t-t')

[a(t), a t(t')7=[a (t), a t(t')]=0 .

Time-dependent operators can be defined piecemeal for the different sections ofthe z-axis . Thus the outgoing mode functions on the left of the cavity given by (11)and (12) combine to generate an operator

aL(t)=(c/27r) 112Jdk(Rka k +Tkak)exp(-ickt-ikz),

(54)

where

t=t-(IzI/c) .

(55)The analogous operator on the right of the cavity is

aR(t)=(c/27r) 112 Jdk (Tkak +Rkak) exp (-ickt+ikz) .

(56)

The commutators of these operators with their Hermitian conjugates are

[aL(t), ai(t' )] = [aR(t), aR(t')l =d(t- t' )

['MO, 4R(t )] = [aR(t ), 4(t ') l = 0,

(53)

(57)

where (17), (21), (22), (24) and (45) have been used . The quantities

fL(t)=-<ai(t)aL(t)> and fR(1)=<aR(t)aR(7)>

(58)have the dimensions of inverse time . For a field excitation of sufficiently narrowfrequency spread, they represent the outward rates of energy flow, or the intensities,measured in numbers of quanta per unit time .

Coherent-state excitations of the continuous distribution of modes Uk(z) aredefined by

1141 > = exp{fdk (akak -akak)}10>,

(59)

where ak is any complex function of k and 10> is the multimode vacuum state . Thesecoherent states have the eigenvalue properties

akl {ak} > = akl l a k} >

(60)

and

a(t)17akf>=a(t)Ilak}>,

(61)

where

a(t) =(c/2n) 1 / 2 Jdk ; e xp (-ickt) .

(62)

For coherent light of very narrow spectral width around the wave-vector k o ,corresponding to the continuum representation of 'single-mode' laser light, we put

ak =(2rrf/c) 112 exp(i4) 5(k-k0),

(63)


235

where f is the intensity or energy flow of the light and 0 is its phase angle . Then from(62)

a(t) =fl 12 exp (- icko t + i4) .

(64)

Coherent states of the modes Uk(z) are defined in a similar fashion .Simultaneous coherent-state excitations of both kinds of mode are denoted

I{ak}{ak}> . The ak and ak functions can both be taken in the form (63) when the wave-vector spread is small compared to the cavity resonance width F given by (36) or (38),and when both functions are centred on the same wave-vector k o , as would be thecase for joint excitation by the same laser source . The operator defined in (54) thensatisfies the eigenvalue equation

aL(t) I {ak}{ ak}> = aL(t)I{ak}{ak}> ,

(65)where

aL(t)= [Rof 1"2 exp (it)) + To f"12 exp (i4')] exp (-ickoi) .

(66)

The energy flow obtained from (58) has the time-independent form

fL=IRof 1"2 eXp (i4))+ To f'112 exp (i4) ' )1 2 .

( 67)The corresponding outwards flow on the right of the cavity obtained with the use ofthe operator defined in (56) is

fR = I To f 1 12 exp (i4)) + Ro f'112 exp (i4)')12 .

(68)The zero subscripts on the mode coefficients in these expressions are a shorthand forko . It follows from (21) and (22) that

fL +fR =f+f',

(69)in accordance with energy conservation .

4 . Photodetector modelAn experimental arrangement of the kind represented in figure 2 is assumed, with

a light source and a detector on either side of the Fabry-Perot cavity . The coincidentinput and output beams could be separated in practice by optical circulators, notshown in figure 2 . The bandwidths of the input light beams are assumed to besufficiently narrow for the frequency dependence of the detector response to beignored . The response is then simply proportional to the appropriate flow rate, orintensity, as defined in (58) . The measured data are the numbers n(r) of photocountsrecorded over repeated time-intervals of duration i . The mean and the secondmoment of the photocount distribution function are accordingly

<n(i)> =J s

dt <at(t)a(t)>

(70)0

and

<n(i)2>=J i

dtJ T

dt' <at(t)a(t)at(t')a(t')> .

(71)0

0

No allowance has so far been made for the photodetector quantum efficiencies,which are often much smaller than unity in practice . It is assumed that the two

236


i f I

f,

I fR

I

I

fLI

f_

f'I ~

II

I

source

detector

Figure 2 . Arrangement of light sources and detectors showing the notation for the opticalenergy flows .

detectors shown in figure 2 have the same quantum efficiency rl . The mean count (70)is scaled to become

<n(r)>=r1J

dt<&t(t)d(t)> .

(72)0

The second-moment expression (71) must first be put into a normally ordered form,with the use of (53), in order for the quantum efficiency to be easily inserted . It isconvenient also to subtract off the square of the mean, and the resulting expressionfor the photocount variance is

<(On(i))2>=tJJ

t dt<at(t)a(t)>0

f fi

t

r+r7 2

dtf

dt'< : at(t)6(t)at(t')d(t') : )-C

dt<at(t)a(t)>]21,

(73)0

0

f0

where the colons denote normal ordering .These expressions for the photocount mean and variance apply as they stand to

both detectors, and it is only necessary to specify a particular detector by insertion ofappropriate subscripts (L or R) . However, an alternative way of processing themeasured data is to form the difference between the photocounts in the two detectorsin each time-interval r . The mean difference photocount is

<nD(z)>=21 f t0dt<aL(t)aL(t) - aR(t)aR(t)>,

(74)J

where the detectors are assumed to lie at equal distances from the cavity so that thetime of detection suffers the same retardation at each . The variance of the differencephotocount, obtained with the use of the commutation relations (57), is

<(OnD( U))2>=t1J t

dt <aL(t)aL(t)+ iR(t)aR(t)>0

+ 1121 J t dtJt

dt'< : [aL(t)aL(t) - aR(t)dR(t)]LaL(t' )aL(t' ) -aR(t~)hR(t)]0

0

-C J t

dt <aL(t)aL(t)-4R(t)aR(t)>]2} .

(75)0

Note that the term linear in 11, sometimes referred to as the shot noise in the detectionprocess, is proportional to the sum of the photocounts at the detectors even thoughthe variance refers to their difference .

detector

source

These expressions take simple forms for coherent excitation of both kinds ofmode, where eigenvalue equations like (65) apply for the left and right destructionoperators . The contribution proportional to n2 in (73) vanishes for such excitations,and the means and variances are given by

<(AnL(t))2> = <nL(T)> = ntfL(76)

((OnR(i))2>=<nR(T)>=nif ,

where the energy flows are given by (67) and (68) . The contribution proportional ton2 in (75) also vanishes, and the mean and the variance of the difference photocountare accordingly

and


237

<nD(?)) = r/i(fL -fR) (77)

<(AnD(ti)) 2> = nT(L+fR) = nt( +f ' ),

(78 )where (69) has been used .

The flow ratesfL andfR provide convenient characterizations of the strengths oflight beams in the continuous wave-vector representation . These quantitiescorrespond more closely to what is measured by photodetectors than do the photonnumbers more commonly used in discrete wave vector representations . In addition,the total photon number is awkwardly infinite for a z-axis of infinite length with anexcitation whose flow at each point has a finite value .

5 . The squeezed vacuum stateThe assessment of the Fabry-Perot interferometer as a length-measuring device

in the subsequent sections considers arrangements in which the left-hand lightsource provides coherent light, while the right-hand source is either absent,corresponding to a vacuum state of the Uk(z) modes, or provides squeezed vacuum-state light . The suggestion that the latter could be advantageous was first made byCaves [6] in his treatment of the limiting resolution of a Michelson interferometer asa gravitational wave detector . A suitable right-hand light source in this case is adegenerate parametric amplifier operated with a vacuum input .

The most complete continuum-mode theory of the degenerate parametricamplifier has been given by Collett and Gardiner [14], and we here quote withoutproof some of the results needed in later sections . The light source is assumed toconsist of an amplifying medium in a single-ended cavity, and the notation of [14] isconverted according to

y ,--*y, y,--+O and e---e exp (i0 s),

(79)

where y is the amplifier cavity damping constant and a is a measure of the amplifierpump intensity . Both parameters y and a are real and have the dimensions of inversetime. The phase of s in (79) is chosen to agree with the squeezed state notation ofCaves [6] . The required mode-operator expectation values are

f

<ak> - 0,

(S0)

1_

1

~aktak ~

c2(k-ko )2 +(iy-e)2 c2(k-ko)2+(zY+e)2}6(k-k')

(81)

238


and

<akak >- c2(k

2E

o)2 +(iy)-E)2 +c2(k ZEko)2+( )+E)2}6(kFk -2k o ) > (82)

where the distribution of wave-vectors in the squeezed-state spectrum is assumed tobe centred on the same wave-vector ko as the left-hand input coherent state describedby (63) or (64) .

The squeezed vacuum light has a spread of wave-vector components of order(?y ±e)/c. It is assumed that this spread is small compared to the Fabry-Perotresonance width F given by (36) or (38) . The right-hand input flow for this model ofthe light source is given by

f'=<a t (t)a'(t)> = 1E 2y/(4Y 2- E 2 ),

(2Y±9)T>>

and the variance then takes the form

where

(83)

where (52) and (81) have been used . The left-hand output flow obtained from (58)with the use of (54), (60), (63), (80) and (81) is

fL=JRo1 2f+I7' 1 2f ,

(84)

and the corresponding result for the right-hand output is

fR = I ToI 2f+ IRol 2f' .

(85)

It is seen by comparison with the flows (67) and (68) for two coherent inputs that thecross-terms are now absent . The energy conservation condition (69) is however stillsatisfied by the flows (84) and (85) .

The photodetector model of the previous section can be applied to the case of asqueezed-vacuum right-hand input . The direct photodetection of squeezed light hasbeen treated by Collett et al . [16], and the same method can be used here with someslight generalization to take account of the Fabry-Perot mode structure. The meanand the variance of the counts at the left-hand detector are given by (72) and (73) withL subscripts attached to the operators, which are then defined in accordance with(54) . The mean count obtained with the use of (17) and (84) is

<nL(i)> = rjr(IRo I2f + I T1 I 2f') .

(86)

The variance is obtained from (73) with the use of (60), (63), (80), (81) and (82) .The photocount integration time r is assumed to be sufficiently long to satisfy theinequality

(87)

<(OnL(Z))2>_<nL(ti)>+Yl2i{(4Y2E

E 2 )2[EY - (4Y2 +e2)cos(2x)]IRo1 2IT ' I 2f

+4YC y+EizY-E

IToI4f

(88)(2Y-E)

2+

th+E) 2]

X=0-20s+argRo-arg To.

(89)

Quantum theory of the Fabry-Perot interferometer 239

This expression may be written more compactly in terms of a squeezing parameters(called r by Caves [6]) defined by

1,y 2

2sinhs=1 2 Y 2 and coshs= 1 2 +E2

IV-E

4y-9

Then (88) becomes

(90)

<(AnL(T))2>=nt(IR0I 2f+ITo1 2f)+q 2 T{(exp (2s) sin 2x+exp (- 2s) cos 2 x-1)1R0 1 2 1 T0 1 2f

+4[exp(s)(exp(s)+1)+exp(-s) (exp(-s)+1)]1T0 1 4f'},

(91)

where (17) and (24) have been used . The photocount mean and variance for theright-hand detector are obtained from (86) and (91) by simply interchanging Roand To .

A very similar calculation based on (74) and (75) provides the mean and thevariance for an experiment that takes the difference between the left- and right-handphotocounts as its measured data. The mean is found to he

<'ID( .[)> = r1T(IRol 2- IT01 2)(f-f')

(92)and the variance is

<(OnD(T))2> = riT( f +f') + yj 2T{4(exp (2s) sin 2x+ exp (- 2s) cos 2 x - 1)IRoI 2 1 To 1 2f+4[exp(s)(exp(s)+1)+exp(-s)(exp(-s)+1)](IR0 1 2 -ITo I 2 ) 2f'} .

(93)Note that both variances (91) and (93) are minimized for choices of phase angles suchthat x is zero or an integer multiple of it . With suitable values of the other parametersin these expressions, the variances can be reduced below their values for a vacuumright-hand input where f' = 0 and s=0. The reduction of the photodetection noise byreplacement of the vacuum right-hand input by squeezed vacuum-state light isconsidered in §6.3 .

6 . Photocount length uncertaintySmall changes in the length L of the Fabry-Perot cavity produce small changes in

the mean photocounts at the two detectors, and these can in principle be used tomeasure the small distortions of the interferometer caused by gravitational waves .The limiting resolution is determined by the intrinsic uncertainty in the photocount,characterized by the photocount variance . This resolution is calculated in the presentsection for detection schemes that use the left-hand or the right-hand detector or thedifference between the two photocounts . The left-hand input is always coherentlight, while the right-hand input is either vacuum or squeezed vacuum-state light .

The uncertainty AL in cavity length caused by the photocount uncertainty isobtained from

AL =<(An(T))2>"2I d<n(T)>/dLI

(94)

240


with subscripts L, R or D as appropriate to denote the detection scheme employed . Itfollows from (67) and (68) or (86), and the result

(dIRo1 2/dL)+(dIT0I 2/dL)=0

(95)

obtained from (21), that

(d(nL(T)>/dL) + (d<nR(T)>/dL) = 0

(96)

for the two kinds of right-hand input . The length uncertainties for left and rightdetection are therefore in the ratio of the square roots of their photocount variances .

The derivatives in (95) are obtained straightforwardly with the use of (15), (17)and (20) . The general results are however quite complicated algebraically and wefirst consider the case of identical cavity mirrors, where the subscripts on r and t canbe dropped. Then

6.1 . Zero right-hand inputWithf'=0, the means and variances obtained from (76) with the use of (67) and

(68) are

and((OnL(T)) 2) =<nL(T)>=11TIRoI 2f (100)

<(OnR(T))2)=<nR(i)>=gt1TO1 2f.

(101)

The corresponding quantities for difference detection obtained from (77) and (78)are

and<nD(T)> = IIT(IRo1 2 - I To1 2)f

<(AnD(T))2> =qT.f.

(102)

(103)

Consider first the case where the left-hand detector is used to measure the changein cavity length . The length uncertainty obtained from (94) with the use of (97), (98),and (100) is

AL,

(It1 4 +4Ir1 2 s in 2 0) 3 '2

(104)-4ko(rlTf) 1/2 Ir1 ItI 4Icos0I

This quantity and the mean photocount from (100) are plotted as functions of 0 infigure 3, where an unrealistically large value of Iti 2 is taken for ease of drawing . It isseen that the minimum length uncertainty occurs at the null in the Fabry-Perotreflected intensity where

(ALL)mi.=~tl 2 /4ko(r/tf)' /2 1r1 for sin 0=0 .

(105)

sin 20IR,I 2 = 44Ir12

ItlTTI2 (97)20 1ItI +4IrI sin I

° a

2

201Its +4Ir1 sin

and

8k0Ir1 2 It1 4 sin 0 cos 0(98)

dIRDI 2 dI To j 2

\ \heredL dL (ItI4 +4Ir1 2 sine 0) 2 '

(99)0= k0L + arg r .

Y

Yc

Figure 3 . The mean photocount and the length uncertainty for measurements with the left-hand detector . The mirrors are identical with

IrI 2 =0 .9 and It1 2 =0. 1 .

The left-hand photocount variance vanishes for these values of the angle 0 .According to (105), the length-measuring resolution of the Fabry-Perot inter-ferometer can be improved without limit by reduction of the mirror intensitytransmission coefficient

It1 2 or the optical wavelength 2ir/k o , or by increase in theintensity f of the coherent light source. We shall find in § 7, however, that limits areimposed when the effects of radiation pressure fluctuations are taken into account . Aresult equivalent to (105) has been obtained by Yurke et al . [17] .

The corresponding length uncertainty for measurements that use the right-handdetector is obtained from (94) with the use of (97), (98) and (101) as

AL,(It14 +41r1 2 sine 0) 3/2=8ko(iirf )t" 2 IrI 2 ItI 2 Isin 0 cos 0l

(106)

The 0 dependences of this quantity and the mean photocount from (101) are shownin figure 4 . The minimum length uncertainty occurs at angles slightly displaced fromthe positions of the Fabry-Perot transmission maxima . These angles are easily foundby differentiation of (106), and for highly reflecting mirrors they are given by

sin20 ;:Zt~ It14/8 for ItI 2 <<1, (107)

where

(ALR)min ~ 27 112 1t1 2/8ko(rhf ) 112 (108)

and the mean photocount is

<ne(z)> x 2rhf/3 . (109)

242


dJ4

Ys

Figure 4 . Same as figure 3 but for measurements with the right-hand detector .

The minimum length uncertainty in this case is a factor of 2 .6 larger than theuncertainty (105) for measurements by the left-hand detector .

Finally, the length uncertainty for difference detection is obtained from (94) withthe use of (97), (98), (102) and (103) as

OL_

(tI4 +41rI 2 sin e 0) 2D 16ko(gif ) 112 jrj 2 jtI 4jsin 0 cos 0~

(110)

The 0 dependences of this function and the mean difference photocount from (102)are shown in figure 5 . The minimum length uncertainty in this case occurs at anglesgiven by

The minimum length uncertainty is a factor of 1 . 5 larger than the uncertainty (105)for measurements by the left-hand detector .

6.2 . Non-identical mirrorsWith different mirrors at the two ends of the Fabry-Perot cavity the simple

expressions (97) no longer apply, and in general the mode coefficients must be takenin the forms given by (15), (17) and (20) . Whereas the cavity with identical lossless

sin 2 0m NItI4/12 for I tI 2 <<1, (111)

where

(OL°)min,'t~ 21 t1 2 /27 112ko(rhf)1/2 , (112)

and the mean difference photocount is

<n°('[)> -tlzf/2 . (113)


243

03

02

Ja.r

Y

s0 .1

Go

Figure 5 . Same as figure 3 but for measurements with the difference in photocount for thetwo detectors .

mirrors has 100 per cent transmission and zero reflection at the resonant wave-vectors k n , this is no longer the case with different mirrors and some reflected lightremains even at resonance . The effects of non-identical mirrors are particularlystriking for length determinations that use the left-hand detector, where thenumerator of (94) no longer has a zero to cancel the zero that continues to occur in thedenominator at the resonant wave-vectors . The corresponding length uncertainty isthus infinite on resonance, similar to the behaviour for right-hand and differencedetection shown in figures 4 and 5, and quite different from the left-hand behaviourfor identical mirrors shown in figure 3 .

The general expressions for the length uncertainties obtained with the use of(15), (17) and (20) are quite complicated . However, since the minimum uncertaintiesoccur close to the resonant wave-vectors, it is permissible to use the Lorentzianapproximations to the mode coefficients with forms given by (35) and (36). Thus forexample, in the case of right-hand detection, the length uncertainty (106) isgeneralized for different mirrors to

(It 1 I 2 +It 2 1 2 )[(k 0L-k0L) 2 +(FL) 2 ] 3 / 2

ALIt_

4ko(qTf) 1 / 2 I t11 I t2Ir'LI k0L-knLI

(114)

where (40) has been used, and the L-independent quantities k0L and FL are given by(37) and (38) . Minimum uncertainty occurs for

I k0L - knLI =FL/21 /2 ,

(115)where

(ALR)min =271/2 (It1I 2 +lt2I 2 ) 2 /32ko( , rf) 1J2 It1I lt21 .

(116)

244

M. Ley and R . I oudon

This reduces to (108) for identical mirrors, and it shows that the uncertainty for non-identical mirrors can be obtained by making the replacement

ItI 2 _(It1 I 2 +It2I 2 ) 2 /41tjI It21 .

(117)

It is seen with the use of Cauchy's inequality that the quantity on the right-hand sideis larger than the average of the intensity transmission coefficients of the two mirrors .The length uncertainty becomes infinite when one of the mirrors is made perfectlyreflecting, as would be expected on physical grounds, since the photocount isindependent of cavity length in this case .

Results similar to (114) and (116) apply for the left-hand and difference detectionschemes .

6.3 . Squeezed vacuum-state right-hand inputCaves [6] first suggested that the length-measuring resolution of a Michelson

interferometer could be improved with the use of a squeezed vacuum-state input .The Michelson resembles the Fabry-Perot interferometer in having two kinds ofinput mode, corresponding to the two input faces of the beam-splitter at the centre ofthe interferometer . With only one of the inputs excited, an analysis similar to thatcarried out above shows that minimum length uncertainty occurs for measurementsmade at the nulls in the interference fringes . The nulls are of course characterized asthe interferometer settings for which none of the incident light, and correspondinglynone of its photon-number fluctuations, reach the detector . The residual photocountlength uncertainty at the nulls can be ascribed to the vacuum fluctuations that enterthe interferometer via the unexcited input channel .

Similar arguments can be applied to the Fabry-Perot interferometer . Considerthe simple case, treated earlier in the present section, of left-hand detection with avacuum right-hand input . The photocount variance (100) can be written in the form

`;(OnL(T)) 2>=t1 2TIRol 4f+i IRoI 2f(1 - gIRo12),

(118)and for perfect detection (ij = 1) this reduces to

<(OnL(T)) 2 >=TIRoI4f+TIRo1 21ToJ 2f.

(119)

The two terms on the right are interpreted as contributions to the photocount noisethat arise, respectively, from the beating of the reflected input coherent light of meancount TIR0 I 2f with its own reflected vacuum fluctuations of magnitude 'IRO 12 andwith the transmitted vacuum fluctuations of magnitude ZI To12 from the unexcitedright-hand input . The minimum length uncertainty (105), at an interferometersetting for which IR0I = 0, arises entirely from the second term on the right of (118) or(119), the residual zero in this term being cancelled by a zero of the same order in thedenominator of (94) . It should therefore be expected that the minimum lengthuncertainty can be further reduced by the use of squeezed vacuum light at the right-hand input .

The above argument suggests that squeezed input light reduces the minimumlength uncertainty significantly in conditions where this occurs at, or close to, a nullin the interference fringes . It is seen by reference to figures 3 to 5 that thisrequirement is satisfied for left-hand detection and approximately for differencedetection, but is not at all satisfied for right-hand detection . Detailed expressions forthe length uncertainties are obtained by substitution of (86) and (91) or (92) and (93)into (94) . The Fabry-Perot mirrors are assumed henceforth to be identical so that the


245

mode coefficients and their derivatives are given by (97) and (98) . The resultingexpressions are nevertheless quite complicated and we first consider some numericalillustrations of the effects of introducing squeezed light .

One obvious consequence of a non-zero right-hand input is the removal of thenulls in the interference fringes at the left-hand detector, and this effect is illustratedin figure 6 . The difference photocount, however, retains its zero-crossing, as isillustrated in figure 7. The removal of the nulls at the left-hand detector has aconsequence similar to that of using non-identical mirrors, discussed in § 6.2, in thatthe minimum length uncertainty no longer occurs at the resonant wave-vectors . Thiseffect is illustrated in figure 8, which clearly shows the shift in minimum uncertaintyaway from 0=0 for f' :A 0. The shift destroys to some extent the beneficial influenceof the squeezed vacuum input, which as explained above is particularly effective forangles 0 such that JR0 1 2 vanishes. The length uncertainty increases for the largervalues off', corresponding to more heavily squeezed vacuum-states, on account ofthe positive contributions of the terms in (91) linear in f', which begin to dominatethe terms proportional to f. Figure 9 shows the analogous variation of lengthuncertainty for right-hand detection with various squeezed vacuum-state inputs,and as is suggested by the qualitative discussion of the previous paragraph, little orno reduction in uncertainty is produced . Finally, figure 10 shows the correspondingresults for difference detection . The introduction of squeezed vacuum-state light at

Figure 6. Variation of mean photocounts at the two detectors obtained from (86) withITf=1000 and the values of qTf ' shown against the curves. The cavity mirrors areidentical with lrl 2 =0 . 9 and It1 2 =0. 1 .

10

115

10go

Figure 7 . Variation of mean difference photocount for the same conditions as figure 6 .

Figure 8 . Variation of length uncertainty for left-hand detection with iitf=1000 and thevalues of >7tf'shown against the curves . The squeezing parameters are respectivelys=0,1,2 and 3 ; sinx=0, tl=1, 1rJ 2 =0 .9, and It1 2 =0 . 1 .

001

rcJaY

O I5

Figure 9. Same as figure 8 but for measurements with the right-hand detector .

001

J4s

II/

10

5

10

Figure 10 . Same as figure 8 but for measurements with the difference in photocount forth,two detectors .

248

M. Lev and R . Loudon

the right-hand input here shifts the minimum in the length uncertainty towards theangle 0 of the zero-crossing in the interference fringes (cf. figure 7), where thesqueezing is expected to be particularly effective . The curves in figure 10 show thatsignificant reductions in the minimum length uncertainty are indeed achieved . Thecancellation of the mode coefficients in the final term of the difference photocountvariance (93) has an important influence in limiting the damage caused by the termslinear in f for the more heavily squeezed vacuum states .

Following the above remarks, we consider in detail only the most promisingarrangement that uses a squeezed vacuum state input in conjunction with differencephotodetection . It is assumed that

f»f',

(120)

and the photocount mean from (92) is approximately

<no(r)> qT(IRo12-IToI2)f.

(121)

For identical mirrors, the use of (23) simplifies the phase angle defined in (89) to

x=0-20 s +21t .

(122)

The variance (93) is minimized by a choice of phase angle such that sin x=0 andcos x=1, when it takes the approximate form

<(Onn(T))2)',~OTf{1 +4q(exp( - 2s) - 1)IRo1 2 IT01 2} .

(123)

The length uncertainty obtained from (94) with the use of (97) and (98) is

{(ItI4 +41r1 2 sin e 0) 2 + 16q(exp ( - 2s) - 1)Irl2 Itl4 sin e 01"2(1t14+41r 12 sin e 0)AL,-

16ko(gTf ) 112 Ir1 2 It1 4lsin 0 cos 01

(124)

This expression reduces to (110) in the absence of any squeezing, when s = 0, or forlow detection efficiency, q-*0 .

In the presence of squeezing, it is convenient first to consider the limit of verylarge squeezing, s-> oc, and perfectly efficient detection, r~=1, when (124) reduces to

OL_I Itl 8 16IrI 4 sin4 01AL,

16k o(Tf ) 112 Ir1 2 1t1 41sin 0 cos 01

(125)

The length uncertainty therefore vanishes for angles 0 m that satisfy

Isin 0m1= It1 2 /21rI,

(126)

where

IRO1 2 =IT012= 2 and <-n(T)>=0.

(127)

The shift in the minimization condition from angles that satisfy (11) for zerosqueezing to angles that satisfy (126) for large squeezing agrees with the behaviour ofthe numerical results illustrated in figure 10 .

For intermediate values of the squeezing parameter s and quantum efficiency 11,the angles 0m that minimize the length uncertainty must be found by differentiationof (124) . This procedure produces some complicated algebra, which we do notreproduce here . The results simplify considerably however for squeezing para-meters s greater than unity and quantum efficiencies q not much smaller than unity,


249

when the minimization angles continue to satisfy (126) to a good approximation .Minimum length uncertainty thus continues to coincide with the zero-crossings inthe difference fringe intensity, and substitution of (126) into (124) gives

(OLD)min(1+lexp(-2s)-t1)1J2ltl2

for ItI 2 <<1 .

(128)2ko(tiTf )

1 / 2

For perfectly efficient detection, this result simplifies further to

(OLD) min - ItI 2 exp ( - s)/2ko(Tf ) 1J2 for ?1=1 .

(129)

The length uncertainty can thus in principle be reduced to any arbitrary level by theuse of sufficient squeezing in the vacuum-state light in the right-hand input . Withincrease in the squeezing parameter, it is of course also necessary to increase theintensity of the coherent left-hand input in order that the inequality (120) continuesto be satisfied . The purpose of the inequality is partly to ensure that the final term inthe variance expression (93) proportional to f should be relatively small, and thisobjective is greatly helped by the coincidence of minimum length uncertainty withzero-crossings in the difference fringe intensity where (127) is satisfied .

Similar calculations can be performed for the other detection arrangements, andthe results confirm the qualitative features of the numerical calculations shown infigures 8 and 9 . The contribution linear in f' for the photocount variance (91) in left-hand detection has a larger prefactor than for difference detection, and it is necessaryto satisfy the inequality (120) more strongly in order to make this contributionnegligible and thus take advantage of the squeezing reduction of the contributionlinear inf. When the stronger inequality is satisfied, the left-hand minimum lengthuncertainty is approximately

(ALL)min:ItI2 exp ( - s)/4ko(Tf )112 for q =l,

(130)

similar to (129) . The corresponding result for right-hand detection is

(LtLe)minHtJ2/2ko(Tf) 1/2 for t1=1 .

(131)

These minimum values are not shown by the curves in figures 8 and 9, particularlyfor the higher values of the squeezing parameter, because the coherent energy flow fis not sufficiently large to justify the neglect of the variance terms linear in f' .

7 . Radiation-pressure length uncertaintyThe minimum length uncertainties calculated in the previous section can be

made arbitrarily small by suitable combinations of small mirror transmission Its,high-intensityfin the left-hand input, and large-squeezing s in the right-hand input .However, we shall show in the present section that all of these factors tend to producelarge-intensity fluctuations in the interior of the Fabry-Perot cavity, and that theassociated radiation-pressure fluctuations on the cavity mirrors prevent arbitraryreductions in the minimum length uncertainties .

The effects of radiation-pressure fluctuations in a Michelson interferometer havebeen calculated by Edelstein et al. [4] and by Caves [5, 6], and we follow the generalapproach of these authors by calculating the radiation-pressure length uncertainty asa separate contribution, unconnected with the photocount length uncertainty of theprevious section. It has been shown in the Michelson case [7] that both contributionscan be included in a single unified calculation of the quantity that is actually

250


observed, the photocount fluctuation at the detector . However, the main results ofthe two methods of calculation are the same, and it is simpler to make a separatecalculation of the radiation-pressure length uncertainty .

With the notation for intensity components in the three regions of the z-axisshown in figure 2, and for a narrow spread of optical excitation around the wave-vector ko , the mean forces on the two mirrors in the positive z direction are

F1=hko(f+fL -f+ -f-)(132)

F2 = hko(f+ +f- -fR-f') .If the compliances of the mirror mountings are respectively S 1 and S2 , the meanchange in cavity length produced by these forces is

l=S2F2-S 1F1 .

(133)The two compliances are assumed to be of similar magnitude, with

S 1 =S+8S and S2 =S-SS,

(134)

and the length change takes the form1=kkOS(2f++ 2f- -f-f '-fR-fL)+hkobS(fR+f'-f-fL) .

(135)

It has been shown in the previous section that the minimum length uncertaintiesoccur for angles Bm that are close to the cavity resonances, where the mode strengths(16) and (18) inside the cavity take high values, particularly for low mirrortransmission coefficients . The internal intensities f+ and f_ are correspondinglymuch larger than the input intensities f andf' or the output intensities fL andf, It isseen from (135) that the mean radiation-pressure length change is approximately

l: 2hk0S(f+ +f_),

(136)

and the contribution proportional to 6S is relatively insignificant . The differencebetween the compliances can therefore be ignored .

The radiation-pressure length uncertainty bl is given approximately by thefluctuation in the length change whose mean value is given by (136), and it isnecessary to specify the mode operators whose expectation values determine theinternal intensities . As in the derivations of the output energy flows (84) and (85), weassume that the spread of wave-vectors in the optical excitation is much smaller thanthe cavity resonance width h . Then analogous to (58),

f+(t) =<(Ioat(t)+J'a t(t))(Ioa(t)+Joa'(t))>f_(t)=<(Joat(t)+Io a't(t))(Joa(t)+Ioa'(t))> .

(137)

The cavity mirrors are assumed to be identical for the remainder of the presentsection, and the primed and unprimed mode coefficients in (137) then become thesame. It follows from (16) and (18) that for a high-Q cavity with highly reflectingmirrors

11012'&Vol2 ~ltl 4 + l4sine 6'

(138)

similar to (97) but with lrl set equal to unity, and with 9 defined by (99) . Also

argJo-argIo =k0L+argr^--nxc

(139)


251

close to the nth cavity resonance, where (33) has been used . Thus (137) reduces to

f+(t) ~f-(t)^ IIo12<(at(t)±a^'t(t))(a(t)±i (t))>,

f+(t) , f_(t),II012(f+f') .

(140)

and for excitations in which the left-hand input is coherent light while the right-handinput is squeezed vacuum-state light, use of (64), (80) and (83) gives

(141)

It makes no difference for these excitations whether the + or the - signs are used in(140) and we henceforth retain only the + signs . The mean length change (136) isnow

lz4hk 0SII0 1 2(f+f') .

(142)

The radiation-pressure length uncertainty is obtained from the variance of thecavity length change given by (136) with the energy flows represented by theoperator combination given in (140) . It is convenient to introduce the time durationio of an experimental observation of the cavity length, and to average the varianceover the observation time . Thus

(6L) 2 =16h2koS2II0 I4

x T J odtJ o

dt'<(at(t)+a't(t))(a(t)+a(t))(at(t')+a't(t'))(a(t')+a'(t'))>0 1 '0

'01

-C

dt<(at(t)+at(t))(a(t)+6'(t))>]2)))~ .

(143)0,

The mean length change (142) is unaffected by the time-averaging . The variance isevaluated by steps similar to those that lead from the general photocount variance(73) to the explicit expression (91) . With the observation time 'r. assumed to satisfy aninequality similar to (87), the variance is

(SL) 2 =16h2k2S2II01 4i

{(exp(2s)cos2 X+exp(-2s)sin2 X+1)f0

+['exp(s)(exp(s)+1)+'exp(-s) (exp (-s)+ 1) + 2] f'},

(144)

where x is defined in (122) .In order to compare this expression with the minimum photocount length

uncertainties calculated in the previous section, we evaluate (144) for the sameconditions assumed there, namely

f>>f', sinx=0 and cosx=l,

(145)

when it reduces to(5L)2 =16ft2k0S2II0I4 (exp (2s) + 1)f/;.

(146)

It is seen from (138) that 1101 is proportional to 1/1t1 at the angles 0for minimumphotocount length uncertainty, given for example by (107) or (126) . Thus for a largesqueezing parameter s, the radiation-pressure length uncertainty from (146) has theproportionality

bLocexp (s)f 1 J 2 11t1 2 .

(147)

252

M. Lev and R. Loudon

This contrasts with the proportionalities(LL)m ; n rx 1t1 2 exp (-s)/f 1/2

(148)

of the minimum photocount length uncertainties given by (129) or (130) . Theradiation-pressure fluctuations accordingly limit the improvements in lengthresolution that can be obtained by increasing s or f, or by decreasing ItI .

The final expression for the Fabry-Perot length uncertainty is found by squaringand adding the photocount and radiation-pressure contributions . For differencedetection with a squeezing parameter s greater than unity and I rI x 1, substitution ofthe angle 0m from (126) into (138) and insertion of the result into (146) gives

(6L,)2;:t 4h2koS2 exp (2s)fl It1 4 T o .

(149)

We take the photocount uncertainty from (129) for the limit of perfectly efficientdetection, and the combined length uncertainty is then

CIti4 exp (-2s) + 4h 2koS 24exp (2s)f 1/2 .

(150)4k otf

ItI To

This quantity has a minimum value

(2hS) 1/2/(TTo ) 1/4

(151)

for a coherent input intensity

tI 4 ex

2s) T1/2fmin _ I

4 k 2S

a

(152)o

(

The overall minimum length uncertainty (151) is a form of the standard quantumlimit that arises in other schemes for high-resolution length measurement . Itdepends only on the compliance of the mirror supports and on the photocountintegration and total observation times . However, the input intensity (152) needed toachieve the optimum length resolution can be greatly reduced by the use of highlyreflecting mirrors and a highly squeezed vacuum-state input . Similar conclusionsapply for left-hand detection alone, where the photocount length uncertainty is givenby (130) . For right-hand detection alone, where the photocount length uncertainty isgiven by (131), the minimum length uncertainty of order (151) is achieved only in theabsence of squeezing, where the right-hand input is an ordinary vacuum state, andthe required coherent input intensity is of the order of (152) but with the squeezingexponential removed .

8. ConclusionsThe fringes of a Fabry-Perot interferometer are much more sensitive to changes

in cavity length than are the fringes of a Michelson interferometer to changes inrelative arm length . This is illustrated for example in figure 3, where, even for themodest reflectivity of 0 . 9, the fringe intensity falls to half its maximum value for achange in k0L amounting to about 3° . By contrast, the corresponding angle for aMichelson interferometer, with the usual cosine dependence on the difference of armlengths, is 90° . It might therefore be expected that an order of magnitudeimprovement in sensitivity could be achieved by the use of Fabry-Perot inter-ferometers instead of Michelson interferometers in gravitational wave detectors .


253

The sensitivity of the Michelson interferometer can however be improved by theuse of a multiple-reflection arrangement in which the light in each arm is reflected btimes by the end mirror before the two beams are combined to form interferencefringes. The multi-reflection Michelson interferometer still has cosine fringes butany shift in the mirror position produces a change in optical path length that isamplified by a factor b . An analysis of the length resolution of a multi-reflectionMichelson interferometer [6] produces a length uncertainty that has the same overallform as the Fabry-Perot result (150), except that Jti 2 is replaced by 1/b. Thisreplacement is perhaps not surprising since 1/iti 2 provides a measure of the numberof reflections that take place in the Fabry-Perot cavity before the light emergesthrough the mirror . It should however be stressed that the two interferometersoperate in quite different fashions; the sharp fringes of the Fabry-Perot inter-ferometer result from the superposition of all the light beams in the multiplyreflected set, whereas the different orders of reflected beam in the Michelsoninterferometer are spatially separated in a zig-zag pattern . The Fabry-Perotinterferometer basically has a simpler structure than the multi-reflection Michelsoninterferometer .

The final result (151) for the minimum length uncertainty of the Fabry-Perotinterferometer is no different from that found for other measurement schemes .Schemes do differ however in the feasibility of achieving the minimum value inpractice. For Michelson interferometric techniques with practicable laser intensitiesf, the photocount length uncertainty greatly outweighs the radiation-pressure lengthuncertainty, For the Fabry-Perot interferometer, the coherent laser intensity (152)required to achieve the optimum length resolution is greatly reduced by the Itt 4factor, of order 10 -6 or less in practice, which is an inherent property of theinterferometer. The additional exp (- 2s) reduction factor produced by the use of asqueezed vacuum-state input is the same for both kinds of interferometer . Thesefactors in principle facilitate the achievement of the minimum length uncertainty(151), which occurs at the coherent intensity for which the photocount and radiation-pressure contributions are equal .

The detailed form of the radiation-pressure contribution is affected by themeasurement strategy adopted in the search for cavity length changes, which is inturn influenced by the spectral characteristics of the length-changing forces appliedto the cavity . The simple model used here has mirrors that respond instantaneouslyto the radiation-pressure force and no explicit assumptions have been made aboutthe time-dependences of the length changes that must be detected . In applications togravitational wave detection, the resolution can be optimized by suitable restrictionof the detection bandwidth to frequencies present in the wave and the choice ofappropriate mirror damping and restoring forces . A variety of minimum lengthuncertainties is found for the different spectral distributions of gravitational force[18] . These results together with (151) are in accord with the basic quantum theory ofmeasurement (see [19] and the earlier references therein) . We note that in (151) and(152), the observational time-duration To associated with the radiation-pressurefluctuations has a minimum value equal to the photocount integration time T, but it isof course longer than T for a series of photocount measurements .

The limits discussed above refer to a somewhat idealized system and we shouldemphasize two of the assumptions on which they rely . Thus, it was assumed in § 5that the spectral distribution of the squeezed vacuum-state light is much narrowerthan the Fabry-Perot resonance width t . This essentially requires that the cavity of

254


the degenerate-parametric-amplifier squeezed-light source should have narrowerresonances than the Fabry-Perot length-measuring cavity . The requirement isdifficult to satisfy in practice since Itl 2 , and hence F'=1t12/2L, need to be as small aspossible to achieve a manageable coherent input intensity (152) . The technology ofsqueezed light generation is in any case in its infancy, and sources with the requiredproperties are not currently available .

The final results for the limiting length resolution also rely on the assumption ofunit photodetector quantum efficiency, made in the transition from (128) to (129) .When rj is not equal to unity, the former expression must be used, and the minimumlength resolution no longer tends to zero in the limit of infinite squeezing parameter .Indeed the term that involves the squeezing parameter in (128) can be neglectedunless i is sufficiently close to unity that

1-rl<rlexp(-2s) .

(153)

Thus any attempt to improve the interferometer sensitivity by the use of squeezedvacuum-state light is worth while only for highly efficient photodetection . A similarconclusion applies to the Michelson interferometer [6] .

In summary, we have evaluated the quantum theory of the Fabry-Perotinterferometer and have shown that it has properties similar in outline, but differentin detail, from those of a Michelson interferometer in terms of its potential for high-resolution length measurement . The interferometric detection of gravitationalwaves remains of course a formidable experimental problem .

Note addedIn a recent paper, Knoll et al . [20] have considered the Fabry-Perot inter-

ferometer as an example of a spectral filter that adds unavoidable quantum noise toits output . We briefly show how the results of the present paper conform to theirgeneral formalism .

For a high-Q cavity close to resonance at k=k,,, where the Lorentzianapproximation (35) is applicable, (56) can be transformed to

The expression (154) has the same general structure as equation (3) of [20] . With noexcitation of the right-hand input, &R(t) represents the right-hand output, withcontributions from the transmitted left-hand input, filtered in accordance with(155), and a noise operator/'(t) that is independent of the input . As Knoll et al . point

4(f) := f

dt'T(t"-t')a(t')+7(t), (154)

where the transmission response function is defined by

(155)T(t) = B(t)F'cI T Lax exp (- ick„t-cFt),

with the step function

(0 fort<0(156)B(t)= j

l 1 fort > 0,

and

7(t)=(c/2tc) 1 12 JdkR~k kexp (-ickt) . (157)

Quantum theorv of the Fabry-Perot interferometer 255

out, the latter is needed to preserve boson commutation properties, and it degradesthe spectral-filtering characteristics of the interferometer that would be expected onthe basis of a purely classical theory .

AcknowledgmentM. Ley thanks the Science and Engineering Research Council and the British

Telecom Research Laboratories for financial support in the form of a CASEstudentship .

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General Relativity and Gravitation (Cambridge University Press) .[3] MEYSTRE, P ., and SCULLY, M. O . (editors), 1983, Quantum Optics, Experimental

Gravitation and Measurement Theory (New York: Plenum) .[4] EDELSTEIN, W. A., HOUGH, J ., PUGH, J . R., and MARTIN, W ., 1978, J . Phys . E, 11,

710-712 .[5] CAVES, C. M ., 1980, Phys . Rev . Lett ., 45, 75-79 .[6] CAVES, C. M ., 1981, Phys . Rev . D, 23, 1693-1708 .[7] LOUDON, R., 1981, Phys . Rev. Lett., 47, 815-818 .[8] DREVER, R. W . P., et al., see [3], pp. 503-514 .[9] HoucH, J ., et al ., see [3], pp. 515-524 .[10] HOUGH, J ., 1985, Nature, Lond., 316, 576-577 .[11] BASEIA, B ., and NUSSENZVEIG, H. M ., 1984, Optica Acta, 31, 39-62 .[12] PENAFORTE, J . C ., and BASEIA, B ., 1984, Phys . Rev. A, 30, 1401-1406 .[13] GARDINER, C . W., and SAVAGE, C. M ., 1984, Optics Commun ., 50, 173-178 .[14] COLLETT, M . J ., and GARDINER, C. W ., 1984, Phys . Rev. A, 30,1386-1391 .[15] LOUDON, R., 1983, The Quantum Theory of Light (Oxford: Clarendon Press) .[16] COLI.FTT, M . J ., LOUDON, R., and GARDINER, C . W ., 1987, J . mod. Opt., 34 (to be

published) .[17] YURKE, B ., MCCALL, S . L., and KLAUDER, J . R., 1986, Phys . Rev . A, 33, 4033-4054 .[18] CAVES, C. M ., 1983, Proc . Int. Symp. Foundations of Quantum Mechanics, Tokyo,

pp. 195-205 .[19] Ni, W.-T ., 1986, Phys . Rev. A, 33, 2225-2229 .[20] KNOLL, L., VOGEL, W ., and WELSCH, D . G ., 1986, J. opt . Soc. Am. B, 3, 1315-1318 .

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