The Quantum Zeno Effect – Watched Pots in the Quantum World

GENERAL ⎜ ARTICLE The Quantum Zeno Effect – Watched Pots in the Quantum World

Anu Venugopalan

In th e 5th cen tu ry B C , th e p h ilosop h er an d logi- cian Zeno ofElea posed severalparadoxesw hich rem ained unresolved for over tw o thousand ¯ve hundred years. In recent times,the Zeno e®ect m ade an intriguing appearance in a rather un- likely place { a situation involving the tim e evolution of a q u an tu m sy stem w h ich is su b ject to Anu Venugopalan is on the faculty of the Centre for `observations'over a period of tim e. In 1977, B Philosophy and Founda- M isra and E C G Sudarshan published a paper tions of Science, New on the quantum Z en o e® ect, called \T he Z eno's Delhi, on leave from GGS paradox inquantum theory"1 . T h e ir fa s c in a tin g Indraprastha University, resu lt revealed th e b izarre w ork ings of th e q u an - Kashmere Gate, Delhi. Her primary research interests tu m w o rld . In th e fo llo w in g a r tic le , th e q u a n - are in the areas of tu m Z en o e® ect is d escrib ed an d a b rief ou tline Foundations of Quantum ofsom e ofthe w ork follow ingM isra and Sudar- mechanics, Quantum s h a n 's p a p e r is g iv e n . Optics and Quantum Information. 1. In tro d u ction T he quantum Z eno e®ect gets its nam e from the G reek philosopherZeno w ho lived inthe5th century B C inthe 1 B Misra and E C G Sudarshan, G reek colony of E lea (now in southern Italy). Z eno w as The Zeno’s paradox in quantum k n o w n to b e th e m o st b rillia n t d isc ip le o f P a rm e n id e s, theory, Journal of Mathematical Physics, Vol.18, No.4, pp.756– a very prom inent¯gureoftheEleaticSchoolofphiloso- 763, 1977. p h ers. A ristotle is su p p osed to h ave cred ited Z en o w ith having invented the m ethod ofthe `dialectic',w here tw o sp eakers altern ately attack an d d efen d a th esis. Z en o is Keywords also cred ited w ith inven ting th e argu m en t form `red u c- Zeno paradox, unstable sys- tio a d a bsu rd u m '. H owever,Zeno ism ost fam ous for tems, survival probability, quan- hisfour paradoxes of m otion w hich he developed as ar- tum evolution, quantum mea- gu m en ts against th e th en ex isting n otion s of sp ace an d surements, continuous measurements, quantum Zeno ef- tim e. A d escription of th e q u an tu m Z en o e® ect d o es fect, quantum anti-Zeno effect. notnecessarilyrequirea priorknow ledge oftheclassical

52 RESONANCE ⎜ April 2007 GENERAL ⎜ ARTICLE p arad ox es of Z en o. H ow ever, as an interesting h istori- Aristotle is supposed calreference, w e w illbrie°y describe particular versions to have credited of th e fou r classical p arad ox es b efore em b ark ing on a Zeno with having d escription of th e q u an tu m Z en o e® ect itself. T h e p ara- invented the method doxespopularlyarecalled:(i)A chillesand thetortoise, of the ‘dialectic’, (ii) th e arrow , (iii) th e d ich otom y, an d (iv) th e sop h isti- where two speakers cated stad ium . W e w ill b rie° y d escrib e th em h ere. alternately attack 1.1 A chilles and the T ortoise and defend a thesis. A chilles has to race a tortoise. Since A chilles isob- v iou sly faster th an h is op p on en t, th e tortoise is given a head start.Letussupposethattheracebeginsatt =0, w hen A chillesislocated at x = 0, an d th e tortoise, w ith its h ead start stan d s at x = x 0 (Figure1a).Zeno argued that the runningA chillescouldnevercatch up w ith the tortoise b ecau se h e m u st ¯ rst reach w h ere th e tortoise started . F or instan ce, w h en A ch illes reach es x 0 at tim e t0 , th e tortoise h as alread y p rogressed to x 1 (>x0 ). B y thetimeAchillesreachesx 1 ,attime t1 , th e tortoise has craw led up to x 2 . T hus, w hen ever A chilles reaches x i, th e tortoise w ou ld h ave m oved to x i+1,and Achilles wouldneedtimeti+1 ti b efore h e gets to th at p oint (by w h ich tim e th e to rto¡ ise w o u ld h a v e m o v e d o n a h e a d !). Zeno argued that for A chilles to reach the tortoise (or to overtake it),he m ustperform an in¯nitesum ofsuch tim e increm en ts, ti+1 ti,or spatialincrem ents,x i+1 x i fo r a ll i upto . If space¡ and tim e are considered to¡ be continuous,these1 increm entsw ould tend to zero duration , an d if th ey are con sidered d iscrete, th e increm en t w ould be of ¯nite duration. (Note that in the continuous case, the increm ents ten d to zero but w illnot be actually zero forany ¯nite i). Z en o argu ed th at if sp ace and time are continuous, an in¯nite sum of elem ents tendingtowardszero length (duration) m usthave a to- tal of zero len gth (d u ration ). A ltern atively, if sp ace an d tim e are discrete,then an in¯nite sum of ¯nite elem ents m ust be of in¯nite length (duration). Since A chilles is `seen ' to overtake th e tortoise, th e ab ove argu m en ts fail.

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T hus,the seem ingly absurd conclusion that follow s is th at b oth sp ace an d tim e can n either b e con tinu ou s n or d iscrete. T h is com p els u s to con sider th e n otion th at sp ace an d tim e are illusory { h en ce, so is everyth ing w e see! 1.2 TheArrow Im agine an arrow ° y ing th rou gh sp ace (Figure1b). T im e is con sidered to b e m ad e u p of `instan ts'. T h ese instan ts arede¯ned asthesm allestm easureand they areindivis- ible.A tany instantoftime,when thearrow isobserved, it m u st b e seen to o ccu p y a sp ace eq u al to itself. If it is `seen ' to `m ove' at an y instan t, it m ean s th at th e ob - server can d ivide an instan t into a tim e w h en th e arrow w as `here'and a time w hen thearrow w as `there'.T his w ould m ean that the instant of time consists of parts, w h ich v io la te s its b a sic d e ¯ n itio n (th a t o f it b e in g in d i- v isible). T h u s, Z en o asserted th at th ere are n o instan ts of tim e w h en th e arrow d o es m ove. T h is, again, lead s to theconclusionthatthearrow isalwaysatrestandthat a ll m o tio n is illu so ry .

Figure 1. (a) Achilles and the Tortoise, (b) The Arrow, (c) The Dichotomy, (d) The Sophisticated Stadium.

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1.3 TheDichotomy Zeno argued that ImaginethatAchilleswantstorun from pointA topoint infinite numbers of B(Figure 1c). B efore h e can cover h alf th e d istan ce to small spatial th e en d , h e m u st cover th e ¯ rst q u arter. B efore th is, separations in a h e m u st cover th e ¯ rst eigh th , an d b efore th at th e ¯ rst finite time would sixteen th , an d so on . B efore A ch illes can cover an y d is- be impossible. tance at allhe m ustcoveran in¯nitenum berofsm aller spatialseparations.Zeno argued that coveringthesein- ¯nite num bers of sm allspatial separations in a ¯nite time w ould be impossible. T hus,itcan be concluded th at A ch illes can n ever get started . B u t A ch illes is `seen ' to m ove.T hus,once again,theargum ent com pelsus to conclude thatallm otion isan illusion. T hisparadoxical argum ent is called `the dichotom y' because it involves rep eated ly sp litting a d istan ce into tw o p arts. It con - tains som e of th e sam e elem ents as th e A ch illes an d th e Tortoise paradox,but w ith a m oreapparent conclusion of m otion lessn ess. 1.4 T he Sophisticated Stadium T his is the last of Zeno's four paradoxes. L et us sup- p ose th at sp ace an d tim e are d iscrete in n atu re an d th at m otion con sists in o ccu py ing d i® eren t sp atial p oints at d i® e re n t tim e s. F o r sim p lic ity , c o n sid e r o b je c ts m o v - ingataminimalspeed,v , of one fundam ental spatial distance per fundam entaltem poralduration. C onsider ninepersons m oving roughly collinearly (Figure 1(d )). Personsa i are all station ary, w h ile p erson s bi movepast th em to th e left, w ith th e velocity, v .Atthesame ¡ tim e, p erson s ci m ove to th e righ t p ast th e a iswithve- lo c ity v .Say,attimet = 0 thecon¯guration isasshow n in Figure1d(i).O ne fundam entalunitoftime later,the con¯guration in Figure 1d(ii) is achieved . O ne can see th at c3 h as p assed b y b2 ,buttherew asneveran instance when c3 waslinedup with b2 . T hu s, it can b e argu ed th at th ere is n o tim e at w h ich th e actu al p assing o ccu rs, and hence, it never happened! A nother w ay to bring

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It was only after the attention to thisparadoxicalsituation isto note that in development of the Figure 1d(ii), each B has passed twice as m any C s as calculus of A s.T hus,one m ight concludethat ittakesrow B twice infinitesimals by as lon g to p ass row A as it d o es to p ass row C . H ow ever, Leibniz and Newton, th e tim e for row s B an d C to reach th e p osition of row the concept of A isthesame.So,itappearsthathalfthetimeisequal to tw ice th e tim e! functions, limits, continuity, infinite T he four paradoxes, sum m arized above in their barest series, and form s,foxed and confused m athem aticians and philoso- convergence, that a phers for over two m illenia. Itw as only after the de- satisfactory resolution velopm ent of the calculusof in¯nitesimalsby Leibniz to Zeno’s paradoxes an d N ew ton , th e con cep t of fun ction s, lim its, continu - came about. ity, in¯nite series,and convergence, that a satisfactory resolution to Z en o's p arad oxes cam e ab ou t. H ow ever, even today,inspiteofourfam iliarityw ith thesem odern m athem atical ideas and concepts, there is a continuing debateabout the validity ofZeno'sparadoxesand their variou s resolution s. W e en d ou r b rief intro d u ction to the `classical'paradoxes of Zeno at thispoint and w ill d escrib e th e Quantum Zenoe®ect{themainsubjectof th is article, in th e n ex t section . 2. T he Q uantum Zeno E®ect T h e Z en o p arad ox in q u an tu m system s w as b rou gh t into focus in the 1960s by L eonid A K hal¯n,w orking in the form erU SSR ,and by E C G Sudarshan and B aidyanath M isra,w orkinginU SA duringthe1970s.M israand Su- darshan'spaperentitled \T he Zeno'sparadox inQ uan- tu m T h eory " in th e Journal of M athem atical P hysics ¯ rst intro d u ced th e n am e \Z en o's p arad ox " for th e e® ect studied. Q uantum Zeno E®ect (QZE) isa term m ore com m only used these days to describe sim ilar situations in various quantum system s. In order to understand th e essen ce of th e Q Z E , it is n ecessary to ¯ rst b rie° y review som e basic concepts of quantum m echanics and q u an tu m m easu rem en t.

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2.1 Quantum M easurement Quantum mechanics Q u an tu m m ech an ics is cu rren tly accep ted as th e m ost is currently accepted elegant and satisfying description of phenom ena at the as the most elegant atom ic scale. Since our larger, fam iliar `m acro' w orld and satisfying is eventually com posed of elem ents of the `m icro' w orld description of w hich isdescribed by quantum m echanics,quantum the- phenomena at the ory isalso,inevitably,thefundam entaltheoryofnature. atomic scale. T hough stunningly pow erful,the quantum m echanical v iew of th e w orld h as com p elled u s to resh ap e an d re- vise our ideas of reality and notions of cause,e®ect and m easurem ent. W ithout going into too m any details of the various conceptual di± culties of quantum m echanics,w e w illonly focus on the quantum m echanical description of measurementwhichisofdirectrelevanceto th e d escription of th e Q Z E . T he quantum m echanicaldescription ofa system iscon- tained in its w avefu n ction or state vector, Ã ,which lives in an ab stract `H ilb ert sp ace'. T h e d yj n ami ics of th e w avefu n ction is govern ed by th e S ch rÄodinger equation : d i¹h Ã = H Ã ; (1) dtj i j i whereH isthe H am iltonian operator,and the equation islinear,determ inisticand thetime evolution governed b y it is unitary. U nitary evolutions preserve probabili- ties. E x am p les of u n itary tran sform ation s are rotation s and re°ections. D ynam ical variables or observables are represented in quantum m echanics by lin ea r H e rm itia n operators, w hich act on the state vector. A n op erator, A^, corresp on d ing to a d y n am ical qu an tity, A ,is associated w ith eigenvalues a i an d corresp on d ing eigen- vectors, ® i ,which form a com plete orthonorm alset. A ny arb itraryfj ig state vector, Ã ,ingeneral,can be represented b y a linear su p erp ositionj i of th ese eigen vectors, or, for that m atter, a com bination of any orthonorm al set of b asis vectors in H ilb ert sp ace. T hu s, on e can w rite Ã =§c ® . A b asic p ostu late of q u antu m m ech an - j i ij ii

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A basic postulate of ics regarding m easurem ent is th at an y m easu rem en t of quantum mechanics th e q u an tity A can only yield one o f th e e ig e n v a lu e s a i, regarding but the resultisnot de¯nite in the sense that di®erent m easu rem en ts for th e q u an tu m state Ã can yield dif- measurement is that j i any measurement of feren t eigen values. Q u antu m th eory p red icts on ly th at th e probability of obtaining eigenvalue a is c 2 . Q uan- a quantity A can only i j ij tum theory de¯nesthe exp ecta tio n v a lu e of th e op erator yield one of the A^ as: eigenvalues. A^ = Ã A^ Ã =§a c 2 : (2) h i h j j i ij ij In term s of th e d en sity m atrix½ ^ = Ã Ã ,an equivalent form u la for th e ex p ectation value is:j ih j

A^ = Trace A^½^ : (3) h i f g A n additional postulate of quantum m echanics isthat the m easurem ent of an observableA ,w hich yields one 2 of the eigenvalues a i (w ith p rob ab ility ci ) culminates w ith th e red u ctio n or collapse of th ej statej vector Ã j i to th e eigen state ® i . T his m eans that every term in th e linear su p erp ositionj i van ishes, ex cep t on e. T h is reduction isa non unitary process and hence in com plete con trast to th e u n itary d y n am ics of q u antu m m ech an - ics p red icted by th e S ch rÄo d in g e r e q u a tio n a n d th is is w herethecruxoftheconceptualdi±cultiesencountered inquantum theorylies.Fornow wejustacceptthisasa b asic p ostu late of q u antu m th eory (also called th e pro- The QZE (or jec tio n p o stu la te ) an d go on to d escrib e th e Q Z E . paradox) was the name given by 2.2 TheResultofM israandSudarshan Misra and T he Q ZE (or paradox) w as the nam e given by M isra Sudarshan to the and Sudarshan to the phenom enon of the in h ib itio n of phenomenon of tran sition s b etw een q u an tu m states b y freq u en t mea- the inhibition of surem en ts. For theirstudy,they considered the decay transitions of an u n stab le state, su ch as an u n stab le p article, like between quantum a rad ioactive n u cleu s. T h e classic m o d el of any sy stem states by frequent d ecay is an exp on en tial fu n ction of tim e in m ost situ- measurements. ations. W e are allfam iliar w ith the radioactive decay

58 RESONANCE ⎜ April 2007 GENERAL ⎜ ARTICLE la w The decay of a ¸ (t t0 ) N (t)= N (t0 )e ¡ ¡ ; (4) quantum system where N (t) is the num ber of nuclei that have not de- can show a cayed after tim e t,and ¸ is a con stan t w h ich d ep en d s deviation from the on th e p rop erties of th e sp ecies of n u clei. W h ile th e d e- familiar exponential cay of a q u antu m sy stem is sim ilar to th is classic m o d el decay law. of exp on en tial d ecay, th ere h ave b een th eoretical stu d - ies th at sh ow th at in certain tim escales (sp eci¯cally, for very short and very long tim es as m easu red from th e instant of p rep aration of th e state of th e system ), th ere can be a deviation from the fam iliar exponentialdecay la w . In fa c t, re c e n tly th is th e o re tic a l d e v ia tio n fro m th e exp onential decay law has also been con¯rm ed exp eri- m entallyinquantum tunnellingexperimentsw ith ultra cold atom s by a group at the U niversity ofT exas,U SA . It is in th ese sp ecial tim e regim es th at w e see m an ifestation s of th e Q Z E . C on sider th e d ecay of an u n stab le q u an tu m state. L et Ã 0 be the (undecayed) state ofthe sy stem at t =0andÃ (t) b e th e state at an y later tim e t. T h e evolution of th e state is govern ed b y a u n itary operator,U (t), w h ere

iH t U (t)= e ¡ ; (5) (h ereh ¹ =1)and Ã (t) = U (t) Ã ; (6) j i j 0 i H b eing th e H am ilton ian of th e sy stem . A s d iscu ssed in th e p rev iou s section , an y ob servation th at th e state has not decayed w illcausea collapse (reduction) ofthe w avefu n ction to th e u n d ecayed state. T h e survivalprob- a b ility , P (t), i.e., th e p rob ab ility th at th e sy stem is still in the undecayed state,w illbe the m odulussquared of th e survivalam plitude and can be w ritten as: P (t)= Ã U (t) Ã 2 : (7) jh 0 j j 0 ij Now,(7)canbeexpandedas: P (t)= 1 t2 ( Ã H 2 Ã Ã H Ã 2 )+ ::: (8) ¡ h 0 j j 0 i¡ h 0 j j 0 i

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The short-time If ¢ H = Ã H 2 Ã Ã H Ã 2 ; (9) quantum decay is q h 0 j j 0 i¡ h 0 j j 0 i not exponential in th en th e su rv ival p rob ab ility in th e sh o rt tim e lim it can time, but quadratic. b e rew ritten as:

P (t) 1 t2 (¢ H )2 + :::: (10) ¼ ¡

Ifw e de¯ne ¿Z =1=¢ H as the Zeno time,thisgives: t2 P (t) 1 2 + ::: (11) ¼ ¡ ¿Z w h ich , for sh ort tim es, can b e w ritten as: t2 P (t) (1 2 ): (12) ¼ ¡ ¿Z T h e ab ove ex p ression sh ow s th at th e sh ort-tim e q u an - tu m d ecay is n ot ex p on en tial in tim e, b u t quadratic. N ow ,let us suppose that one m akes N equally spaced m easurem ents overthetimeperiod[0;T ]. If ¿ is th e tim e interval b etw een tw o m easu rem ents, th en T = N¿. Letusassum e thatthem easurem entsarem ade attimes T=N ,2T=N ,3T=N , ... ,(N 1)T=N and T and are in - stantaneous. So, essentially¡ this describes an alternate seq u en ce of u n itary evolution s (each lasting for a tim e ¿ ) follow ed b y a collap se (th e b asic p ostu late of q u an - tu m m easu rem ent). T h e su rv ival p rob ab ility after N m easu rem en ts, or after tim e T can b e w ritten as: T 2 P N (T )= [P (¿ )]N =(1 )N : (13) 2 2 ¡ N ¿Z Itcanbeseenthatinthelimitofcontinuous m easure- ments, i.e ., w h e n N , !1 lim P N (T )= 1: (14) N !1 Thusth e p ro ba b ility th a t th e sta te w ill su rv iv e fo r a tim e T goes to 1 in the lim it N . T hism eans that continu ou s m easu rem en ts actu!1 ally p revent th e system from

60 RESONANCE ⎜ April 2007 GENERAL ⎜ ARTICLE ever decaying! So, m uch like the m otionless arrow in Continuous Z eno's paradox, the system never decays, or a `w atched measurements pot never boils'. N ow ,can w e see thishappening in a actually prevent real ex p erim en t? U n fortu n ately, in sp on tan eou s d ecay the system from this e®ect is very di± cult to observe for reasons that ever decaying! w e w illbrie°y discussina latersection. N o experiment h as b een ab le to p rob e th is regim e to ob serve th e inh i- b ition of th e d ecay of an unstableparticle lik e a ra d io a c - tive n u cleu s, as yet. H ow ever, as m en tion ed earlier, re- cently there have been experim ental groups w hich have rep orted th e ob servation of th e Q Z E in u n stab le system s com prisingof trapped ultra cold atom s w hich undergo quantum m echanical tu n n ellin g . W e w illdiscuss these later. F irst, w e w ill d escrib e an earlier ex p erim ent carried ou t b y Itan o et alin1990 ofthe experim entalgroup h ead ed b y W inelan d at th e N ation al In stitute of S tan - dardsand Technology,B oulder,C olorado. T hisw asthe ¯ rst instan ce w h en a m an ifestation of th e Q Z E w as su c- cessfu lly d em on strated ex p erim en tally. T h is is d iscu ssed in the next section. 2.3 ExperimentalM anifestations of the Q ZE 2.3.1 T he E xperim ent of Itano et al: Follow ing an original proposal by C ook, Itano et al,attheNa- tionalInstitute of Standardsand Technology,B oulder, C olorad o, ex p erim en tally d em on strated th e o ccu rren ce The experiment of ofthe Q ZE in in d u ced transitions between two quantum Itano et al tested the states.U nlike the casestudied by M israand Sudarshan, inhibition of the th is is a situation w h ere th ere is n o sp on tan eou s d ecay induced radio of an u n stab le sy stem b u t an induced transition betw een frequency transition tw o states of a sy stem . T h e ex p erim ent of Itan o et al between two tested th e inh ibition of th e ind u ced rad io freq u en cy tran - hyperfine levels of a sition between two hyper¯ne levelsofa Beryllium ion, caused by frequentm easurem entsof the levelp opulation Beryllium ion, caused usingopticalpulses.T he experimentcan be understood by frequent as follow s (Figure2): C on sider a tw o-level sy stem , w ith measurements of the th e levels lab elled as 1 an d 2. A ssu m e th at th e sy stem level population using can be driven from level 1 to level 2 by applying optical pulses.

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Figure 2. Quantum Zeno Effect in Induced Transi- tions between energy levels – the experimental system of Itano et al.

a reson a n t rad io frequ en cy p u lse. A ssu m e th at it is p ossible to m ake instan tan eou s m easu rem en ts of th e state of th e sy stem , i.e., to ascertain w h eth er th e sy stem is in level1 orinlevel2. In ordertoobserve thelevelpopula- tion s, level 1 is con n ected b y an op tical tran sition to an additionallevel3 such thatlevel3 can decay onlyto level 1. Spontaneous decay from level2 to level1 isnegligible. T h e m easu rem ents are carried ou t (d u ring th e evolution under the resonant radio frequency pulse) by driving th e 1 3 tra n sitio n w ith N eq u ispaced sh ort op tical pulsesand! observingthepresence (or absence)ofspon- tan eou sly em itted p h oton s from level3 to level 1. S u ch a situation w as created in a real exp erim en tal sy stem w ith a trap p ed B ery llium ion w h ere ap p rop riate en ergy levels of th e ion cou ld b e ch osen to corresp on d to th e 1;2and 3 levels described above. In recent tim es, trapp ed ions an d atom s h ave b ecom e very p op u lar sy stem s for carry - ingout m any experimentsthat testfundam entalissues in quantum m echanics. T hey are considered `clean'sys- Trapped ions and tem s th at can b e ob served for lon g p eriod s of tim e an d atoms are isolated from noise. M oreover, their energy levels can considered ‘clean’ be easilym anipulated w ith appropriateradiofrequency systems that can and opticalpulses. be observed for Suppose the ion isin level1 at a time t =0.Anrf long periods of ¯eld having resonance frequency − =(E E )=¹h )is 2 ¡ 1 time and isolated ap p lied to th e sy stem an d th is creates a state w h ich is a from noise. coherent superposition of states 1 an d 2. T h e d yn am ics

62 RESONANCE ⎜ April 2007 GENERAL ⎜ ARTICLE of a tw o-level sy stem in th e p resen ce of reson an t d riving The dynamics of a ¯eld isw ell-studied and understood. T he frequency − two-level system in is called the R abi frequency. A n on-resonance `¼pulse' the presence of isa pulse ofduration T = ¼=− and takes the ion from resonant driving le v e l 1 to le v e l 2 . If P (t)istheprobabilityattime 2 field is well-studied t for the ion to b e at level 2, then P (T )= 1.This 2 and understood. w ould be the situation w hen no `measurem ent pulses' are applied. In the experiment,N m easurem ent pulses are applied (w hich connect level1 to level3 through an op tical p u lse each tim e), within tim e T ,i.e.,attimes ¿k = kT =N ;k =1;2;3;:::;N . N ote that the dynam ics of th e tw o-level sy stem d riven b y th e reson an t rf ¯ eld is u n itary an d can b e d escrib ed q u antu m m ech an ically u sing th e B loch vector rep resen tation . T h e n onu n itary projection postulate (or th e collapse induced by quantum m easu rem en t) is incorp orated each tim e a m easu rem en t ism ade. Itiseasy to show that at the end ofN mea- su rem en ts, i.e., at th e en d of th e rf p u lse at tim e T ,the probability P 2 (T ), w h ich corresp on d s to th e population of level2 isgiven by: 1 P (T )= [1 cosN (¼=N )]: (15) 2 2 ¡ For large N , i.e ., in th e lim it o f continuous m easure- ments,one can seethat

1 N P 2 (T )= lim [1 cos (¼=N ) 0: (16) N 2 !1 ¡ ¼ C learly, th e continu ou s m easu rem ents d escrib ed ab ove in h ib it th e ind u ced tran sition from 1 to 2, m ak ing th e system `freeze' in level1. T his e®ect show ed itselfup in th e real ex p erim en tal ob servation s of Itan o et al.Thus, although itw as not seen in the decay of a unstableparticle, the experim ent of Itano et al w as th e ¯ rst real d em on stration of th e Q Z E .Interestingly, th e exp erim en t wasfollowedbyaslewofpaperswheremanyissueswere raised regard ing th e actu al d y n am ics of th e m ech an ism explored in the experim ent by Itano et al.Theexperi- m en t w as critically an alyzed from th e p oint of v iew of

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Many physicists assert qu an tum m easurem en ts and questions w ere raised re- that the QZE is simply gard ing w h eth er or n ot th e collapse p ostu late p lay s any a consequence of the role at allin the outcom e of the experim ent. M any unitary dynamics of p h y sicists assert th at th e Q Z E is sim p ly a con seq u en ce conventional quantum of the unitary dynam ics of conventional quantum m e- chanicsand need notinvolve thenon unitary collapseof mechanics and need q u an tu m m easu rem en t. H ow ever, since th e p ro jection not involve the non p ostu late of con ven tion al q u antu m m easu rem en t th eory unitary collapse of also su ccessfu lly d escrib ed th e ou tcom e of th is ex p eri- quantum m en t, it is valid to see th e exp erim en t of Itan o et al as measurement. a d em on stration of th e inh ibition of tran sition d u e to frequentm easurem ents,ortheQ ZE.In thenextsection w e d escrib e on e m ore ex p erim en tal m an ifestation of th e QZE. 2.3.2 T he E xperim ent of K w iat et al: In 1995, Paul K w iat and hisgroup at the U niversity of Inns- bruck realized a version of the Q ZE in the laboratory usingthepolarization directionsofsinglephoton states. T h eir ex p erim ent w as b ased on an ex am p le ¯ rst su g- gested by A sher P eres in 1980. C onsider plane polarized ligh t. It can h ave tw o p ossible p olarization d irection s, say, `vertical'an d `horizon tal'. W e k n ow th at w h en su ch a beam passes through an optically active liquid (e.g., su gar solution ) its p lan e of p olarization is rotated by a sm all an gle (w h ich d ep en d s on th e con cen tration of thesugar solution,for exam ple).C onsidera photon di- rected th rou gh a series of su ch \rotators" so th at each slightly rotates its p olarization d irection so th at an ini- tially vertically polarized photon ends up horizontally p olarized . A t th e en d of th is series of rotators, th e p h o- ton en cou n ters a p olarizer. A p olarizer is a d ev ice th at transm itsphotonsw ith one kindofpolarization butab- sorb s p h oton s w ith th e p erp en d icu lar p olarization . A n id e a l N ic o l p rism a c ts a s a p o la riz e r (o r a n a ly z e r). N o w letussupposethatanexperimentissetupwithsixro- tators, each of w h ich rotates th e p lan e of p olarization of a vertically p olarized p h oton by 15 o .Attheendof

64 RESONANCE ⎜ April 2007 GENERAL ⎜ ARTICLE

th is series is a p olarizer w h ich tran sm its on ly vertically Figure 3. Six rotators turn polarized light, w hich is then detected by a photon de- the polarisation by 15 detector (Figure 3).Itisobviousthatintheaboveset grees at each stage such up,the photon w illnever get to the detector as itspo- that a vertically polarized larization w ill h ave tu rn ed b y 90 o after p assing th rou gh photon changes to a hori- th e six rotators an d b ecom e h orizon tal. T o im p lem en t zontally polarized one. the Zeno e®ect, P aul K w iat and his colleagues sought to in h ib it th is step w ise rotation of th e p olarization , or th e evolution of th e p olarization state from th e vertical to th e h orizon tal, by measurementsof the polarization state. K w iat et al realized th is b y intersp ersing a verti- calpolarizer betw een each rotator(Figure3).Ifthe¯rst rotator rotates th e p lan e of p olarization b y an an gle ® , th en th e vertical p olarizer kep t after it w illtran sm it th e photon w ith a probability cos2 ® , an d th e original vertical p olarization w ou ld h ave b een restored (th is can b e recogn ized as th e w ell-k n ow n M alus'law ). A t th e secon d rotator th e p olarization is on ce again tu rn ed by ® and it th en en cou n ters th e secon d p olarizer w h ere it w ill b e tram sm itted w ith a p rob ab ility cos2 ® an d th e vertical p olarization w ill b e, on ce again, restored . T h is p ro cess repeats tillthe photon com es to the ¯nal polarizer. If 0 2 6 2 ® =15;(cos ® ) = 3 . T hus an incident photon has tw o th ird ch an ce of b eing tran sm itted th rou gh all six If the first rotator inserted p olarizers an d m ak ing it to th e d etector. It can rotates the plane b e easily seen th at if on e increases th e n u m b er of stages, of polarization by decreasingtherotation angleateach stage,theprobabil- an angle α, then ity of tran sm itting th e p h oton to th e d etector increases. the vertical Ifthere w ere an in¯nite num ber of stages,the photon polarizer kept after w ouldalways getthrough and hence therotation ofthe it will transmit the plane of polarization w ould be com pletely inhibited { th e Z en o e® ect! In th e actu al ex p erim ent, K w iat et al photon with a 2 α. created single p h oton states u sing a n on linear crystal. probability cos

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T hus,like theexperiment ofItano et al,the experiment ofK w iat et aldem onstrated thesuppression ofevolution in a d riven tw o-state sy stem th rou gh freq u ent m easu re- m en ts. W h at ab ou t th e Q Z E in u n stab le system s? 2.4 TheResultofKurizkiandKo®man{ The A nti-Z en o E ® ect In the previous section w e have seen experim ental evi- dences of the Q ZE in induced transitions between two q u an tu m states. A n atu ral q u estion th at arises, th en , is w hether the Z eno e®ect can be used (in a real exp eri- m ent) som e day to `freeze'radioactive nuclear decay. For thepastthree decadesitseem ed that theanswerto the q u e s tio n w a s a `y e s ', p ro v id e d o n e h a d th e e x p e rim e n - tal tech n ology an d sop h istication to p erform su ccessive, `freq u ent en ou gh ' m easu rem ents. H ow ever, recen t w ork by G ershon K urizkiand A braham K o®m an attheW eiz- m ann Institute of Science, Israel,has show n that such a freezing m ay actually not be possible at all. K urizki andKo®manhavearguedthatthereisan`Anti-Zeno E®ect'which infactenhances thedecay ofunstablepar- ticles instead of inhibiting it! A ccording to theircal- cu lation s, th e ab ility to `freeze' th e evolution th rou gh frequent m easurem ents depends on the ratio between th e `m em ory tim e' of th e d ecay, an d th e tim e interval b etw een successive m easurem ents. E very decay p rocess hasa`memorytime'.Thismemorytimeisthetimefol- low inga quantum event in w hich the particlecan still retu rn to its initial state. In th e case of rad ioactive decay, for instance, the m em ory tim e is the p eriod in w hich theradiation has not yetescaped from theatom , allow ing th e sy stem to `rem em b er' its state p rior to th e d ecay. T yp ically, th is m em ory tim e for an u n stab le p ar- ticleislessthan one billionth ofa billionth ofa second. K u rizk i an d K o® m an argu e th at freq u en t m easu rem en ts in th is tim e scale (if it w ere p ossible) w ou ld cau se m ore p articles to b e created . T h is w ou ld interfere w ith, an d essen tially d estroy th e original sy stem , m ak ing it m ean -

66 RESONANCE ⎜ April 2007 GENERAL ⎜ ARTICLE ingless to ask w hether the decay has frozen or not. O n the other hand, if the tim e interval betw een m easure- m en ts is lon ger th an th e m em ory tim e (i.e.,ob servation s are n ot fast en ou gh for th e `ex p ected ' Q Z E ), th e rate of decay in creases and one w ould have the AntiZenoef- fec t. W hilew e w illnot go into thedetailsoftheirwork, w e can state th at th e su rp risin g co n c lu sio n o f th e re - searchofKurizkiandKo®mannisthattheAnti-Zeno e®ect (i.e.,the increase of decay through frequent m easurem en t) can occur in all processes of decay, w hile the Z eno e® ect w hich w ould slow dow n and even stop decay requires conditions thatare m uch rarer. W hilethe predictions ofK urizkiand K o®m an are yet to b e ex p erim entally veri¯ed on an u n stab le sy stem like a radioactive nucleus,recent experim ents by M ark R aizen an d h is colleagu es at th e U n iversity of T ex as, A u stin h ave d em on strated th e q u an tu m Z en o an d th e q u antu m A ntiZeno e®ects in the tunnelling behaviour of cold trap p ed atom s. R aizen 's team trap p ed so d ium atom s in a `lig h t w a v e '. S u ch a sy ste m , if le ft a lo n e , w ill slow lydecay as individualatom s escape through quantum m echanical tunnelling through an energy barrier w hich wouldbe classicallyinsurm ountable.T hrough in- genious experim entaltechniques,the team show ed that thetunnellingrateslowed dow n dram aticallyw hen they `m easu red ' th e sy stem every m illion th of a secon d { th e Q Z E ! W h en th ey m easu red th e sy stem every ¯ ve m illion th of a secon d , th e tu n n elling rate increased { th e Figure 4. Quantum Zeno q u an tu m A n ti Z en o e® ect! It is interesting to n otice Effect: Interspersing a thehappy coincidence that thisspectacular experimen- polariser after each rotator taltestoftheQZE whichisclosestinspirittotheorig- inhibits the polarisation inalproposalofM isra and Sudarshan w as perform ed at state from changing.

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Acknowledgements th eUn iversity of T ex as, A u stin { th everysameplace The author would like to thank from w hereM isra and Sudarshan published theirwork, Ragothaman Yennamalli and alm ost th ree d ecad es ago. Vivek for help with the figures. 3. C onclusions M uch w aterhas°ow n underthebridge since Zeno w on- dered about A chilles and the tortoise at the daw n of c iv iliz a tio n . T h o u g h m a th e m a tic ia n s h a v e so lv e d th e classicalparadoxesofZeno long ago by introducingthe concept of real num bers, lim its, continuity and calculus to describe quantities ofduration and distance,theno- tion of freezing m otion by continuous observation has tu rn ed ou t to b e a very real e® ect in th e stran ge w orld ofquantum physics. Spectacular exp erim ents bear testi- Address for Correspondence Anu Venugopalan m on y to th e reality of th is e® ect in th e q u an tu m d om ain Centre for Philosophy and and the¯eldsofatom opticsand coldtrapped ions con- Foundations of Science tinue to spring up tantalizingnew surprises every day. Darshan Sadan, E-36 W h ile in th e classical w orld, A ch illes overtakes th e tor- Panchshila Park toise an d all is w ell w ith th e w orld, in th e m y steriou s New Delhi 110 017, India. Email: la n d o f th e q u a n tu m , w a tch e d p o ts sto p b o ilin g (o r b o il [email protected] faster, m ayb e!) an d th e gh ost of Z en o con tinu es to m ake itspresence feltinunimaginablyinterestingways.