Download Quantum theory of high-resolution length measurement with a Fabry

JOURNAL OF MODERN OPTICS,
1987,
VOL .
34,
NO .
2, 2 2 7-255
Quantum theory of high-resolution length measurement
with a Fabry-Perot interferometer
M . LEY and R . LOUDON
Physics Department, Essex University,
Colchester C04 3SQ, England
(Received 17 November 1986)
Abstract. The quantum limits on measurements of small changes in the length
of a Fabry-Perot cavity are calculated . The cavity is modelled by a pair of
dissimilar mirrors oriented perpendicular to a one-dimensional axis of infinite
extent . The continuous spectrum of spatial modes of the system is derived, and
the electromagnetic field is quantized in terms of a continuous set of mode
creation and destruction operators . Coherent state and squeezed vacuum-state
excitations of the field are characterized by energy flow, or intensity, variables .
The determination of small changes in the cavity length by observations of fringe
intensity is considered for schemes in which the cavity is simultaneously excited
by coherent and squeezed vacuum-state inputs . The contributions to the limiting
resolution from photocount and radiation-pressure length uncertainties are
evaluated . These properties of the Fabry-Perot cavity are compared with the
corresponding results for the Michelson interferometer .
1.
Introduction
Interest in the limiting resolutions of interferometers for measurements of small
changes in length has been greatly stimulated in recent years by the development of
optical methods for the detection of gravitational waves [1-3] . Most of the detailed
theoretical work on the limiting length resolution has been concerned with the
Michelson interferometer [4-7], but practical systems that use the Fabry-Perot
interferometer are also being developed [8-10] . The main content of the present
paper is a study of the quantum theory of the Fabry-Perot interferometer and its
application to the measurement of length . The interferometer is here treated in
isolation, and we do not consider its incorporation into a gravitational-wave
detecting system .
The Fabry-Perot cavity is modelled by a pair of plane high-reflectivity mirrors
oriented perpendicular to a one-dimensional axis . No boundaries are placed on the
axis, and the spatial modes of the cavity system accordingly have a continuous
distribution of wave-vectors . The mirror reflectivities are in general allowed to be
different, and the mode structure derived here generalizes earlier work [11, 12] in
which one of the mirrors was taken to be perfectly reflecting . The electromagnetic
field is quantized by the association of creation and destruction operators with these
spatial modes . For a spatial axis of infinite extent, it is natural to work with the energy
flow, or intensity, of the field rather than the photon-number variables often used in
quantum optics theory . The flow variables also correspond more closely to what is
measured in experimental determinations of fringe intensity, and we express the
results 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 a
beam of coherent light with a narrow spread of frequencies . It has been pointed out
by Caves [6] that the length resolution of a Michelson interferometer can in principle
be improved by the injection of squeezed vacuum-state light through the normally
unused input channel . We accordingly consider the effects of simultaneous
excitation of the Fabry-Perot cavity through its other mirror by a beam of squeezed
vacuum-state light obtained from a degenerate parametric amplifier .
Small changes in the cavity length produce small changes in the Fabry-Perot
fringe intensities . We treat length measurement schemes in which photodetectors
are 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 length
determination is produced by two factors . The first of these is the uncertainty or
fluctuation in the photocount rate that occurs for the coherent input light . Its
magnitude can in principle be reduced without limit by increasing the intensity of
the coherent input and by increasing the degree of squeezing of the auxiliary input
light . However, both these increases have the counter-effect of increasing the second
contribution to the length measurement inaccuracy, which is caused by fluctuations
in the cavity length associated with fluctuations in the radiation pressure . An
appropriate balance between the two contributions produces a minimum length
uncertainty equal to a standard quantum limit that has the same value for a range of
length-measuring schemes .
The main results of the paper are summarized in its final section, where a
comparison is made of the length-measuring capabilities of the Fabry-Perot and
Michelson interferometers .
2 . Cavity model and field modes
The optical system is treated as purely one-dimensional with plane-wave
propagation parallel to the z-axis . The Fabry-Perot interferometer consists of two
partially reflecting mirrors whose planes are at right angles to the z-axis . The details
of 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 of
thickness e and real dielectric constant x, taken in the limit where (-+O and K--* 00 in
such a way that
µ=xe
(1)
remains finite . The appropriate limits of standard results for a dielectric slab then
give 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 a
symmetrical mirror,
Ir12+It12=1
(3)
rt*+r*t=0 or argr-argt=2n .
(4)
and
They also have the additional properties
t-r=1, t+r=exp(2iargt)
(5)
Quantum theory of the Fabry-Perot interferometer
229
and
sin (arg r) = Its,
sin (arg t) = Irk,
cos (arg r) = - I rl,
cos (arg t) = tl .
(6)
For the usual Fabry-Perot limit of highly reflecting mirrors where kµ>>1,
Jri2 ~Z_'1 - ( 4 /k 2 µ 2 ),
JtJ 2 ~4/k2µ 2 ,
(7)
and the optical phase changes on reflection and transmission are approximately
argr : n- j tj
and
argt x 21 7r-1tI .
(8)
The Fabry-Perot cavity, represented in figure 1, has different mirrors of
characteristic 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 , are
defined 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 the
electric field has a time-dependence exp (- ickt) and a spatial variation described by a
mode function Uk (z) that satisfies the wave equation
(d 2 Uk (z)/dz 2 ) + k 2 K(z) Uk (z) = 0 .
(10)
There are two solutions for each wave-vector magnitude k . We choose them so that
one mode, with function Uk (z), is purely outgoing on the right of the cavity at
positive z, while the other mode, with function Uk (z), is purely outgoing on the left of
the cavity at negative z . These modes correspond respectively to illumination of the
cavity from the left and from the right . The spatial dependences of the two kinds of
mode are taken to be
exp (ikz) + Rk exp (- ikz)
Uk(z)= Ik exp (ikz) + J, exp(-ikz)
Tk exp (ikz)
I
Uk i
Rk
I
I
I4
I
Ik
Jk
Ik
'
Jk
W
i
-oo<z<-4L
-ZL<z< 1 L
1L
2 <z 00
(11)
Tk
I
I
I
I -fL
I
I
I
I
Nz
I
R k'
Figure 1 . Geometry of the Fabry-Perot cavity showing the two kinds of mode and the
notation for the mode coefficients .
230
M . Ley and R . Loudon
and
Tk exp(-ikz)
UU(z)=
-oc<z<-zL
I'k exp(-ikz)+Jkexp(ikz) -2'L<z<zL
exp (- ikz) + Rkexp (ikz)
(12)
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 mirror
because of the finite change in the time-derivative of the electrical displacement
across the infinitesimal mirror thickness . The resulting boundary condition at the
left-hand mirror is
(dUk( -!L )/dz) - (dUk( - iL + )/dz)=k 2 u 1 Uk(-iL) .
(14)
Similar boundary conditions apply at the right-hand mirror and for the second kind
of mode function . After some mildly tedious algebra, the mode function coefficients
are found to be
Rk ={r l exp (-ikL) +r 2 exp (ikL+2iargt l )}/D k ,
(15)
Ik =t l /D k, Jk -t l 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 )}/D k ,
(19)
Dk =1 -r l r 2 exp (2ikL) .
(20)
where
The coefficients Rk and Tk agree with the usual expressions for the amplitude
reflection and transmission coefficients of a Fabry-Perot cavity . It is not difficult to
verify that they satisfy the conditions
IRkI2+ITkI2=IRkI2+ITkI2=1
(21)
Rk Tk+RkTk =0,
(22)
argR k -arg Tk +argRk-arg Tk=x
(23)
and
or
as required of all lossless optical systems . The equality (17) of the transmission
coefficients is a consequence of the time-reversal symmetry of the system, and it
leads, in view of (21), to
IRkI
= IRkI .
(24)
In addition, it can readily be shown that
1 - IRkI 2= ITkI 2- IJkI 2= IIk1 2- IJkI 2= ITkI 2 ,
(25)
Quantum theory of the Fabry-Perot interferometer
231
Iklk _ J k Jk = Tk
(26)
Rk T k = Jklk -Ik Jk = - Tk Rk .
(27)
Mid
The standard Sturm-Liouville form of orthogonality integral for the eigenvalue
equation (10) is
J
K(z)Uk(z)Uk(z) dz= k2 1 k,2 lim [Uk(z) dd z) - d dz z) Uk (z)] Z Z,
(28 )
where the expression on the right is obtained by partial integration after substitution
from (10) for Uk(z) and Uk(z) respectively . The integral can thus be evaluated by
insertion of the explicit mode functions from (11) . With the use of (21) and a standard
representation of the delta function, retention only of the dominating term in the
limit of large Z gives the result
f
K(Z) Uk(z)Uk(z) dz=2n8(k-k') .
( 29)
oo
It can similarly be shown that
K(Z) Uk(z)
Ukf(z) dz=2nb(k-k'),
( 30)
f7
and the mode normalization is therefore determined . The orthogonality condition
K(z) Uk (z) Uk*(z) dz = 0
(31)
f _'O oo
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 Tk1 2 of order unity for mirrors whose constants y, and µ2 are not too different . For
wavevectors k close to k, (32) is approximately
IIk1 2
~IM
,axT2/[(k-k„)2+h2],
( 35)
I'=(1-Ir1I Ir2I)/2L(Iril Ir2I) 1 ' 2 .
(36)
where
For a high-Q cavity with highly reflecting mirrors where (7) and (8) are valid, the
resonant wave-vectors are
k„ : [(n-1)ir+il tl I +ijt2I ]/L
(37)
232
M. Lev and R . Loudon
and the linewidth from (36) is
2C',: (It1 2 +It21 2 )/2L .
1
(38)
The internal resonant modes correspond to standing waves of the cavity, and the
linewidth can be attributed to the rate of loss of energy by transmission through the
end mirrors . The other mode coefficients also take Lorentzian forms similar to (35)
with appropriate maximum values denoted
1
J
I ti Ma.
e Z' IIlmax =41t I 2 1(It i 2 +It21 2 ) 2
(39)
and
ITIm
-IRlm,nz-41t,1 2 1t21 2
/(It,l
2 +lt21 2 ) 2 .
(40)
The Lorentzian approximation (35) holds over the high-transmission ranges of
wave-vector for the high-Q cavity, and the sharpness of the transmission maxima is
of course the feature responsible for the outstanding practical importance of the
Fabry-Perot interferometer . The maximum transmission mode strength (40) cannot
exceed unity, but the mode strengths (39) inside the cavity take very high values
when the mirror reflectivities are close to 100 per cent .
Gardiner and Savage [13] have considered the relation between the input and
output fields of a Fabry-Perot cavity . They use quantized fields with different
creation and destruction operator pairs for the different spatial regions of the modes
shown 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 cavity
modes are represented by a system of discrete internal standing waves coupled
through the mirrors to continua of external modes . Each external mode is coupled in
this model to all of the internal standing waves, but the connections between input
and 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 of
the optical cavity in this case . The other mode function, Uk (z), exactly reproduces
the standing-wave spatial dependence derived by Baseia and Nussenzveig [11] when
account is taken of their different coordinate origin, different normalization, and a
difference 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
,=r 2 =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)
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)
Quantum theory of the Fabry-Perot interferometer
233
3.
Field quantization
The 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, ak
and akt, ak respectively . With k taken to be a continuous variable, they satisfy the
commutation relations
Lak, 61] = Lak, akt] = 6(k - k'),
(45)
Lak, ak ] = Lak, ak'] = 0.
With a unit quantization area in the xy-plane and a single linear polarization E, the
usual procedure for quantization of the electromagnetic field [15] produces a vector
potential operator of the form
A(z, t)=A + (z, t)+A - (z, t),
(46)
A + (z, t)=E' dk(h/47tE O ck) 112 [hk Uk (z)+d' U' (z)] exp (-ickt)
J0
(47)
A (z, t)=[A + (z, t)]f .
(48)
where
and
When the Fabry-Perot mirrors are removed, and the mode functions have the planewave spatial dependences (44), the vector potential reduces to a one-dimensional
form 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)=ic
dk(hck/47cc0) 112 [OkUk(z)+O Uk(z)] exp (- ickt) .
(49)
J0
If the zero-point contributions are ignored, the Hamiltonian can be written
dzCO K(z)t - ( z, t) •
H=2
J
E
+ ( z, t),
(50)
~_ 00
and this reduces with the use of the orthogonality relations (29)-(3 1) to the expected
form
0 khk(k k +k) .
(51)
J0dc000'0
The photodetection rates considered in the following section depend upon timedependent operators defined by
H=
at(t)=(c/27t) 112 jdk a kexp(ickt)
(52)
&(t)=(c/271)1/2 Jdk~ k exP(_ickt),
and similar relations for the primed operators . The ranges of integration extend from
0 to oc but it is a good approximation to take a range - oo to oo for narrow-bandwidth
234
M. Ley and R . Loudon
excitations . It can then be shown with the use of (45) that the above operators satisfy
the commutation relations
[a(t), at(t')l =[a'(t), a't(t')]=8(t-t')
(53)
[a(t), a t (t')7=[a (t), a t (t')]=0 .
Time-dependent operators can be defined piecemeal for the different sections of
the 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) 112
dk(Rk a k +Tkak)exp(-ickt-ikz),
(54)
J
where
(55)
t=t-(IzI/c) .
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
(57)
= 0,
where (17), (21), (22), (24) and (45) have been used . The quantities
and fR(1)=<aR(t)aR(7)>
fL(t)=-<ai(t)aL(t)>
(58)
have the dimensions of inverse time . For a field excitation of sufficiently narrow
frequency 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) are
defined by
f
dk (akak -akak)}10>,
(59)
{
where ak is any complex function of k and 10> is the multimode vacuum state . These
coherent states have the eigenvalue properties
1141 > = exp
akl {ak} > = akl l a k} >
( 60 )
a(t)17akf>=a(t)Ilak}>,
(61)
a(t) =(c/2n) 1 / 2 Jdk ; e xp (-ickt) .
(62)
and
where
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)
Quantum theory of the Fabry-Perot interferometer
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 wavevector 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 the
case for joint excitation by the same laser source . The operator defined in (54) then
satisfies the eigenvalue equation
aL(t) I { a k}{ a k}> = aL(t)I {ak}{ ak}> ,
(65 )
1"2 exp
(66)
where
aL(t)= [Rof
(it)) + To f"12 exp (i4')] exp (-ick o i) .
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 of
the operator defined in (56) is
fR = I
1
To f 1 2 exp (i4)) + Ro f'112 exp (i4)')12 .
(68)
The zero subscripts on the mode coefficients in these expressions are a shorthand for
k o . It follows from (21) and (22) that
fL +fR =f +f',
(69)
in accordance with energy conservation .
4.
Photodetector model
An 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 coincident
input and output beams could be separated in practice by optical circulators, not
shown in figure 2 . The bandwidths of the input light beams are assumed to be
sufficiently narrow for the frequency dependence of the detector response to be
ignored . The response is then simply proportional to the appropriate flow rate, or
intensity, as defined in (58) . The measured data are the numbers n(r) of photocounts
recorded over repeated time-intervals of duration i . The mean and the second
moment of the photocount distribution function are accordingly
dt <at(t)a(t)>
<n(i)> =
J
(70)
0
s
and
dt' <at(t)a(t)at(t')a(t')> .
(71)
dt
J i0 J 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
<n(i)2>=
T
236
M. Ley and R . Loudon
i
f
source
I
I
f,
I
fLI
detector
fR
detector
I
f_
I ~
f'
I
I
I
source
Figure 2 . Arrangement of light sources and detectors showing the notation for the optical
energy flows .
detectors shown in figure 2 have the same quantum efficiency rl . The mean count (70)
is scaled to become
<n(r)>=r1
dt<&t(t)d(t)> .
J
(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 is
convenient also to subtract off the square of the mean, and the resulting expression
for the photocount variance is
<(On(i))2>=tJ
t dt<at(t)a(t)>
J 0
i
+r7 2
f
t
f dt
0
dt'< : at(t)6(t)at(t')d(t') :
f
0
)-C
]2 1,
r
dt<at(t)a(t)>
(73)
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 of
appropriate subscripts (L or R) . However, an alternative way of processing the
measured data is to form the difference between the photocounts in the two detectors
in each time-interval r . The mean difference photocount is
f
(74)
dt<aL(t)aL(t) - aR(t)aR(t)>,
J t0
where the detectors are assumed to lie at equal distances from the cavity so that the
time of detection suffers the same retardation at each . The variance of the difference
photocount, obtained with the use of the commutation relations (57), is
<nD(z)>=21
U
<(OnD( ))2>=t1
dt <aL(t)aL(t)+ iR(t)aR(t)>
J t0
+
112
dt'< : [aL(t)aL(t) - aR(t)dR(t)]LaL(t' )aL(t' ) - aR(t~)hR(t)]
1 J t0 dtJt0
-
dt <aL(t)aL(t)-4R(t)aR(t)>]2} .
(75)
C J 0t
Note that the term linear in 11, sometimes referred to as the shot noise in the detection
process, is proportional to the sum of the photocounts at the detectors even though
the variance refers to their difference .
Quantum theory of the Fabry-Perot interferometer
237
These expressions take simple forms for coherent excitation of both kinds of
mode, where eigenvalue equations like (65) apply for the left and right destruction
operators . 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 to
n2 in (75) also vanishes, and the mean and the variance of the difference photocount
are accordingly
(77)
<nD(?)) = r/i(fL -fR)
and
<(AnD(ti)) 2 > = nT(L+fR)
= nt(
+f ' ),
(78)
where (69) has been used .
The flow rates fL and fR provide convenient characterizations of the strengths of
light beams in the continuous wave-vector representation . These quantities
correspond more closely to what is measured by photodetectors than do the photon
numbers 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 an
excitation whose flow at each point has a finite value .
5.
The squeezed vacuum state
The assessment of the Fabry-Perot interferometer as a length-measuring device
in the subsequent sections considers arrangements in which the left-hand light
source provides coherent light, while the right-hand source is either absent,
corresponding to a vacuum state of the Uk(z) modes, or provides squeezed vacuumstate light . The suggestion that the latter could be advantageous was first made by
Caves [6] in his treatment of the limiting resolution of a Michelson interferometer as
a gravitational wave detector . A suitable right-hand light source in this case is a
degenerate parametric amplifier operated with a vacuum input .
The most complete continuum-mode theory of the degenerate parametric
amplifier has been given by Collett and Gardiner [14], and we here quote without
proof some of the results needed in later sections . The light source is assumed to
consist of an amplifying medium in a single-ended cavity, and the notation of [14] is
converted 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 amplifier
pump intensity . Both parameters y and a are real and have the dimensions of inverse
time . The phase of s in (79) is chosen to agree with the squeezed state notation of
Caves [6] . The required mode-operator expectation values are
(S 0)
< ak> - 0,
1_
~ aktak ~
fc2(k-k o)2 +(iy-e) 2
1
c2(k-ko)2+(zY+e)2}6(k-k')
(81)
238
M. Ley and R . Loudon
and
<akak >
2E
- c2(k
o)2 +(iy)-E)2 +c2(k ZEko)2+(
)+E)2}6(k
Fk -2k o ) > (82)
where the distribution of wave-vectors in the squeezed-state spectrum is assumed to
be centred on the same wave-vector ko as the left-hand input coherent state described
by (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-Perot
resonance width F given by (36) or (38) . The right-hand input flow for this model of
the light source is given by
f'=<a t (t)a'(t)> = 1E 2y/(4Y 2- E 2 ),
(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
(85)
f R = I ToI 2f + IRol 2f' .
It is seen by comparison with the flows (67) and (68) for two coherent inputs that the
cross-terms are now absent . The energy conservation condition (69) is however still
satisfied by the flows (84) and (85) .
The photodetector model of the previous section can be applied to the case of a
squeezed-vacuum right-hand input . The direct photodetection of squeezed light has
been treated by Collett et al . [16], and the same method can be used here with some
slight generalization to take account of the Fabry-Perot mode structure . The mean
and the variance of the counts at the left-hand detector are given by (72) and (73) with
L 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(IR o I 2f +
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 the
inequality
(87)
(2Y±9)T>>
and the variance then takes the form
E
<(OnL(Z))2>_<nL(ti)>+Yl2i{(4Y2
E2)
2[EY - ( 4Y 2 +e 2 )cos(2x)]IRo1 2 IT ' I 2f
+4YC
y+E
2+ izY-E
(2Y-E)
th+E) 2]
IToI 4f
(88)
where
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 +E 22
sinhs=1 2 Y 2 and coshs= 1 2
(90)
4y-9
IV-E
Then (88) becomes
<(AnL(T)) 2 >=nt(IR0I 2f+ITo1 2f)
+q 2T{(exp
(2s) sin 2 x+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 the
right-hand detector are obtained from (86) and (91) by simply interchanging R o
and To .
A very similar calculation based on (74) and (75) provides the mean and the
variance for an experiment that takes the difference between the left- and right-hand
photocounts 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 2 T{4(exp (2s) sin 2x+ exp (- 2s) cos 2 x - 1)IRoI 21 To 12f
+4[exp(s)(exp(s)+1)+exp(-s)(exp(-s)+1)](IR0 1 2 -IToI 2) 2f'} .
(93)
Note that both variances (91) and (93) are minimized for choices of phase angles such
that x is zero or an integer multiple of it . With suitable values of the other parameters
in these expressions, the variances can be reduced below their values for a vacuum
. The reduction of the photodetection noise by
right-hand input where f' = 0 and s=0
replacement of the vacuum right-hand input by squeezed vacuum-state light is
considered in §6 .3.
6.
Photocount length uncertainty
Small 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 to
measure 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 present
section for detection schemes that use the left-hand or the right-hand detector or the
difference between the two photocounts . The left-hand input is always coherent
light, while the right-hand input is either vacuum or squeezed vacuum-state light .
The uncertainty AL in cavity length caused by the photocount uncertainty is
obtained from
AL
=<(An(T))2>"2
I d<n(T)>/dLI
(94)
240
M . Ley and R . Loudon
with subscripts L, R or D as appropriate to denote the detection scheme employed . It
follows from (67) and (68) or (86), and the result
(dIRo1 2 /dL)+(dIT0I 2 /dL)=0
(95)
(d(nL(T)>/dL) + (d<nR(T)>/dL) = 0
(96)
obtained from (21), that
for the two kinds of right-hand input . The length uncertainties for left and right
detection 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 we
first consider the case of identical cavity mirrors, where the subscripts on r and t can
be dropped . Then
sin 20
IR,I 2 = 44Ir12
ItI +4IrI sin 20 1
Itl 2
I TTI 2 ° Its
a +4Ir1 sin 20 1
(97)
and
dIR DI 2
dL
dI To j 2
dL
8k0Ir1 2 It1 4 sin 0 cos 0
(ItI4 +4Ir1 2 sine 0) 2 '
(98)
\ \here
0= k 0 L + arg r .
(99)
6 .1 . Zero right-hand input
With f'=0, the means and variances obtained from (76) with the use of (67) and
(68) are
((OnL(T)) 2 ) =<nL(T)>=11TIRoI 2f
(100)
and
(101)
<(OnR(T)) 2 )=<nR(i)>=gt1TO1 2f.
The corresponding quantities for difference detection obtained from (77) and (78)
are
<nD(T)> = IIT(IRo1 2 - I To1 2 )f
(102)
<(AnD(T))2> =qT.f.
(103)
and
Consider first the case where the left-hand detector is used to measure the change
in cavity length . The length uncertainty obtained from (94) with the use of (97), (98),
and (100) is
2 0) 3 '2
(It1 4 +4Ir1 2 s in
AL, (104)
4ko(rlTf) 1/2 Ir1 ItI 4 Icos0I
This quantity and the mean photocount from (100) are plotted as functions of 0 in
figure 3, where an unrealistically large value of Iti 2 is taken for ease of drawing . It is
seen that the minimum length uncertainty occurs at the null in the Fabry-Perot
reflected intensity where
(ALL)mi
.=~tl 2 /4ko(r/tf)' /2 1r1
for
sin 0=0 .
(105)
Y
Y
c
Figure 3 . The mean photocount and the length uncertainty for measurements with the lefthand detector . The mirrors are identical with 2 =0 . 9 and 2 =0. 1 .
IrI
It1
The left-hand photocount variance vanishes for these values of the angle 0 .
According to (105), the length-measuring resolution of the Fabry-Perot interferometer can be improved without limit by reduction of the mirror intensity
transmission coefficient 2 or the optical wavelength 2ir/k o , or by increase in the
intensity f of the coherent light source . We shall find in § 7, however, that limits are
imposed when the effects of radiation pressure fluctuations are taken into account . A
result equivalent to (105) has been obtained by Yurke et al . [17] .
The corresponding length uncertainty for measurements that use the right-hand
detector is obtained from (94) with the use of (97), (98) and (101) as
0) 3/2
4 +41r1 2 sine
AL,
=
(106)
8ko (iirf )t" 2 IrI 2 2 Isin 0 cos 0l
It1
(It1
ItI
The 0 dependences of this quantity and the mean photocount from (101) are shown
in figure 4 . The minimum length uncertainty occurs at angles slightly displaced from
the positions of the Fabry-Perot transmission maxima . These angles are easily found
by differentiation of (106), and for highly reflecting mirrors they are given by
sin 20
;:Zt~
for
It1 4 /8
ItI 2 <<1,
(107)
where
(ALR)min ~ 27 112
1
t1 2 /8k o (rhf )
112
(108)
and the mean photocount is
<ne(z)> x 2rhf/3 .
(109)
242
M. Ley and R . Loudon
d
J
4
Y
s
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 the
uncertainty (105) for measurements by the left-hand detector .
Finally, the length uncertainty for difference detection is obtained from (94) with
the use of (97), (98), (102) and (103) as
(tI 4 +41rI 2 sin e
0) 2
OL_
D 16k o (gif ) 112 jrj 2 jtI 4 jsin 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 angles
given by
sin 2
0m NItI 4/12
for
I tI 2 <<1,
(111)
where
1
(OL°)min,'t~ 21 t1 2 /27 112 ko(rhf ) /2 ,
(112)
and the mean difference photocount is
<n°('[)>
-tlzf/2 .
(113)
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 mirrors
With 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 taken
in the forms given by (15), (17) and (20) . Whereas the cavity with identical lossless
243
Quantum theory of the Fabry-Perot interferometer
03
02
J
a
.r
Y
s
0 .1
Go
Figure 5 . Same as figure 3 but for measurements with the difference in photocount for the
two detectors .
mirrors has 100 per cent transmission and zero reflection at the resonant wavevectors k n, this is no longer the case with different mirrors and some reflected light
remains even at resonance . The effects of non-identical mirrors are particularly
striking for length determinations that use the left-hand detector, where the
numerator of (94) no longer has a zero to cancel the zero that continues to occur in the
denominator at the resonant wave-vectors . The corresponding length uncertainty is
thus infinite on resonance, similar to the behaviour for right-hand and difference
detection shown in figures 4 and 5, and quite different from the left-hand behaviour
for 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 uncertainties
occur close to the resonant wave-vectors, it is permissible to use the Lorentzian
approximations to the mode coefficients with forms given by (35) and (36) . Thus for
example, in the case of right-hand detection, the length uncertainty (106) is
generalized for different mirrors to
ALIt
(It
_
1 I 2 +It 2 1 2)[(k 0L-k0L) 2
+(FL) 2 ] 3 / 2
(114)
4ko(qTf) 1 / 2 I t11 I t2I r'LI k0L-knLI
where (40) has been used, and the L-independent quantities k 0 L and FL are given by
(37) and (38) . Minimum uncertainty occurs for
I
k0L - knLI =FL/2 1 /2 ,
(115)
where
(ALR)min =271/2 (It1I 2 +lt2I 2 )2/32 ko( ,
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 nonidentical 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 side
is 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 perfectly
reflecting, as would be expected on physical grounds, since the photocount is
independent of cavity length in this case .
Results similar to (114) and (116) apply for the left-hand and difference detection
schemes .
6 .3 .
Squeezed vacuum-state right-hand input
Caves [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 of
input mode, corresponding to the two input faces of the beam-splitter at the centre of
the interferometer . With only one of the inputs excited, an analysis similar to that
carried out above shows that minimum length uncertainty occurs for measurements
made at the nulls in the interference fringes . The nulls are of course characterized as
the interferometer settings for which none of the incident light, and correspondingly
none of its photon-number fluctuations, reach the detector . The residual photocount
length uncertainty at the nulls can be ascribed to the vacuum fluctuations that enter
the interferometer via the unexcited input channel .
Similar arguments can be applied to the Fabry-Perot interferometer . Consider
the simple case, treated earlier in the present section, of left-hand detection with a
vacuum right-hand input . The photocount variance (100) can be written in the form
`;(OnL(T)) 2 >=t1 2 TIRol 4f+i IRoI 2f(1 - gIRo1 2),
(118)
and for perfect detection (ij = 1) this reduces to
<(OnL(T)) 2 >=TIRoI 4f+TIRo1 2 1ToJ 2f.
(119)
The two terms on the right are interpreted as contributions to the photocount noise
that arise, respectively, from the beating of the reflected input coherent light of mean
count TIR0 I 2f with its own reflected vacuum fluctuations of magnitude 'IRO 12 and
with the transmitted vacuum fluctuations of magnitude ZI To1 2 from the unexcited
right-hand input . The minimum length uncertainty (105), at an interferometer
setting 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 the
denominator of (94) . It should therefore be expected that the minimum length
uncertainty can be further reduced by the use of squeezed vacuum light at the righthand input .
The above argument suggests that squeezed input light reduces the minimum
length uncertainty significantly in conditions where this occurs at, or close to, a null
in the interference fringes . It is seen by reference to figures 3 to 5 that this
requirement is satisfied for left-hand detection and approximately for difference
detection, but is not at all satisfied for right-hand detection . Detailed expressions for
the 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
Quantum theory of the Fabry-Perot interferometer
245
mode coefficients and their derivatives are given by (97) and (98) . The resulting
expressions are nevertheless quite complicated and we first consider some numerical
illustrations of the effects of introducing squeezed light .
One obvious consequence of a non-zero right-hand input is the removal of the
nulls in the interference fringes at the left-hand detector, and this effect is illustrated
in figure 6 . The difference photocount, however, retains its zero-crossing, as is
illustrated in figure 7 . The removal of the nulls at the left-hand detector has a
consequence similar to that of using non-identical mirrors, discussed in § 6 .2, in that
the minimum length uncertainty no longer occurs at the resonant wave-vectors . This
effect is illustrated in figure 8, which clearly shows the shift in minimum uncertainty
away from 0=0 for f' :A 0 . The shift destroys to some extent the beneficial influence
of the squeezed vacuum input, which as explained above is particularly effective for
angles 0 such that JR0 2 vanishes . The length uncertainty increases for the larger
values off', corresponding to more heavily squeezed vacuum-states, on account of
the positive contributions of the terms in (91) linear in f', which begin to dominate
the terms proportional to f . Figure 9 shows the analogous variation of length
uncertainty for right-hand detection with various squeezed vacuum-state inputs,
and as is suggested by the qualitative discussion of the previous paragraph, little or
no reduction in uncertainty is produced . Finally, figure 10 shows the corresponding
results for difference detection . The introduction of squeezed vacuum-state light at
1
Figure 6 . Variation of mean photocounts at the two detectors obtained from (86) with
ITf=1000 and the values of qTf ' shown against the curves . The cavity mirrors are
identical with lrl 2 =0 . 9 and It1 2 =0. 1 .
10
11
10
5
go
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 the
values of >7tf'shown against the curves . The squeezing parameters are respectively
s=0,1,2 and 3 ; sinx=0, tl=1, 1rJ 2 =0 . 9, and It1 2 =0 . 1 .
001
Jrc
a
Y
I
5
O
10
Figure 9 . Same as figure 8 but for measurements with the right-hand detector .
001
J
4
s
I
I/
5
Figure 10 .
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 the
angle 0 of the zero-crossing in the interference fringes (cf . figure 7), where the
squeezing is expected to be particularly effective . The curves in figure 10 show that
significant reductions in the minimum length uncertainty are indeed achieved . The
cancellation of the mode coefficients in the final term of the difference photocount
variance (93) has an important influence in limiting the damage caused by the terms
linear in f for the more heavily squeezed vacuum states .
Following the above remarks, we consider in detail only the most promising
arrangement that uses a squeezed vacuum state input in conjunction with difference
photodetection . It is assumed that
(120)
f»f',
and the photocount mean from (92) is approximately
<no(r)>
(121)
qT(IRo12-IToI2)f.
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 and
cos 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
e
e
12
e
{(ItI 4 +41r1 2 sin 0) 2 + 16q(exp ( - 2s) - 1)Irl 2 Itl 4 sin 01"2(1t14+41r sin 0)
AL,-
16ko(gTf ) 112 Ir1 2 It1 4 lsin 0 cos 01
(124)
This expression reduces to (110) in the absence of any squeezing, when s = 0, or for
low detection efficiency, q-*0 .
In the presence of squeezing, it is convenient first to consider the limit of very
large squeezing, s-> oc, and perfectly efficient detection, r~=1, when (124) reduces to
OL_
AL,
I Itl 8 16IrI 4 sin 4 01
16k o (Tf ) 112 Ir1 2 1t1 4 1 sin 0 cos
(125)
01
The length uncertainty therefore vanishes for angles 0 m that satisfy
Isin 0m1= It1 2 /21rI,
(126)
where
IRO1 2 =IT01 2= 2
and
.
<-n(T)>=0
(127)
The shift in the minimization condition from angles that satisfy (11) for zero
squeezing to angles that satisfy (126) for large squeezing agrees with the behaviour of
the numerical results illustrated in figure 10 .
For intermediate values of the squeezing parameter s and quantum efficiency 11,
the angles 0 m that minimize the length uncertainty must be found by differentiation
of (124) . This procedure produces some complicated algebra, which we do not
reproduce here . The results simplify considerably however for squeezing parameters s greater than unity and quantum efficiencies q not much smaller than unity,
Quantum theory of the Fabry-Perot interferometer
249
when the minimization angles continue to satisfy (126) to a good approximation .
Minimum length uncertainty thus continues to coincide with the zero-crossings in
the difference fringe intensity, and substitution of (126) into (124) gives
(OLD)min
(1+lexp(-2s)-t1)1J2ltl2
2ko(tiTf ) 1 / 2
for ItI 2 <<1 .
(128)
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 the
use of sufficient squeezing in the vacuum-state light in the right-hand input . With
increase in the squeezing parameter, it is of course also necessary to increase the
intensity of the coherent left-hand input in order that the inequality (120) continues
to be satisfied . The purpose of the inequality is partly to ensure that the final term in
the variance expression (93) proportional to f should be relatively small, and this
objective is greatly helped by the coincidence of minimum length uncertainty with
zero-crossings in the difference fringe intensity where (127) is satisfied .
Similar calculations can be performed for the other detection arrangements, and
the results confirm the qualitative features of the numerical calculations shown in
figures 8 and 9 . The contribution linear in f' for the photocount variance (91) in lefthand detection has a larger prefactor than for difference detection, and it is necessary
to satisfy the inequality (120) more strongly in order to make this contribution
negligible and thus take advantage of the squeezing reduction of the contribution
linear in f. When the stronger inequality is satisfied, the left-hand minimum length
uncertainty is approximately
: ItI 2 exp ( - s)/4ko(Tf )112
(ALL)min
for q =l,
(130)
similar to (129) . The corresponding result for right-hand detection is
2
(LtLe)minHtJ /2ko(Tf) 1/2
for
t1=1 .
(131)
These minimum values are not shown by the curves in figures 8 and 9, particularly
for the higher values of the squeezing parameter, because the coherent energy flow f
is not sufficiently large to justify the neglect of the variance terms linear in f' .
7.
Radiation-pressure length uncertainty
The minimum length uncertainties calculated in the previous section can be
made arbitrarily small by suitable combinations of small mirror transmission Its,
high-intensityf in 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 produce
large-intensity fluctuations in the interior of the Fabry-Perot cavity, and that the
associated radiation-pressure fluctuations on the cavity mirrors prevent arbitrary
reductions in the minimum length uncertainties .
The effects of radiation-pressure fluctuations in a Michelson interferometer have
been calculated by Edelstein et al. [4] and by Caves [5, 6], and we follow the general
approach of these authors by calculating the radiation-pressure length uncertainty as
a separate contribution, unconnected with the photocount length uncertainty of the
previous section . It has been shown in the Michelson case [7] that both contributions
can be included in a single unified calculation of the quantity that is actually
250
M. Ley and R . Loudon
observed, the photocount fluctuation at the detector . However, the main results of
the two methods of calculation are the same, and it is simpler to make a separate
calculation of the radiation-pressure length uncertainty .
With the notation for intensity components in the three regions of the z-axis
shown in figure 2, and for a narrow spread of optical excitation around the wavevector 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 mean
change in cavity length produced by these forces is
l=S2 F2 -S 1 F1 .
(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 form
1=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 uncertainties
occur for angles B m that are close to the cavity resonances, where the mode strengths
(16) and (18) inside the cavity take high values, particularly for low mirror
transmission coefficients . The internal intensities f + and f_ are correspondingly
much larger than the input intensities f and f' or the output intensities fL and f, It is
seen from (135) that the mean radiation-pressure length change is approximately
l : 2hk 0 S(f+ +f_),
(136)
and the contribution proportional to 6S is relatively insignificant . The difference
between the compliances can therefore be ignored .
The radiation-pressure length uncertainty bl is given approximately by the
fluctuation in the length change whose mean value is given by (136), and it is
necessary to specify the mode operators whose expectation values determine the
internal intensities . As in the derivations of the output energy flows (84) and (85), we
assume that the spread of wave-vectors in the optical excitation is much smaller than
the cavity resonance width h . Then analogous to (58),
f+(t) =<(Ioat(t)+J' a t(t))(Ioa(t)+Joa'(t))>
(137)
f_(t)=<(Joat(t)+Io a't(t))(Joa(t)+Ioa'(t))> .
The cavity mirrors are assumed to be identical for the remainder of the present
section, and the primed and unprimed mode coefficients in (137) then become the
same . It follows from (16) and (18) that for a high-Q cavity with highly reflecting
mirrors
11012'&Vol2 ~ltl 4 4sin e
+l
6'
(138)
similar to (97) but with lrl set equal to unity, and with 9 defined by (99) . Also
argJ o -argIo =k 0L+argr^--nxc
(139)
Quantum theory of the Fabry-Perot interferometer
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))>,
(140)
and for excitations in which the left-hand input is coherent light while the right-hand
input is squeezed vacuum-state light, use of (64), (80) and (83) gives
f+(t) , f_(t),II012(f+f') .
(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) is
now
lz4hk 0 SII0 2 (f+f') .
1
(142)
The radiation-pressure length uncertainty is obtained from the variance of the
cavity length change given by (136) with the energy flows represented by the
operator combination given in (140) . It is convenient to introduce the time duration
io of an experimental observation of the cavity length, and to average the variance
over the observation time . Thus
(6L) 2 =16h 2 koS 2 II0 I4
x
d
dt'<(at(t)+a't(t))(a(t)+a(t))(at(t')+a't(t'))(a(t')+a'(t'))>
T0 J o'0 tJ o'0
1
1
-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 is
evaluated 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 an
inequality similar to (87), the variance is
(SL) 2 =16h 2 k 2 S2 II01 4
{( exp(2s)cos 2 X+exp(-2s)sin 2 X+1)f
i
0
+['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 same
conditions 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 minimum
photocount length uncertainty, given for example by (107) or (126) . Thus for a large
squeezing parameter s, the radiation-pressure length uncertainty from (146) has the
proportionality
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) . The
radiation-pressure fluctuations accordingly limit the improvements in length
resolution 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 squaring
and adding the photocount and radiation-pressure contributions . For difference
detection with a squeezing parameter s greater than unity and I rI x 1, substitution of
the angle 0m from (126) into (138) and insertion of the result into (146) gives
(6L,) 2 ;:t 4h 2koS2 exp (2s)fl It1 4 T o .
( 149)
We take the photocount uncertainty from (129) for the limit of perfectly efficient
detection, and the combined length uncertainty is then
Iti4
C
exp (-2s) + 4h 2 koS 24exp (2s)f
4k o tf
ItI To
1/2 .
(150)
This quantity has a minimum value
(2hS) 1/2 /(TTo) 1/4
(151)
for a coherent input intensity
2s) T
fmin _ I
tI 4 ex
4 k 2o S
( a
1/2
(152)
The overall minimum length uncertainty (151) is a form of the standard quantum
limit that arises in other schemes for high-resolution length measurement . It
depends only on the compliance of the mirror supports and on the photocount
integration and total observation times . However, the input intensity (152) needed to
achieve the optimum length resolution can be greatly reduced by the use of highly
reflecting mirrors and a highly squeezed vacuum-state input . Similar conclusions
apply for left-hand detection alone, where the photocount length uncertainty is given
by (130) . For right-hand detection alone, where the photocount length uncertainty is
given by (131), the minimum length uncertainty of order (151) is achieved only in the
absence of squeezing, where the right-hand input is an ordinary vacuum state, and
the required coherent input intensity is of the order of (152) but with the squeezing
exponential removed.
8.
Conclusions
The 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 in
relative arm length . This is illustrated for example in figure 3, where, even for the
modest reflectivity of 0 . 9, the fringe intensity falls to half its maximum value for a
change in k 0 L amounting to about 3° . By contrast, the corresponding angle for a
Michelson interferometer, with the usual cosine dependence on the difference of arm
lengths, is 90 ° . It might therefore be expected that an order of magnitude
improvement in sensitivity could be achieved by the use of Fabry-Perot interferometers instead of Michelson interferometers in gravitational wave detectors .
Quantum theory of the Fabry-Perot interferometer
253
The sensitivity of the Michelson interferometer can however be improved by the
use of a multiple-reflection arrangement in which the light in each arm is reflected b
times by the end mirror before the two beams are combined to form interference
fringes . The multi-reflection Michelson interferometer still has cosine fringes but
any shift in the mirror position produces a change in optical path length that is
amplified by a factor b . An analysis of the length resolution of a multi-reflection
Michelson interferometer [6] produces a length uncertainty that has the same overall
form as the Fabry-Perot result (150), except that Jti 2 is replaced by 1/b . This
replacement is perhaps not surprising since 1/iti 2 provides a measure of the number
of reflections that take place in the Fabry-Perot cavity before the light emerges
through the mirror . It should however be stressed that the two interferometers
operate in quite different fashions ; the sharp fringes of the Fabry-Perot interferometer result from the superposition of all the light beams in the multiply
reflected set, whereas the different orders of reflected beam in the Michelson
interferometer are spatially separated in a zig-zag pattern . The Fabry-Perot
interferometer basically has a simpler structure than the multi-reflection Michelson
interferometer .
The final result (151) for the minimum length uncertainty of the Fabry-Perot
interferometer is no different from that found for other measurement schemes .
Schemes do differ however in the feasibility of achieving the minimum value in
practice . For Michelson interferometric techniques with practicable laser intensities
f, the photocount length uncertainty greatly outweighs the radiation-pressure length
uncertainty, For the Fabry-Perot interferometer, the coherent laser intensity (152)
required to achieve the optimum length resolution is greatly reduced by the Itt 4
factor, of order 10 -6 or less in practice, which is an inherent property of the
interferometer . The additional exp (- 2s) reduction factor produced by the use of a
squeezed vacuum-state input is the same for both kinds of interferometer . These
factors in principle facilitate the achievement of the minimum length uncertainty
(151), which occurs at the coherent intensity for which the photocount and radiationpressure contributions are equal .
The detailed form of the radiation-pressure contribution is affected by the
measurement strategy adopted in the search for cavity length changes, which is in
turn influenced by the spectral characteristics of the length-changing forces applied
to the cavity . The simple model used here has mirrors that respond instantaneously
to the radiation-pressure force and no explicit assumptions have been made about
the time-dependences of the length changes that must be detected . In applications to
gravitational wave detection, the resolution can be optimized by suitable restriction
of the detection bandwidth to frequencies present in the wave and the choice of
appropriate mirror damping and restoring forces . A variety of minimum length
uncertainties is found for the different spectral distributions of gravitational force
[18] . These results together with (151) are in accord with the basic quantum theory of
measurement (see [19] and the earlier references therein) . We note that in (151) and
(152), the observational time-duration To associated with the radiation-pressure
fluctuations has a minimum value equal to the photocount integration time T, but it is
of course longer than T for a series of photocount measurements .
The limits discussed above refer to a somewhat idealized system and we should
emphasize two of the assumptions on which they rely . Thus, it was assumed in § 5
that the spectral distribution of the squeezed vacuum-state light is much narrower
than the Fabry-Perot resonance width t . This essentially requires that the cavity of
254
M. Ley and R . Loudon
the degenerate-parametric-amplifier squeezed-light source should have narrower
resonances than the Fabry-Perot length-measuring cavity . The requirement is
difficult to satisfy in practice since Itl 2, and hence F'=1t12/2L, need to be as small as
possible to achieve a manageable coherent input intensity (152) . The technology of
squeezed light generation is in any case in its infancy, and sources with the required
properties are not currently available .
The final results for the limiting length resolution also rely on the assumption of
unit 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 minimum
length 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 neglected
unless i is sufficiently close to unity that
1-rl<rlexp(-2s) .
(153)
Thus any attempt to improve the interferometer sensitivity by the use of squeezed
vacuum-state light is worth while only for highly efficient photodetection . A similar
conclusion applies to the Michelson interferometer [6] .
In summary, we have evaluated the quantum theory of the Fabry-Perot
interferometer and have shown that it has properties similar in outline, but different
in detail, from those of a Michelson interferometer in terms of its potential for highresolution length measurement . The interferometric detection of gravitational
waves remains of course a formidable experimental problem .
Note added
In a recent paper, Knoll et al . [20] have considered the Fabry-Perot interferometer as an example of a spectral filter that adds unavoidable quantum noise to
its output . We briefly show how the results of the present paper conform to their
general formalism .
For a high-Q cavity close to resonance at k=k,,, where the Lorentzian
approximation (35) is applicable, (56) can be transformed to
4(f) :=
f dt'T(t"-t')a(t')+7(t),
(154)
where the transmission response function is defined by
T(t) = B(t)F'cI T Lax exp (- ick„t-cFt),
(155)
(0 fort<0
B(t)= j
l 1 fort > 0,
(156)
k kexp (-ickt) .
7(t)=(c/2tc) 1 12 JdkR~
(157)
with the step function
and
The expression (154) has the same general structure as equation (3) of [20] . With no
excitation of the right-hand input, &R (t) represents the right-hand output, with
contributions 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
Quantum theorv of the Fabry-Perot interferometer
255
out, the latter is needed to preserve boson commutation properties, and it degrades
the spectral-filtering characteristics of the interferometer that would be expected on
the basis of a purely classical theory .
Acknowledgment
M . Ley thanks the Science and Engineering Research Council and the British
Telecom Research Laboratories for financial support in the form of a CASE
studentship .
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