Quantum Zeno Effect Induced by Quantum-Nondemolition Measurement of Photon Number

PHYSICAL REVIEW A VOLUME 45, NUMBER 7 1 APRIL 1992

Quantum Zeno effect induced by quantum-nondemolition measurement of photon number

M.J.Gagen and G.J.Milburn Department of Physics, Uniuersity of Queensland, St Lu.cia $072, Australia (Received 4 November 1991)

Measurements performed on a system will alter the dynamics of that system, and in the strong- measurement limit, can theoretically freeze the free evolution completely. This is the quantum Zeno effect. We utilize quantum-nondemolition photon-counting techniques to realize the Zeno effect on the evolution of either a two-level haynes-Cummings atom interacting with a resonant cavity mode, or on two electromagnetic modes configured as a multilevel parametric frequency converter. These systems interact with another electromagnetic cavity mode via a quadratic coupling system based on four-wave mixing and constructed so as to be a nondemolition measurement of the photon number. This mode is then coupled to the environment through the cavity mirrors. This measurement is shown to be a measurement of system populations, which generates the desired Zeno effect and in the strong-measurement limit will freeze the free dynamics of the system. PACS number(s): 42.50.Md, 03.65.Bz

I. INTRODUCTION possible to describe a more general class of measurements which are not arbitrarily accurate. In Ref. [3] one of us Unlike classical mechanics, measurements are given a showed that the effect of a sequence of inaccurate, instan- special status in quantum mechanics; it cannot be as- taneous measurements on the two-level system could be sumed that the effect of measurement on the system can described by a Markov master equation for the system be made arbitrarily small. This is a direct result of the state. In an appropriate limit, defining an efficient mea- noncommutativity of the operators that represent the surement, the initial-state occupation probability decays physical variables in quantum mechanics. From the very exponentially linearly in time with a very slow rate. This earliest days of quantum mechanics it was necessary to is a manifestation of the quantum Zeno effect in a more supplement quantum theory with additional postulates, realistic way. generally referred to as projection postulates, which de- There has only been one attempt to verify the quan- scribe the effect of measurement. The essential feature tum Zeno effect [7]. The main difficulty to be overcome in of these postulates is their nonunitary character. The searching for an experimental realization of the quantum projection postulates describe a very general, highly ide- Zeno effect is arranging for the measurement time scales alized class of measurements, corresponding to instan- to be much shorter than the time scales of any other taneous, arbitrarily accurate readouts. The probability nonunitary effect such as dissipation. Recently we pro- distributions for such measurements are given directly by posed a scheme based on Rydberg atoms in a microwave projecting the quantum state of t, he system onto the ap- cavity which may provide just this situation [8]. In this propriate eigenstates of t, he operator corresponding to the paper we discuss two other quantum-optical models, a measured quantity, without any reference to the details parametric frequency conversion model and a Jaynes- of the measurement scheme. This degree of generality Cummings model, which offer some hope of experimental suggests that such measurements should be seen more as realization. In addition the parametric frequency conver- part of the interpretation of quantum theory than as a sion model requires more than two levels to describe the realistic description of the measurement process. free dynamics. We thus are provided with the opportu- It has been known for some time that such idealized nity of studying how the quantum Zeno effect is manifest measurements can dramatically alter the free dynamics in multilevel systems. Aspects of the Zeno effect in mul- of systems with a discrete spectrum [I, 2]. In particular tilevel systems were recently discussed by Peres [9]. a two-level system undergoing coherent oscillations be- In the haynes-Cummings model we assume that the tween the two st;ates, subjected to a sequence of projec- atom is initially prepared in the excited state and the tive measurements, can be frozen in the initial state [3], field in the vacuum state. The coupling between the the so-called quantum Zeno effect. This extreme result atom and the field is such that the atom periodically is a direct consequence of t, he extreme idealization such emits one photon into and absorbs one photon from the projective measurements embody; real measurements are cavity mode. The objective is to monitor the initial state neither instantaneous nor arbitrarily accurate. Recently of the atom by watching for this photon to appear in the the theory of measurement has been supplemented by cavity mode. If the Zeno effect occurs the probability mathematical results which enable these restrictive as- of finding the photon in the cavity mode should become sumptions to be discarded[4 —6]. In particular it is now a very slowly increasing function of time. In the para-

45 5228 1992 The American Physical Society 45 QUANTUM ZENO EFFECT INDUCED BY QUANTUM-. . . 5229 metric conversion model we one mode frequency prepare - = —,'(II)(21+ 12)(ll), in a vacuum state and the other mode in an Nz photon number eigenstate. As the evolution proceeds these Nz (II) &2[ —l2) (I I) (2 2) photons are exchanged between each of the two interact- = — ing modes. We monitor the initial state by measuring th.. o. —,'(12) (21 II) (Il) photon number in the initially unexcited cavity mode. For the second rotation system we consider the cou- In the effect measurement free dynam- studying of on pling of two electromagnetic field modes in a nonlinear ics it is useful reduce the effect measurement to of to crystal. To obtain this interaction we might configure the least possible disturbance me- permitted by quantum a second-order susceptibility as a parametric frequency chanics. In this respect quantum nondemolition measure- converter [11,12]. We have ments seem particularly interesting. It is the purpose of an ideal quantum nondemolition measurement to yield a h]c Hbg = (atb+ ab'). (2 3) determinate sequence of measured results [10]. For this 2 to be possible it is essential that an initial accurate mea- Here b is the same cavity mode given above, but the surement of some physical quantity does not feed noise two-level atom is replaced by a further cavity mode with into the system in such a way that it is coupled back into annihilation operator a. The coupling constant, K is pro- the measured variable. Such a situation would produce portional to the second-order susceptibility. a subsequent measurement result which could not be re- A simple way to describe the dynamics of the two- lated in a deterministic way to the initial result. Vari- mode parametric frequency converter is to introduce the ables which avoid this noise feedback are referred to as Hermitian operators [14, 15] quantum-nondemolition (QND) variables. Constants of motion are of course QND variables, provided such vari- J = -'(atb+ abt), ables remain constants of the motion in the presence of the interaction with the measuring device. This latter cri- — J5' = (atb aha) (2.4) terion is referred to as back-action evasion. In this paper we will consider QND measurements of photon number J = -'(ata —btb). of an intracavity mode. Such measurements do not lead to any systematic damping in the system but do drive These operators satisfy the usual su(2) Lie algebra of diffusion processes in dynamical variables which do not [J~, Js] = i J„together with the cyclic permutations of commute with the photon number, such as phase. We z, y, and z. The Casimir invariant for this group is will use a QND measurement of the photon number in an optical cavity to monitor the evolution of a system Np (Np (2.5) away from the initial state. '-2 2"

where N& is the total number of photons in the a and b II. FREE DYNAMICS modes, —ata+ btb. We consider two quantum-optical models, one based on Np (2.6) the nondegenerate parametric frequency convertor [11, It is also of interest to note that the photon number 12], the other on the Jaynes-Cummings model [13].The operator for the b mode can be written in terms of these first of these models describes the interaction of two in- angular momentum operators. We have tracavity field modes via a second-order nonlinear suscep- tibilty. The second describes the interaction between a nb = 2(Np —2J,). (2.7) single cavity mode and a two-level atom, in the rotating- Using the above notation we can write the Hamiltonian wave and dipole approximation. As we shall show it is of the rotational systems as possible to describe the free dynamics of both these systems in terms of the precession of an abstract angular momentum vector, and we thus refer to both systems (2.8) collectively as the rotation system. We consider first a Jaynes-Curnrnings two-level atom interacting with a cavity of the radiation field described where we write JC to refer to the Jaynes-Cummings two- by [13] level atom model, and FC to refer to the frequency converter model. We can h]c readily show for the frequency converter model Hgg = (ho++ bto (2.1) that the operator N& (the total photon number occu- 2 ), pancy of both modes) is a constant of motion as it com- mutes with the above Hamiltonian. For the Jaynes- where b (bt) are the annihilation (creation) operators of Cummings model the equivalent conserved quantity is the b-mode cavity field, and ~ is the dipole coupling given in terms of the operator 0', + ny. strength of the atom-field interaction. The cr's are the For the Jaynes-Cummings two-level atom with one usual operators of the su(2) Lie algebra defined by photon (Nz —1), the basis states are given by the usual 5230 M. J N ANDD G- 3. MII.BURN

1 Sp» eigennstates of th e o, op he precession of the a g omentum ababout the J axis. above Haamiltonian h. = xthe n therefore th —-mode syste momentu asis vectors, is in the va = 1-, ).~. 1o)b (2 9) nitial st t —40 —2)a~am I1}b ec recurr e initial The numbe f e angular um vector es one corn le u ion. No e initial st t

. A important p io n eatures w' e t earance of t e ect in such systems. T ny — er system foor e & land%, = monitor th i ia state on t, h e b mode. In Fig. 1wwe plot thheh probabili e system t' hhe initial sta s ". e scaled ime for tw III. MEAS UREMENT MODEL tion b the e s iscusse above it itor th

p t er inside t y co ert number me as' he eigenst represent, th ment „/2 and m = 0, 1,2, . . . , 2j we c t, }li we cou pel the rot ation 'c number sta tes of mo des a and I5 this is s st ird fiel q coupling sc f. — = —m I3 . .. — lm)b. (2.10) Ij, j m) l2jj m), plays the t stage of the me "d We prepare the fre converter s i the 6 th ' m od i th e vacuum e a mode'e in a numbber state m er in the other p p otons: I@IN = }o,b—lj j) ~) IO)b. (2.11) However thisis is sstill not a m e l the dete t -mode e ree dynamics of tth'is system is described b a pe y Th is cou l

course we do phng to th m y th output rn e o 0.8

0.6 e ector mode. H ynamics of the sys em alon 0,4 step in the calcul- ' ain an effec q r e reduc ge that wee ssee O.R e appearanc oft th 0.0 0 10 15 asured t a amiltonia n in the in ter actionn picture is FIG. S. Thheh free--evolution l tot —IIIloto IIQND + +dRMp 2 , and t e case N, =40 +

requency can- where H,,ot describes the fr describe th evasion to uency converter . H esetz = l. bes 'h o P g e etector ), =( II, ). o the cavity QUANTUM XENO EFFECT INDUCED BY QUANTUM-. . . 5231

We introduce a QND-type coupling between the b where ln„(t)), denote coherent states of the c mode with mode and a third cavity mode, with annihilation operator c. The coupling is via a third-order susceptibility. In ~ (t) = (I —c "") (3 7) the interaction picture we have hg Here we see that the c mode has been driven into a HctND = b b(c+ c (3.2) superposition of coherent states. In turn, the damping of 2 ), the c mode has introduced the exponential factor depen- where we choose g to be real. This could be realized dent on (n —m)~ which will damp the coherences of the as a four-wave-mixing interaction with one of the modes b mode density matrix in the number basis. This result — treated classically [16 18], in which case g is proportional is typical of measurement schemes which destroy coher- to the third-order nonlinear susceptibility. ences and leave the system in a mixture of states in the The above Hamiltonians when combined give a com- measurement pointer basis, in this case the b mode num- plicated but still straightforward unitary evolution, and ber states. In the limit yt )& 1 the coherence decay is thus do not describe a measurement. We now need to exponential with the decay rate given by g (n —m)2/2p. couple the detector mode c to the many-mode field ex- This would seem to indicate that a good measurement ternal to the cavity. Ordinary photon-counting measure- corresponds to having y~t/y very large where t is the ments are made directly on the field which leaves the measurement time. Indeed in the Appendix we show that cavity. The detector mode and external field comprise in this limit the signal-to-noise ratio for the measurement the total measuring device. We shall return to this point in Sec. IV. The interaction between the detector field and the external field is described by the interaction Hamil- tonian (a) I2) IE f Hdamp = cI' + c I' (3.3) where I' are bath operators describing the many-mode external field. In Fig. 2 we give a diagrammatic repre- QND sentation of the system-detector-environment complex. We are now in a position to write down the evolution equation for the state of the system and detector. To do this we use standard techniques [19] to eliminate the dynamics of the external field, yielding a master equation

g A A p = [H«q, p] —ty[hyX„p] + —(2cpc —n, p —ph, ),

(3 4) where n, is the number operator for the c mode and X, = &(c+ ct) is the quadrature phase operator for the (b) c mode. Here we have treated the dissipation resulting FC from the coupling of the c mode to the environment under the Markov approximation. To describe the dominant; eH'ect of the coupling to QND the measuring device we first consider the case in which oA' ~ = 0, i.e., we turn the free dynamics. The solution to the master equation in this case was obtained by Walls, Collet, and Milburn [16]. When e = 0 then ng is a constant of the motion as there is no mechanism to shift photons from the a to the 6 mode. They considered an initial state with the b mode prepared in an arbitrary FIG. 2. The schematic experimental arrangement is state and the c mode prepared in a vacuum, shown for the Jaynes-Cummings two-level atom system (a), and for the two-mode frequency converter system (b). In (a) p~. (o) = ln) ~(~l Io).(oI (3.5) ).p, the two-level atom is tuned to the cavity mode b In (b). m n, the cavity supports two modes u and b interacting through a and obtained for the density matrix at time t, second-order susceptibility so as to be configured as a parametric frequency converter (shown as FC). In both systems then, mode b is coupled to another electromagnetic cavity = —m— 1 ———e pq, (n I (t) ) p„exp 72 ) mo de, c via a quadratic coupling system based on four-wave mixing and constructed so as to be a nondemolition measurement of the quadrature phase (shown as QND). Photon num- l~ (t)).(~ (t)l mode @ ' (3.6) ber counting is then performed on this third c (shown (~-(t)l~-(t)). as PD). 5232 M. J. GAGEN AND G. J. MILBURN 45 is very good. For yt )& 1 with large damping rate a good To obtain these equations we discarded all terms involv- measurement means that g2t )& 2p. As well, for large y ing the matrix elements l0), {'2l and l'2), (ol. Here we see we are motivated to consider the possibility of adiabat- the essence of the adiabatic approximation in that for ically eliminating the detector mode from the dynamics large y, p~ and p& are heavily damped. and obtaining an evolution equation for the system alone. Using Eq. (4.2) we have for the reduced density oper- When It E 0 the above analysis suggests the mecha- ator nism by which the Xeno effect is manifest. The damping —2X- of the c mode rapidly eliminates the off-diagonal coher- proc = [H-t P.ot] [&—b P+l (4.4) ences in the photon number distribution and it is pre- ~ 2 cisely these coherences that need to "build up" in order — where p+ —pq + p& . The adiabatic approximation for the unitary rotation of the free evolution part of the is made —— — This system to occur. by setting p& p& 0. approximation amounts to a slaving of the fast variables for large damping (pt )& 1) to the slow ones. If we substitute these IV. ADIABATIC ELIMINATION results into Eqs. (4.3) we obtain OF THE DETECTOR MODE ~ = F(i+) = [H-~, (4 5) In this section we treat the large-damping limit (yt » (i++'r i+I), In this limit we make an ansatz for the state of the 1). where c mode, and by adiabatically eliminating this mode obtain a reduced master equation for the evolution of the 'x- A= [nb —p—b] (4.6) remaining system —the two-level atom and b mode in the 7 Jaynes-Cummings case, or the a and b modes in the frequency converter case. and p = 2(hy) ~. We invert this equation to obtain We are interested in the large-damping regime on the c mode where we are rapidly and destructively counting p+=F '(A) all the photons that are generated in the c mode. We —A ~ tp[Hrot i A] p [Hrot ~ [Hrot ~ A]] (4 7) expect, based on this, that the c mode is never able to evolve very far away from the vacuum. This provides the We expand p+ to lowest order in y/y and substitute these basis of the adiabatic method we are going to consider. results into the reduced master equation (4.4) to obtain There are various schemes to do this and we choose to follow the method used by Mortimer and Risken [20]. -2- proc = [Hrotq prot] [rtbr [rtb& prot)j ~ (4 8) For large p the c mode is heavily damped and remains 227 close to the vacuum state lo), (ol. A photon number ex- In this equation we see an additional g term arising. As pansion of Eq. (3.6) indicates that this is readily satisfied we have p && g, yt && 1 and y2t && y this allows the good in the limit In what follows we allow for small y && y. measurement limit. For the frequency converter we can deviations from the vacuum in setting write ny in terms of J, and obtain pt.t = pop lo), (ol+ p, g II),(ol+ g lo), — — pt (ll+ P2eli). {Il, PFc '"[J* PFc] rf J., [~., PFc]] (4 9) (4. 1) where I' = g2/2y is the Zeno measurement parameter. The last term in Eqs. (4.8) and (4.9) describes the where po, p~, p&, and p~ are operators with respect to effect of the measurement on the in the limit of either the a and b modes in the frequency converter case, system very good measurements. This double commutator term or to the b mode and o operators in the two-level atom case. We also note that the reduced density operator for leads directly to the system becoming diagonalized in the basis which diagonalizes n~. Note that the rate of decay the a and b modes is is the same as that given by the ~ = 0 long-time solution

prot —trc(ptot) —po + p2 ~ (4 2) given at the end of the preceding section.

When we insert Eq. (4. into the total system master 1) V. REDUCED SYSTEM DYNAMICS equation (3.4) and collect terms in lo), (ol, ll), (ol and lO), (1l and ll), (ll we obtain We now consider the effect of the measurement on either the 3aynes-Cummings two-level atom or the N, level = [Hroti Po] [nbp& pl~tb] + VP2) frequency converter system. We firstly note the equiva- Po f 2 lence of both systems when N, = 2 or equivalently, when —2 2X - - 'y = [H ot [rtbpo P2 N& —1. In the selected basis for the Jaynes-Cummings P I- p&] b] 2» h 2 two-level atom, Eq. (2.9), the free evolution term has (4 3) matrix elements equivalent to J . The measurement op- — erator np has the matrix elements — . We readily 6 i = [H.ot, p ] — [nbp2 p»b] — p~, 2I J, & 2 2 see that for N, = 2 the master equations for both systems are identical in their respective basis states. We P2 = [H-t, P2] — [nbpi —pinb] —VP2 & 2 first consider the frequency converter case and recover 45 Q UANTUM ZEN Q EFFECT IINDUCED BY QUANTUM-. . . 5233

1 atom resu its when —2.' As well we , . &~ npt«ah t for the ayne„s-gnmming;, tpm we 'ann obtain the aatomic leve o ulatjons an d ccoherences imm ediately . pop rom the bas t tes abPve. 1.o The effectt ofo measurem ent pn the e dynamic pf a o.g two-le e mode} equi va 1 nt tp ours as alrea dy been d» 0.6 cussed ;in Rf. l~] The apprpac emplpyed in that pa- ' s insigght tp the N& & 2 system. We consider the p mentumt vector d o.p fined to ha ve compo nent, s z(t) = = The aatjpn of mptip h components z(&) i(J~ ) . eq noft esec is readily obtaine d f,om the adiabafia jc mas eer equatipn, see Ref. [3],

(-r o o) (~) The initial-state p yp robabilityroba for the (5.1) I' 0. This app liesies equally we ll to th

co orl =Ower t times we see the pro ab'l'ti i t (ii to the 'mi ' I/N, = 2. Note also thee cessation o os magnitude o e where e = 1. ior. nl or ship between p; &t, the ini ia-1-. ity and z(t). We have = 0 to the exponen- ' b h f 1 e e I' the ' 't I't/2(-At/2 th 1 y(t) = ——e e ' pp e e strong, continuous me constrain (5 2) N system of firs -o I e I' A-t/2At/2 -At/2 '(') = 2n' t/2[ f ( (eAt/2 + avior can be o The matrix elements o p () ( 2e the vector undder goes — — I,t = (i,i &+ &(Pli,i + (5.5) ' or' ' exponentiae ll y decays to th e Thee first tw o coupled equations we have = —xK JZ ~ -re Pii t&(JI,2P2, I Pi, 2 z(t) = je t/j. (5.3) (5.6) r slow exponnential decay. ==—— — — — Pi,, 2 t&[JI,2(P2,„—2 Pi,P, ,i ) Jss,2P,2 ,] rP, , r later re feren th tf g' ra ofd ale I' ~ , w h effective s re ion. We aso no e fo I') 0 an t "1""f'h e vector is 0,0 represents .a .sys.t, ..., bl, d 'b d r h o = s stem can bee readily an, =— — pi, i(&) r"[2J (pi, i p2, 2 + s,2, si t rmsofth g (5.7) initial state (I) is given y At t=0) P22, (o) = pi,i s(O) —= ps,si3 i(0) =Oandthus i = —,'+ p*(t) zg) (5.4) tt2(N, —1) . pl, l(t)(t=O— )( (5.8) F the effect of the measuremment term 21 ' =21 1 1 t n. We plot hat t e ev wa from the ini'tialia state oc- suremen t p arameters I'.. Ofiin i h factor is t e in 5234 M. J. GAGEN AND G. J. MILBURN

that we the i.O z I I I I I I I I I measurement I require parameter I to scale I proportionally to N, in order to obtain a Zeno effect for . re- large N, This behavior is evident in the numerical 0.8 sults. For semiclassical systems with N, ~ oo, the Zeno effect becomes unobservable. In Fig. 4 we show the initial-state population p;(f) in O.e the large N, limit. We take N, = 10 and vary I from 0 to 2. The free evolution of the system appears as the 0.4 curve I' = O. Here we see a similar picture to t e, = 2 case. Damped oscillations occur for small I but ie ou 0.2 for large I'. In all cases it is apparent that the long time limit of the population is tending to 1/N, , and for large (c), the initial state is trapped exhibiting the Zeno eA'ect. T e 0.0 I i , 0 5 10 15 choice of N, = 10 in Fig. 4 is illustrative of the system evolution for all large X, . In particular, all oscillations FIG. 5. We compare the rate of decay of the snstial-state~ ~ ~ results of Eq. (5.2). We see in Fig. 5 a plot of the initial- occupancy probability for three systems with (a) N, = 2, (b) state occuppancy probability for systems wit N, = 10, and (c) N, = 15 and for the value I' = 2r. In this 's exhibit 10 and 15 and for the value I' = 2K. In this p loto aall lot all oscillations have ceased, and t e p opulations a oscillations have ceased, and the populations exhibiti i a smooth exponential decay to their respective long-time value smooth exponential decay to their respective long-time of 1/N, .

The dominant feature in Fig. 5 is the enhanced rate We investigate the range of applicability for this discrete model now. proac h we expexpect that the initial rate of decay is propor- The detector operates in the limit, pt 1 and y 2tt see » » '2y. The resolution time for the detector is the time in this confirmed in Fig. 6 where we compare the rate o which the b-mode photon coherences are destroyed and initial-state for three decay of the occupancy probability is given by with K = 1. We do see similar initial rates of decay for 27 &c —. (5.9) these systems. This result then indicates that the size o » X2 the measurement parameter neededd to roduce a given p This detector resolution time should be substantially Zeno effect, as shown in the rate of decay of the initial shorter than typical evolution times in the system o in- state population, must scale linearly with N, . terest. We gain some insight into the typical evolution In an experiment it will eventually be impossible to time from the free evolution of the system. In this case (with I' = 0) the initial-state occupancy probability is size of I' is limited by the resolution times and accuracy of readily found to be [14] the detector [4—6]. We therefore expect a divergence e- tween this discrete model, which features arbitari y large I', and the experimental behavior of large N, systems. 1.0

0.8

O.S 0.8

O 6 0.4

0 ~

0 p 0.2 (c)

z I 0.0 I z z t z I t z 0 5

FIG. 6. We compare the rate of decay of the initial-state~ ~ FIG. 4. The initial-state occupancy probability for the occupancy probability for three systems with (a) (N, , I ) case N, && 2 subject to increasingly strong measurement (2, 2), (b) (N. , I') = (10, 20), azzd (c) (N, , I') = (15, 28). We 0. This case only applies to the frequency converter. predict similar rates of decay for these systems given the result For I' = 0 we recover the free evolution. For long times we I' —K (N. —1). With r. = 1 we do see roughly similar rates see the probability tending to the limiting value 1/N, Note.of deca.y for ea,ch pa.ir ~S.- I"). In each case the population also the cessation of oscillations for I = 2~ wherehere z = l. tends to 1/N .. 45 QUANTUM ZENO EFFECT INDUCED BY QUANTUM-. . . 5235

( ) - 2(N, —1) est. We show a major feature of these reduced dynamics p;(t) = cos i —i (5.10) is a loss of coherence in the b-mode photon number ba- &2). sis. This basis is the pointer basis of the system, and this as shown in Fig. 1 for the N, = 2, 40 case. The initial- loss of coherence is the mechanism that enables the Zeno state occupancy becomes an increasingly narrow peak for eRect. We present the dynamics of the system of interest for large N, with a half width of pq q(t) = 2 where various measurement regimes and for various numbers 2 of levels in the system. The dynamics feature a smooth (5.11) (N, —1)~~ transition in the initial-state population from the usual free evolution Rabi Bopping to an exponential decay of We note that for large N, we have t ~ 0. We require probability for strong measurements. For systems with a that the resolution time of the detector t, needs to be large number of levels (N, ) the dynamics are similar but substantially shorter than t to manifest the zeno eRect. we must scale the measurement strength with N, in or- This requirement permits us to detect the system in some der to get equivalent rates of decay. We use this scaling state before it has appreciably evolved away from that to demonstrate the regime for which this discrete level state. For this time to be much less than t we need model will not adequately model real systems by consid- ering the finite resolut;ion time of our detection process. ~ (N, —1) « 2I' . (5.12) This gives a heuristic limit to the size of the systems N, APPENDIX: SIG NAL- TO-NOISE RATIO that we would expect to be adequately modeled by this discrete model. The behavior shown in an experiment In this appendix we consider the system-detector in- beyond this limit will show the cessation of the Zeno teraction with the free evolution turned off e = 0. We effect for increasing N, . For increasing N, systems will make use of the results of Sec. III and perform selective increasingly show free-evolution-type behavior over short measurements on the c-mode photon number and derive times. This is in effect a semiclassical limit and thus we the relevant signal-to-noise ratio (SNR) of interest. expect the Zeno effect to be unobservable in this limit. In t,he long-time limit pt )& 1 the c mode settles in a superposition of coherent states of the form — ~ iyn/p). (There is also a short-time non-Markovian limit yt « 1 VI. CONCLUSION which we do not consider here. ) Once this equilibrium has been reached then photons are measured in the c In this paper we show an experimental arrangement mode as fast as they arrive. We expect therefore a to demonstrate the quantum Zeno effect. We consider linearly increasing photon count in the long-time limit. two possible systems of interest. The first is a Jaynes- These results will be demonstrated below. Cummings two-level atom interacting with a cavity mode For the selective measurement we quote results from with annihilation operator b. The second is an N, -level Ref. The probability of a readout of m pho- system made from two cavity modes, with annihilation [18]. getting tons at time t is operators a and b, interacting via a parametric interaction configured as a frequency converter. For these sys- = e tems there is a direct correlation between the populations P(m, t) ) ", "Pg(n, 0) of the system of interest and the b-mode photon number. +=0 We employ this correlation to measure the initial-state system population. (A1) We perform this measurement by establishing further correlations between the 6 mode and another cavity where mode, with annihilation operator c. This interaction is 2n2 t —— «'~ — «'~ — . via ~ a quadratic coupling configured so as to be a quan- z„= ~ (e 1)(e 3) (A2) tum nondemolition measurement of the b-mode photon number. Subsequent photon counting measurements on the c mode are a continuous readout of the populations In the limit 7t && 1 we have of the system of interest. We calculate the c-mode pho- 2I'tn~) ton count signal-to-noise ratio and show that it is indeed z„= a good measurement of the b-mode photon number, and where we write I' = g /2p. In the long-time limit we can hence of the system populations. We show that for strong evaluate the average c-mode photon count in time t, m, measurements, the system populations are frozen. This the variance in that count V(m), and the signal-to-noise is the quantum Zeno effect. ratio of interest. We have In the strong-measurement limit, the c mode is heavily damped by the photon counting and we are justified m = 21't(n~)g p) in adiabatically eliminating the c mode. We do this by (A4) treating the c mode as being only slightly perturbed from V(m) = (2I't) Vg(n~) + 21't(n~) g p, the vacuum. This elimination gives a reduced master equation describing the dynamics of the system of inter- where Vb(n~&) is the variance of n&s evaluated in the initial 5236 M. J. GAGEN AND G. J. MILBURN 45 b-mode we state. Here see the expected time dependence (nb)~, p of m in the long-time measurement limit. In the case V( ') (A6) where the b mode is initially in a photon number state, then Vt(nq2) = 0. Here we see that the c-mode photon number readout is a We define the c-mode photon count signal-to-noise ra- good measurement of the b-mode photon number distri- tio (S) to be bution. Thus, for a given time, a "good" measurement corresponds to large I'. —2 For the case where the b mode is initially in a photon S= ' (A5) V(m) number state or when (2I't)2'(nt, ) « 2I't(nt) t p we have and therefore, for sufBciently large I't, (2I't)~Vt(n&) && S; = 2I't(nt2)t p. (A7) 2I't(nq)t p we have

[1] B. Misra and E. C. G. Sudarshan, J. Math. Phys. 18, (1969). 757 (1977). [12] G. J. Milburn, A. S. Lane, and D. F. Walls, Phys. Rev. A [2] A. Peres, Am. J. Phys. 48, 931 (1980). 27, 2804 (1983). [3] G. J. Milburn, J. Opt. Soc. Am. 5, 1317 (1988). [13] E. T. Jaynes and F. W. Cummings, Proc. IEE 51, 89 [4] K. Kraus, States, Effects and Operations: Fundamental (1963). Notions of Quantum Theory (Springer, Berlin, 1983). [14] B. Yurke, S. L. McCall, and J. R. Kaluder, Phys. Rev. A [5] C. M. Caves and G. J. Milburn, Phys. Rev. A 36, 5543 33, 4033 (1986). (1987). [15] B. C. Sanders, Phys. Rev. A 40, 2417 (1989). [6] A. Barchielli, Nuovo Cimento B 74, 113 (1983). [16] D. F. Walls, M. 3. Collet, and G. J. Milburn, [7] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. 3. Phys. Rev. D 32, 3208 (1985). Wineland, Phys. Rev. A 41, 2295 (1990). [17] G. J. Milburn and D. F. Walls, Phys. Rev. A 28, 2646 [8] G. J. Milburn and M. J. Gagen (unpublished). (1983). [9] A. Peres, Phys. Rev. D 39, 2943 (1989). [18] G. 3. Milburn and D. F. Walls, Phys. Rev. A 30, 56 [10] C. M. Caves, K. S. Thorne, K. W. P. Drever, N. D. Sand- (1984). berg, and M. Zimmerman, Rev. Mod. Phys. 52, 341 [19] W. H. Louisell, Quantum Statistical Properties of Radia (1980). tion, (Wiley, New York, 1973). [11] J. Tucker and D. F. Walls, Ann. Phys. (N.Y.) 52, 1 [20] I. K. Mortimer and H. Risken (private communication).