P H Y S I C 11, R E Y I E W D V O L U M E 1 8 , N U M B E R 6 1 5 S E P T E M B E R 1 9 7 8

Analysis of quantum-nondemolition measurement

W. 6. Unruh Department of Physics, University of British Columbia, Vancouver, B.C., V6T I W5

(Received 28 February 1977)

The quantum-nondemolition measurement process suggested by Braginsky and co-workers is analyzed and shown not to work because of the fundamentally linear coupling between the oscillator and the measuring apparatus. A second-order coupling would work but the effect is probably too small for experimental realization.

Recently Braginsky el u L . ' - ~ ( re fe r red to here - a f te r a s B) , in examining the ultimate l i n i t s to the sensitivity of gravitational radiation detection, proposed a scheme by which it would be possible to detect the energy of the electromagnetic modes in a resonant cavity without disturbing the energy of the system. The method was claimed to be applicable even to the ground s t a t e of the cavity.

Although the claim that it is in p r i n ~ i p l e possible to measure the s ta te of the resonator without d i s - turbing i ts energy significantly will be shown to he t rue , the technique suggested in B i s in princi- ple unable to do so. The plan of this paper will be to f i r s t descr ibe and then analyze the B proposal, and then to descr ibe a technique in which it is possible to achieve the above aim.

1. ANALYSIS OF FIRST-ORDER DETECTION

which electrons a r e detected in the f o r m e r minima will be proportional to the squared intensity of the field (Fig. 2).

The s u c c e s s of this design i s based on two sepa- r a t e claims4:

(a) The sys tem can be a r ranged s o that the elec- t rons which a r e not detected will have a negligible effect on the s ta te of the system.

(b) Those electrons which a r e detected have a s m a l l probability of having disturbed the detector. The arguments presented in B to support the above c la ims a r e in genera l non-quantum-mech- anical in nature (except f o r the occasional u s e of d i sc re teness of energy levels, etc.). Although a quantum investigation supports asse r t ion (a ) , it does not support (b). The arguments fo r the pos- sibility of realizing par t (a) given in B a r e e s - sentially c lass ica l and will be reproduced here.

In going through the capaci tor , the part ic le feels

Since I will be re fe r r ing to s o m e detai ls of the a fo rce in the y direction due to the field, designa-

B scheme, I will f i r s t descr ibe the essent ial fea- ted with the abs t rac t symbol q, such that the in-

t u r e s of that scheme. teraction i s of the f o r m

An electron~agnet ic resonant cavity, idealized as a n L-C circui t , is postulated in which a n e lec t ron~agne t ic field exis ts in s o m e quantum state. The resonator IS assumed to have only one mode, of frequency u, and the possible s ta tes therefore differ only in the number of quanta in this mode. The capaci tor is assumed to be a spl i t capacitor. An electron beam i s f i red between the plates of one of the capaci tors and is then focused by arl e lectron lens sys tem, where the de tec tors are located. The beam which has not been stopped by the detectors i s then focused through the second capaci tor (Fig. 1).

The idea behind the scheme is that in passing through the f i r s t capacitor the electrons a r e given s o m e ver t i ca l momentum by the field within the capacitor. The detectors a r e placed a t the minima of the diffraction pat tern produced by the finite width of the capacitor ( o r of a s l i t within the capacitor) when the field within the capacitor i s zero. The presence of a field within the capacitor will shift the diffraction pat tern, and the ra te a t

where LY represen ts a coupling constant (dependent on the charge of the part ic le and on the relation between q and the electr ic field). h(x) here gives the x dependence through the capacitor of the field. Equation (1.1) is assumed to be derivable f r o m a Lagrangian of the f o r m

DETECTOR CAP. LENS PLANE LENS CAP

I

SAMPLE

PATH

FIG. 1. The proposed Braginskii detection scheme.

A N A L Y S I S O F Q U A N T U M - N O N D E M O L I T I O N M E A S U R E M E N T

no subtleties there which would invalidate the argument.

Consider the Lagrangian

FIG. 2. Interference pattern in the detector plane. where @(x) i s the unlt s tep function which i s 0 fo r x <O. This sys tem can be quantized and leads t o a

where the relation between q and E is such that the Schrgdinger equation describing i t s quantum be- Lagrangian f o r q takes this form. havior. If we expand the wave function in t e r m s

The equation for q i s then of the s t a t e s $ j ( q , t ) of the f r e e harmonic osci l la- to r , we obtain a wave function of the f o r m

f r o m which we obtain a n equation for the r a t e of where the part ic le functions q j obey the equation change of energy in the cavity,

34 1 a2&, i _ _ F = - - - at C aqj, (t ) @ ( x ) ~ ( L - x ) $ ~ ,

dq 2 a x 2 - dE . d [($) + y2qi] = OL ;ir ,qXj dt 2 dl (1.4)

(1.9)

o r where we define

Lf h(x) i s effectively a 6 function a t x = 0 and x = L (of unit a rea) , we obtain

where t = 0 , T a r e the t imes a t which the electron ( 0 , j # l i l .

p a s s e s through the two capaci tors , and v, i s the velocity in the x direction. Expand the in a perturbat ion s e r i e s in a, such

If T is chosen to be a multiple of the period of that $ j =C ,$:'&here $'" i s of o rder c u r . Assume the resona tor , and the lens sys tem is designed s o that the only nonzero zeroth-order t e r m i s fo r j that v ( T ) = -y(O), then the above expression for = n , with A E i s z e r o ( a s the resonator i s in f r e e oscillation between T = 0 and t = T , the "velocity" dq/dt will be periodic).

The above i s a c lass ica l argument. Will the conclusion be valid if the whole sys tem i s t reated quantum mechanically ? A s the above sys tem with i ts l enses , etc., and i t s two-dimensional nature i s difficult to analyze, a much s impler one- dimensional sys tem can be studied, which will display the essent ial features. A two-dimensional sys tem will then be studied to show that there a r e

The initial s ta te therefore has the osci l la tor in the nth s ta te , and the part ic le has velocity k toward the osci l la tor and originates f r o m x = -m.

The only nonzero f i r s t -o rder t e r m s a r e $:::

a s the interaction through q connects only these s t a t e s to $,"' to f i r s t o rder in a. The Green ' s function solutions f o r these a r e given by

1766 W . G . U N R U H 18 -

where

F o r x> L, this becomes

while f o r x < 0

The f o r m e r represen ts the wave exiting f rom the f a r s ide of the interaction region while the l a t t e r represen ts reflection by the region. If the velocity /z i s chosen s o that ( k - K).L= Lw k = wT i s a n in- tegral multiple of 2 n , the numerator in Ec;. (1.i4) will then be zero.

F o r the reflected wave (1.15) no such s imple con- dition applies, a s the condition that ( k + K ) L be a multiple of 2n is a lmost impossible to real ize in pract ice Tor l a rge k. However, this reflected t e r m is very smal l when compared with the t rans - mitted wave anyway [i.e., i s down by a factor of o r d e r W ( K ' 2) which is assumed to be smal l ] .

Classical ly , the osci l la tor obeys the equation

and

Therefore ,

When the part ic le i s within the osci l la tor (i.e., 0 < x . : L ) the equilibriumposition of the osci l la tor i s shifted to the point q= 0, but during th i s t ime q i s s t i l l a periodic function with period 2a/w. Therefore , if T i s chosen to be a multiple of this per iod, the change in the osci l la tor energy caused by the part ic le 's t ravers ing the osci l la tor is zero. The c lass ica l condition for the osci l la tor t o r e - main in i ts s t a t e a f te r passage of the par t i c le i s the s a m e a s the quantum condition-namely, that wT be a n integral multiple of 2n. There i s no c lass ica l analog to the reflected wave.

This resul t gives us confidence that a quantum analysis of the setup in B would duplicate the c lass - ical insofaras satisfying the requirement (a ) to reasonable precision. (Even classical ly perfect compensation i s not possible as the length of the part ic le 's path and the velocity of the part ic le depend in dctai l on the unknown value of thc f i ~ l c l a s the pa i t i c lc p a s s e s t l ~ e f i r s t capacitor,)

This leaves par t (b) to worry about, and the above example demonstrates the problem. To f i r s t o r d e r in a , one has ei ther a wave which has not been aiiected by the osci l la tor a t a l l (i!,"') o r one has a wave ($2; ) which has been affected, but which has in the p rocess changed the s ta te of the osci l ia tor with certainty [i.e., the re a r e no f i r s t -o rder t e r m s of the f o r m :,"'(x,!)]. Any experimental techniclue rllust be designed s o a s to re jec t those part ic les which would have been there even if the interaction had never taken place, and these a r e exactly the part ic les represented the t e r m s $:'. If this rejection i s perfect (e.g.. by looking in the minima of the diffraction pattern of the unaffected part ic les) , the only part ic les one can detect a r e those which have with certainty affected the s ta te of the oscillator. Therefore, although it i s t rue that a par t ic le which passes only through the left capacitor has a low prob- ability of exciting the capaci tor , those which a r e deterleri have a probability of nearly unity of affecting the s ta te of the oscillator. As the f i r s t - o rder t e r m s will in general dominate those part i - c les which a r e detected, when compared with higher-order t e r m s , the probability of a l ter ing the s ta te by the measurement p rocess i s very high.

The reason f o r this behavior l i es in the property of the harmonic osci l la tor that the expectatton value of the coordinate y in any energy eigenstate i s zero. Another way of looking a t the problem i s to wr i te y in t e r m s of annihilation and creat ion opera tors fo r quanta of the osci l la tor

Any interaction that proceeds v ia q must do s o by annihilating o r creat ing a quantum in the oscilla- tor. Since the two processes will be about equal u priori , the expectation value for the energy may change litt le, but the fluctuations in the energy will be increased. This accountsfor the classical argu- ments which would imply very litt le change in the average energy.

The essent ial resu l t obtained f rom the analysis s o f a r i s that fo r the one-dimensional scheme proposi.d, t h ~ nondemolition detection will not work because the interaction i s f i r s t or_der in the coordinate q of the oscillator. T h e B scheme

18 - A N A L Y S I S O F Q U A B T C T M - N O N D F , \ I O L I T I O N M E A S U R E M E N T 176i

proposed, however, i s a two-dimensional scheme. In one dimension the s t a t e of the detecting part ic le can change only when accompanied by a change in its energy. In two dimensions the direct ion of the beam can change without necessar i ly being a c - companied by a change in energy. One might expect that the electron could bounce off of the electr ic field without changing the energy of the osci l la tor , in analogy with recoi l less scat ter ing in solid s ta te physics.

Fur thermore , the above analysis has examined only the f i r s t -o rder t e r m s in the perturbation se r ies . In the two-dimensional c a s e , it may be that the higher-order t e r m s a r e responsible fo r the "recoi l less scattering" and their effect will dominate the f i r s t -o rder t e rms .

11. ANALYSIS OF TWO-DIMENSION.4L SCHEME

To show that this does not happen. I will now analyze a two-dimensional model of the experi- ment proposed by Braginsky.

The model i s defined by the Lagrangian

where ltjx) will be taken to be given by

To connect with the actual sys tem proposed by Brag insky,

where C is the capacitance (0.3 pF) , w i s the ang- u la r frequency of the osci l la tor (2 X 101n/sec), 2d i s the separat ion of the plates m) , V is the voltage a c r o s s the capaci tor , and H T is the m a s s of the electron. In the analytic expressions, nz and i7 will be taken to be unity and will bereinser ted into expressions when necessary for numerical evalua- tion.

The above model s e e m s to me to be a reasonable model for the actual physical setup. There a r e a number of t e r m s neglected here. The electr ic field inside the capacitor is assumed to be "rigid", i.e., the shape of the modes between the plates of the capacitor is assumed to be independent of the position of the electron going through the cap- aci tor . The force on the electron due to i ts own image charges i s neglected. (These would be independent of the s ta te of the field in the cavity anyway, and would thus not affect the nleasur- ability of that field.) The mode s t ruc ture within the cavity i s assumed to have the f o r m lz(x)v, which

negiects the effects of fringing f ie lds a t the edges of the capacitor. Using a more con~pl ica ted ex- p ress ion f o r the shape of the mode would not a l t e r the conclusions reached here , but would com- plicate the mathematics. All re lat ivis t ic effects a r e neglected, including the effects of any mag- netic fields within the capacitor.

A quantity which opera tes a s a n effective couplirig constant is

The second-order t e r m s will be found to be of o r d e r f f 2 l e s s than the f i r s t -o rder t e rms . This quantity must be kept l e s s than unity fo r a per - turbation analysis to apply a t all. F o r l a r g e r values, a l l o r d e r s in a perturbation expansion would become equally Likely, a p ~ o v i , and would almost certainly not resul t in a "nondemolition" measurement .

I will a s s u m e that the incident xnomentum of the electron is p with a spread 6p (such that 6p/p

and such that p2 >> 2w (i .e. , the energy of the electron i s much g r e a t e r than the energy of a quantum of the oscillator).

The wave function f o r the electron osci l la tor sys tem is written a s

where O j i s the j -quanta eigenstate of the f r e e oscillator. The electron wave functions obey

The solutions fo r \kj can be expanded in a perturba- tion s e r i e s in E ,

I will be interested in the Four ie r t rans form of \k j in the region x > L. The Four ie r t rans form of 9ir' in this region will be denoted by a?), where

[ t h e integral i s defined f o r x < L by taking the analytic extension of 4'' (.u, v ) fo r va!ues of r > L t o values of x < L].

The zeroth-order solution for the electron

W . G . U N R U H 18 -

wave function is taken to be

In o rder to mimic the effect of the finite width of the capaci tor , +"' is taken to be given by

where S(kl) i s some unit norm function strongly peaked a t kl"p and of width 6 p (e.g., a normalized Gaussian).

The detection scheme envisaged is to measure the number of e lectrons which a r e sca t te red into the f i r s t minimum of the diffraction pattern. (The lens s e r v e s to focus electrons with the s a m e wave number in the y direct ion onto a s ingle spot on the detector plane.) I will in the following there - fo re calculate the probability of an electron sca t - ter ing with y -wave-number n / d (i .e. , into the f i r s t minimum of the unperturbed diffraction pattern).

The t e r m s which will interest m e a r e the lowest- o rder t e r m s which correspond to scat ter ing into

The only t e r m which contributes to the scat ter ing into X = n/d i s the t e r m proportional to a&'O' /a~. One finally obtains

The argument of the s in can be rewri t ten a s n T / r where T is the t rans i t t ime through the capacitor and T i s the period. If T is chosen s o a s to min- imize P t l ' ( i . e . , T = T), the spread 6 p in p a l so becomes important and one obtains

In the scheme envisaged by Braginsky, however,

that direct ion by having al tered the s ta te of the osci l la tor and the lowest-order t e r m s which have not a l t e red the s ta te of the oscillator. Theprob- ability densi t ies for these two processes a r e , respectively,

where * stands for n * l . h he lower-order t e r m 2 ~ e ( $ F ' l j r ' ~ ' *) , fo r nondisturbative scat ter ing i s z e r o a t the y -wave number of interest s ince +"' i s z e r o there.]

The f i r s t -o rder amplitudes $2' a r e given by the equation

After some manipulation this becomes

T - 4 7 , and one obtains

One mus t now calculate the second-order t e r m $:' in o rder to est imate the probability P"' of an electron scat ter ing into y-wave number 71 'd without al ter ing the s ta te of the oscillator. Only if P"' >> P"' can one be said to have performed a quantum nondemolition measurement . If P"' >> P"' , any electron detected will have almost certainly al tered the s ta te of the oscillator. (In calculating $2' only those t e r m s which depend on n a r e of impor- tance a s the other t e r m s cannot give any informa- tion a s to the s ta te of the oscillator.) The full expression for the second-order t e r m i s

18 - A N A L Y S I S O F Q U A N T U M - N O N D E M O L I T I O N M E A S U R E M E N T

Because I)'O)(v,r/d) = 0, only those terms in the above expression which depend on derivatives of I)('' with respect to its second argument will con- tribute to pC2'. The terms proportional to 81)'01/

8X in general have an amplitude which is smaller than the amplitude of $2' by a factor of the order of

These cannot significantly contribute to P"'. (Except for one term, they all have a similar dependence on L a s do I),'". Therefore chosing L so a s to minimize P(,, will also minimize these terms.)

The only significant term is therefore the term

Using the assumed form of I)"' this term gives a contribution to pC2' of

(note that choosing L s o a s to minimize P"' would also minimize F '~ ') . The ratio of P'2) to P (') is given by

2 2 n + l N--

CtZw a2

-- 1 x 10'3(2n + 1) << 1 for low n . (2.21)

This implies that only of the order of 0.1% of the electrons detected a t the diffraction minima will be electrons which have not altered the state of the oscillator.

The quantum-mechanical analysis presented above is confusing when one attempts to think classically. After all no energy is needed to alter the direction of the electron. A rather more qualitative argument can show how this effect ar ises quantum mechanically.

Let us write the beam after it passes through the first capacitor {with zero field in the capacitor) in the form

where

- X 2

and

(i.e., the above represents the a wave of energy k2/2 diffracted through a slit of width 2ci in the y direction). The effect of an elect1 ic field in the slit is to lowest order to change value of the X at which the maximum occurs. With a time varying field, the resultant wave becomes something like

e-i(rf2't j H ( h - a C O S W ~ ) ~ ' ~ ' eihYdX ,

which represents an oscillating diffraction pat- tern. However, expanding H(X - a coswt ) in powers of f f , the above becomes

The first term is the diffraction pattern of the un- disturbed wave, while the second represents to the lowest order the oscillating pattern. Note that it is made of particles not with energy k2/2 but of (k2/2 i 0). The particle must have absorbed or given up an energy of w to produce an oscillating diffraction pattern. [ ~ o t e that a t x = 0, the y dependence of the first-order term is

which is just the type of term expected for the form of first-order perturbation at the capacitor with an interaction of the form ffb(x)yq.]

The problem ar ises because the interaction is of first order in q. If these first-order effects could be arranged to all cancel out there could be a hope for perturbing the particle without also perturbing the energy of the oscillator. Alternatively, it would be much simpler to design the experiment so that all first-order effects a r e automatically zero-namely, by making the interaction second- order in q. An interaction term of the form q2 can be written in terms of annihilation and creation operators a s

The first two terms represent an interaction which can measurably alter the state of the particle while leaving the energy of the oscillator un- changed. If the experiment can be designed so that the time-dependent terms have very little effect, one will succeed in measuring the state of the oscillator without altering its energy.

W . G . U N R U H

111. SECOND-ORDER DETECTION The equation for 4, i s

The construction of a n example of such a sys tem i s straightforward. Consider the Lagrangian

L = + ( X I 2 + a q 2 f ( x ) 4- i(q2 - wZq2) , ( 3 . 1 )

where q represen ts the abs t rac t coordinate of the osci l la tor , a is a coupling constant, and f ( x ) gives the spat ial dependance of the interaction of the part ic le with the oscillator. A physical r e a - lization of the above sys tem would be obtained by letting the osci l la tor be an electromagnetic mode in a cavity, where q will be related to the max- imum electr ic field in the cavity, and q 2 f ( x ) is the squared e lec t r ic field along the path of the particle. The coupling constant a will be related to the polarizability of the part ic le in question. Such a specific physical realization is not necessary for the mat te r of principle being discussed, however.

F o r simplicity 1 will a ssume in any specific ca l - culations that f ( ~ ) is of the fo rm

f ( x ) = (g xB(x)B(L/2 - x )

i.e., f ( x ) has a triangular shape. (Classically this gives a force on the part ic le which i s constant in the regions O<x<L '2 and L / 2 < x < L.)

The quantization of this sys tem i s s t raightfor- ward. If we expand the wave function for the com- bined part ic le-osci l la tor sys tem in t e r m s of the wave functions fo r the f r e e osci l la tor with f r e - quency w we obtain a wave function of the f o r m

where $ j ( q , t ) is the normalized wave function for the jth s ta te of the f r e e harmonic oscillator. The wave equation f o r the sys tem can now be written a s

where 1 define

Let us define the initial conditions s o that the osci l la tor is in the pure s t a t e j = n , and the par t i - c le is coming into the interaction region with ve l - ocity k f r o m x = -to. The boundary conditions a r e therefore that near x = -to, only +, ( x , t ) has a n ingoing par t which I will a s s u m e is of the fo rm

To lowest o r d e r in a we can neglect the second s e t of t e r m s on the right-hand side. As q2,, i s independent of t , the problem reduces in this o r d e r to the one-dimensional scat ter ing problem for X , = e ikZtI2 +,. One obtains

If the initial velocity i s l a r g e enough, this can be solved by a WKB approximation to give (for x > L )

where A is approximately one, and

Here T i s the t ime for the part ic le to t r a v e r s e the distance L in which the interaction occurs. Such a phase shift i s measurable by interferometry techniques. Subtracting f r o m $, the fo rm it would have had if the interaction had been z e r o , namely, +',0)= e m i P t / 'e ikX IG, we obtain

in the interference region. The flux of incident par t ic les represented by $,

i s k / 2 n while the flux of part ic les in the inter- ference region i s

1 e" - 1 1 'k I2n . (3 .12)

The rat io of par t ic les detected in the interference region to the number of par t i c les sen t through the interaction region i s the rat io of these fluxes o r

where the value of q2,, , i .e. , (n /w) , has been used.

In o rder to determine the s t a t e (i.e., n ) a c - curately, the e r r o r in n introduced by the d i sc re te

18 - A N A L Y S I S O F Q U A N T U M - N O N D E M O L I T I O N M E A S U R E M E N T

Poisson nature of the detection process should be l e s s than 4:

The total number of par t ic les which must be sen t through the osci l la tor should therefore obey

(note that N i s not dependent on which s t a t e the osci l la tor i s in). T h i s experiment differs f r o m that of Braginsky et nl. in that the detection of one of the part ic les N, does not imply that the oscilla- t o r has changed state.

The above analysis has been concerned only with those part ic les fo r which the s tate of the osci l la tor remains unchanged in the nth s tate . One must now show that the total probability that the s ta te i s changed by sending through N part ic les is very smal l , and that the flux of par t ic les which have changed the s ta te of the osci l la tor and which en te r the interference region i s small . (If mos t of the flux in the interference region were due to part ic les which had changed the s ta te of the de- tector , then the conditional probability that the detector had changed s ta te given that a par t ic le had been detected in the interference region would be v e r y large.)

T o demonstrate that such a situation i s possible, I will calculate the f o r m of the functions with j # n to f i r s t o rder in a. T o lowest o rder in a , the equation f o r $, i s

i k x - aq2,(t) f(x)e - ( k Y / 2 ) t - .

4 - 5 '

qZjn i s nonzero only for j = n * 2, and f o r these t e r m s it takes the f o r m

F o r o r d e r of magnitude es t imates the square roots will be replaced by their approximate value n.

These equations fo r $,,, may be solved by Green ' s function techniques. If we define fo r

We can wr i te the solutions fo r e i ther of /I,,, a s

The total flux in each of these modes leaving the osci l la tor divided by the incident ilw 1s t hus

where I have used K = k arid have assumed that over the spread 6k of n wave packet the exi;o:ien- t ia l minus one in the second t e r m averages out to of o rder unity. The second rerrn represen ts part ic les refiecied f rom the oscillator. region and one thus need not w c r r y about detecting the111 in the interference region, but only w e d consider whether the total probability of transition to a n n * 2 quantum s ta te be s m a l l for A' part ic les passing through the oscillator. As Nz 2w2 ' a Z T 2 , the probability of t ransi t ion to the TZ i 2 s ta te due to the second t e r m in Eq. (2.20) is of o rder n 2 ( k ~ i - ' . F o r la rge enough k o r L , this can he made a s s m a l l a s desired. Note, however, that fo r given k , the probability of excitation increases with increasing n. The f i r s t t e r m i s more interesting.

We can rewri te

The probability that the osci l la tor will be excited by s o m e one of those of N part ic le which goes through the osci l la tor is then given by

(This a l s o is the probability that one of t h e N, par t ic les detected i s a par t ic le which has disturl;ed the oscillator. The two needed conditions a r e

U N R U H - 18

therefore equivalent.) By making d T very l a r g e (i .e. , by making L very large) this t e r m can be made a s smal l a s deslred. I-iowever, the re i s a m o r e efficient procedure. If we let ( k - K ) L ' 2 = wT be a n integer multiple of 2i7, then the t e r m in the numerator goes to zero. This can strictly be t rue only f o r one value of k since T depends inversely on k. However, ~f k differs f rom the required value k, , by 613, then

and the probability of exciting the osci l la tor be- comes

(Because iz - K differs fo r the t ransi t ions to n + 2 and n - 2, one can never choose L and k s o a s to satisfy the required relation :or both transitions. One can take this into account by saying that 6k"k i s a t least 2w:fi2. One can therefore make the probability of transition extremely small.)

Fur thermore , the exact expression for the t r a n s - ition probability depends onthe* form assumed for f (x). If we define a function f , (x ) such that

the power 4 which occurs in the various transition probabilities is replaced by 2p. The phase shift 6 i s , however, left unchanged. By choosing p sufficiently large, t!ie probabilities can be fur ther suppressed.

This behavior can be clarified by examining the classical equation for g

The part ic le therefore acts on the osci l la tor by changing the frequency of the osci l la tor , Since f ( x ) = J ( t 'u). by designing f s o that the effective osci l la tor frequency (w' - af ) ' I2 changes slowly with t ime, the s t a t e of the quantum osci l la tor will adiabatically t rack the frequency shift r a ther than jl;;:~p~ng to another stare. By ei ther increasing L for 3 fixed k , o r by increasing ii. fo r a fixed k and L , one brings the s y s t e m n e a r e r this adiabatic limit.

This suggests other possible techniques fo r mea-

suring the s ta te of a n electromagnetic mode in a cavity. F o r example, a smal l needle conductor can be introduced into the cavity on a torsion f iber such that the natural frequency i? of this needle i s much l e s s than the frequency of the cavity. The dipole moment induced in this needle will couple the needle to the square of the elec- t r i c field. The interaction will of the f o r m a02q2 (for s m a l l angles @ f r o m t h e direction of the E field). This represen ts a frequency -f requency coupling in that the presence of energy in ei ther sys tem a l t e r s the frequency of the other. Since the needle's period is much longer than the field's, the field will respond quasiadiabatically to the motion of the needle, while the needle will r e - spond to the average energy in the cavity. If the frequency of the needle can be monitored, it will give a n indication of the energy in the electro- magnetic field.

A useful physical realization of the above schemes i s extremely difficult, especially be- cause of the second-order nature of the interactions necessary. Also, if effective interference ex- periments w e r e to be c a r r i e d out. one would like relatively long coherence lengths f o r the part ic les being used. This suggests the use of photons (which a r e a l so useful because of the i r z e r o m a s s and of the constancy of their velocity). Unfortuna- tely, the d i rec t photon-photon coupling in vacuum is exkvenzely weak. However, if a mate r ia l could be placed within the cavity whose optical propert ies depended on the square of the applied field, it could be used to provide a coupling between the optical photons used a s detectors and the modes in the cavity. (Alternatively by using photons direct ly on a t ransparent b a r in such a way a s to induce a second-order coupling between the photons and the cavity mode of interest one could dispense with paramet r ic amplif iers , etc.)

ACKNOWLEDGMENTS

I would like to thank R. Drever fo r a stimulating colloquium a t the University of BritishColumbia which a roused my lnterest in this topic. I would a l s o like to express my thanks to G. Opat and to the la te R. Burgess fo r many c r i t i ca l and s t imula- ting discussions and to various other members of the Department of Physics a t U.B.C. who listened to and helped alleviate my confusion while I worked on this problem.

'\ . U. Braginsky aiid Yu. I. Vorontsov, Usp Flz. Nauk. 111, 41 (1974) [Sov. Phys.-Usp. 17, 644 (1975)l.

1- V R. Braginsky, Yu. I . Vorontsov, and V. D. Kriv-

tienkov, Zh . Ekep. Teor. Fiz. - 68, 55 (1975) [Sov.

Phys.-JETP 41, 28 (1975)l. 3 ~ . 5. E5raglns6, in Top~rs In Thporet~cal and Erpcrt-

rnf>?~taZ Graz ztatton Phvstcs, edited by V. De Sabbata and J. Ureber (Plenum, New York, 1977).

DETECTOR

SAMPLE ELECTRON PATH

FIG. 1. he proposed Braginskii detection scheme.

FIG. 2. Interference pattern in the detector plane.