Ph 103c: The Physics of LIGO 18 & 20 May 1994

LECTURE 16

Squeezed Light and its Potential Use in LIGO

Lecture by H. Jeff Kimble

Assigned Reading:

TT. C. M. Caves, "Quantum mechanical noise in an interferometer," Phys. Rev. D, 23, 1693- 1708 {1981}.

UU. D. F. Walls, "Squeezed states of light," Nature, 306, 141-146 {1983}. VV. M. Xiao, L. A. Wu, and H. J. Kimble, "Precision measurement beyond the shot-noise

limit," Phys. Rev. Lett., 59, 278-281 {1987}.

Suggested Supplementary Reading:

I. H. J. Kimble, "Quantum fluctuations in quantum optics-Squeezing and related phe­nomena," in Fundamental Systems in Quantum Optics, eds. J. Dalibard, J. M. Rai­mond, and J. Zinn-Justin, {Elsevier, Amsterdam, 1992}, pp. 545-674.

m. "Squeezed States of the Electromagnetic Field," Feature Issue, J. Opt. Soc. Amer. , B4, 145(}-l741 {1987}.

n. "Squeezed Light," Special Issue, J. Modem Optics, 34, 709-1020 {1987}. o. "Quantum Noise Reduction," Special Issue, Appl. Phys. B, 55, 189ft". {1992}. p. S. Reynaud, A. Heidman, E. Giacobino, and C. Fabre, "Quantum fluctuations in

optical systems," in Progress in Optics, XXX, ed. E. Wolf {Elsevier, 1992}, pp. 1- 85.

A Few Suggested Problems:

1. Detection of Modulation in a Squeezed State. An electromagnetic field propagates through a medium whose transmission coefficient is given by t = toe-or(I), where ')'{t} = ')'0 cos{not} {Le., sinusoidally modulated absorption with amplitude ')'0 and frequency no}.

a. Assuming that ')'0 « 1 and that the input field is in a coherent state {with frequency» no}, derive an expression for the minimum detectable value of ')'0, for a fixed input energy flux (IEI12) and a fixed bandwidth B = Af {corresponding to a photo diode integration time f = 1/ B}.

b. If the input field instead is in a squeezed state, derive an expression for the minimum detectable amplitude ')'0. lllustrate in a "ball-and-stick" sketch the dependence of your answer on the orientation of the squeezing ellipse.

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2. Squeezed Vacuum in an Interferometer. In Part IV of Kimble's lecture transparen­cies, he sketches a calculation of the minimum detectable phase deviation ';0 when a coherent state is put into one port of the Mach-Zehnder interferometer shown be­low, and either the vacuum state or the squeezed vacuum state is put into the other port. His answer was ';0 = 1/v'N for the vacuum state, and ';0 = (1 + {S)I/2/v'N for the squeezed vacuum, where N is the total number of available photons, S is the squeeze factor (-1 < S :5 0), and { < 1 is the efficiency of the squeezing. Show, in a phasor diagram, the relative phase relationships for the fields that emerge from the outputs, and from your diagrams infer that to achieve the above optimal sensitivities with readout at output #1, the unperturbed position of mirror A should be adjusted so that the phase difference between the two paths along the two anna is <1>0 = 11"/2. More specifically:

a. Show the orientation of the squeezing ellipses relative to the coherent amplitudes for each of the two fields Ea , Eb that contribute to the total field EI at the output #1.

b. Show how these two fields with their fluctuations sum to give a resultant EI that (for <1>0 = 7r}2) produces noise in the photodetector below the standard shot-noise level 1/v'N and a signal proportional to the phase deviation ';0'

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c. Note that for an efficiency ~ -+ 1 and for perfect squeezing S -+ -1, the above analysis and diagrams predict that the minimum detectable phase deviation be­comes arbitrarily small , So -+ O. Show that , in fact , if the interferometer system is perfectly lossless, and So is modulated so So = ~o cos{Oot), the minimum detectable modulation amplitude ~o is actually ~o ~ 1/ N. Calculate the cor­responding length sensitivity ~z for the displacement of mirror A. Estimate t'he laser power required to achieve the sensitivity of the advanced LIGO, if this limit could be achieved.

d. In the above discussion it was tacitly assumed that the interferometer mirrors are so massive that light pressure fluctuations do not disturb them significantly. Suppose now that mirror A has a finite , small mass and is free to move in response to light pressure, and that we apply a feedback force to the back of the mirror, to counteract the time-averaged light-pressure force on its front . Show, using the phasor diagrams of parts a. and b., that when we improve our measurement of So (and hence of the mirror position z) by increasing the amount of squeezing, we increase the random light-pressure perturbations of the mirror, thereby enforcing the uncertainty principle. Relate this result to the standard quantum limit for sensing the position of the small mass, and thence to the curve labeled "Quantum Limit" in the plots of LIGO noise sources that were shown in earlier lectures. [For a quantitative analysis , in the context of a Michelson interferometer, see C. M. Caves, PhYJ . Rev. D, 23, 1693 (1981). In this problem you are supposed to be ignoring the possibility of going beyond the standard quantum limit as discussed by Jackel and Reynaud, EUTOphYJ . Lett. , 13, 301 (1990))

--

3

Lecture 16 Squeezed Light and its Potential Use in LIGO

by Jeff Kimble, 18 & 20 May 1994

Kimble lectured from the following transparencies, which Kip has annotated a bit. Kimble's lecture came in two parts, one covering the second half of the class on 18 May; the other covering the full class on 20 May.

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/ pJlYSIC;\L REVIEW 0 VOLUME 23. NUMDER 8 IS APRIL 1981

Quantum-mechanical noise in an interferometer

Carlton M. Caves W. K. K~/logg Radwtioll Laboratory. ColIJonri4 [,utitut~ ofTKltnolou. PastJd~"o. Califomia 9111J

IReceived 15 AUllust 1980)

The intencrQmders now beinll devdopcd to dctcet IIravil.liona! waves work by measurin, thc relativc positions of widely separated muses. T1Ito'O fundamental SOU~ or quantum-mechanical noise dctcrmine the sensitivity or such an intcrfcromctcr: Iii nuctuations in number or output photons Iphoton-cou.ntin, CfTOI') and Iii) nuctuations in rJidialron pressure on the masses lradiation.pressure erTorl. Because: of the low power 0( availabk continuous-..... vc tas.en. the Ktlsitivity or curuntly planned intcrferometcn will be limited by pboc:on<ountin, CTTOr. This ~pcr presents an analysis of the two types or quantum-mechanical noise, and it proposes a new technique-thc "squetted· state" lechniqu~that aUows one to decrease 1M pholon<ounlini efTor while iDcreasiD, the: radiation-pressure crror, or m versll. The kcy requirement of the squeezed.Slalc lechnique is that the state of tbe liahl cn1crinl the: inlmeromctcr's norma.lly unuwd input pen must be not the vacuum, as in a Sl.aAdard ioterfC'f"Ometer. but ralher a wsqucacd stalc"-a stale whost uncertainlks in the two quadraturc phases arc ~ Squeezed states can be lIeneraled by a variety or nonlinear optical processa. indudinll descncntc parametric ~

I. INTRODUCTION

The task of detecting gravitational radiation is driving dramatic improvements in a variety of technologies for detecting very weak forces.1

These improvements are forcing a careful exami­nation of quantum- mechanical limits on the ac­curacy with which one can monitor the state of il macroscopit body on which a weak force acts.z'

One promising technology uses an interferometer to monitor the relative positions of widely sepa­rated masses. This paper analyzes the quantum­mechanical Umits on the performance of inter­ferometen, and it introduces a new technique that might lead to improvements in their sensi­tivity.

The prototypal interferometer for gravitational­wave detection is a two-arm, multirenection ~tichelson system, powered by a laser (see Fig. 3 beloW) . The intensity in either of the interfero­mete r 's output ports provides information about the difference z, %2 - z, between the end mirrors ' positions relative to the beam spUtter, and changes in z reveal the passing of a gravitational q\'e. The first interferometer for gravitational­wave detection was built and operated at tbe Hughes Research Labor:Ltories in ~-talibu, Califor­nia, in the early 1970's (Ref. 3); this first effort Ta.S small-scale and had modest sensitivity. Now ~.eral groups around the world are developing interferometers of greatly improved sensitivity.41 A long-range goal Is to construct large-Bcale in­terferometers, with baselines 1-1 km, in order to achieve a strain sensitivity o.z / 1-10'"'21 for frequenc-ies from about 30 Hz to 10 kHz. This sensitivity goa! is based on estimates (or the strength of gravitational waves that pass the

23

Earth reasonably often.1

It has been known for some time that quantum mechanics limit! the accuracy with which an interferometer can measure z-or, indeed, the accuracy with which any position-8ell8ing device can determ~ the position of a. free mus.2• S• 1

In a measurement of duration T, the probable error in the interferometer'! determination of z can be no smaller than the Of standard quantum limit":

("_>SOL =(Vir /m~/ ' • (1.1)

where m is the maaa of each end mirror l (~zlsQL -6 x 10·'1 cm for In -lOS g, T -2 x 10~ sec). The validity of the standard quantum limit is unques­tionable, resting as it does solely on the Heisen­berg uncertainty principle applied to the quantum­mechanical evolution of a free ma.ss.

The standard quantum limit {or an interfero­mete r can also be obtained from a more detailed argument'· ··10 that balances two sources of error: 0) the error in determining z due to nuctuatlons in the number of output photons (photon-counting error) and (ti) tbe perturbation of z during a measurement produced by fluctuating radiation­pressure forces on the end mirrors (radiation­pressure error). As the input laser power P increases, the photon-counting error decreases, while the radiation-pressure error increases. Minimizing the total error with respect to P yieJds a minimum error of order the standard quantum limit and an optimum input power" 11

(1.2)

at which the minimum error can be achieved, Here w is the angular frequency of the light, and b is. the number of bounces at each end mirror.

1693 () 1981 The American Physical Society

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PHYSICAL REVIEW

LETTERS

\'01.1 'II -; 1 29 AUGUST 1983 NI ' ''flltll 'J

Contractive States and the Standard Quantum Limit for Monitoring Free-Mass Positions

Horace P. Yuen Dt!ptfrlme"t of ElecJrlcal Engineering and Computer Seinee. Norl/uDuten Unit'ersily, Evanston. Illinois 60201

(Received 21 JUDe 1983)

The ramlliar mltllmum-uncertalnty wave packets for muses are generalized In analogy with the two-photon coherent states of the nd.iatiOfl field. The free evolution of a sub­cbss of these states, the contractive states. leads to a narrowing of the position uncer­tainty in contnst with the usual minlmum-uncertaJnty wave packets. As a consequence "the st3nd:aro quantum Hmlt for monitoring the posltloruJ of a Cree mass can be breached. Further Implications on qU!lntum nondemol1t1on measurements are discussed.

PACS numbers : 03.IiS.Bt: , 04.80 ... %

There has been considerable recent interest in ascertaining and achieving the fundamental quan­tum limits on signal processing and precision measurements, in particular for applications to optical communica.tion,.sl·3 and gravitational­wave detection. -c.. A major result of this work is that one can beat the so-called standard quan­tum limit for amplitude measurements on har­moniC oscillators. However, for the gravitation­al-wave interferometer! it is usually supposed?" that the resolution is limited by the "standard quantum limit" (~L) for measuring the poSitions of a free mass ... · 5 In this paper it is shown that the laUer SQL is also not generally valid; it can be breached by a speCific quantum meaSlJrement without special preparation of the free-mass quan­tum state. Toward this end I will describe a class of generalized minimum-uncertainty wave packets for masses, to be called twisted coherent states, which are also of interest in their own r ight. The breakdown o[ the ~L [or free-mass position measurements demonstrates the [act that back ac­tions [rom a conjugate observable do not neces­sarily, at least in accordance with the principle of quantum mechanics, limit the accuracy of sub­sequent measurements on an observable.

The evolution of a free mass is given by X(t)

= X{O) + P{O) 1/ 111, so that the position fluctuation at time t is

(u'{ t)) =(U'{O» + (dP'{O» I'/ m'

+ (U{O)dP{O) +dP{O)AX{O» I/m. (l)

In the previous derivation"·' of the general SQL for mOnitoring free -mass poSitions, it is implic­itly assumed that the , = 0 state of the mass (or the state after measurement) Is such that the last term in (1) either vanishes or is positive. Under this assumption the uncertainty principle can be applied to minimize (I) at any time' with the resulting SQL

(2)

On the other hand, it is clear that (AX'{'.,» = 0 : the initial state is an eigenstate of the self-ad joint operator X {O)+P{O)t,,/m. Thus, the last term in (I) can surely be negative and the SQL is not generally valid_ However, (AX'{ t» = 0 im­plies (AP'{t» ="" so that ( P'{O)/ 2m) = (P'{I)/ 2m) =«), i.e., an infinite average energy is needed to produce such a state.3 A more realistic descrip­tion can be developed as (allows.

For an oscillator of mass III and frequency w, the twisted or two-photon coherent states {TCS)I.l

© 1983 The American Physical Sociely 719

PHYSICAL REVIEW LETTERS . . ,

I ilK!) are the elgenstates of IlO + va 1':

(I1D + va ')11110) = (jUl + 10 *)111,.,),

1111' -Ivl'= I, (3)

where a is the annihilation operator of the oscil­lator mode. Here we adopt them ID yield a class of states for a mass 1ft with position X and mo­mentum P. Define the following operator a on the Hilbert space of states for the mass:

a.X(mw!2A), /'+iP/ (2lim..,),/" [a,a'}=I, (4) I

where.., Is now an arbitrary parameter with lIDit sec·'. The Iwisled cohere;;1 slales (TeS) 111 .... ..,) of a mass are defined to be the eigenstates of l1a +va'. 1111' -I vi';' i ;''In aiIalogy with' (3) but with a given by (.). ':Tbe .tree-mass Hamlllxlnla.n can be expressed

H = P'/2m = hw!a'a -10'"" la"+tl. (5)

The wave function (xl 1110..,) , Xix) =x l .. ), can be found through Eq. (3.24) of R~. 1. Within the ebelce of a constant pb;P,se It Is given by

.'. I "- . '

I -[ m", ]'/' {mw l+iE [ (2Ii )'/' J' (2mW)'!> ' [" .( 211)'1' J} (Xl1lOw) - II I I' exp --2Ii ~I x- - a, +1 -II a. x- - a, , lr IJ.-II J!-V nlW . mw

I .

(6)

'where

('I)

The wave functions (6) constitute a generalization of the usual mlnimum-uncertalnty wave packets treated in every quantum mechanics textbook, which are given by (6) with E=O. In the context 0( OSCil­lators, "squeezing" is obtained when v. 0 inl",va), and ~ Is related to the direction ol. mlnlmum squeez. Ing. As will be seen in the following, when E> 0 the x-dependent phase in (6) leads to a na.rr<nring 0(

(l>X'( t)) from ( t.X'(0» ckIring free "olutlon, In direct contrast with the weil-known spreading 0(

(t.X'( t» for minimum-uncertainty wave packets.'· Because of this behavior, mass states (6) with E > 0 will be ca.11ed ccmtl'oclive states . \

The first two moments of (6) are .. - . , . . ~! "'I I' , ~ .,! •

(X) '(l1vaw IXI"va.., ) = (211 / mw),/'a" (p) = (2Iitnw)'i2a ..

("X') .«X - (X })2) = nt/ mw, (l>P' ) = 2 lim W!) ,

t~ll1- vl '/4, ~. I ,, +vI'/4; t1l=(1+4E')/ lG ;'1

("Xt>.P} =ifi/ 2 - FJi, (t.Pl>X) =-ifi/2- Eli, (P'/ 2m) =/r..,(a,'+~).

The average mass energy (12) is finite when "', a .. and I vi are finite. From (9)-(10) it follows that the minimum-uncertainty product (t..X'}{t.P ') =/r ' / 4 is achieved if and only if E=O.

The position flucruation for a free mass start­ing in an arbitrary TCS (6) is immediately ob­Iained from (1) and (9)-(11),

m(l>X'(t)} / 2li = tI.., - EI + ~w t'. (13) ,\, - , -

1/ 16""1 - - - -

(8)

(I)

(i9)

(11) 'Ii

(q) .",

ir.

II ~" O, (l>X'(t )} increases monotonically. In contrast to this usual situation, Eq. (13) Is plotted in Fig. 1 for contractive states at t = 0 (i.e., for E> 0). The minimum f1uctualion 1 /16"'~ can be made arbitrarily small even for fixed w by letting ~ (and thus also (I/} ) become arbitrarily large. The lime I. at this fluctuation level is I. = Ei2TJ'" so that m(l>X'(t.)}/21i/~= 1/ 8E. II (l>X'(I )} is minimized with respect to w at any givan t simi-

o L. __________ ~ __________ _1_ _____

720

o t~ z~

FIG. 1. The position noctuatiOa of a c:;od:r a:;Uve st:J.te !rom (13) ; t _ = t /2,,~ '. - ·0 -.rt.n: t -e.

Vot.UMf. ~l . NI 'Mllflt t} PHYSICAL REVIEW LETTERS 27 FnUlUAItY 19R4

Comment on "Contractive States and the Standard Quantum Limit for Monitoring free-Mass Positions"

In a recent Letter,1 Yuen has considered the so­colled twlsled or two-photon coherent states to 8hO'W' that the Cree evolution o[ certain of such st1tes (contractive states) leads to a narrowing d. the position uncertainty .. ave packets. It Is the purpose of this Comment to stress that this nuro .. lng property has nothing to do with Yuen's coheren,t states and has an almost twenty-year­old history. The general criterion for narrowing of the free-motion position uncertainty has been oi>Wned In some of the standard t."tbooks on

: ~m mechanics where the feJtlowlng expres­sloa has been derived2

:

(A.r(t» = ( "i'(O» + ({ "P(O» /11/')1' , +2tjdx[.<-(i(0»]j(x). (1)

Itt this equation, j is the standard quantum mec.h­anical probability current of the initlal wave [unc­Hon.. H we assume (without loss of generaHty) Ibllnitially <'« 0» = 0, a narrowing of ( "x'(t)) is obtained if and only if

(2)

There is, o( course, an inHnile number of states ttw. saUs(y tillS condition.. As an example, the , -l! '

Initial wave function

~(x O)=f(:r).~(- ilA,1 x") ".1.2 •• .. (3) • . -r , II 2" , .

,. ~ " 1 '" .... ' with A~ arbitrary e!ililpiex' tlbiiibe.ri and with f(:r) a real L' DOrmallzAble tlliietlon Itads to the nar­rowing eUect. The wave funcllon with" = 2 Is especially Instructh'.. For any 1(") Ibla waYe

. functton. which has a contractl.e phase similar to Yuen's wave puket. Is DOt a twisted' ,herent state but nevertheless leads to a nar"",!", eUect.

A general discussion on how to realtz..' ~:rperl­mentally the Initial ...... e function (3) and'bow to oblaln the narrowing effect (Including Ibe one giv­en by Yueo) .. as presented by Lamb In 1969.'

This research has been partially supported by the U. S. Department of Energy aDd the U. S. Of­fice of Naval Research.

K. W6dklewlcz -, '.-,

Recelved 07 November 1983 PACS numbers: 03.65.Bz, M.80.+z

(lItperm.:I.Dent address. IH. P. Yuen. Pbys. Rev. Lett. g, 719 (19! ! :-: 2see. (or example. K. Gottfried, Q_dnbl:m ··~clltJ7"cS

(BenJamln, New York. 1966), p.21, Eq. (41J ~ 'w. E. Lamb, Jr .• Phys. Today!!, No. 4, '23 (1969).

See pp. 25 and 26 (or deta.tls.

'.

© 1934 The American Phys;cal Sociely 187

VOLUME 52, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FI.BRUARY 1984

Yuen Responds: Contractive twisted coherentstates (TCS) comprise the first explicit cl,ass ofstates that was shown to lead to a narrowing of thefree-mass position fluctuation (MP(t)), as faras I know. It makes little sense to say that theyhave nothing to do with such narrowing. Nowherein my paper is it stated or implied that thesestates are the only ones leading to such narrow-ing, or that they are somehow essential for thatpurpose. In fact, when I first mentioned the pos-sibility of such narrowing I used the eigenstate ofthe self-adjoint operator X+Pt/m as an example,which is strictl. y speaking not a TCS. Among allthe possible states that exhibit such narrowing,contractive TCS form a natural generalization ofthe usual minimum-uncertainty wave packets. Inaddition, the time duration of their contractionand the associated (~'(t)) can be convenientlyparametrized. Call. ing such states "contractivestates" when other states may also contract islike calling TCS "squeezed states" when they arenot the only states that exhibit squeezing.

A main objective of my paper is to give a meas-urement scheme that can directly monitor the po-sitions of a free mass in a quantum-nondemol. i-

tional way. For this purpose, the states whichcontract are to be the ones in which a free masswould be left after a certain measurement, with-our additional intervention. For a discussion ofthis point see Caves in Ref. 8 of my paper. Forcontractive TCS such a measurement is the onedescribed by ly. va~)(p, varsy as discussed in mypaper; the possible realization of this measure-ment I merely stated without proof because ofspace l.imitation. Thus, contractive TCS turn outto be essential. in my quantum-nondemolitionalposition measurement scheme. Dr. %odkiewiczdid not give a measurement which would leavethe free mass in his more general contractivestates. On the other hand, I would be very sur-prised if TCS are essential in al.l possibl. e quan-tum-nondemolitional. position measurements.

Horace P. YuenDepartment of Electrical Engineeringand Computer ScienceNorthwestern Univer sityEvanston, Illinois 60201

Received 21 November 1983PACS numbers: 03.65.Bz, 04.80.+z

788

VOLUME 52, NUMBER 19 PHYSICAL REVIEW LETTERS 7 MAY 1984

Comment on "Contractive States and theStandard Quantum Limit for MonitoringFree-Mass Positions"

Recently, Yuen has published a very interestingpaper' in which he gives a non-QND method forbeating the standard quantum limit when measuringthe position of a free mass (QND stands for quan-tum nondemolition). The technique utilizes theso-called two-photon coherent state (TCS).

There is a difficulty with his repeated measure-ment scheme, however. The reason for this Com-ment is to call attention to this difficulty, and alsoto show that TCS can be used to make finite-energyQND-type measurements.

Consider the following recapitulation of Yuen'spaper. At t =0, an arbitrary free mass state ~tit) isprepared into a TCS, ~p, vnto) = ~n), by interactionof the system with a generalization of the two-meter detector of Arthurs and Kelly, followed bysubsequent meter reduction. This measurementcan be described in the Gordon and Louisell termi-nology (which Yuen prefers) by ~a) (ot~. It is im-

portant to note that the acutal state ~n) which ob-tains after meter reduction is only probabilisticallydetermined, depending on the overlap between ~u)and

~ Q) . Finally, one may also look upon this statepreparation as the measurement of the non-self-adjoint operator A (0), where

A (t) = A (x(t),p(t))= (p, + v) (mto/2') tt'x(t )

+i (p, —v)p(t)/(2ttmto)tt'

Thus at t = 0 the system is found in some eigenstate~a) of A (0) with eigenvalue n = n t+ in&.

As shown by Yuen, from the measured eigen-value one can read off the mass's position(x(0)) = (2t/mto)' nt and momentum (p(0))= (2tmto) ti'n2, with uncertainties (hx'(0) )= 2ig/tm to and (Ap (0) ) = 2t m top, where tc and

q are functions of p„v. He also shows that as t goesfrom 0 to a time 2t, the position uncertainty(Ax2(t ) ) first decreases, and then increases to itst = 0 value. So far, no difficulties.

However, at t = 2t Yuen calls for another mea-surement on the system, presumably of A (2t ).But we should not describe this measurement by( u) (ct ( as Yuen does, but as

~

u') (n' (, assuming in

general that o. & o, '. This assumption is correctsince one finds [A (0),A (2t )] =i (lit, +v) totW 0. One can easily show by calculating(o. ~A (2t ) ~n) and the nonzero (tr. (bA2(2t ) ~~)that the system will "jump" to a range of statescentered about n't = ut + (m co/2tt ) ' (p (0)) 2t~/m,o.2=o.2. The width of the range of states and themagnitude of the effect that it has on Yuen's propo-sal are difficult to calculate because of the non-self-adjoint nature of A (2t ). Nevertheless this"back-action" mechanism will contaminate themeasurements, and has not been included inYuen's scheme. A complete evaluation of the ex-tent of this difficulty will presumably involve alengthy calculation of the "meter-interaction-and-reduction" type pioneered by Caves.

It is interesting to note that one could achieve acontinuous QND-like measurement with the opera-tor A (0). This is because the operatorsx(0) =x(t) —p(t)t/m and p(0) =p(t) areseparately QND. Measurement of A (0) has the ad-ditional desirable property that the resulting statehas finite energy, quite properly one of Yuen'smotivations for examining TCS in his original pro-posal. However, with x(0) mixing position andmomentum operators, measurement of A (0) couldno longer be described as a position measurement,a view also taken of Yuen's scheme by Caves re-cently. 4

Robert LynchUniversity of Petroleum and MineralsDhahran, Saudi Arabia, andBlackett Laboratory'Imperial CollegeLondon SW7 2BZ, United Kingdom

Received 30 January 1984PACS numbers: 03.65.Bz, 04.80.+z

~'~Address during 1983—1984 sabbatical leave.tH. P. Yuen, Phys. Rev. Lett. 51, 719 (1983).2E. Arthurs and J. L. Kelly, Jr. , Bell Syst, Tech. J. 44,

725 (1965).3J. P. Gordon and W. H. Louisell, in Physics of Quan

tunt Electronics, edited by P. L. Kelly et ai. (McGraw-Hill,New York, 1966), pp. 833-840.

4C. M. Caves, "A Defense of the Standard QuantumLimit for Free-Mass Position" (to be published).

1984 The American Physical Society 1729

VOLUME 52, NUMBER 19 PHYSICAL REVIEW LETTERS 7 MAY 1984

Yuen Reponds: I am grateful to Dr. Lynch for pro-viding me with the opportunity to clarify certainpoints in connection with the standard quantumlimit (SQL), quantum nondemolition (QND), and

my paper. Terminology and concepts such as backaction, position measurement, QND, etc. , arefraught with ambiguity and imprecision, both in theQND literature and in my own paper. Before theyare definitively cleared up, it is important to attendto the actual content of a result rather than its ver-bal representation.

Using the notations of my paper, ' I would like tofirst describe more accurately my principal resultsas they relate to SQL and QND: (1) The previousderivation of the SQL for monitoring free-mass po-sitions is not generally valid. In particular, a specif-ic realizable measurement described by ~p, vnc0)& (pvnt0 ~

could leave the free mass in a contractivetwo-photon coherent state (TCS). The SQL canthen be broken to an arbitrary degree in a secondsufficiently accurate position measurement. Notethat the SQL would not be broken by the same~pvnt0) (avoca~ measurement. As explained later,another one with ~WM) having a smaller associatedposition fluctuation (&P~~AX2 0"~) is required, forexample, ~ p, 'v'nt0) (p, 'v'neo with

~ p,' —v'

~( ~p,—v performed at t —t (2) Th.e ~p, vut0)

x (pvnt0 measurement can be used to monitor thefree-mass positions in an arbitrary sequence of mea-surement, with a limitation ir/m on the ratio of theposition resolution to the time lapse between twomeasurements. No such limitation exists for themeasurement ~p, vnc0) (p, 'v'nc0'~, of which no reali-zation is known, however. If such measurement isindeed realizable, there can be no quantum limit ofany kind on position monitoring.

Dr. Lynch's main point appears to result from acombination of both his confusion about approxi-mate simultaneous measurements and my overlycondensed presentation as a Letter. [There are alsoa number of misprints in my paper, some of whichhave been corrected in the erratum. ' I have sincefound four more: "this work" should read "theseworks" in the second sentence of the paper; 1+ i(should read 1+i 2( in Eq. (6); f/2qr0 should read(/2qt0 in the figure caption; and ii(/mes shouldread 4i(/geo in line 12 of the second column of p.721.] The measurement described by ~ p, vnc0)x (p, v n r0 ~, the A (0) measurement in Lynch's ter-minology, is an approximate simultaneous measure-ment of position and momentum: o. being a vari-able whose real and imaginary parts provide themeasurement readings corresponding to the positionand momentum estimates. The state after a mea-

surement with reading n is just ~pv, neo); it is notprobabilistically determined. In the case of point (2)above corresponding to that discussed by Lynch,the same ~@voce) (pvnco~ measurement is made at

t =0 and t = 2t while the mass state has evolved.It is important to note that (b, X2) gives only thestate contribution to the position fluctuation in ameasurement; it is the fluctuation observed in aperfect or "exact position measurement. " In an"approximate measurement" there would be addi-tional fluctuation from the measurement itself. Inmy paper, both the state and measurement contri-butions to the position fluctuation have been in-

cluded through the resolution factor 4tg/mes in-stead of 2tg/mes Thi.s resolution value is obtainedfrom the probability

~(pvn'co~pvnto) ~; it makes no

sense to set a'=o;. [It turns out that this doublingof the position uncertainty exactly cancels out thefactor of 2 advantage of Eq. (15) compared to theSQL. This explains why a second measurementwith ~tilM) having lower (b, X2) is required forbleaching the SQL.] The momentum reading needsnever be made; it has no effect on the position fluc-tuation during the sequence of measurements.Thus, whatever "back action" there is has alreadybeen accounted for.

The ~p, vat0) (p, vnc0~ measurement without theo.2 reading is emphatically a position measurementon all grounds: physical, formal, and the purposeof such measurements. The o.

&reading indicates the

free-mass position before and after measurementswithin prescribed uncertainties. It is mathematical-ly equivalent to an exact position measurement inthe presence of meter-reading fluctuation, as far asthe measurement probability is concerned. There isno reason to call an expression a "quantum limit"if it does not cover this kind of approximate posi-tion measurements which serve the purpose ofmonitoring the mass positions. The SQL is meantto apply to all conceivable measurements.

Horace P. YuenDepartment of Electrical Engineering

and Computer ScienceNorthwestern UniversityEvanston, Illinois 60201

Received 21 February 1984PACS numbers: 03.65.Bz, 04.80.+z

t H. P. Yuen, Phys. Rev. Lett. 51, 719, 1603(E)(1983).

~R. Lynch, preceding Comment [Phys. Rev. Lett. 52,1729 (1984)]

1730

VOLUME 54 15 APRIL 1985 NUMBER 15

Repeated Contractive-State Position Measurements and the Standard Quantum Limit

Robert LynchPhysics Department, Universt ty ofPe'troleum d'c Minerals, Dhahran, Saudi Arabia

(Received 23 April 1984)

It is shown that if the "standard quantum limit" is taken in a predictive sense, then a repeatedmeasurement scheme involving contractive states, recently proposed by Yuen, does not break thislimit.

PACS numbers: 03.65.Bz

Recently Yuen' has proposed a scheme to beat the"standard quantum limit" (SQL) on free-mass posi-tion monitoring by means of contractive states. Therehave been several unpublished responses2 to Yuen'sproposal which seek to defend the SQL on generalgrounds.

A Comment3 I wrote takes a different view. A qual-itative point made was that even assuming the validityof the framework adopted by Yuen, he has not fullyconsidered the impact of measurements in his scheme.It is the purpose of this paper to flesh out the argu-ments of that Comment, and to show that if the SQLis taken in a predictive sense, Yuen's proposal fails tobeat this limit precisely because of such measurementcorrections.

Before turning to the detailed discussion it isworthwhile reviewing the reasoning which leads to theSQL. Suppose at t =0 one places a free mass approxi-mately at the origin, with the intent of measuring itssubsequent position in time (to see if a weak force isacting on it, for example. ) How often, and how close-ly, should one monitor the particle's position? If it isdecided to make a measurement every t seconds, onemust make t short enough to counter any possiblespreading of the wave function. On the other hand,each measurement to precision Ax produces a varianceof momentum Ap (by means of the uncertainty princi-ple), which then feeds back into the uncertainty of theposition, b, x(t), at the time of the next measurement.An analysis4 of this "back action" leads to the SQL,b, x(t) ~ (it t/m)'l'.

Yuen seeks to beat the SQL by means of "contrac-tive states, " and a measurement formalism based on

the work of Gordon and Louisell. 5 The contractivestates are the so-called "two-photon coherent states"(TCS), ~p, vctto). For a full discussion of these statesthe reader is referred to Yuen's original paper' andthe references therein. Here I simply recall that thisstate may be taken to represent a free particle of massm, whose expectation values of position and momen-tum are (x) = (2t/mto)'l2Re(n), (p) = (2ltmco)'l2x lm(n), with variances (b, x2) = 2t(/mes, (Ap2)=2h'm cog. Here to is an arbitrary parameter, and(=/p, —v/'/4, q =—/p, +v/'/4, subject to ip, [' —/v/'

6

As Yuen has shown, for values of the parameter(=—Im(iM, 'v) & 0, the ~p, vnco) states are contractive,that is, the initial position variance (b, x2) narrowsunder free evolution for a time t =g/27ico. Thisresult is cleverly exploited by Yuen to avoid spreadingof the wave function. The idea then is to make a sharpposition measurement at time t = t when the positionuncertainty is a minimum, while leaving the system ina ~p, vnto) state after the measurement, ready to un-dergo another contraction.

In the Gordon-Louisell terminology such a measure-ment is described by the projection operator,

~ p v aco) (p, 'v'a t0' ~.7 This notation is somewhat

abstract —in fact, it is not clear that such measure-ments are physically possible, in the sense of aHamiltonian realization, for example. Nevertheless, ifone assumes the existence such measurements, and ifone considers a system initially in the state ~P),then according to the Gordon-Louisell theory a~pvato) (p, 'v'ohio'~ measurement yields the value o. '

and the corresponding state ~pvn'to) aft, er measure-

1985 The American Physical Society 1599

VOLUME 54, NUMBER 15 PHYSICAL REVIEW LETTERS 15 ApRIL 1985

ment, with probability density 1(v'v'n'~'Iy) I2/~.Consider the following sequence of events: The sys-

tem starts in the state Ipvncu) at t=0, and evolvesfreely until time t; at that time I p vnco) (p, 'v'n~'I mea-surement is made. What are the statistics of such ameasurement? For our purposes, it is physically moretransparent to be able to label the states Ip, vncu)resulting from the measurement by their expectation

value of momentum and position, (x') and (p'),respectively, than by the values of Re(n') = (mes/2t)' (x') and Im(n') = (p')/(2hmcu)'t . In terms ofthese variables the probability density P((x'), (p');(x), (p) It) for a measurement at time t to yield astate whose position and momentum expectationvalues are (x') and (p'), respectively, given that thestate had position and momentum expectation values(x) and (p), respectively, at t = 0, is then

(3)

(6)

P((x'&, (p');(x&, (p) It) =1(p, 'v'n'~'Ig(t))I /h = I(p, 'v'n'~'lexp( —iHt/t) Ipvncu& I /h, (I)

where H =P /2m. The factor of 1/h results from the Jacobian of the transformation such thatd2n' d (x') d (p')/2V.

With this probability function it is found after some calculations that the measurement at time t produces the fol-lowing expected values and variances:

(( '))=„( ') ( ')( ') =( )+( ) / (2)

(&(x')') = „d(x') d(p') (x')'P ((x') )—'=2t('/m~'+ (2t/m) ((/~ —gt+~~t2),

((p'» = J"d (x') d (p'& (p') P = (p), (4)

(&(p')') = JI d(x') d(p') (p')'P —((p') )'=2tmo)'q'+2f mcus, (5)

((x') (p'» = ((p') (x'&) =J d(x'&«p'& &x'& (p'&P = —«+&')~+2t n t+ (p& (x&+ (p)'t/m

Here (', q', and g' are defined exactly as the corre-sponding g, q, and ( with all quantities primed.

It is worth noticing two features of Eqs. (2)—(6).The first is that since (6 (x') ) and (b, (p') ) in (3)and (5) are in general nonzero, we conclude that themeasurement "demolishes" the initial state Ip, vncu).That is, if we have an ensemble of systems allprepared in the state Ip, vnco) at t= 0, and perform themeasurement at time t, over the ensemble we will ob-tain values of (x') and (p') in a range given by (3)and (5), centered on (2) and (4).

We also note that if we choose the measurementtime to be t = t, then as is shown in Ref. 1, thesecond term on the right-hand side of (3) is aminimum. Then by appropriate choice of the valuesof (', cu' we can make the magnitude of the first termas small as we wish, leading to an overall (A(x')2)which can beat the SQL. As explained above, this is inessence the scheme espoused by Yuen.

But does this tell the whole story? One difficulty isthat this claim is being made on the basis of a singlemeasurement. 9 The effects of back action, however,are seen on the second measurement. Furthermore,(3) and (5) do reveal the presence of back action. For

example, if (' is chosen to be small in (3) in order todefine the particle's position sharply, the'n the spreadof momentum values in (5) becomes large, since (', q'satisfy an uncertainty-type relation, ('7l'= (I+4/'2)/16 ~ —,', .

The back action in this case differs from that whichobtains when one measures the particle's positiononly. Here we are making simultaneous approximateposition and momentum measurements, and eventhough the latter's value is disturbed by the measure-ment, one obtains the disturbed value as a result ofthe measurement. With the measured values of (x')and (p') in hand one is then able to predict the posi-tion (x") which will be found at the next measure-ment at t=2t by means of (2), to arbitrary sharp-ness, as discussed above.

However back action does prevent one from beingable to predict the value of (x") to better than theSQL prior to making the two measurements. To seethis consider the probability density 9' that measure-ments at times t, 2t will yield values (and the corre-sponding states) (x'), (p') and (x"), (p"), respec-tively. We have

~=P((x'). (p') (x). (p) It)P(&x"). (p");(x'). (p') It)

The expected outcome of the second position measurement is

((x")) =„d(x') d(p') d(x") d(p") (x")Q= (x) +2(p) t/m.

The second line follows easily from the first if one does the (x"), (p") integrals first, using Eqs. (2) —(6). The

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VOLUME 54, NUMBER 15 PHYSICAL REVIEW LETTERS 15 APRIL 1985

variance of the measured value is found after some calculation to be

(& (x") ) =„d(x') d(p') d(x") d(p") (x")2~ —((x")) 2= (2tt/m)I (t),

where I'(t) =I t(t)+12(t) andI

r t(t) —= 2('/co' —g't+ q'to't',

I,(t) =—2 i(/~ (—t +qt'I —gt +3qtot'

(10a)

(lob)

i.e. , 1.2 times the SQL.Besides being an interesting result in itself, this

predictive limit implies a real shortcoming of Yuen'sproposal, at least when position monitoring is used forweak-force detection. Because the position of the par-ticle jumps around in a random way at each measure-ment, one must decide at the end of each measure-ment interval whether the measured position indicatesthe presence or not of a force. Since one is unable topredict the particle's position after a number of mea-surements to better than the SQL, this then limits theability of the scheme to detect the accumulated effectof a weak force over a long time base to this limit.

In summary, then, one may say that the use of con-tractive states in repeated position measurements issubject to the SQL in a predictive sense, which makesthe technique less than straightforward compared to,say, quantum nondemolition4 techniques. The -reasonis that ~p, vapo) (p, 'v'neo'~ measurements demolish thestate. Then in the repeated measurement of the posi-tion of the particle the resulting back action causes themeasured position values to jump around in a randomway. This underlines the prime importance given toavoiding state demolition in quantum nondemolitiontheory.

The author thanks the University of Petroleum andMinerals Research Committee for their support, andProfessor T. W. B. Kibble and his group at ImperialCollege for their hospitality during his sabbatical yearat which time this work was undertaken.

If we now take the measuring time to be t, 2t, andminimize the value of (A(x") 2) with respect to theparameters (,q, co, g', q', to', subject to the constraint(7)= (1+4/ )/16 (and, as noted above, the corre-sponding primed equation), we find after some calcu-lation

(b, (x")') ~ —,'(1+&2)(t/m)(2t ) (t=t )

—1.2(lt/m) (2t ),

tH. P. Yuen, Phys. Rev. Lett. 51, 719 (1983).C. M. Caves, to be published; B. L. Schumaker, to be

published. A Comment has also appeared by K. Wodkiewicz[Phys. Rev. Lett. 52, 787 (1984)] which is not relevant tothe issues discussed here.

3R. Lynch, Phys. Rev. Lett. 52, 1729 (1984).4See, for example, C. M. Caves et al. , Rev. Mod. Phys.

52, 341 (1980), especially p. 359.sJ. P. Gordon and W. H. Louisell, in Physics of Quantum

Electronics, edited by P. L. Kelley et al. (McGraw Hill, NewYork, 1966), pp. 833—840.

6W. G. Unruh (private communication) has pointed outthat a contractive state may be viewed simply as aminimum-uncertainty wave packet propagated backwards intime. Thus, given a minimum-uncertainty wave packetP(x) at t=0, with position and momentum uncertaintiessatisfying Axb, p =f/2, Yuen's contractive state (x~pvncu, )is then exp(iHt /f)P(x). Hence the discussion in this pa-per could be carried out in the context of these more fami-liar wave functions. Alternatively the treatment could bepresented in terms of the also somewhat simpler notation of"squeezed states. " See C. M. Caves, Phys. Rev. D 23, 1963(1981), particularly pp. 1965 f. However, in the spirit ofYuen s paper, I will stick to his notation. Calculations are inthe Schrodinger picture.

7Here I only consider the repeated~ p vnco) (tL'v'ace'~ mea-

surement schemes, which are prominently featured inYuen's original paper (Ref. 1). A new point raised by Yuenin his author's reply [Phys. Rev. Lett. 52, 1730 (1984)] tomy Comment, that a single

~ p, vary) (p, 'v'neo~ measurementat t —t could break the SQL, is not particularly surprising.It has long been realized that special state preparation anddetection could rival quantum nondemolition measure-ments. See C. M. Caves, in Quantum Optics, ExperimentalGravitation and Measurement Theory, edited by P. Meystreand M. O. Scully (Plenum, New York, 1983), p. 605; W. G.Unruh, ibid. , pp. 637—645.

8The calculation may be accomplished in a straightforward(and probably inelegant) way by taking explicit wave func-tions for ~iLvnco), ~p'v'n'cu') as in, (6) of Ref. 1. The timedependence can then be included by a Green's function.See E. Merzbacher, in Quantum Mechanics, 2nd Ed. (Wiley,New York, 1970), p. 163. The details are tedious and notparticularly enlightening, so are not given here.

9In view of Eqs. (4) and (6), the ~p, vary) at t =0 cannotbe, in general, the outcome of an initial ~p, vncu) (p, 'v'o. co'~

measurement without state selection, a possibility we specifi-cally exclude —see Ref. 7.

1601

VOLUME 54 10 JUNE 1985 NUMBER 2)

Defense of the Standard Quantum Limit for Free-Mass Position

Carlton M. CavesTheoretical Astrophysics, California Institute of Technology, Pasadena, California 9II25

(Received 6 April 1984)

Measurements of the position x of a free mass I are thought to be governed by the standardquantum limit (SQL): In two successive measurements of x spaced a time r apart, the result of thesecond measurement cannot be predicted with uncertainty smaller than (tr/m)'t2. Yuen has sug-gested that there might be ways to beat the SQL. Here I give an improved formulation of the SQL,and I argue for, but do not prove, its validity.

PACS numbers: 03.65.Bz, 06.20.Dk

Conventional wisdom'2 holds that in two successive measurements of the position x of a free mass m, the resultof the second measurement cannot be predicted with uncertainty smaller than (hr/m)' 2, where r is the timebetween measurements. This limit is called the standard quantum limit (SQL) for monitoring the position of a freemass.

The standard "textbook" argument for the SQL runs as follows. Suppose that the first measurement of x att = 0 leaves the free mass with position uncertainty Ax(0). This first measurement disturbs the momentum p andleaves a momentum uncertainty hp(0) ~It/2bx(0). By the time r of the second measurement the variance of x(squared uncertainty) increases to

(Ax) (r) = (hx) (0) + [(Ap) (0)/m ]r ~ 2bx(0)bp(0)r/m ~tr/m.

The standard argument views the SQL as a straightforward consequence of the position-momentum uncertaintyprinciple Ax(0)hp(0) ~ , It. —

Yuen3 has pointed out a serious flaw in the standard argument. Between the two measurements the free massundergoes unitary evolution. In the Heisenberg picture the position operator x evolves as

x(t) = x(0) +p(0) t/m.

Thus the variance of x at time r is given not by Eq. (1), but by

(b,p) (0) 2 (x(0)p(0)+p(0)x(0)) —2(x(0)) (P(0))m m

(2)

(3)

The standard argument assumes implicitly that the last term in Eq. (3) is zero or positive. Yuen's point is thatsome measurements of x leave the free mass in a state for which this term is negative. He calls such states contrac-tive states because the variance of x decreases with time, at least for a while. As a result, the uncertainty Ax(r)can be smaller than the SQL. Yuen3 4 concludes that there are measurements of x that beat the SQL. My con-clusion is different: The flaw lies in the standard argument, not in the SQL. In this Letter I give a new, heuristicargument for the SQL, formulate an improved statement of the SQL, and analyze a measurement model that sup-ports the heuristic argument.

1985 The American Physical Society

VOLUME 54, NUMBER 23 PHYSICAL REVIEW LETTERS 10 JUNE 1985

The heuristic argument is based on including the ef-fect of the imperfect resolution o. of one's measuringapparatus. If the free mass is in a position eigenstateat the time of a measurement of x, then cr is definedto be the uncertainty in the result; thus, roughlyspeaking, the measuring apparatus can resolve posi-tions that are more than o. apart. I assume that themeasuring apparatus is coupled linearly to x, so that ingeneral the variance of a measurement of x is the sumof o.2 and the variance of x at the time of the measure-ment. Consider now two measurements of xat timest = 0 and t = r, made with identical measuring appara-tuses. In the absence of a priori knowledge, the resultof the first measurement is completely unpredictable.

Nonetheless, the first measurement does yield a valuefor x. Since this value does not tell one the positionbefore the measurement, it is hard to see what onecould mean by a "measurement of x with resolutiona" unless one means that the measurement deter-mines the position immediately after the measurementto be within roughly a distance o. of the measuredvalue. s Therefore, I assume that just after the firstmeasurement, the free mass has position uncertaintyhx(0) ~ o;this-assumption implies that an immediaterepetition of the same measurement would yield thesame result within approximately the resolution o..The variance of the second measurement (t=a) isgiven by

b, ,'=a'+ (Ax)'(r) ~ (Ax)2(0)+ (Ax)2(v) 2bx(0)bx( ) ~tr/m (4)

According to this argument, the SQL is a consequenceof the uncertainty principle

bx(0)hx(r) ~ —,' i([x(0),x(~)]) i =br/2m, (5)

provided that o ~ (bx) (0). Contractive states donot vitiate this argument; even if the free-mass stateafter the first measurement is a contractive state suchthat b, x(7 ) & (h ~/m), the SQL is valid.

Yuen uses a measurement model developed by Gor-don and Louisell, 6 which includes the measuring-apparatus resolution. How then can he contend that itis possible to violate the SQL? The answer lies in theassumption o~hx(0), -which links the uncertainty b, 2

in the second measurement to the position uncertaintyb, x(0) just after the first measurement. An easy wayto circumvent this link, pointed out by Yuen, 4 is touse measuring apparatuses which are not identical.The first measurement, performed with an apparatusof poor resolution o.t ~ Ax(0) && (tr/2m)', isdesigned to leave the free mass in a contractive statesuch that b, x(r) « (h~/2m)'l2; the second measure-ment, performed with an apparatus of good resolutiono2 « (hr/2m)' 2, has uncertainty A2= [o22+ (hx)2x (r)]' « (hr/m)', which violates the SQL. Twosuch measurements should be regarded as a singlemeasurement process, because in a sequence of mea-surements one would repeat the entire process, not theindividual measurements separately. The first rnea-surement is a preparation procedure for the second; itputs the free mass in a state that becomes a neareigenstate of position at the time of the second mea-surement. It is obvious that a measurement of x canhave arbitrarily small uncertainty if one is allowed anarbitrary prior preparation procedure. Although thepossibility of two such measurements may be impor-tant, more important is to sharpen the formulation ofthe SQL—to rule out this case to which the SQL clear-ly cannot apply.

With the preceding discussion in mind, I formulate

the SQL as follows: Let a free mass m undergo unitaryevolution during the time ~ between two measurements of'its position x, made ~ith identical measuring apparatuses;the result of the second measurement cannot be predictedwith uncertainty smaller than (h r/m)'t . For this formu-lation to be true in general, the uncertainty in thesecond measurement must be understood to be anaverage uncertainty, averaged over the possible resultsof the first measurement; the averaging procedure ismade explicit in the model considered below [see dis-cussion preceding Eq. (14)]. This improved formula-tion of the SQL is still an important restriction, be-cause it applies to a class of real experiments. 2 Inthese experiments one has available a particular tech-nique for measuring x, which is used to make a se-quence of measurements on a single free mass. Theobjective is to detect some external agent (e.g. , aforce) that disturbs x. The relevant question is howsmall a disturbance can be detected or, equivalently,how well one can predict the result of each measure-ment in the absence of the disturbance. The improvedSQL addresses precisely this question. Notice that theimproved SQL explicitly disallows any tinkering withthe free mass during the interval between measure-ments; the free mass must evolve unitarily with nostate preparation and no modification of its Hamilton-ian.

Yuen would not agree even with the improvedversion of the SQL, because he believes that thereare measurements that violate the assumptiono~b, x(0).3 4 Sp-ecifically, he suggests that there areways to measure x which have good resolutiono- « (fr/2m)' 2, but which leave the free mass in acontractive state with Ax(0) » (tr/2m)'t » o-

such that Ax(r) « (tr/2m)' . Although contrac-tive states are essential to this scheme, they are notenough to invalidate the SQL; also required are mea-surements of resolution o. that do not determine the

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VOLUME 54, NUMBER 23 PHYSICAL REVIEW LETTERS 10 JUNE 1985

position just after the measurement to be within o- ofthe measured value. Yuen states his suggestion in thenotation of the Gordon-Louisell formalism, whichcan describe formally measurements with b,x(0) )o. -

The existence of this formal description, however,does not guarantee that such measurements can berealized. Gordon and Louisell simply assume the ex-istence of certain measurements [Eqs. (22) and (23)of Ref. 6] without demonstrating that all such mea-surements can be realized. The measurements sug-gested by Yuen are among those for which no realiza-tion is known. 3 4

I turn now to a simple model of measurements of x.Within the model one can demonstrate the validity ofthe SQL, but the model by no means provides a gen-eral proof. The model is, however, sufficiently generalthat it clarifies the meaning of the SQL and indicateswhere one might seek violations. I work in theSchrodinger picture; operators are denoted by caret.

The first task is to model the initial measurement ofx; the model that I employ is like that of Arthurs andKelly. 7 The free mass is coupled to a "meter, " a one-dimensional system with coordinate Q and momentumP, which can be regarded as the first stage of a macro-scopic measuring apparatus. The coupling is turned onfrom t= —r to t =0 (~ && r), it is described by aninteraction Hamiltonian KxP (K is a coupling con-stant), and it is treated in the impulse approximation{the coupling is so strong that the free Hamiltonians ofthe free mass and the meter can be neglected). Thecoupling correlates g with x. When the interaction is

turned off at t = 0, one "reads out" a value for g,from which one infers a value for x. The readout of gcan be viewed as an ideal measurement of g made bythe subsequent stages of the measuring apparatus.

At t = —~, just before the coupling is turned on, thefree-mass wave function is P(x), and the meter isprepared in a state with wave function 4&(g). The to-tal wave function is Wp(x, g) = P(x)4( g); expectationvalues and variances with respect to Irp(x, g) are dis-tinguished by a subscript 0. For simplicity I assumethat ( g ) p

= (P) p= 0. At the end of the interaction

time (t = 0) the total wave function becomes

0'(x, g) =y(x)e(g —x) (6)

(units such that Kr =1); expectation values and vari-ances with respect to W(x, g) are distinguished by hav-ing no subscript. The expectation value of Q at t=0is (g) = (x)p. Thus the result of the first measure-ment —the inferred value of x—is the value g ob-tained in the readout of Q. The expected result is

(g) = (x) p, and the variance of the measurement isthe variance of Q at t = 0:

5', = (Ag)'= 0-'+ (dkx)'. (7)

Here o is the resolution of the meter, defined byo =—(&Q)p= fdg Q IC'(Q) I . Notice that the vari-ance (7) has the form assumed in Eq. (4)—a conse-quence of using an interaction Hamiltonian KxP that islinear in x;

The free-mass wave function $(x l g) just after thefirst measurement (t = 0) is obtained (up to normali-zation) by evaluating 'Ir(x, g) at Q = g:

q (x I g) =+(x,g)/[P(g) ]'i'= P(x)@(g—x)/[P(g) ]'i', (8)

P(g) —= J dxlW(x Q) l

= ' dxlQ(x) l l@(g—x) l . (9)

Notice that P(g) is the probability distribution to obtain the value Q as the result of the first measurement. Ex-pectation values and variances with respect to Q(x l g) are distinguished by a subscript Q.

During the time ~ until the second measurement the free mass evolves unitarily. The second measurement isdescribed and analyzed in exactly the same way as the first (assumption of identical measuring apparatuses). Theexpected result is the expectation value of x at time r, which can be written as

(x (~) ) g——J dx y'(x l g) [x+ (h r/im ) (0/Bx) ]y (x l g), (10)

x (7 ) —= x+ pr/m.

(13)

The result g of the first measurement is known, and the meter wave function C (Q) is under one's control, butthe free-mass wave function P(x) before the first measurement is presumably not known. Nonetheless, I assumeknowledge of Q(x) so that (x(~)) y can be calculated exactly. Then the unpredictability of the second measure-ment is characterized by its variance

b, 220= o-2+ [Ax(~) ]g~, (12)

[5x (~ ) ]g' =„I dx y'(xl g ) [x + (h 7./im ) (rl/8x) —(x (r ) ) g ]'y (x l Q ) .

A simple case, corresponding to the argument leading to Eq. (4), occurs when one has almost no a priori know-ledge about x before the first measurement —i.e. , when Q(x) = lQ(x) le'~t ~ is such that iQ(x) l varies slowly onthe scale set by o.. Then one finds that P(xlg) =4(g —x)e'~t"), which implies (x) &= g and (b, x)0 o-2;——

VOLUME 54, NUMBER 23 PHYSICAL REVIEW LETTERS 10 JUNE 1985

the variance (12) of the second measurement becomes

S,'0= (Sx)~+ [ax(r)]~ ~ ~([xx(r)])~~=h. r/m.

An arbitrary P(x) requires more care. It is possible to find Q(x) and 4(g) such that b, 20 (tr/m for somevalues of Q. Since one cannot control the outcome of the first measurement, a reasonable way to characterize theunpredictability of the second measurement is to average A2 0 over all values of Q, weighting each value by its pro-bability P(Q):

b2= J dg P(Q)hz y=cr + ([x(r) —(x(r)) 0] ), (14)

([x(r) —(x(r)) 0] ) = J dx dg + (x,g) [x+ (fr/im) (r)/r)x) —(x(r)) 0] %"(x,g). (i5)

dg P(Q) (bx)g~= ((x —(x) &)2) =rr2 —((Q —(x) &) ) ~ a (i6)

((x —(x) g)') =„dx dg(x —(x) g)'i+(x, g) i'.

Equation (16) is the analog of the assumption o- ~ b, x(0) in the heuristic argument leading to Eq. (4). By notingthat

(17)

The notation (x(r) ) 0 emphasizes that (x(r) ) 0 is a function of the operator Q. The average variance of x justafter the first measurement satisfies

((x) -) = J dgP(Q)(x) g= (x) ((x(r)) -) = J dg P(Q) (x(r)) g= (x(7)),one can write the inequality

522~ [b, (x —(x) 0) ]'+ [b, (x(r) —(x(r)) 0) ]'—I&[ —

& )~, ( ) —& ( )&~]&l = I([, ( )l&l=h / . (i8)

Thus the average variance obeys the SQL.Both the heuristic argument and the model just con-

sidered require an essential assumption —that themeasuring apparatus is coupled linearly to x. Withinthe context of a linear coupling, the model is quitegeneral, since it allows the meter to be prepared in anystate. Linear coupling does apply to the specific caseYuen describes in Refs. 3 and 4, which involvesGaussian free-mass contractive states that he calls"twisted coherent states"; any nonlinear coupling to xwould destroy the Gaussian character of these states.To seek violations of the SQL, one should considernonlinear couplings —e.g. , Kf(x)P. A word of cau-tion: The interaction Kf(x)P describes directly mea-surements of the quantity y = f(x); interpreting andanalyzing such measurements as measurements of x isdifficult.

This work was supported in part by a Precision Mea-surement Grant from the National Bureau of Stand-ards (NB83-NADA-4038) and by the National ScienceFoundation (AST-82-14126) .

~V. B. Braginsky and Yu. I. Vorontsov, Ups. Fiz. Nauk114, 41 (1974) [Sov. Phys. Usp. 17, 644 (1975)].

2C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sand-berg, and M. Zimmermann, Rev. Mod. Phys. 52, 341(1980).

3H. P. Yuen, Phys. Rev. Lett. 51, 719 (1983).4H. P. Yuen, Phys. Rev. Lett. 52, 1730 (1984).5For an excellent discussion of this point, see E. Schro-

dinger, Naturwissenschaften 23, 807, 823, 844 (1935) [En-glish translation by J. D. Trimmer, Proc. Am. Philos. Soc.124, 323 (1980)].

6J. P. Gordon and W. H. Louisell, in Physics of QuantumElectronics, edited by P. L. Kelley, B. Lax, and P. E. Tan-nenwald (McGraw-Hill, New York, 1966), p. 833.

7E. Arthurs and J. L. Kelly, Jr. , Bell. Syst. Tech. J. 44, 725(1965).

sWhen both ~Q(x) ~2 and ~C&(Q) ~2 are Gaussian functions,Eq. (8) shows that (bx)g~= [(Ax)0 2+ a 2] ~~ o2', thus,in this case, the SQL follows directly from the argumentleading to Eq. (4), with no need for any averaging.

2468

PHYSICAL REVIEW

LETTERS

VOLUME 60 I FEBRUARY 1988 NUMBER 5

Measurement Breaking the Studard Quota. Limit for Free-Mass Position

Masaaao Ozawa Ihpanm<1fl of MaliumaIia. CoIlq. of c..-.. Edw_ H"f/OYQ U • ....uy. Ha_a 46~. iapall

~ 2 Joiy 19C7)

M e.'It.pIicit iAteractioa-.llamiltoaiaa reallzatioe of .a measart1De1lt of the frce·mass pJlSition with the ror.vwing prope:rties is ~ (1) ne probability distn"butioa 01 tbe readouts is exactly the same as the free-MaSS posit .. distribttt-iotl ,;.st herOIc the measuremenL (2) The measuremeat leaves tbe free mass i. a coMractiYe stale just after the measurement. It is shown that this measurement breaks the standard quut.m limit (or the free·m.ass position in the sense sharpened by the recent coatrovcny.

PACS numbers: OJ.6S.Bz., 04 .80.+ z

For monitoring the position of a free mass such as tbe gravitational·wavc interferometer, I it is usually SUP'"

posed2.) that the predictability of the results is limited by the so-ealled standard quantum limit (SQLl. In the re­cent controversy, 4-1 started with Vuen's proposal" of a measurement whicb beats the SQL, the meaning of the SQL has been much clarified and yet no onc bas given a general proof nor a counterexample for the SQL. Re· eently, Nit succeeded in constructing a repeated­measurement scheme to monitor the free-mass position to an arbitrary accuracy. However, it is open whether th is scheme beats tbe SQL in the sense sharpened by the recent OOIItrooersy. In particular, the following problem remains oper. Can we realize a high-precision measure­ment which leaves the free mass in a contractive state?

In the present paper, I shall give a model nf measure­ment of a free-mass position which breaks the SQL in its most serious formulation. An explicit form of the system-meter interaction Hamiltonian will be given and it will tk shown that if the meter is prepared in an ap­propriate contractive state· then the measurement leaves the free mass in a contract",e state and the uncertainty of the prediction for the Igett identical m...urement de­creases in a given duratiaR to a desir'ed extent. Thus Yuen's original proposal' is fdy reafized. This result will open a new way to ;m arl7Ptrarify accurate 000-

quantum-noudemolition 1IJO'II'it:-orift.g for graYitational wave detection .od other rehted Bekls S1lCh as optical

communications. The precise formulation of tbe SQL is given by Caves '

as follows: Let a free mass m undergo unitary evolution during the time T between two measurements of its posi­tion x, made with identical measuring apparatus; the re­sult of the second measurement cannot be predicted with uncertainty smaller than (hT/m) l/2 in average O\'er all the first readout values. Caves' showed that the SQL holds for a specific model of a position measureD.1 ;,\, due to von Neumann 10 and be also gave the following ;:.:uris­tic argument for the validity of the SQL His point is the notion of the imperfect resolution (7 of Doe' S measur­ing apparatus, His argument runs as follows: The first assumption is tbat the yariancc of the mea51lf'C"RJent of x is the sum of (72 and the variance of x at the time of the measurement; this is the case when the measuring ap­paratus is coupled linearly to x. Tir. stcrmd assumption is that just after the first measurement, the free mass has position uncertainty Ax(O):s <1. Under these conditions. he derived the SQL from the uncertainty relation Ax(O)Ax ( r) 2:: fir/ 2m.

However. his definition of the resolution of a measure­ment is ambiguous. In fact, he used three diITerent definitions in his paper: (I) tho uncertainty in the resu lt, (2) tbe position uncertainty after the measurem.,t, and (3) the uncertainty of the meter before the fro' ,",ure­menl These three notions are essentially diffen lit, al­though they are the same for von Neumann's model. I

© 1988 The American Physical Society 385

1.. •• •

,..-' . , .l r,'

VOLUME 67, NUMBER IS PIIYSICAL REVIEW LETTERS .." .", Q6-0BER 1991 .,. . " ; ., '

Does a Congnation Law Limit Position Measuremeats?

Masanao Ozawa III L)'nran LAboralm')' 01 Physic's. IInfT'tJrd U";'''rTJiry. Cambridgt. MtUSDc#lus,IIJ 021 J8

(Received 5 March 1990)

The demon.!Olratiom: of Wi,ncr and othen;. thai obscrvables which do not commute with additive con­served quanlilia: cannot be maJured precisely •• re reexamined. A proposed new formuillion of the cl3im is shown to be Yalid for observable. .. with a continuous spectrum whenever the conserwd qaantities :are hounded. flOwnef. a couftlermodel ~ con.o;tructcd. and it soZgests that the position an be measu'red as rrecisely as the momut.m even though the measuring interaction conserres the total tiDeir momen­tum.

PACS numbcn: O).65,8z "" ,,:

the instrument and the obsel"'f'Cr measures In observ1\b!c B of the instrum.nt to get lite outcome of this measu_.­menl. In the H.isenberg picture, 'We can writ. Ai:.) -Ael. B(0)-18B. A!t)-U'U8))U, Ind 8(r) -U'{I8B)U. By saying that Ihis tne8sure_ is _ ~XQcl measurement of the observable A it is me::.t: III:at tbe measur.ment satisfies (;) the stalisticol r ...... for tbe probability distribution of the outaJll1e 0( .1 ;::; ment (R.f. III I, pp. 200 and 20)) and (ii) the " i ? Ii ity hypothesis: If an ob .. ",able I.t _rrd /wi« ill :sw:­ctSsion In ,h~ SQm~ individual ""tM. tltm *t ftf Me same ralue each time (Ref. II II. p. 335).

When does the measuring interaction U lift aD euet measurem.nt of A? A simple but seneral conditiooo sufficient for it is as follows: Tllut Is 'Sol11t st/f-adjoin, optrator N. caUtd th~ noist 0/Nrator. in 1illor which U. A. and B satisfy the relations

U'{I8BlU-A81 + 18N,

U.'(A8IlU-A81. . . ~

rp , .-i

Recently, there has been considerable interest in the analysis of fundamental quantum limits on measurements of unquanlized quantities. such as positions of masses and amplitudes or harmonic oscillators. [or applications. in particular, to optical communications [11 and gravitation­al wave detection (21. A major achievement in this area is that we have breached the two types or quantum limits posed previously, the so-called standard quantum limit [or amplitude measurements on harmonic oscillators (3] and the so-called standard qwantum limit ror monitoring or rree-mass positions .4.SJ. However. it has long been claimed by several authors 16-101 that obsm-abl .. which do not commut~ witll additi« cons~rrvd quantitj~s can­not b~ m~asur~d pr~cistly. This limit. which will be called the coru~rrQ';on-law-inductd quantum limit (CQL) for measurements. implies that the conservation law of the linear momentum limits the accuracy of posi­lion measurements. Allhough implications of the CQL in measurements of the spin components have been exam­ined in detail (91. those in position measurements have not been discussed seriously. An obvious difficulty for Lei .. be the initial stat. of the object" .nd ~ th. inb,1 discussions about the limits on position measurements lies state or the instrument. Note Hhlt ".e can assume in the fact that the position observable has a continuous without any loss or generality that 0' is in the spectrum or spectrum and that any observable wilh a continuous spec- the noise operator N; otherwise, repI.~ 8 in Eq. (I) by trum cannot be measured with absolute precision. wheth· 8-l..1 ror any l.. in the spectrum or N: ,In our rormula· er it commutes with the additive con~erved quantiti~ or tion. a B measurement at time t gives the outcome or an not. Thus. if one would claim the CQL for position mea- A measurement at time 0: i.e. t a 8fr) measurement gives surements in a physically meaningful way. the claim the outcome or an A(O) measurement in the initial would imply thai the .ccur.cy of position m ... urements Heisenberg stal ... 8~. Th.n. if we prepare the inslru-has an apparent limitation compared with th •• ccuracy of menl in Ihe .ig.nstat. of N for tlte .ig.nvalu. 0, i. •.• mom.nlum measur.m.nts. In this L.ller Ihe v.lidity of N~ -0, it follows .asily from Eq. (J) that tlte outcome of th. CQL is examined from this point of view and it is the B(r) m.asurem.nt has the same probabilily distri"--.hown Ihal th. CQL is not g.n.rally valid for position tion as the A (0) measurem.nl. A. to lite repelt.",",,, measurem.nts. hypothesis, th. first measurem.nl is tlte A{O) ....-re-

We ~hall first give a rigorous statement of the CQL. ment and the second measurement is an A rneasert.ie it Seppose that an observable (<<If-adjoint operator) A of a ot the tim. just aft.r the objccl ·.ystent is Iqlerated fnoo" qoranlum sy<tem. c.lled .n "bjut, represent.d by a Hil- the first measuring instrum.nt, so thll the IIUn is;" 'J.

bert spac. U ,. is aclu.lly m ... urable hy a measuring in- A(r) measur.m.nl. On th. aliter hind, Eq. U) _t~ strwm.nl. Then w. can d=ribe th. interaction betw •• n that th. A(O) m.asur.m.nt and lbe A(r) " t the object and the instrument by quantum mechanics in give the same value. Thus a rncanrin, iDtaa:tiua IIJIII.&. principl • . Let 'If, be tlte Hilbert spac. of the instrum.nt fying Eqs. (I) and (2) gives.n euct in II I ..... system. The interaction is supposed to be turned on dur- obsernble A. Indeed. the conditi .... :li(ot......, ill !he in~ a finite time ;"ternl rrom time t -0 to t - T and re~ conventional approach to mea .. ~u.:~;'.diIcrete otJ-r=nted b" ___ .,. .operator U on 7( , 87f,. Just .fter sembles by a suilabl. r.llbelin, ott~ ... wMre I"" inler:oction is t1mled ofT the object is separ.led from the wnita.,. U is giv.n by Ih. retillo.. (}(" ... {.) -"m

.,'" 1956 © 199 1 The American Physal Society ·' :'; .(¥~;i> .

. \ .< , w,;, I , ~.~#Wn4 ~:. .

EIJROPHYSICS LETTERS 15 Oetober Ill!?':'

. Europhy •. ·Letl., 13 (4), pp. 301-306 (1990)

. , . £ -

r' . "':Qiiantum Limits in Interferometric Measurements. <~;~i: .:~:., . ':' ~.;' -M. -T. 'JAEKEL(*) and S. REYNAUD(**)

:' ' (0)'. Laborntoi"" de Phy3ique Tlr.eoritpU de l'Ecol. Ncrrm.ak Supmeur.(') . ~:~ 7: .!4 "'" Llwmond, F·75f31 Pam Cedez 05 ;:.,.-.~ (*o,- 'Lalxnutoi"" de Speclroscopie Hertzienne(U), UniveniU Pinn el Mam Curie

~ place JuosWu, F-75f5f Pam Cedez 05

(reeeived 25 June 1990; accepted in final form 7 August 1990)

PACS. 42.50 - Quantum optio, PACS. 06.30 - Measun!ment of basic nriabIes. PACS. 03.65 - Quantum theory; __ ics

Allotrad. - Quant_ noioe Iitnita Iht IIOI>Oitmty ol iDterfelometrie ~nto. It is genenlly _ted that it JeacIs to an ultimate oenoitivity, lht .standard quantam limit>. Using a aemi' . • ) analysis of quantum noiae, we ahow that a judicious WJe of aqueezed states alIo ... one in p:-inciple to push the ""naitivity beyond this limit. This general method eouJd be apptied to large-tlC3le interferometers ror gravitational wave detection.

Quantum noise ultimately limits the sensitivity in interferometric detection of gravi­tational waves [1-3). A gravitational wave is detected as a phase difference between the optical lengths of the two arms. It seerns accepted that there exists a .standard quantum limit. (SQL), equivalent to an ultimate detectable length variation:

(6z)SQL = y7i;iM', (1,

where M is the mass of the mirror!' ond ~ the measurement time [4). The SQL can be derived by considering that the positior< .,.) and %(1. + ~), which are noncommuting observable., are measured [5). This interpretatiuu of SQL has given rise to a long controversy [6).

Alternatively, the SQL can be lInrl.'""tood by considering the quantum noise as a sum of two contributions. Photon counting noise corresponds to fluctuations of {he number of photons detected in the two output ports, while radiation pressure noise sterns from the random motion of the mirrors which ; ... nsitive to the Ouctuations of the numbers of photons in each arm. The sum of these two contributions leads to an optimal sensitivity given by expression (I). This limit is reached for very large laser power which is not presently achievable.

(I) Unit~ propre du Centre National de Ia Recherche Scientifique associ~ l I'Ecole Normale Supt!rieure et ~ I'Universit~ de Paris-Sud.

(n ) UnM de I'Ecoie Normale Supmeure et de I'Universi~ Pierre et Marie Curie, 1!SOCi~ au Centre National de Ia Recherche Scientifique.

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