The in 2012 , David J. Wineland

The 2012 was awarded jointly to Serge Haroche and David J. Wineland "for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems"

David J. Wineland, U.S. citizen. Born 1944 in Milwaukee, WI, USA. Ph.D. 1970 Serge Haroche, French citizen. Born 1944 in , . Ph.D. from , Cambridge, MA, USA. Group Leader and NIST Fellow at 1971 from Université Pierre et Marie , Paris, . Professor at National Institute of Standards and Technology (NIST) and University of Colorado Collège de France and Ecole Normale Supérieure, Paris, France. Boulder, CO, USA www.college-de-france.fr/site/en-serge-haroche/biography.htm www.nist.gov/pml/div688/grp10/index.cfm

A laser is used to suppress the ion’s thermal motion in the trap, and to electrode control and measure the trapped ion.

lasers ions

Electrodes keep the beryllium ions inside a trap.

electrode electrode

Figure 2. In David Wineland’s laboratory in Boulder, Colorado, electrically charged or ions are kept inside a trap by surrounding electric fields. One of the secrets behind Wineland’s breakthrough is mastery of the art of using laser beams and creating laser pulses. A laser is used to put the ion in its lowest energy state and thus enabling the study of quantum phenomena with the trapped ion.

Controlling single in a trap Serge Haroche and his research group employ a diferent method to reveal the mysteries of the quantum world. In the laboratory in Paris photons bounce back and forth inside a small cavity between two mirrors, about three centimetres apart. The mirrors are made of superconducting material and are cooled to a temperature just above absolute zero. These superconducting mirrors are the world’s shiniest. They are so refective that a single can bounce back and forth inside the cavity for almost a tenth of a second before it is lost or absorbed. This record-long life-time means that the photon will have travelled 40,000 kilometres, equivalent to about one trip around the Earth.

During its long life time, many quantum manipulations can be performed with the trapped photon. Haroche uses specially prepared atoms, so-called Rydberg atoms (after the Swedish Johannes Rydberg) to both control and measure the microwave photon in the cavity. A Rydberg has a radius of about 125 nanometers which is roughly 1,000 times larger than typical atoms. These gigantic doughnut-shaped Rydberg atoms are sent into the cavity one by one at a carefully chosen speed, so that the interaction with the microwave photon occurs in a well controlled manner.

The traverses and exits the cavity, leaving the microwave photon behind. But the interac- tion between the photon and the atom creates a change in the phase of quantum state of the atom: if you think of the atom’s quantum state as a wave, the peaks and the dips of the wave become shifted. This phase shift can be measured when the atom exits the cavity, thereby revealing the presence or absence of a photon inside the cavity. With no photon there is no phase shift. Haroche can thus measure a single photon without destroying it.

THE NOBEL PRIZE IN PHYSICS 2012 THE ROYAL SWEDISH OF HTTP://KVA.SE 2(6) Photons bounce back and forth inside a small cavity between two mirrors for more than a tenth of a second. Before it disappears the photon will have travelled Rydberg atoms – roughly 1,000 times a distance of one trip around the Earth. larger than typical atoms – are sent through the cavity one by one. At the exit the atom can reveal the presence or absence of a photon inside the cavity.

2.7 cm

superconducting niobium mirrors

microwave photons

Figure 3. In the Serge Haroche laboratory in Paris, in vacuum and at a temperature of almost absolute zero, the microwave photons bounce back and forth inside a small cavity between two mirrors. The mirrors are so reflective that a single photon stays for more than a tenth of a second before it is lost. During its long life time, many quantum manipulations can be performed with the trapped photon without destroying it.

With a similar method Haroche and his group could count the photons inside the cavity, as a child counts marbles in a bowl. This may sound easy but requires extraordinary dexterity and skill because photons, unlike ordinary marbles, are destroyed immediately by contact with the world outside. Building on his photon counting methods, Haroche and collaborators devised methods to follow the evolution of an individual quantumReferences state, step-by-step, in real time. Paradoxes of Quantum mechanics describes a microscopic world invisible to the naked eye, where events occur contrary NY Times: “French and U.S. Win Nobel Prize” to• our expectations and experiences with physical phenomena in the macroscopic, classical world. Physics in• theNobelprize.org… quantum world has some inherent uncertainty or randomness to it. One example of this contrary behaviour is superposition, where a quantum particle can be in several diferent states simultaneously. We do not• normallyParticle think control of a marble in asa beingquantum both ‘here’ world and ‘there’ at the same time, but such is the case if it were a quantum marble. The superposition state of this marble tells us exactly what probability the marble has of •beingMeasuring here or there, and if we Manipulating were to measure Individualexactly where Quantumit is. Systems

Why• Phys do we Rev. never A become(1979): aware“Laser of theseCooling strange of Atoms”facets of our world? Why can we not observe a superposition of quantum marble in our every-day life? The Austrian physicist and Nobel Laureate (Physics• Phys 1933 Rev) Erwin Lett Schrödinger(1989): “Laser battled Cooling with this to question. the Zero Like Point many otherEnergy pioneers of Motion” of quantum theory, he struggled to understand and interpret its implications. As late as 1952, he wrote: “We never experiment• Phys Rev with Lett just one (1995): electron “Demonstration or atom or (small) of . a Fundamental In thought-experiments Quantum weLogic sometimes Gate” assume that we do; this invariably entails ridiculous consequences...”. • Rev. Mod Phys (2003): “Quantum Dynamics of Single Trapped Ions” In• orderSci Americanto illustrate the(2008): absurd “Quantum consequences Computing of moving between with Ions” the micro-world of quantum physics and our every-day macro-world, Schrödinger described a thought experiment with a cat: Schröding- er’s cat is completely isolated from the outside world inside a box. The box also contains a bottle of deadly cyanide which is released only after the decay of some radioactive atom, also inside the box.

THE NOBEL PRIZE IN PHYSICS 2012 THE ROYAL SWEDISH ACADEMY OF SCIENCES HTTP://KVA.SE 3(6) French and U.S. Physicists Win Nobel Prize By DENNIS OVERBYE Published: October 9, 2012

Now are able to direct experiments and catch nature in the act of being quantum and thus explore the boundary between quantum reality and normal life. Their work involves isolating the individual nuggets of nature — atoms and the particles that transmit , known as photons — and making them play with each other.

1522Dr. Wineland’sD. workJ. WINKLAND has focusedAND WA onYNE theM. materialITAN 0 side of where20 matter meets light. His prize is the fourth Nobel awarded to a associated with the excited electronic states, but we assume that we a resonant electric dipole transition (frequency, haveNationala narrow-band Institutelaser which can ofexcite Standardsthe andv,) in someTechnologyconvenient spectral overregion thewith pastradia- 15 years for work atom from one of the ground states to only one tive linewidth y/2w (full width at half-intensity excitedinvolvingelectronic state. the Thetrappingrelevant levels andare measuringpoints). Now of supposeatoms.that Dr.we irradiate Winelandthese and his showncolleaguesin Fig. 1. trap charged berylliumatoms atoms,with monochromatic, or ions, directed,in an electriclow intensi- field and cool Without laser. irradiation, the atoms eventually ty tuned near, ' but slightly lower than, reachthemthermal withequilibrium speciallyby collisions tunedor inter- lasersthe soresonance that theyfrequency. are Webarelyassume thatmoving,the in- which is action with the background blackbody radiation. tensity is well below that which would cause satu- Therefore,anotherthe ratio wayof the ofnumber sayingof atoms theyin are very,ration (theverycase cold.of saturation is treated in Ref. state 2 to those in state 1 is given by the Boltz- 10), and that the thermalizing collision rate y, mann law: between atoms is much less than the natural line- width y, but is larger than the optical absorption N2/N~ = exp —(E, — )/As T [ E, j, 1522 rate (y»y, » absorptionD. J. rate,WINKLANDsee Sec.ANDVF).WA YNE M. ITAN 0 20 where k& = Boltzmann's constant, T = temperature, Those atoms of a particular velocity class moving excited electronic states, but we assume that we a resonant electric dipole and and N„N, are the respective energies against the radiation are Doppler shifted toward transition (frequency, E„E, have a narrow-band laser which can excite the v, in some convenient spectral region with radia- and numbers of atoms in the For sim- the resonant ) states. atom from one of the frequencyground statesv,toandonlyscatterone the incomingtive linewidth y/2w (full width at half-intensity plicity we assume that the ground state has only excitedlightelectronicat a higherstate. rateThe relevantthan thoselevelsatomsare movingpoints).with Now suppose that we irradiate these two nondegenerate energy levels; hence statistical shown thein Fig.radiation1. which are Doppler shifted awayatomsfromwith monochromatic, directed, low intensi- ' weight factors are absent in Eq. (1). Withoutresonance.laser. irradiation,For eachthe scatteringatoms eventuallyevent, the tyatomradiation tuned near, but slightly lower than, reach thermal equilibrium by collisions or inter- the resonance frequency. We assume that the in- If we now apply the laser radiation to the atoms, receives a momentum impulse kk (k is the photon action with the background blackbody radiation. tensity is well below that which would cause satu- Atoms excited to wave in the optical pumping occurs. are Therefore, thevector)ratio of the numberabsorptionof atomsprocess.in Forrationan (the case of saturation is treated in Ref. PHlevelYSICAL3, but can decayRE VIE%into either Aground state. VOLUMEstate atom2 to thosewhichin 20,stateis moving1 isNUMBERgivenagainstby the theBoltz-4radiation, 10),thisand that the thermalizingOCTOBERcollision rate1979y, If we neglect the finite frequency widths of the mann impulselaw: retards its motion. This retardationbetweencan atoms is much less than the natural line- and of the this also be in width y, but is larger than the optical absorption laser optical transition, process N2/N~ = exp described—(E, — )/As T terms of radiation pres- [ E, j, rate (y»y, » absorption rate, see Sec. VF). continues until all of the atoms are in level 1. Lasersure. '"coolingThe averageof atomsmomentum per scatter ing 1522 D. J. WINKLAND AND WA YNE whereM. ITANk&0 = Boltzmann's constant,20 T = temperature, Those atoms of a particular velocity class moving They remain there until another process (say, the event transferred to the atom the reemitted and E„E,and N„N, are the respective energiesby against the radiation are Doppler shifted toward excited electronic states, but we assume that we a resonant electric dipole transition (frequency, collisions) depopulates this level. However, in the and numbersphotons ofisatomszero,in thebecausestates. ofForthesim-randomness theofresonantthe frequency and scatter the incoming have a narrow-band laser which can excite the v,) in some convenient spectralD. regionJ. Winelandwith radia- v, pumping process,atom from oneN,of/N,the ground-O, statesand,to onlyviaoneEq. (1),tivewelinewidthplicityy/2wphotons'(fullwe widthassumeat directionshalf-intensitythat the ground(if westateneglecthas onlyterms of lightsecondat a higher rate than those atoms moving with 1522 excited —electronic Frequencystate. TheD.relevantJ.andWINKLANDlevelsTimeare Standardspoints).AND NowtwoWAsupposeGroup,nondegenerateYNEthatM.weNationalirradiateITANenergythese0 Bureaulevels; ofhenceStandards,statistical 20Boulder,the radiationColoradowhich80303are Doppler shifted away from may say thatshownTin Fig.0 also.1. In this simple example,atoms with monochromatic,order indirected,iv~/c,low intensi-where v is the atom velocity and c weight factors' are absent in Eq. resonance. For each we see that Withoutwe canlaser.coolirradiation,an internalthe atoms eventuallydegree of ty radiation tuned isnear,thebut speedslightly lowerof light).than, The(1). average net effect then scattering event, the atom excited reachelectronicthermal equilibriumstates, butby collisionswe assumeor inter-that wethe resonanceaIfresonantfrequency.we now Weapplyelectricassumethethatdipolelaserthe in-transitionradiation to(frequency,the atoms, receives a momentum impulse kk (k is the photon freedom ofactionthe withatomthe background(the hyperfineblackbody radiation.structure) tensityby is well belowis thatthatWaynewhichthewouldatomiccauseM. satu-Itano*velocity is changed an amount have a narrow-band laser which can excite the opticalv,) in somepumpingconvenientoccurs. spectralAtoms areregionexcitedwith toradia-by wave vector) in the absorption process. For an optical pumping.Therefore, Inthe ratioprinciple,of the numberweof atomscouldin continueration (the case ofAv—saturation= h k/Mis treatedperin scatteringRef. event, where M is the atom fromstate one2 to thoseof thein stateground1 is givenstatesDepartmentby thetoBoltz-only one of10),Physics,and thatleveltivethe Haruardthermalizinglinewidth3, but cancollisiony/2wUniversity,decay(fullrateintoy,widtheitherCambridge,at half-intensityground state.Massachusettsatom which02138is moving against the radiation, this this excitedopticalmannelectronicpumpinglaw: state.processThe and,relevantusinglevelscollisionsarebetween atomsIfpoints).weis atomicmuchneglectNowless thanthemass.supposethefinitenaturalWhenfrequencythatline-wevirradiateandwidthsk areofthesetheantiparallel,impulsethis retards its motion. This retardation can width y, but is larger (Receivedthan the optical absorption16 May 1979) N2/N~ = exp —(E, — )/As T to transfershown in kineticFig. 1. energy[ E, to thej, internal degreerate (y»y,laseratoms» absorptionleadsandwithofrate,monochromatic,totheseeaopticalnetSec. VF).cooling,transition,directed,providedthis processlow iv+intensi-iv) & ivi.also be described in terms of radiation pres- = ' of freedom,Withoutwherelaser.couldk& = Boltzmann'sirradiation,reduce constant,thethetranslationalTatomstemperature,eventuallytem-Those atomscontinuestyofradiationa particular(Seeuntiltunedvelocityall2.ofnear,classtheIn movingaatomsbut slightlyare inlowerlevel than,1. sure. '" The average momentum per scatter ing and Variousand areaspectsthe respectiveof energiesthe laser coolingthe of atomsFig.are) investigatedpracticaltheoretically.cooling experimentMore generally,it the authors reach thermalE„E,equilibriumN„N, by collisions or inter-against Theytheradiationresonanceremainare Dopplertherefrequency.shifteduntiltowardanotherWe assumeprocessthat(say,the thein- event transferred to the atom by the reemitted perature ofandthenumbersgas ofifatomstherein thewerestates. sufficientFor sim- isola-the resonant frequencywouldv, andbescatterdesirablethe incomingto irradiate the atoms from all action with theinvestigatewe backgroundthea blackbodyprocess radiation.through which collisions)tensitythe kineticis welldepopulatesenergybelow thatthisofwhichlevel.a collectionwouldHowever,causeofinsatu-resonantthe photonsabsorbersis zero,canbecausebe reducedof the randomness of the tion from theplicity assumeof thethat environment.ground state has onlyThe light at a higher ratesidesthan thosewithatomsradiationmoving with Therefore,two nondegeneraterestthe ratio ofenergythe levels;numberhenceofstatisticalatoms inpro-the radiationpumpingrationwhich are(theprocess,DopplercaseshiftedofN,saturation/N,away-O,from and,thatis treatedviacoveredEq.in(1),Ref.thewe entire photons'lower directions (if we neglect terms of second weightbyfactorsirradiatingare absent in Eq.these(1). absorbers resonance.with near-resonantFor each scattering event,electromagneticthe atom radiation.' The process is described here as cessstateof laser2 to thosecoolingin statediscussed1 is givenbelowby theisBoltz-very may10), sayhalfand thatthatof Tthethe—0thermalizingDoppleralso. In thisprofile.collisionsimple Alternatively,example,rate ordernar- in iv~/c, where v is the atom velocity and c If weanti-Stokesnow apply the laserspontaneousradiation to the atoms,Ramanreceivesscattering.a momentum impulse kk (k mechanisms,is the photon andy, limits are discussed for both free similarmann tolaw:opticalthe opticalpumping occurs.pumpingAtoms arecaseexcitedexceptto thatwave vector)webetweeninseetherow-bandabsorptionthatCoolingatomswe process.canislasermuchcoolForanschemeslessan internalthan themightdegreerates,naturalbeof line-employedis wherethe speed of light). The average net effect then the translationallevel 3,andbut canbounddecay intoatoms.either ground state. atom whichfreedomwidthis movingthey, againstofbuttheistheatomlargerradiation,(thethanthishyperfinethe opticalstructure)absorptionby is that the atomic velocity is changed by an amount If we= neglect—degreesthe finite— frequencyof freedomT widths of arethe opticallyimpulse retards its motion.laserThis frequencyretardation can is swept from some very low N2/N~ exp [ (E, E,)/As j, opticalrate pumping. In principle, we could continue Av—= h k/M per scattering event, where M is the pumped directly.laser and of the optical transition, this process also be described value(y»y,in terms »toof radiationabsorptiona valuepres-approachingrate, see Sec.theVF).rest frequency. ' where k&continues= Boltzmann'suntil all of theconstant,atoms are inT level= temperature,1. sure. '" ThethisThoseaverageopticalatomsmomentumpumpingof aperparticularscatterprocessing and,velocityusingclasscollisionsmoving atomic mass. When v and k are antiparallel, this They remain there until another process (say, the event transferred Thisto the atomrequirementby the reemitted is substantially relaxed if the and and are the respective energies toagainsttransferthe kineticradiationenergyare toDopplerthe internalshifted degreetoward to a net cooling, provided iv+ iv) & ivi. E„E,collisions)B. LaserN„N,depopulatescoohngthisoflevel.free atomsHowever, in the photons is zero, because of the randomness of the and numberspumping ofprocess,atomsN,in/N, the-O, and,states.via Eq.For(1), wesim- photons' directionsofthefreedom,resonant(if we neglectcouldfrequencytermsreduceof secondv,theandtranslationalscatter the incomingtem- (See Fig. 2.) In a practical cooling experiment it — ABSORPTION 1522 D. WINKLAND AND WA plicityYNE wemayM.assumesayITANthat Tthat0 also.the Ingroundthis simplestateexample,has only order 20in iv~/c,peraturelightwhereat avofishigherthethe atomgasratevelocityif therethanand thosewerec sufficientatoms movingisola-with would be desirable to irradiate the atoms from all J. Assume thatwe see wethat wehavecan coolan anunboundinternal degreegasof of atomsis the speed of light). The average net effect then two nondegenerate energyI. INTRODUCTIONlevels; hence statistical tionthe fromradiationthe restwhichof arethe environment.anti-StokesshiftedThe pro-spontaneoussides with radiationHamanthatscattering;covered the entirethislower (or resonantfreedomabsorbersof the atom (thein hyperfinegeneral)structure)whichby possessis that the atomic velocity is changed by an amountDoppler away from ' weight factorsoptical pumping.are absentIn principle,in Eq.we could(1). continue Av—= h k/M cessperresonance.scatteringof laserevent,ForcoolingwhereeachMdiscussedscatteringis the lastbelowevent,conceptis verythe atomis thehalfoneof theprimarilyDoppler profile.usedAlternatively,here be-nar- excited electronic states, but we assume that we a resonant thiselectricoptical pumpingdipoleprocesstransitionand, using collisions(frequency,atomic mass. When v andthek are antiparallel, this row-band If we now apply the laser radiationelectronicto the atoms, similarreceivestoa momentumoptical pumpingimpulse casekk exceptis thethatphoton laser schemes might be employed where to transfer kinetic energy to the internal degreeexcited state leads to a net cooling, provided iv+ iv) & ivi. (k have a narrow-band laser which can excite the v,In) inthesomepastconvenientfew years,spectral regionthere withhasradia-been theincreasingtranslational degrees of freedomcause areofopticallyits generality.the laser frequencyIt shouldis sweptbefrommentionedsome very low optical pumpingof freedom, occurs.could reduceAtomsthe translationalare excitedtem- to (See Fig. 2.wave) In a practicalvector)coolingin theexperimentabsorptionit process. For an ' atom from one of the ground states to only one tive levellinewidth3, peraturebut y/2wcanof thedecay(fullgas ifintotherewidtheitherwereatsufficientgroundhalf-intensityisola-state. would be desirablepumpedatom whichto directly.irradiateis movingthe atoms fromagainstall the radiation, this value to a value approaching the rest frequency. tioninfromthethe useof theofenvironment.near-resonant sides photonwith scat- that Raman new interest rest The pro- radiation that covered the entire lower cooling by This requirementscatteringis substantiallyis not relaxeda if the excited electronic state. The relevant levels are points).If we neglectNowcess ofsupposelaserthe finitecoolingthatfrequencydiscussedwe belowirradiatewidthsis very ofthesethe half of the Dopplerimpulseprofile.retards' its motion.REEMISSIONnar- This retardation can B.Alternatively,Laser coohng of free atoms teringlaser toandsimilarcoolof theto theopticalaopticalcollectionpumpingtransition,caseLaserexceptdriventhisofthatprocesstransitionatoms,row-bandions,laseralsoschemesbeordescribedmight be employedin whereidea; "lumino-refrigeration" was hypothesized shown in Fig. 1. atoms with monochromatic, directed, low intensi- terms of radiation pres- ABSORPTION ENERGY the translational degrees of freedom are optically the laser frequency is swept from some very low continues until all of the' atoms are in level 1. sure.Assume'" thatThe weaveragehave anmomentumunbound gasper ofscatteratomsing Without laser. irradiation, the atoms eventually .ty radiationpumpedtuneddirectly.Thisnear,interestbut slightlyislowermotivatedthan,value to a valuein approachingpart bythe rest frequency. ' as early as 1950 by Kastler. ' We also note that They remain there until another process (say, theThis requirement(oreventresonantistransferredsubstantiallyabsorbersrelaxedto theifintheatomgeneral)by thewhichreemittedpossess reach thermal equilibrium by collisions or inter- the resonance frequency.B. Laser Wefree atomsassume that the in- the collisions) depopulatesneed coohngthistooflevel.reduceHowever,first-in theand photonssecond-FIG.is zero,2. Qualitativebecause ofotherdescriptionthe randomnesscoolingof radiation-pressureof the are for exam- practical ABSORPTION electronic excited state processes possible; action with the background blackbody radiation. tensity is wellAssumebelowthat we thathave whichunbound would cause satu- pumping process, N, /N,hyperfine-O,an and,statesgasviaof atomsEq. (1), we photons'cooling.directionsIn the(ifabsorptionwe neglect process,terms of secondthe atomic velocity order Doppler(orOIresonant absorbersshiftsin general)in ultra-high-resolutionwhich possess one could use optical followed Therefore, the ratio of the number of atoms in rationmay(thesay casethat Tof—0saturationalso. In thisissimpletreatedexample,in Ref. order isinchangediv~/c, where(reducedv is fortheple,atomk v& velocity0) by an andamountc &v=8k/M. pumping by electronic excited state FIG. 1. Levels of interest in a REEMISSION state 2 to those in state 1 is given by the Boltz- spectroscopy10),weandseethatthat thewe canthermalizingandcoolinan partinternalhypotheticalcollisionbydegreetherateofalkali-estheticy, is appealthe Inspeedthe reemissionofoflight). Theprocess,averagecollisionalthenetaverageeffect relaxation,thenchange in velo- as discussed in Sec. II, like atom. Optical pumping into state 1 occurs while Laser driven transition mann law: betweenfreedomatomsof the atommuch(the hyperfine structure) by is thatENERGYcitythe isatomiczero. velocityThereforeis changedin the overallan amountscattering pro- is less than the natural line- REEMISSION by ' controllingdriving the 2 3thetransitionpositionswithLasera drivenlaser.andtransition velocities of a col-the or cooling by collisionally aided fluorescence. optical pumping.ENERGY In principle, we could continue Av—= hcess,k/M per scatteringkinetic energyevent,canwherebe reduced.M is the = — — width y, but is larger than the optical absorption N2/N~ exp [ (E, E,)/As Tj, this optical pumping process and, using collisions atomic mass. When v and k are antiparallel, this FIG. 2. Qualitative description of radiation-pressure lectionrate (y»y,of»atomicabsorptionparticlesrate, see Sec.to withinVF). FIG.the2. Qualitativelimitsdescription of radiation-pressure The process described in this paper is, however, hyperfine states & cooling. In the absorption process, the atomic velocity to transfer kinetic energyhyperfineto the internal degree cooling. In leadsthe absorptionto a net cooling,the atomic provided iv+ iv) ivi. = = states process,OI velocity where k& Boltzmann's constant, T temperature, Those atoms of a OIparticular velocity class moving is changed imposedof freedom,by couldquantumreduce thefluctuations.translational tem- isRecentchanged (reduced(See pro-Fig.for k v&2.0)) byInan aamountpractical&v=8k/M.morecooling experimentdirect andit does not(reducedrelyforonk v&atom-atom0) by an amount &v=8k/M.col- and and are the respective energies against the FIG.radiation1. Levels of interest in a hypothetical alkali- In the reemissionFIG.process,1. Levelsthe averageof interestchange in velo-in a hypothetical alkali- In the reemission process, the average change in velo- E„E, N„N, perature likeofatom.the Opticalgas ifpumpingarethereDopplerintowerestate 1sufficientoccursshiftedwhile isola-towardcity is zero.wouldThereforebein desirablethe overall scatteringto irradiatepro- the atoms from all posals anddriving experimentsthe 2 3 transition with a laser.using narrow-bandlike atom.tunableOptical pumping intolisionsstate 1 occurstowhilealter thecityatomis zero. kineticTherefore inenergy.the overall scattering pro- and numbers of atoms in the states. For sim- the resonant frequency and the cess, the kinetic energywithcan be reduced. tion from the rest of thev,environment.scatter The pro-incoming drivingsides the 2radiation3 transitionthat coveredwith a laser.the entire lower cess, the kinetic energy can be reduced. plicity we assume that the ground state has only laserslightcessat aofsuggesthigherlaser coolingratethatthandiscussedsuchthosebelowatomscontrolis verymovingmaywithsoonhalf ofbecomethe Doppler profile. ' Alternatively,The papernar-is divided as follows. In Sec. II we two nondegenerate energy levels; hence statistical athereality.similarradiationto thewhichItopticalis arenotpumpingDopplerdifficultcaseshiftedexceptto awaythatimaginefrom row-bandthat thelaser schemes mightdescribebe employedthewheregeneral aspects of the cooling pro- weight factors are absent in Eq. (1). resonance.the translationalFor eachdegreesscatteringof freedomevent,are opticallythe atom the laser frequency is swept from some very low conceptspumped directly.and techniques which are beingvaluedevelopedto a value approaching cessthe restandfrequency.treat' the problem combining simple If we now apply the laser radiation to the atoms, receives a momentum impulse kk (k is the photon This requirement is substantially relaxed if the optical pumping occurs. Atoms are excited to maywave vector)have applicationinB.theLaserabsorptioncoohng of freeinprocess.atomsa varietyFor ofan areas not classical and quantum ideas. Section III introduces level 3, but can decay into either ground state. atom which is moving against the radiation, this ABSORPTION initiallyAssumeanticipated.that we have an unbound gas of atoms the concepts and notation of the quantum-mechani- If we neglect the finite frequency widths of the impulse retards its motion. Current(or resonant absorbers iningeneral)theThis whichretardationpossess can cooling be- which is then to free atoms laser and of the optical transition, this process also be describedinterestin terms of radiationpossibilitypres- of cal treatment applied electronic excited state continues until all of the atoms are in level 1. gansure. with'" Theindependentaverage momentumproposalsper scatterto ingreduce the tem- in Sec. IV and bound atoms in Sec. V. In order to They remain there until another process (say, the peratureevent transferredof a gasto the ofatomneutralby the reemittedatoms' or ions which make the problem somewhat more tractable, we in REEMISSION collisions) depopulates this level. However, the photons is zero, because of theLaserrandomnessdriven transition of the boundENERGY "trap"' with near- which ex- pumping process, N, /N, -O, and, via Eq. (1), we arephotons' directionsin an(ifelectromagneticwe neglect terms of second limit the discussion to simple systems may say that T —0 also. In this simple example, resonantorder in iv~/c,laserwhereradiation.v is the atom Thisvelocitymethodand of cooling hibit the salient features of the process. c FIG. 2. Qualitative description of radiation-pressure we see that we can cool an internal degree of is the speed of light). The average net then has subsequently beenhyperfine statesincorporatedeffect intocooling.the in-In the absorption process, the atomic velocity freedom of the atom (the hyperfine structure) by is that the atomicOI velocity is changed by an amount is changed (reduced for k v& 0) by an amount &v=8k/M. FIG. 1. Levels of interest in a hypothetical alkali- In the SIMPLE DESCRIPTION OF THE COOLING PROCESS optical pumping. In principle, we could continue terestingAv—= h k/M perschemesscattering forevent,trappingwhere M ofis theparticles reemissionusing process, the averageII. change in velo- like atom. Optical pumping into state 1 occurs' while city is zero. Therefore in the overall scattering pro- this optical pumping process and, using collisions near-resonantatomicdrivingmass.the 2 When3 transitionopticalv and withk areafields.laser.antiparallel,The thisfirstcess,demon-the kinetic energy can be reduced. A. Analogy with optical pumping to transfer kinetic energy to the internal degree stration4leads to a netofcooling,coolingprovidedusingiv+theiv) basic& ivi. techniques of freedom, could reduce the translational tem- (See Fig. 2. In a ) practical cooling experiment it of have perature of the gas if there were sufficient isola- describedwould be desirablehere towasirradiatemadethe foratomsa fromslightlyall modified The basic features the cooling process tion from the rest of the environment. The pro- situation;sides with radiationspecifically,that coveredthethe magnetronentire lower motion of been outlined previously (Refs. 1, 2, 5, 6, 7). The ' cess of laser cooling discussed below is very anhalf electronof the Dopplerboundprofile.in aAlternatively,Penning trapnar- was "cooled" attempt is made here to describe the qualitative similar to the optical case except that row-band laser schemes pumping might be employedsidebandwhere of the problem more how- the translational degrees of freedom are optically bythe lasera techniquefrequency calledis swept frommotionalsome very low excita- aspects completely; pumped directly. tion,value to' whicha value approachingis formallythe equivalentrest frequency. to' the laser ever, the general problem becomes quite com- coolingThis requirementof atoms.is substantiallyCoolingrelaxedof ionsif thebound in an plicated, and therefore several limiting cases B. Laser coohng of free atoms electromagneticABSORPTIONtrap was more recently demon- will be treated. Assume that we have an unbound gas of atoms which that in optical we have a (or resonant absorbers in general) which possess strated. " The cooling is potentially First, recall pumping achievable should permit of unprece- way of drastically altering the temperature of a electronic excited state dented resolution and accuracy. specific degree of freedom in an atom or mole-

As discussedREEMISSIONbelow, the technique can variously cule. Assume, for example, that we have an Laser driven transition ENERGY be described in terms of radiation pressure, mo- alkali-like atom which has ground-state "hyper- tional sideband excitation, optical pumping, or fine" structure. This atom can also have many FIG. 2. Qualitative description of radiation-pressure hyperfine states cooling. In the absorption process, the atomic velocity OI is changed (reduced for k v& 0) by an amount &v=8k/M. 1979 The American Physical Society FIG. 1. Levels of interest in a hypothetical alkali- In the reemission process, the average change in velo- like atom. Optical pumping into state 1 occurs while city is zero. Therefore in the overall scattering pro- driving the 2 3 transition with a laser. cess, the kinetic energy can be reduced. VOLUME 62, NUMBER 4 PHYSICAL REVIEW LETTERS 23 JANUARY 1989

trap axes onto the 282-nm beam axis are calculated to be 1.5 I I I I I I I I I I I I I [ p„=0.03, p~ =0.64, and p, =0.33. Since we make the differences between the x, y, and z frequencies bigger than directions are cooled simultaneously. 1/r„all VOLUME 1.62,0—NUMBER 4 PHYSICAL REVIEW LETTERS 23 JANUARY 1989 However, in the analysis, we assume that the probing ab- A sorption strength is due only to the ion's motion in the y trap axes onto the 282-nm beam axis are calculated to be 1.5 I I I I I I I I I I I I and z directions. Since the x axis is nearly perpendicular V I [ to the 282-nm beam, no meaningful statement abut the p„=0.03,0.p~5—=0.64, and p, =0.33. Since we make the differences between the x, and z frequencies bigger energy in this degree of freedom can be made. By y, than 1/r„all directions are cooled simultaneously. neglecting the contribution of the x motion to the side- 1.0— However, in the analysis, we assume that the probing ab- band strength we overestimate (n, ) for the y and z direc- A sorption strength0— is due only to the ion's motion in the y tions. In order deduce the and z oscillations to (n„) for y and z directions. Since the x axis is nearly perpendicular V from our data (Fig. 2), an assumption about the energy I I I I I I I I I I I I I I I 0.5— to the 282-nm beam,0 no meaningful50 100statement 150abut the distribution between the two directions has to be made. energy in this degree of freedom can be made. By VOLUME 62, NUMBER 4 PHYSICALD cia y Time (mREVIEWs) LETTERS 23 JANUARY 1989 If we assume temperature equilibrium between the y and neglecting the contribution of the x motion to the side- z degrees of freedom, both contain an energy correspond- bandFIG. strength3. Vibrationalwe overestimatequantum (n,number) for the(n„,y )andfor z thedirec-axial — '~ — 0— ing to (n, ) =(1 SL/SU) 1=0.051+ 0.012 quan- motiontions. (co.In /2norder=4.to66deduceMHz)(n„)as fora functionthe y andof ztimeoscillationsdelay be- I I I I I I I I I axes onto the 282-nm beamtween theaxisend of thecalculatedsideband cooling andbe probing. A linear 1.5 I I I I I I II I I I I I I I I I I ta. Therefore, for the trapy and z degrees of freedom, the from our dataare(Fig. 2), an assumptionto about the energy I [ 0 50 100 150 ion is in the n, =0 state 95% of the time.=0.The corre-and extrapolation=0.distribution33. ofSincebetweenthe datathewepointstwomakedirections(circles)the hasto zeroto bedelaymade.time p„=0.03, 64, D y Time (m s) p~ p, & cia sponding temperature given by kz T = 6 ro, /In (1+ 1/ yieldsIf we (n.assume(triangle)temperatureconsistentequilibriumwith the betweentheoreticalthe yexpecta-and tion. (n, )) is T=47+'3 pK.differencesFor any other betweenenergy partition,the x, y,z degreesand ofz freedom,frequenciesboth containbiggeran energy correspond- FIG. 3. Vibrational quantum number (n„,) for the axial — '~ — (n„) and T for one degreethanof freedom1/r„allwould bedirectionsless than areing tocooled(n, ) =(1 simultaneously.SL/SU) 1=0.051+ 0.012 quan- motion (co./2n =4.66 MHz) as a function of time delay be- tween0—the end of the sideband cooling and probing. A linear these values. Independent of the energy distribution, for ta. Therefore, for the y and z degrees of freedom, the 1. However, in the analysis, we assumeion is in thethat=0thestateprobing95% of theab-time. The corre- extrapolation of the data points (circles) to zero delay time both degrees of freedom the temperature is much lower amounts to d, v/vn, 10 (Ref. 8). It can be made sub-A ( yields (n. & (triangle) consistent with the theoretical expecta- sponding ion'stemperature given inby kz T = 6 ro, /In (1+ 1/ than the 194-nm Dopplersorptioncoolingstrengthlimit and isthedueion only stantiallyto the lower bymotionadiabatically theloweringy the potential tion. (n, )) is T=47+'3 pK. For any other energy partition, spends most of its time in the harmonic-oscillator well depth after the ion is cooled into the ground state.V and z directions. Since the x (n„)axisandisTnearlyfor one degreeperpendicularof freedom would be less than ground-stateVOLUME 62,level.NUMBER 4 PHYSICAL REVIEWWith ourLETTERSexperiment, the absorption of a23singleJANUARYquantum1989 to the 282-nm beam, no meaningfulthese values. statementIndependent ofabutthe energythe distribution, for 0.5— The theoretical sideband cooling limit gives a value of of energy at a (tunable) frequency in the MHz range both degrees of freedom the temperature is much lower amounts to d, v/v 10 (Ref. 8). It can be made sub- ) 10 . However, energysince the in this in degreethe experi- of freedomcan be detectedcanwithbean made.efficiency of nearly 100%. With ( (n, = LaserprobingCooling to the Zero-Pointthan theEnergy194-nmofDopplerMotion cooling Bylimit and the ion stantially lower by adiabatically lowering the potential ion's ment is done at saturatingneglectingpower, the themeasuredcontribution(n„) cor- ofappropriatespendsthe mostx motioncouplingof its totimetothethein theside-motionharmonic-oscillator(for example, well depth after the ion is cooled into the ground state. responds to the of theF. ionDiedrich,at the 'endJ. C.of theBergquist,prob- Waynevia oneM. Itano,of the andendcaps),D. J. Winelanda similar apparatus could serve energy band strength we overestimate ground-state(n, ) forlevel.the and z direc- With our experiment, the absorption of a single quantum ing interval, whichTime isandtypicallyFrequency15Division,ms afterNationalthe endInstituteof the of Standardsas aTheverytheoreticalandsensitiveTechnology,sidebandspectrumy Boulder,coolinganalyzer.Coloradolimit Ingives80303anothera valuepossi-of of 0—energy at a (tunable) frequency in the MHz range sideband cooling, and tions.external Inheatingordermightto deducehave(Receivedoc- (n„)28 bleJuly(n,forapplication,)1988)= the10 .y However,andthe motionz oscillationssinceofthea trappedprobing chargedin the experi-particle can be detected with an efficiency of nearly 100%. With curred. In order to checkfromfor externalour dataheating(Fig.processes, an assumptioncouldment isbedonedampedataboutsaturatingby couplingthepower,energyittheelectronically'measured (n„) cor-to a appropriate I couplingI I I toI theI I ion'sI I motionI I (forI I example,I I ' 2), we extended r„upA tosingle100trappedms and measuredHg+ ion wasSL,cooled/SU asbya scatteringsecondrespondslaserlaser-cooledradiationto the energythation wasofin theatunedseparateion toat thethetrap,resolvedend therebyof the prob-reduc- via one of 0the endcaps), 50a similar apparatus100 could serve150 function of r„. lowerWe determinedmotionaldistributionsidebanda heatingof betweenthe ratenarrow(dueS]/2-theap-D5/2twotransition.ingingdirectionstheinterval,firstThechargedwhichdifferenthasis particle'stypicallyabsorptionto be 15made.kineticstrengthsms afterenergyonthetheendto nearof thethe as a very sensitive spectrum analyzer. In another possi- D cia y Time (m s) parently to pickupupperofandstrayIflowerwenoisesidebandsassumefields ataftertemperatureradiocoolingfrequen-indicatedequilibriumthatzero-pointsidebandthe ion wascooling,energy.betweenin theandgroundResonantexternalthestateyheatingofexcitationandits confiningmightofhavethe oc-first ble application, the motion of a trapped charged particle cies) of (n„)=6/s.well approximatelyIf we assume95%theof theheatingtime. is due to particle'scurred. Inmotionorder tocouldcheckthenfor beexternaldetectedheatingveryprocesses,sensitively could be damped by coupling it electronically' to a z degrees of freedom, both containwe extendedan energyto 100correspond-ms and measured FIG.as a 3. secondVibrationallaser-cooled quantumion in a separatenumbertrap, thereby(n„,) reduc-for the axial thermalization of the ion to room temperature —by noise by its influencer„upon the laser-cooled ion. SuchSL,/SUa device PACS numbers: 32.80.Pj, 32.30.Jc, 35.—10. d ' '~ function— of r„. We determined a heating rate motion(due ap- ing the first charged particle's kinetic energy to near the be- VOLUME 62, NUMBER 4 PHYSICALatREVIEWfrequency LETTERSm„ the ingheatingto time(n, )constant=(123 JANUARYisSL/SU)951989h. might 1=0.be useful051+in mass0.spectroscopy.012 quan- (co./2n =4.66 MHz) as a function of time delay to of noise fields at radio frequen- zero-point energy. Resonant excitation of the first This rate varied slightly with cu„around ro, =3 MHz, parentlyIn summary,pickupwe havestray realized laser coolingtweenin thethe end of the sideband cooling and probing. A linear ta. Therefore, for the y and cies)z degreesof (n„)=6/s.of Iffreedom,we assume thetheheating is due to particle's motion could then be detected very sensitively trap axes onto the 282-nm beam axis are calculated tobutbe was substantially1.5 higherI I I I forI I I co,I ~2.I I 5I MHz.I From resolved sideband regime for the first time. The kinetic The subject of laser coolingI of ions[ and neutral atoms confiningthermalizationpotentialof the(aboution to95%roomof thetemperaturetime here)extrapolationby thenoisefun- by itsofinfluencethe dataon thepointslaser-cooled(circles)ion. Suchto zeroa devicedelay time p„=0.03, =0.64, and =0.33. Since we makethesethe theionmeasuredis in the) is consistentn, =0 statewith 95% ofofthea trappedtime. atomicTheioncorre-was reduced to a value p~ p, heating data, (n, in- energy ' differences between the and z is currently of great experimental and theoretical damentalat frequencylimitm„of laserthe heatingcooling timefor aconstantconfined isparticleyields95 h. has(n. & might(triangle)be useful consistentin mass spectroscopy.with the theoretical expecta- x, y, frequencies biggerthe theoretical' limit atspondingthe end of thetemperaturesideband coolinggivenwherebyit spentkz Tmost= 6ofro,its /Intime(1+in the1/ground-state level than 1/r„all directions are cooled simultaneously. terest. It has been applied to high-resolution spectros- beenThis reached.rate varied slightly with cu„around ro, =3 MHz, In summary, we have realized laser cooling in the period. We also1.0—measured (n, ) for a single degree of of its confining well. To the extent that the iontion.is in the However, in the analysis, we assume that the probing ab- 5 ' resolved sideband regime for the first time. The kinetic copy, low-energyA collisions,(n, )) quantumis T=47+'3jumps, andpK.photonFor butanyOurwasotherexperimentssubstantiallyenergywerehigherperformedpartition,for co, ~2.with aMHz.single FromHg+ ion's freedom directly by changing Uo to —25 V in order to zero-point energy state of motion, this realizes for the sorption strength is due only to the motion in the antibunching.y In all cooling experiments done so far, ionthesestoredheatingin adata,Paul the(rf)measuredtrap' "(n,which) is hadconsistentro, /2tr with=2.96 energy of a trapped atomic ion was reduced to a value and z directions. Since the x axis is nearly perpendicular the radialV and axial(n„) sidebandand T forfrequencies.one degreeThe offirstfreedomtime the wouldfundamentalbe lesslimitthanof laser cooling for a splitthe oscillation frequency co, of the particle in its the theoretical limit at the end of the sideband cooling where it spent most of its time in the ground-state level to the 282-nm beam, no meaningful statement abut the 0.5— MHz (see Fig. I). In order to optimize the cooling, a 282-nm radiation was thesetuned sovalues.that only Independentthe first axial boundperiod.the particleWe alsoanddistribution,measuredthe ideal (n,of)anforforisolateda single atomicdegree parti-of of its confining well. To the extent that the ion is in the energy in this degree of freedom can be made. Byconfining well was less than the linewidth I of the cool- of two-stageenergyprocess was used. First, the ion was cooled to sideband at co,/2' =4.66 MHz was cooled and probed. clefreedomat rest to within the quantum-mechanicalto —25 V in orderlimits toim- zero-point state of motion, this realizes for the neglecting the contribution of the x motion to the side-ing transition. This bothconditiondegreesalso appliesof freedomto free-atom the neartemperaturethe directlyDoppler bycoolingischangingmuchlimit'Uolower(T=AI /2ktt, amountswhere ktt to d, v/v energy10 (Ref. It can be made sub- From the results, shown in Fig. 3, we calculate (n„) posed bythe theradialsurroundingand axialapparatus.sideband frequencies. The first time the(fundamental limit 8).of laser cooling for a band strength we overestimate (n, ) for the y and z direc-experiments, where0— co, 0. The lowest temperatures T issplitthe Boltzmann constant) by scattering light of wave- tions. In order to deduce (n„) for the and z oscillations=0. 045 at the end of the cooling194-nmperiod, con- ' We acknowledge the support of the U.S. Air Force y have049~0.been obtained thanin recentthefree-atom experiments,Doppler coolinglength282-nm 194radiationnmlimitonwastheandtunedstrongtheso thationonly thetransiti-onfirststantiallyaxial boundlowerparticleby andadiabaticallythe ideal of an isolatedloweringatomic theparti- potential I I I I I I I I I I I I [(A) I I I Sii2 P~I2 from our data (Fig. 2), an assumption about the energy Office Scientific Research and the Office of Na- im- sistentwhere withkinetic the 0theoretical50 belowcooling100 limit.150 The sidebandof at co,' /2' =4.66 MHz was cooledU.S. and probed. cle at rest to within the quantum-mechanical limits distribution between the two directions has to be made. energiesspendsnear or mostthatofcorrespondingits time inin Fig.the1(a)].harmonic-oscillatorAt the Doppler cooling limit,well(n, ) =depth12 after the ion is cooled into the ground state. confinement of the axial motionD y isTimegiven(m s) by the spread of valFromResearch.the results,F. D.shownthanksin theFig.Deutsche3, we calculateForschungsge-(n„) posed by the surrounding apparatus. If we assume temperature equilibrium between the andto the recoil of a single photoncia from an atom at rest have (T=1.7 mK) for each degree of freedom. In the next ythe zero-point wave functionground-statez(rms) = 2.level.4 nm. meinschaft=0.049~0.for045financialat the endsupport.of the Wecoolingthankperiod,C.WithWieman,con- our experiment,We acknowledge thethe absorptionsupport of the ofU.S.aAirsingleForce quantum z degrees of freedom, both contain an energy correspond-been achievedFIG. 3. Vibrational(T= 1 pK).quantumIn numberthis Letter(n„,) forwethereport,axial for stage of cooling, the 194-nm radiation was turned off '~ Na- ing to (n, ) =(1 —SL/SU) —1=0.051+ 0.012 quan-For ourmotiondata,(co./2n the=4.66uncertaintyMHz) as a functionin theof timesecond-orderdelay be- J.sistentBollinger,with andtheJ. Helffrichtheoretical for coolingcommentslimit.and sugges-The Office of Scientific Research and the U.S. Office of the first time, laser coolingTheof atheoreticalsingle bound atomsidebandin the coolingand the narrowlimit S~y2-givesD5g2a electricvalue quadrupoleof oftransitionenergy at a (tunable) frequency in the MHz range ta. Therefore, for the y and z degrees of freedom, Dopplerthe tweenshift theis enddominatedof the sidebandby thecoolinguncertaintyand probing. inA(n„)linearand tionsconfinementon the ofmanuscript.the axial motionWe isthankgiven byR. theCookspreadandofR. val Research. F. D. thanks the Deutsche Forschungsge- resolved sideband regime I co„. An ion has been [(B) in Fig. 1 (a)] was driven on the first lower sideband ion is in the n, =0 state 95% of the time. The corre- extrapolation of the (n,data )points=«(circles)10 to. zeroHowever,delay time sincethethezero-pointprobingwave functionin thez(rms)experi-= 2.4 nm. can be detectedmeinschaft withfor financialan efficiencysupport. WeofthanknearlyC. Wieman,100%. With cooledyieldsso (n.that& (triangle)it occupiesconsistent withthe thegroundtheoretical stateexpecta-of its — sponding temperature given by kz T = 6 ro, /In (1+ 1/ frequencyFor our modata, co,the. Sinceuncertaintythe naturalin the lifetimesecond-orderof the J. Bollinger, and J. Helffrich for comments and sugges- T=47+'3 confiningtion. potential mostmentof theistime.done at saturating power, the measured (n„) cor- appropriate405 coupling to the ion's motion (for example, (n, )) is pK. For any other energy partition, DopplerD5(2 stateshiftlimitsis dominatedthe maximumby the uncertaintyscatter rateinto(n„)approxi-and tions on the manuscript. We thank R. Cook and R. (n„) and T for one degree of freedom would be less than in re- ' The idea of laser coolingrespondsthetoresolvedthe energysideband of thematelyion at2 Ithe( D5I2)end=6ofphotons/s,the prob-' a coolingviatimeoneof of the endcaps), a similar apparatus could serve these values. Independent of the distribution, for energy gime is as follows: Let the rest frequency of the atom's at least 6 s is required to reach (n„)=0 for all degrees of both degrees of freedom the temperature is much lower amounts to d, v/v ing10 interval,(Ref. 8). It canwhichbe made issub-typically 15 ms after the end of the as a very sensitive spectrum analyzer. In another405 possi- cooling transition be( mo. If the atom oscillates at fre- freedom. This time even is than the 194-nm Doppler cooling limit and the ion stantially lower by adiabatically lowering the potential becomes longer, or cooling spends most of its time in the harmonic-oscillator quencywell~,depthin itsafterconfiningthesidebandion is cooledwell, intothecooling,theatom'sgroundabsorptionandstate. externalprevented,heatingif externalmightheatinghaveis present.oc- Therefore,ble application,in the motion of a trapped charged particle ground-state level. and emissionWith our spectrumexperiment,curred.(asthe absorptionviewedIninoforderthea singlelaboratory)quantumto checkhas fororderexternalto enhanceheatingthe sidebandprocesses,cooling rate, thecouldlifetimebe damped by coupling it electronically' to a The theoretical sideband cooling limit gives a value ofresolvedof energycomponentsat a (tunable)at coo andfrequencycoo+ mco,in the (mMHzan rangeinteger). of the Dy2 state was shortened by coupling it to the fast (n, ) = 10 . However, since the probing in the experi- can be detected with an efficiency of nearly 100%. With laser-cooled ion in reduc- If we irradiate the weatomextendedwith narrow-bandr„up radiationto 100 msdecayingand measuredP3I2 state bySL,398-nm/SU asradiationa [(C)secondin Fig. a separate trap, thereby ment is done at saturating power, the measured (n„) cor- appropriate coupling to the ion's motion (for example, tuned to the first lower sideband at coo —co„ the atom ab- responds to the energy of the ion at the end of the prob- via one of the endcaps),functiona similar ofapparatusr„. couldWeservedetermined1(a)].a Fromheatingthe P3I2ratestate,(duethe ionap-has high probabilitying the first charged particle's kinetic energy to near the sorbs photons of A, (coo —ro, and reemits pho- ing interval, which is typically 15 ms after the end of the as a very sensitivefrequencyspectrum analyzer. )In another possi- fields zero-point Resonant excitation the first sideband cooling, and external heating might have oc-tons ofble averageapplication, energytheparentlymotionAcoo.of a trappedHence,to pickupchargedon theparticleofaverage,stray noise at radio frequen- energy. of curred. In order to check for external heating could be damped coupling it electronically' to a processes,each scattered photoncies)byreducesofthe(n„)atom's=6/s.vibrationalIf en-we assume the (a)heating is due to particle's motion could then be detected very sensitively we extended r„up to 100 ms and measured SL,/SU as a second laser-cooled ion in a separate trap, thereby reduc- ergy by hco„or reduces the atom's vibrational quantum 198 function of r„. We determined a heating rate (due ap- ing the first chargedthermalizationparticle's kinetic energy ofto nearthethe ion to room temperatureH 398noisenm its influence on the laser-cooled ion. Such a device number this by by parently to pickup of stray noise fields at radio frequen- zero-pointn, by l.energy.In Resonantway, we excitationcan obtainof (n„)the «1first and 2 ~ ' cies) of (n„)=6/s. If we assume the heating is due haveto theparticle'satommotionmost couldofatitsthenfrequencytimebe indetectedthe ground-stateverym„sensitivelythe levelheating timer 3/2 constant is 95 h. might be useful in mass spectroscopy. 2 thermalization of the ion to room temperature by noiseof its byconfiningits influencepotential.on the laser-cooledWhenion.(n,Such)«1,a deviceT is no P1/2 ' This rate varied slightly with cu„around ro, =3 MHz, In summary, we have realized laser cooling in the at frequency m„ the heating time constant is 95 h.longer mightproportionalbe useful intomass(n, )spectroscopy.but depends logarithmically =3 we have realized laser in the This rate varied slightly with cu„around ro, MHz,on (n, ).InThesummary,techniquebutof sidebandwas substantiallycoolingcoolinghas previous-higher for 194co, ~2.5 MHz.281.5Fromnm~ resolved sideband regime for the first time. The kinetic but was substantially higher for co, ~2.5 MHz. From resolved sideband regime for the first time. The kinetic .5 nm these heating data, the measured (n, ) is consistent withly beenenergyappliedof a totrappedcooltheseatomicthe magnetronheatingion was reducedmotiondata,to ofa valuetrappedthe measured (A)(n, ) is consistent with energy of a trapped atomic ion was reduced to a value B) Paul Trap+ the theoretical limit at the end of the sideband coolingelectronswherewithit spentan mostrf electronicof its time inexcitationthe ground-stateof thelevelaxial end 1/2 sideband where it spent most of its time in the ground-state level period. We also measured (n, ) for a single degree motionof of thatits confiningwas coupledwell.theTotothetheoreticala extentcooledthatresistor.the ionlimitis Thein theatfinalthe of the cooling — 194 nm freedom directly by changing Uo to 25 V in order temperatureto zero-pointin energythis experimentperiod.state of motion,wasWethislimitedrealizesalso byformeasuredthermalthe (n, ) for a single degree of of its confining well. To the extent that the ion is in the the radial and axial sideband frequencies. The first time the fundamental limit of laser cooling for a FIG. l. (a) A simplified energy level scheme of Hg tt show- split excitation and the energy of the magnetron motion cor- 282-nm radiation was tuned so that only the first axial bound particle and freedomthe ideal of andirectlyisolated atomic parti-changing ingUothe opticalto —transitions25 V ininvolvedorderin ourtosideband zero-pointcooling ex- energy state of motion, this realizes for the responded In experiments by sideband at co,/2' =4.66 MHz was cooled and probed. cle at restto to(n„)»within l.the quantum-mechanicalthe limitsdescribedim- periment. (b) The geometrical arrangement of the laser From the results, shown in Fig. 3, we calculate (n„)here, posedwe achieveby the surrounding(n,split) « I apparatus.bytheopticalradialsidebandandcooling.axial sidebandbeams. The 194-nmfrequencies.Auorescence isThedetected normalfirst totimethe the fundamental limit of laser cooling for a =0.049~0.045 at the end of the cooling period, con-For our Wevalueacknowledgeof m„T282-nmthewassupportaboutradiationof 50the pK.U.S. AirHowever,wasForce tunedto planeso thatof the figure.only Mirrorthe firstM reflectsaxial194-nm radiationboundwhileparticle and the ideal of an isolated atomic parti- sistent with the theoretical cooling limit. Thethe extentOffice thatof Scientificthe particleResearchisandin thethe U.groundS. Officestateof Na-of its transmitting 398-nm radiation. confinement of the axial motion is given by the spread of val Research. F. D.sidebandthanks the Deutscheat co,Forschungsge-/2' =4.66 MHz was cooled and probed. cle at rest to within the quantum-mechanical limits im- the zero-point wave function z(rms) = 2.4 nm. meinschaft for financial support. We thank C. Wieman, The American 403 For our data, the uncertainty in the second-order J. Bollinger, and J.FromHelffrich thefor commentsresults,and1989sugges-shown in PhysicalFig. 3,Societywe calculate (n„) posed by the surrounding apparatus. Doppler shift is dominated by the uncertainty in (n„) and tions on the manuscript.=0.049~0.We thank R.045Cookatand theR. end of the cooling period, con- We acknowledge the support of the U.S. Air Force sistent with the 405theoretical cooling limit. The Office of Scientific Research and the U.S. Office of Na- confinement of the axial motion is given by the spread of val Research. F. D. thanks the Deutsche Forschungsge- the zero-point wave function z(rms) = 2.4 nm. meinschaft for financial support. We thank C. Wieman, For our data, the uncertainty in the second-order J. Bollinger, and J. Helffrich for comments and sugges- Doppler shift is dominated by the uncertainty in (n„) and tions on the manuscript. We thank R. Cook and R.

405 Electromagnetic traps for charged and neutral particles

Wolfgang Paul Physikaiisches institut, Universitat , Bonn,

Experimental physics is the art of observing the struc- single atom, however, one can observe its interaction ture of matter and of detecting the dynamic processes with a radiation field and its own statistical behavior within it. But in order to understand the extremely com- alone. plicated behavior of natural processes as an interplay of a The idea of building traps grew out of molecular-beam few constituents governed by as few as possible funda- physics, , and particle accelerator mental forces and laws, one has to measure the properties physics I was involved in during the first decade of my of the relevant constituents and their interaction as pre- career as a physicist more than 30 years ago. In these cisely as possible. And as all processes in nature are in- years (1950—55) we had learned that plane electric and terwoven,534 one must 8/olfgangseparatePaul:andElectromagneticstudy themtrapsindividual-for charged and neutralmagneticparticles multipole fields are able to focus particles in ly. It is the skill of the experimentalist to carry out clear two dimensions acting on the magnetic or electric dipole : Electromagnetic trapsofforstandardcharged instrumentand neutral particlesand its properties were treated ex- zation of the orbits. If the voltage +0= U+ V cosset is experimentstensively in the literaturein order(Dawson,to get 1976).answers to his questionsapplied betweenun- themomentcaps and ofthetheringparticles.electrode, theLenses for atomic and molecu- iments I am awarded for were done together with disturbedThe two-dimensionalby undesiredquadrupole effects, and it is his ingenuityequations oftomotion larare representedbeams (Friedburgby the same Mathieuand Paul, 1951; Bennewitz and research students or young colleagues in mutual inspira- or the mass filter function of (5). The relevant parameters for r The Eq. Paul, were conceivedthe and tion. In particular, I have to mention H. Friedburg and improve the art of measuring to ever higher motionprecision.correspond to those1954,in the1955)x direction in the realized, improving Configuration is four H. G. Bennewitz, C. H. Schlier and P. Toschek in the (a) generated by hyperbolically field case. the z ThereAlreadyare atmanythe veryexamplesbeginning inof physicsour thinkingshowingabout thatplanehigher Onlyconsiderablyparametersthearemolecular-beamchanged by a method for spectrosco- field of molecular-beam physics, and in conceiving and shaped electrodes linearly extended in the y-direction as factor 2. realizing the linear quadrupole spectrometer and the rf precisiontheis showndynamicinrevealedstabilizationFig. 1. Thenewpotentialof ionsphenomena,weon werethe electrodesawareinspiredofisthe new ideas, py or for state selection. The lenses found application as Accordingly, the region of stability in the a-q map for ion trap H. Steinwedel, O. Osberghaus, and especially ossibility+No/2 if oneof usingapplies ittheforvoltagetrapping+0 betweenions inthea elec-three- or confirmed or dethroned well-established theories.the trap Onhas a diferentwell to theasammoniais shown inas to 7.the hydrogen maser (Townes, the late Erhard Fischer. Later H. P. Reinhard, U. Zahn, dimensionaltrode pairs. field.The fieldWestrengthcalled suchis givena deviceby "Ionenkafig. " shape, Fig. the mass and F. V. Busch played an important role in developing theNowadaysother thehand,word new"ion trap"experimentalis preferred techniques(Berkling, Againconceived range1983).of the storable ions (i.e., ions in the @'o stable region) can be chosen the slope of the operation this field. to1956;answerPaul, specialQsberghaus,x, E,questions= andz, EFischer,=0.in one19S8;fieldFischer,of physics became The byquestion "What happens if one injects charged What are the principles of focusing and trapping parti- 1959). Pp ~o line a/q =2U/V. Starting with operating parameters in cles' Particles are elastically bound to an axis or a coor- very fruitful in other fields, too, be it in ,the tipbiolo-of the stable particles,region, one canionstraporionselectrons,of a single in such multipole fields" led IfTheone potentialinjects ionsconfigurationin the y direction,in theit isionobvioustrap thathas forbeen dinate in space if a binding force acts on them which in- . By lowering the dc voltage one brings the givena constantorinengineering.Eq.voltage(3). ThisNp theconfigurationionsIn willawardingperformis generatedharmonicthe Nobelbyos-an prize to my to the development of the linear quadrupole mass spec- creases linearly with their distance r gy, ions near the axis where their motions are much more hyperbolicallycillations in theshapedx-y plane;ringbutandduetwoto thehyperbolicopposite rotation-sign in q colleagues Norman Ramsey, Hans Dehmelt, andstable.me for trometer. It employs not only the focusing and defocus- F= — allythe symmetricfield theircapsamplitudeas it is shownin the schematicallyz direction willinin-Fig. cr, E„ For it is fre- new6(a).creaseexperimentalFigureexponentially.6(b) gives Thethemethods,viewparticlesof thearethefirstdefocusedSwedishrealizedandtrapAcademy manyindi- applicationsing forcesnecessaryof a tohigh-frequencyknow the electric quadrupole field in other words, if they move in a parabolic potential spectrum of the oscillating catesinwill19S4.beherlost byappreciationhitting the electrodes.for the aphorism the Gottingenquency acting on ions, butions.also exploitsFrom the stability properties of This behavior can be avoided if the is mathematics we learn that the motion of the ions can be N —(ixx +/3y +yz ) . If one brings ions into the trap,appliedwhichvoltageis easily described as a slow their(secular) oscillation with the funda- physicistachievedperiodic. byI3ueGeorgionizingto the Christophinsideperiodica changelow-pressureLichtenbergof the signgas byof theelec-wrote two hun- equations of motion in analogy to the principle of The tools appropriate to generate such fields of force mental frequencies co/2 modulated with a mi- dredtronselectricyearspassingforce,agothroughone ingetsthehisfocusingvolume,notebookandtheydefocusingperform"one inhasthebothsameto do something strongco„,=P„,focusing for accelerators which had just been con- to bind charged particles or neutrals with a dipole mo- Electromagneticthe x and z directions alternatingtrapsin time.forIfchargedthe applied andcromotion,neutrala muchparticlesfaster oscillation of the fre- forced motions as in the two-dimensional case."The only driving ment are electric or magnetic multipole fields. In such newvoltagein orderis given toby aseedc voltagesomethingU plus annew.rf voltageOnV the same one ceived. difterence is that the Geld in z direction is stronger by a quency pageco, if neglects higher harmonics. The frequen- configurations the field strength, or the potential, respec- with the driving co LichtenbergWolfgangsaid:frequencyPaul"I think it is a sad situationstabili- cyindeterminingall factor PIfis onea functionextendsonly ofthetherulesMathieuof two-dimensional focusing to tively, increases according to a power law and shows the factor 2. Again a periodic field is needed for the our No= U V parameters a and q and therefore mass dependent. Its desired symmetry. generally if rn is the number of chemistryElectromagneticPhysikaiischesthat+ cosset,we aretrapsinstitut,unableUniversitatfor chargedto suspendBonn, Bonn,andtheGermanyneutralconstituentsparticles three dimensions, one possesses all ingredients for parti- "" or the order of the is symmetry potential given of thematterequationsfree.of motion" are cle traps. Wolfgang1' Paul thee 0fiOS u=01$9Z will Asz-axisalready mentioned the TodayxPhysikaiisches+ ( U+subjectVinstitut,coscot of)xUniversitat=0,my 1ectureBonn, Bonn, Germanybe the suspension physics or the particle dy- P' f) e -r "cos 807' o namics in 2 of such constituents of rnatter 1 or, in other(4) words, about such focusing devices is very closely related to = oz% / trapsExperimentalfor free( U +physicschargedV cosset is)z theand0 .artneutralof observingparticlesthe struc-withoutsinglema-atom, thathowever,of acceleratorsone can observeor storageits interactionrings for nuclear or parti- For a quadrupole I =4, it gives N-r cos2y; and for Pl I"0 a sextupole =6, one gets 4-r cos3y corresponding terialture ofwalls.matter0 Suchand oftrapsdetectingpermitJithe thedynamicobservationprocesses of isolatedwith a radiationcle fieldphysics.and itsInownfact,statisticalmultipolebehavior fields were used in I At first one that the time-dependent term x,y-plane to a field strength increasing with r and respectively. fZOsight expects alone. r, within it.p Butu= 016VOin order to understand the extremely com- particles,Experimentalof the forceSkt.evencancelsphysicsof outais thesinglein artthe oftimeone,observingaverage.overtheaButstruc-longthis periodsingle atom,of timehowever, molecular-beamone can observe its interactionphysics first. But the two fields have plicated ~behavior of natural processes as an interplay of a The idea of building traps grew out of molecular-beam tureandwouldofthereforematter&beautrueandonlyaccordingofin detectinga homogeneoustothe Heisenberg'sdynamicfield. In processesa periodic with a radiation field complementaryand its own statisticalgoals;behaviorthe storage of particles, even of a Trapping of charged particles few constituents governed by as few as possibleuncertaintyfunda- physics,prin- mass spectrometry, and particle accelerator withininhornogeneousit. But in so%orderfield, tolikeunderstandthe quadrupolethe extremelyfield, therecom-is a alone. in two- and three-dimensional A7 / plicatedciplementalsmallenablebehavioraverageforces andforceusof naturaltolaws,left,measurewhichoneprocesseshasis alwaystotheirasmeasurean ininterplaypropertiesthe directionthe ofpropertiesa withTheextreme-ideaphysicsof buildingI wassingletrapsinvolvedgrewone,outin ofduringofmolecular-beamextremelythe first decadelow energyof my down to the micro- quadrupole fields fewofoftheconstituentstherelevantlower field,governedconstituentsin our caseastowardandfew theirasthepossiblecenter.interactionfunda-There-as pre-physics, careermass spectrometry,as a physicist andmoreparticlethan accelerator30 years ago. In these ly high accuracy.0 by electron-voltelectrons region on the one side and of as many as mentalciselyfore,forcesascertainpossible.187 andconditionslaws,Andone hasexistas alltothatmeasureprocessesenablethe theinpropertiesnatureions toare physicsin- Iyearswas involved(1950—55)in duringwe hadthelearnedfirst decadethat ofplanemy electric and In the electric quadrupole field the potential is quadra- u =0/6'Qg theIntraverserelevantparticular,Skt.the quadrupolethefieldpossibilitywithout hitting tothe observeelec- careerindividualas a physicist morepossiblethan 30ofyearsextremelyIn thesehigh energy on the other. Today tic in the Cartesian coordinates, ofterwoven, oneconstituentsmust separateand theirand interaction1study themas individual-pre- magnetic multipole fields are ago.able to focus particles in trodes; i.zoe. their motion around the axis is stable with — ciselytrappedly. Itasispossible.theparticles, skill Andof theasopensallexperimentalistprocessesup yain newnatureto carrydimensionare in-out clearyearsin (1950atomictwo 55)dimensionswe hadwelearnedactingwill dealthaton theplanewithmagneticelectricthe low-energyandor electric dipolepart. ~'o terwoven,limited oneamplitudesmust separatein x andandz studydirections.them individual-We learned magnetic multipole fields are able to focus particles in (ax +/3y +) z ) . measurements.experimentsrules in orderIUntilto geta answersfew yearsto hisagoquestionsall measurementsun- moment of the particles.At first ILenseswill talkfor atomicabout andthemolecu-physics of dynamic stabili- 2I'p ly. theseIt is the skillfromof the theoryexperimentalistof the Mathieuto carryequations,out clearas two dimensions acting on the magnetic or electric dipole disturbed undesired effects, and it is his lar beams (Friedburg and Paul, 1951; Bennewitz and experimentswerethis typeperformedofbyin differentialorder toongetequationananswersensembleis called.to his questionsof particles.ingenuityun- momenttoThereforeof the particles.zationLenses forof atomicions andin molecu-two- and three-dimensional radio- The Laplace condition 6+=0 imposes the condition 8&0 ZW ZN8 zszMtlz z»o z»» ze'MHz disturbedimprove bytheundesiredart of measuringeffects, and ittoiseverhis ingenuityhigher precision.to lar beamsPaul,(Friedburg1954, 1955)and werePaul,B conceived1951; Bennewitzand realized,and improving n+/3+y=0. There are two simple ways to satisfy this the measured value —for example, the transition proba- frequency quadrupole fields, the quadrupole mass spec- improveThere arethe manyart ofexamplesmeasuring intophysicsever highershowingprecision.that higherPaul, 1954,considerably1955) were conceivedthe molecular-beamand realized, methodimproving for spectrosco- condition. Therebilityprecisionarebetweenmanyrevealedexamplestwonewineigenstatesphenomena,physics showinginspiredofthatanhighernewatomideas,—considerablyis a pyvalueortheformolecular-beamstatetrometer,selection.methodandThe forlensesthespectrosco-ionfoundtrap.applicationIn a secondas part I shall re- (a) a= l = —y, /3=0 results in the two-dimensional (b) precision revealed new new or forwellstate selection. The lenses 4foundt'//////z r as field averagedor confirmedoveror dethronedmanyphenomena,—particles.l72 well-establishedrnME—inspiredTacitlytheories.ideas,one assumespyOn i that////////'~to the ammoniaport onastrappingto the applicationihydrogenof neutralmaserparticles(Townes, with emphasis on an oralltheconfirmedatomsother hand,orshowdethronednewexactlyexperimentalwell-establishedthe sametechniquestheories.statisticalOnconceivedwellbehaviorto the1983).ammoniaif asexperimentto the hydrogenwithmasermagnetically(Townes, stored . — theto answerother hand,specialnew experimentalin onetechniquesfield conceived 1983). The question "What if one (x z ), (2) questions of physics became happens'I /////// ~ ~ injects charged 27" tooneanswerattributesspecial questionsthe resultin onetofieldtheof physicssingle becameatom. On.Thea question "What happensAs in ifmostone injectscases chargedin physics, especially in experimental o very fruitful in other fields, too, be it in chemistry, biolo- trappedparticles, ions or MMrelectrons, in such multipole fields" led very fruitful in other fields, too, be it in chemistry, biolo- particles, ions or electrons,A in suchg multipole fields" led (b) a=/3=1, y= —2 generates the three-dimensional gy, or engineering. In awarding -Pthe/2 Nobel prize to my to the developmentphysics,of thethe linearachievementsquadrupole aremass notspec-the achievements of a gy, or engineering. In awarding the Nobel prize to my to the development of the linear quadrupole mass spec- configuration, in cylindrical coordinates *This C6HLL colleaguescolleagueslectureNormanNorman(a) wasRamsey,Ramsey,deliveredHansHansDehmelt,8 DecemberDehmelt,(&) and me1989,andfor meontrometer.forthe occasiontrometer.It employs Itnotsingleemploysonly theperson,notfocusingonly eventheand focusingdefocus-if he contributedand defocus- in posing the prob- new experimental methods, the Swedish Academy indi- ing forces of a high-frequency electric quadrupole field newofFIG.theexperimental5.presentation(a) Very firstmethods,massof thespectrumthe 1989SwedishofNobelrubidium.AcademyPrizeMassindi-in scan-Physics.ing forces of a high-frequencylems andelectricsomequadrupole ideasfield $0(r —2z ) basic in solving them. All the exper- with 2zo =rp . catescatesningFIR.herwasher1. appreciationachieved(a)appreciationEquipotentialby periodicforlinesfortheforvariationaphorismthea planeaphorismofquadrupolethethe drivingGottingenthefield.frequen-Gottingen(b) acting onactingions, buton alsoions,exploitsbut alsothe exploitsstability propertiesthe stabilityof properties of I o+2zo physicistphysicistcyThev. electrodeParameter:GeorgGeorgstructureChristophu Christoph= U/V,for theLichtenbergatmassu Lichtenberg=0.Alter.164 "Rbwroteandwrotetwo87Rbhun-aretwoful-hun-their equationstheir equationsof motion ofin analogymotion toin theanalogyprincipleto theof principle of dreddredly resolved.yearsyearsagoago(b)in hisinMasshisnotebookdoubletnotebook"one'Kr-C6H„Resolving"onehas tohasdo somethingto do somethingpower strongFIG.focusingstrong6. (a) Schematicforfocusingacceleratorsviewforofacceleratorsthewhichion trap.had just(b)whichCrossbeen hadcon-sectionjustofbeen con- new /6min order=6500to(vonsee Zahn,something1962).new. " On"the same ceived.the first trap (1955). Rev. Mod. Phys. , Vol. 62, No. 3, July 1990 ReviewsnewI in oforderModernto seePhysics,somethingVol. 62,new.No. 3, JulyOn 1990the pagesame page ceived. Copyright 1990 The Nobel Four)dation 531 said: "I think is If one extends the rules two-dimensional LichtenbergLichtenberg said: "I thinkit ita sadis asituationsad situationin all ourin all our If one extends ofthe rules of two-dimensionalfocusing to focusing to chemistry that we are unable to suspend the constituents three dimensions, one possesses all ingredients for parti- chemistryRev. Mod. Phys.that, Vol. 62,weNo.are3, Julyunable1990 the three dimensions, one possesses all ingredients for parti- of matter free. " to suspend constituentscle traps. of matter free. " cle traps. Today QUANTUMthe subject PHYSICSof my 1ecture will be the suspension As already mentioned the physics or the particle dy- of suchTodayconstituentsthe subjectof rnatterof my or,1ecturein otherwillwords,be theaboutsuspensionnamics in suchAs alreadyfocusing mentioneddevices is verythecloselyphysicsrelatedor theto particle dy- trapsof suchfor freeconstituentscharged andof neutralrnatter particlesor, in otherwithoutwords,ma- aboutthat of acceleratorsnamics in orsuchstoragefocusingrings devicesfor nuclearis veryor parti-closely related to terialScientifictrapswalls.for freeSuchcharged trapsAmericanpermitand theneutralobservation (August,particlesof isolatedwithout 2008)ma-cle physics.that ofIn acceleratorsfact, multipoleor storagefields wererings usedfor nuclearin or parti- particles,terial walls.even ofSucha singletrapsone,permitover thea longobservationperiod of timeof isolatedmolecular-beamcle physics.physics Infirst.fact,But multipolethe two fieldsfieldshavewere used in andparticles,thereforeevenaccordingof a singleto Heisenberg'sone, overuncertaintya long periodprin- of timecomplementarymolecular-beamgoals; the storagephysicsof first.particles,Buteventheoftwoa fields have cipleand enablethereforeus toaccordingmeasure theirto Heisenberg'sproperties withuncertaintyextreme- prin-single one,complementaryof extremely lowgoals;energythedownstorageto theof micro-particles, even of a ly high accuracy. electron- region on the one side and of as many as ciple enable us to measure their properties with extreme- single one, of extremely low energy down to the micro- In particular, the possibility to observe individual possible of extremely high energy on the other. Today ly high accuracy. electron-volt region on the one side and of as many as trapped particles opens up a new dimension in atomic we will deal with the low-energy part. measurements.In particular,Until thea fewpossibilityyears ago allto measurementsobserve individual At firstpossibleI will talkofaboutextremelythe physicshigh ofenergydynamiconstabili-the other. Today weretrappedperformedparticleson anopensensembleup aofnewparticles.dimensionThereforein atomiczation ofweionswill indealtwo-withandthe three-dimensionallow-energy part. radio- themeasurements.measured value Until—for aexample,QUANTUMfew yearsthe transitionago all measurementsproba- frequency quadrupoleAt first I willfields,talktheaboutquadrupolethe physicsmass ofspec-dynamic stabili- bilitywere betweenperformedtwo oneigenstatesan ensembleof an atomof particles.—is a valueThereforetrometer, zationand theofionionstrap. inIn two-a secondandpartthree-dimensionalI shall re- radio- averagedthe measuredover manyvalueparticles.—for example,Tacitly onetheassumestransitionthat proba-port on trappingfrequencyof neutralquadrupoleparticlesfields,with emphasisthe quadrupoleon an mass spec- allbilityatomsbetweenshow exactlytwo eigenstatesthe same statisticalof an atombehavior—is ifa valueexperimenttrometer,with magneticallyand the ionstoredtrap.neutrons.In a second part I shall re- one attributes the result to the single atom. On. a As in most cases in physics, especially in experimental averaged over many particles. Tacitly onetrappedassumes that port on trapping of neutral particles with emphasis on an COMPUTINGphysics, the achievements are not the achievements of a all atoms show the same experiment with magnetically *This lecture was deliveredexactly8 December 1989,statisticalon the occasionbehavior singleif person, even if he contributed in posingstoredtheneutrons.prob- ofonethe attributespresentation theof theresult1989 Nobelto thePrizesinglein Physics.atom. On. a trappedlems and someAs inbasicmostideascasesin solvingin physics,them. Allespeciallythe exper-in , the achievements are not the achievements of a *This lecture was delivered 8 December 1989, on the occasion single person, even if he contributed in posing the prob- Reviewsof the ofpresentationModern Physics, ofVol.the62,WITH1989No. 3, JulyNobel1990Prize in Physics. IOlemsNSand Copyrightsome basic1990 ideasThe Nobelin Four)dationsolving them.531 All the exper-

Reviews of ModernResearchersPhysics, Vol. 62, No. 3, Julyare1990 taking the first steps toward Copyright 1990 The Nobel Four)dation 531 building ultrapowerful computers that use individual atoms to perform calculations

By Christopher R. Monroe and David J. Wineland

KEY CONCEPTS ver the past several decades technologi- primes. Multiplying two primes is a simple job Quantum computers can store cal advances have dramatically boosted for computers, even if the numbers are hundreds and process data using atoms, O the speed and reliability of computers. of digits long, but the reverse process—deriving photons or fabricated micro- Modern computer chips pack almost a billion the prime factors—is so extraordinarily difficult structures. These machines may transistors in a mere square inch of silicon, and that it has become the basis for nearly all forms someday be able to perform in the future computer elements will shrink even of data encryption in use today, from Internet feats of computing once more, approaching the size of individual mole- commerce to the transmission of state secrets. thought to be impossible. cules. At this level and smaller, computers may In 1994 Peter Shor, then at Bell Laboratories, The manipulation of trapped begin to look fundamentally different because showed that a quantum computer, in theory, ions is at the forefront of the their workings will be governed by quantum could crack these encryption codes easily because quantum computing effort. mechanics, the physical laws that explain the it could factor numbers exponentially faster than Researchers can store data on behavior of atoms and subatomic particles. The any known classical algorithm could. And, in the ions and transfer informa- great promise of quantum computers is that 1997, Lov K. Grover, also at Bell Labs, showed tion from one ion to another. they may be able to perform certain crucial that a quantum computer could significantly Scientists see no fundamental tasks considerably faster than conventional increase the speed of searching an unsorted data- obstacles to the development computers can. base—say, finding a name in a phone book when of trapped-ion computers. Perhaps the best known of these tasks is fac- you have only the person’s phone number. —The Editors toring a large number that is the product of two Actually building a quantum computer, how-

64 © 2008 SCIENTIFIC AMERICAN, INC. August 2008 [BASICS] environment, yet the electric repulsion among them provides a strong interaction for produc- TRUTH TABLE ing entanglement. And laser beams thinner than A trapped-ion computer a human hair can be targeted on individual A B A’ B’ would rely on logic gates atoms to manipulate and measure the data such as the controlled not stored in the qubits. (CNOT) gate, which con- 0 0 0 0 Over the past few years scientists have per- sists of two ions, A and B. formed many of the proof-of-principle experi- This truth table shows ments in quantum computing with trapped ions. that if A (the control bit) Researchers have produced entangled states of has a value of 0, the gate 1 1 0 0 up to eight qubits and have shown that these leaves B unchanged. But rudimentary computers can run simple algo- if A is 1, the gate flips B, rithms. It appears straightforward (though tech- changing its value from 0 to 1, and vice versa. And nically very challenging) to scale up the trapped- 1 1 1 if A is in a superposition ion approach to much larger numbers of qubits. 0 state (0 and 1 at the same Taking the from classical computers, this time), the gate puts the effort would involve sequencing a few types of two ions in an entangled gates, each made up of only a few superposition. (Their 1 1 1 trapped ions. Scientists could adapt convention- state is now identical 0 al error-correction techniques to the quantum to the one shown in the world by using multiple ions to encode each box on the bottom of qubit. Here the redundant encoding of informa- the opposite page.) tion allows the system to tolerate errors, as long 0 as they occur at a sufficiently low rate. In the end, 0 + 1 a useful trapped-ion quantum computer would 0 0 + 1 1 [BASICS] most likely entail the storage and manipulation environment, yet the electric repulsion among of at least thousands of ions, trapped in complex bors. We can solve this problem, however, by them provides a strong interaction for produc- TRUTH TABLE POWERS OF TWO ing entanglement. And laser beams thinner than arrays of electrodes on microscopic chips. using laser beams that are focused on the partic- A trapped-ion computer The enormous potential of a human hair can be targeted on individual TheA first requirementB for makingA’ a “univerB’ - ular qubit (or qubits) of interest. would rely on logic gates trapped-ion computers lies in the atoms to manipulate and measure the data sal” quantum computer—one that can perform The third basic requirement is the ability to such as the controlled not fact that a system with N ions can stored in the qubits. — (CNOT) gate, which con- all possible0 computations0 is reliable memory. hold 2N numbers simultaneously. devise at least one type of logic gate between Over the past few years scientists have per- 0 0 sists of two ions, A and B. If we put a qubit in a superposition state of 0 and And as N increases, the value of qubits. It can take the same form as classical log- formed many of the proof-of-principle experi- This truth table shows 1, with the ion’s magnetic orientation pointing 2N rises exponentially. ic gates—the AND and OR gates that are the ments in quantum computing with trapped ions. that if A (the control bit) Researchers have produced entangled states of has a value of 0, the gate up and down at the1 same time, it must remain1 building blocks of conventional processors—but 0 0 5 up to eight qubits and have shown that these leaves B unchanged. But in that state until the data are processed or mea- 2 = 32 it must also act on the superposition states unique rudimentary computers can run simple algo- if A is 1, the gate flips B, sured. Researchers have long known that ions to qubits. A popular choice for a two-qubit logic rithms. It appears straightforward (though tech- changing its value from 0 10 to 1, and vice versa. And held in electromagnetic traps can act as very 2 = 1,024 gate is called a controlled not (CNOT) gate. Let nically very challenging) to scale up the trapped- 1 1 1 if A is in a superposition good qubit memory registers, with superposi- us call the qubit inputs A and B. A is the control ion approach to much larger numbers of qubits. 0 state (0 and 1 at the same tion lifetimes (also known as coherence times) 50 bit. If the value of A is 0, the CNOT gate leaves Taking the lead from classical computers, this time), the gate puts the 2 = exceeding 10 minutes. These relatively long life- B unchanged; if A is 1, the gate flips B, changing effort would involve sequencing a few types of two ions in an entangled 1,125,899,906,842,624 quantum logic gates, each made up of only a few superposition. (Their times 1result from1 the extremely1 weak interac- its value from 0 to 1, and vice versa [see box trapped ions. Scientists could adapt convention- state is now identical tion between an ion and its surroundings.0 2100 = above]. This gate is also called a conditional log- al error-correction techniques to the quantum to the one shown in the The second essential ingredient for quantum ic gate, because the action taken on qubit input B world by using multiple ions to encode each box on the bottom of 1,267,650,600,228,229, computing is the ability to manipulate a single (whether the bit is flipped or not) depends on the qubit. Here the redundant encoding of informa- the opposite page.) 401,496,703,205,376 tion allows the system to tolerate errors, as long qubit. If the qubits0 are based on the magnetic condition of qubit input A. as they occur at a sufficiently low rate. In the end, orientation of a trapped ion, researchers can use To make a conditional logic gate between two 0 + 1 a useful trapped-ion quantum computer would oscillating magnetic fields, applied0 0 + 1for 1 a speci- ion qubits, we require a coupling between most likely entail the storage and manipulation fied duration, to flip a qubit (changing it from 0 them—in other words, we need them to talk to of at least thousands of ions, trapped in complex POWERS OF TWOto 1, andbors. vice We versa) can solve or to this put problem, it in a however,superposi by- each other. Because both qubits are positively arrays of electrodes on microscopic chips. using laser beams that are focused on the partic- The first requirement for making a “univer- TheVOLUME enormous potential75, NUMBER of tion25 state.ular Given qubit (orthePH qubits) smallYS ICAL ofdistances interest.REVIEW between theLETTERS 18 DECEMBERcharged,1995 their motion is strongly coupled elec- trapped-ion computers lies in the sal” quantum computer—one that can perform trapped ionsThe —thirdtypically basic requirement a few millionths is the ability of toa trically through a phenomenon known as mutu- fact that a system with N ions can

all possible computations—is reliable memory. N EN meter—deviseit is difficult at least one to localizetype of logic the gate oscillating between al repulsion. In 1995 Juan Ignacio

hold 2 numbers simultaneously.S VOLUME 75, NUMBER 25 PH YS ICAL REVIEW LETTERS 18 DECEMBER 1995 If we put a qubit in a superposition state of 0 and And as N increases, the valuefields of Demonstrationtoqubits. an individual It can take theion,of samea whichFundamental form is as important classical Quantumlog- Logic Gate Cirac and Peter Zoller, both then at the Univer- TIAN S

N I 1, with the ion’s magnetic orientation pointing 2 rises exponentially. becauseic we gates will— oftenthe AND want and to OR change gates one that qubit’s are the sity of Innsbruck in Austria, proposed a way to

CHR M. Meekhof, M. and Wineland up and down at the same time, it must remain C.DemonstrationMonroe,building D.blocksof ofa conventionalFundamentalB.E. processorsKing,QuantumW.—butLogicItano, Gate D. J.

JEN JEN orientation without changing that of its neigh- use this coulomb interaction to couple indirectly in that state until the data are processed or mea- 25 = 32 Nationalit must alsoInstitute act onof theStandards superpositionand Technology, states uniqueBoulder, Colorado 80303 C. Monroe, D. M. Meekhof, B.(ReceivedE. King, W. M.14 Itano,July 1995)and D. J. Wineland sured. Researchers have long known that ions to qubits. A popular choice for a two-qubit logic National Institute of Standards and Technology,"controlled-NOT"Boulder, Colorado 80303 held in electromagnetic traps can act as very 210 = 1,024We demonstrategate theis calledoperation a controlled(Receivedof a nottwo-bit14 July (CNOT)1995) gate. Let quantum logic gate, which, in © 2008 SCIENTIFIC AMERICAN, INC. conjunctionwww.SciAm.comwith simple single-bit operations, forms a universal© 2008quantum SCIENTIFIClogic gate AMERICAN,for quantum INC. SCIENTIFIC AMERICAN 67 good qubit memory registers, with superposi- We demonstrateus callthe operationthe qubitof inputsa two-bit A and"controlled-NOT" B. A is the controlquantum logic gate, which, in computation.conjunction withThe two quantumsingle-bit bitsoperations,are storedforms ina theuniversalinternalquantumand external degreesfor quantumof freedom of a single 50 simple VOLUME 75,logicNUMBERgate 25 PH YS ICAL REVIEW LETTERS 18 DECEMBER 1995 tion lifetimes (also known as coherence times) 2 =trapped computation.atom, Thewhichbit.two Ifquantumis thefirst valuebitslaser areof cooledstoredA is 0,in theto internalthe CNOTzero-pointand gateexternal leavesenergy.degrees ofDecoherencefreedom of a singleeffects are identified trapped atom, which is first laser cooled to the zero-point energy. Decoherence effects are identified exceeding 10 minutes. These relatively long life- for the operation, B andunchanged;the possibility if A is 1,of theextending gate flipsthe B,system changingto more qubits appears promising.2 is as follows: 1,125,899,906,842,624for the operation, and the possibility of extending the system to more qubits appears promising. P3a times result from the extremely weak interac- its value from 0 to 1, and vice versa [see box 2 Input state Output state PACSPACSnumbers:numbers: 89.80.80.+h,+h,03.03.65.—65.w, —32.w,80.32.Pj 80.Pj ~ tion between an ion and its surroundings. 100 above]. This gate is also called a conditional log- )l 2We report = the first demonstration of a fundamental computers and CN gates involving5C a dipole-dipole inter- lo) I l) lo) I l) The second essential ingredient for quantum We report the first demonstrationic gate, becauseof thea actionfundamental taken on qubitcomputers input B and CN gates involving a dipole-dipole inter- - quantum logic gate that operates on prepared quantum action between quantum dots or atomic nuclei [6,7, 11,12] 1,267,650,600,228,229,quantum that operates on prepared quantum action between quantum dots or atomic nuclei 12] Io)l T) —Io)l T) (2) computing is the ability to manipulate a single states.logicFollowinggate the scheme(whetherproposed the bitby isCirac flippedand or not)may dependssuffer from on decoherencethe efforts. The light shifts on [6,7, 11, 401,496,703,205,376 scheme and suffer from Ra man on shifts I states. ZollerFollowing[1], we demonstratethe a controlled-NOTproposed gateby onCiraca atoms locatedmay inside electromagnetictransitiondecoherencecavities haveefforts.been The light I 1) 1) —I 1)I T) qubit. If the qubits are based on the magnetic condition of qubit input A. Detection Zoller pair[1],of wequantumdemonstratebits (qubits).a controlled-NOTThe two qubits comprisegate onshowna to beatomslarge enoughlocated[13,inside14] that electromagneticone could construct cavities have been orientation of a trapped ion, researchers can use To make a conditional logic gate between two (o') Il)l T) Il)l l). pair oftwoquantuminternal (hyperfine)bits (qubits).states and Thetwo externaltwo qubits(quantizedcomprisea quantum showngate whereto bea largesingle enoughphoton prepared[13,14]inthatthe one could construct oscillating magnetic fields, applied for a speci- motional harmonic oscillator)ionstates qubits,of a wesingle requiretrapped a couplingcavity acts betweenas the control qubit [7,15] for the atomic state. The experiment apparatus is described elsewhere and external in two internalatom. Although(hyperfine)this minimalstates systemtwoconsists of only (quantizedtwo However, aextensionquantumto largegatequantumwhereregistersa singlemay bephotondif- prepared [16,17].the A single 9Be+ ion is stored in a coaxial- fied duration, to flip a qubit (changing it from 0 them—in other words, we need them to talk to motionalqubits,harmonicit illustrates oscillator)the basic operationsstates necessaryof a singlefor, andtrappedficult. Ciraccavityand Zolleracts as[1]thehavecontrolproposedqubita very[7,attrac-15] for the atomicresonatorstate. rf-ion trap which provides pseudopotential 2 [17], to 1, and vice versa) or to put it in a superposi- the problems associated with,eachconstructing other. Becausea large bothscale qubitstive quantum are positivelycomputer architecture based on laser-cooled atom. Although this minimal system consists of only two However,1/2 extension to large quantum registers may oscillationbe dif- frequencies of (cu„au~,cu, )/2~ = (11.2, 18.2, quantum computer. ions in which the are associated with in- tion state. Given the small distances between the qubits, it illustrates the basiccharged,operations theirnecessary motion isfor, stronglyandtrapped coupledficult. elecCirac- andqubits Zoller have proposed a very 29.attrac-8) MHz along the principal axes of the trap. We cool [1] I The distinctive feature of a quantum computer is its ternal states of the ions, and information is transferred10& S& the ion so that the n = 0 vibrational ground state is occu- trapped ions—typically a few millionths of a the problems associated with,tricallyconstructing through a phenomenona large scale knowntive as mutuquantum- computer architecture based on laser-cooled ability to store and process superpositions of numbers between qubits through )0&)aux&a shared motional degree of free- pied =95% of the time by employing resolved-sideband [2]. This potential for parallel computing has led to dom. The highlights of theirin whichproposal theare that (i) deco- with in- EN meter—it is difficult to localize the oscillating quantum computer. al coulomb repulsion. In 1995 Juantrapped Ignacioions qubits are associated stimulated Raman cooling in the x dimension, exactly S FIG. 1. Be+ levels. The levels indicated with the discovery that certain problems are more efficiently herence can be small, (ii)energyextension to large registers is fields to an individual ion, which is important The distinctive feature ofCiraca quantum and Peter computerZoller, bothis thenits at thethickternal Univerlinesstates-form ofthe thebasis ions,of the andquantuminformationregister: internalis transferredas in Ref. [16]. The two Raman beams each contain TIAN solved on than on a classical relatively straightforward, and the readout can S a quantum computer (iii) qubit

I = = = ability to store and process superpositions of numbers levelsbetweenare qubitsIS) I 1&throughand I T)a (sharedSiyq)F motional2, mF 2)degreeand of=1free-mW of power at =313 nm and are detuned =50 6Hz because we will often want to change one qubit’s computer [3]. The most dramaticsity ofexample Innsbruckis an algorithmin Austria, proposedhave nearly aunit wayefficiency.= to1, mF = levels, respectively, separated CHR S,y2)F 1) by red of the P~y2 excited state. The Raman beams are [2]. Thispresentedpotentialby Shor [4]forshowingparallelthat computinga quantum computerhas led toIn our implementationcoo/2ndom. =The1.250highlightsofGHz),a quantumand ofCN)aux)theirlogic= proposalgate,S~y2)Fthe= 2, mFare =that0) (i) deco- JEN JEN orientation without changing that of its neigh- use this coulomb interaction to couple indirectly applied to the ion in directions such that their wave-vector should be able to factor large numbers very efficiently. target qubit(separatedlS) is spannedfrom I 1)bybytwo=2.S&I25 MHz);hyperfineexternalgroundvibrational levels the discovery that certain problems are more efficiently herence= can be small, (ii) extension to large registersdifferenceis Bk points nearly along the x axis of the This appears to be of considerable interest, since the states of area singleln) 9Be)0) andion )1)(the(separatedlf = 2,mFby =~,2)/2 andr7= 11.2 MHz). solved on a quantum computer than on a classical Stimulatedrelatively Ramanstraightforward,transitions betweenand (iii)hyperfinethe qubitstatesreadouttrap; canthus the Raman transitions are highly insensitive to security of many data encryption schemes [5] relies on the lF = 1, mF = 1) states, abbreviated by the equivalentS]/2 www.SciAm.com SCIENTIFIC AMERICANare driven through the virtual level = 50 GHz) with motion in the other two dimensions. The Lamb-Dicke © 2008 SCIENTIFIC AMERICAN, INC. © 2008computer SCIENTIFICinability[3]. ofAMERICAN,Theclassicalmostcomputers dramaticINC. to factorexamplelarge numbers.is an algorithm have nearly67 unit efficiency.P~pin (5 spin-1/2 states l J) and l [)) separated frequency by a pair of =313 nm laser beams. Measurement of S is accom- parameter is = 6k xo = 0.2, where xo = 7 nm is presentedA quantumby Shorcomputer[4] showinghosts a registerthat aofquantumqubits, eachcomputerof coo/2~ = 1.250In our6Hz.implementationThe control qubit ofln)aisquantumspanned= CN logic gate, the g plished by driving the cycling I 1) ~ P3y2)F 3, mF 3) = which behaves as mechanical two-level the spread of the n 0 wave function. The carrier should be able to factorquantum large numbers verysystemsefficiently.by the firsttransitiontargettwo qubitquantizedwith o.lS)+-polarizedisharmonicspannedlightoscillatorandby detectingtwostatesS&I2the resultinghyperfine ground and and (ln)) j) )n)) T)) Rabi frequency is Ao2vr = 140 kHz, can store arbitrary superposition states of 0 1. of the trappedion fluorescence.in (lo) and l 1)), separated in frequency This appears to be of considerable interest, since the states of a single 9Be ion (the = =th«edand ~ and blue It has been shown that any computation on a register by the vibrational frequency to, /2' = 11 MHz of thelf 2,mF 2) ()1)l l) )0)l T)) ()0)l g) ~ )1)l T)) security data schemes relies on the = = sideband Rabi frequencies are Ao/2~ = 30 kHz, and of ofqubitsmanycan be brokenencryptionup into a series [5]of two-bit harmonicallyaccordinglF trapped1, tomFion.the followingFigure1) states,1 displaysformat:abbreviatedthe relevant by the equivalent g, operations for example, a series of two-bit the auxiliary transition ()1)l T) ~ )0)l $)) Rabi frequency inability of classical[6], computers to factor large numbers. Be energy levels. Manipulationl betweenand the four basis in spin-1/2A states is J) onl the[)) carrierseparatedtransition. frequency by "controlled-NOT*' quantum logic gates, accompa- (a) vr/2 pulse applied = is g, A, /27r = 12 kHz. The difference frequency of A quantum computer(CN)hosts a register of qubits, eacheigenstatesof spanning = the two-qubit registerThe control(ln)lS) V'~ is „, coo/2~The effect1.250is described6Hz. by the operator qubit(vr/2)ln) spannedthe Raman beams is tunable from 1200 to MHz nied by simple rotations on single qubits [7,8]. The CN lO)l g), lO)l y), ll)l g), ll)l T)) is achieved by applying a 1300 as mechanical two-level in the notation of Ref. I which gatebehavestransforms quantumthe state of two qubits e] and e2 fromsystemspair of off-resonantby the laserfirst beamstwo toquantizedthe ion,1].whichharmonicdrives oscillator withstatesthe use of a double pass acousto-optic modulator and can storeto arbitrary where the operationstates is addi- and (AOM), and the Raman pulse durations are controlled let)le2) le&)le& s E2),superpositions of 0 stimulated1. ofRaman(b)theA transitions2~trappedpulse isbetweeninapplied(lo)basisonandthestates.bluel 1)),Whensidebandseparated in frequency tion modulo 2. Reminiscent of the classical exclusive-OR the difference frequencytransition 6betweenof the beams'T) andis anset near 6 = with additional switching AOMs. Since the Raman beams It has been shown that any computation on a register by the vibrational frequencyI auxiliaryto, /2'atomic= 11(1)MHz of the (XOR) gate, the CN gate represents a computation at the coo (the carrier), transitions are coherently driven between are generated from a single laser and an AOM, broadening of qubits can be broken up into a series of two-bit harmonicallylevel )aux) trapped(see Fig. ion.1). Figure 1 displays the relevant most fundamental level: the "target" qubit le2) is fiipped internal states lS) while preserving ln). Likewise, for 6 = of the Raman transitions due to a finite laser linewidth is is on operationsdepending[6],on theforstate example,of the "control"a qubitseriesle~). of two-bitcon —co, (the(c)Be Aredenergyzr/2sideband),pulselevels.transitionsappliedManipulationarethe coherentlycarrier transition,between the fournegligiblebasis [18]. Experimental realization of a quantum computer re- with a m phase shift relativet5 =to (a), leading to the "controlled-NOT*' (CN) quantum logic gates, accompa-driven between l 1)l 1) and lo)l f), and for con + co, Following= Raman cooling to the )0)l state, but before eigenstates V'spanning— the two-qubit register (ln)lS) [) quires isolated quantum systems that act as the qubits, and (the blue sideband),operatortransitions( zr/2)are ofcoherentlyRef. I 1].driven application of the CN operation, we appropriately nied by simple rotations on single qubits [7,8]. The CN lO)l g), lO)l y), ll)l g), ll)l T)) is achieved by applying a apply the presence of controlled unitary interactions between the between lo)lThe[)~/2and pulsesll)l 1'). inNotestepsthat(a)whenand 6(c)is tunedcause tothe spin IS) tuned and timed Raman pulses to the ion, which can gate transforms the state of two qubits e] and e2 from off-resonant qubits that allow construction of the CN gate. As pointed either sideband,topairundergoofthe stimulated+1/4 andRaman—laser1/4transitionsofbeamsa completeentangleto theRabiion,cycle,which preparedrivesan arbitrary state of the two-qubit register. For out authors if the are not suffi- addi- let)le2) toby le&)le&many s E2),[6,9,where10], thequbitss operation is lS) with ln),respectively,stimulateda crucial partwhileRamanof theleavingtrapped-iontransitionsIn) unchanged.quantumbetweenCNThebasisauxiliarystates. instance,When to prepare a )1)l J) eigenstate, we apply a 7r ciently isolated from outside infIuences, decoherences can tion modulo 2. Reminiscent of the classical exclusive-ORgate. transitionthe differencein step frequency(b) simply reverses6 of thethebeamssign ofis anyset nearpulse6 on= the blue sideband followed by a ~ pulse on destroy the quantum interferences that form the computa- We realize the controlled-NOT gate by sequentially gate, the CN gate represents a computation at the component of the register in the )1)l T) state by inducing a the carrier ()0)l $) )1)l T) )1)l $)). We perform two (XOR)tion. Several proposed experimental schemes for quantum threecoo (thepulsescarrier),of the transitionsRaman beamsareto coherentlythe ion driven between applying complete Rabi cycle from 'T) — measurements to detect the of the register most fundamental level: the "target" qubit le2) is fiipped internal states lS) while)1)lpreserving)0))aux)ln). Likewise,)1)l T). for 6 = population 4714 The auxiliary level )aux) is the Siyz IF = 2, mF = 0) after an arbitrary sequence of operations. First, we depending on the state of the "control" qubit le~). con —co, (the red sideband), transitions are coherently ground state, split from the I J) state by virtue of a measure the probability P(S =)) that the target qubit Experimental realization of a quantum computer re- t5 = Zeemandriven betweenshift of =2.l 1)l5 MHz1) andresultinglo)l f),fromanda for0.18 mT conIS)+is co,in the I J) state by collecting the ion lluorescence quires isolated quantum systems that act as the and applied magnetic field (see Fig. 1). Any component of when o.+-polarized laser radiation is applied resonant qubits, (the blue sideband), transitions= are coherently driven = = the quantum register in the In) )0) state is unaffected with the cycling I P3gz)F 3, mF 3) transition the presence of controlled unitary interactions between the between lo)l and ll)l 1'). Note that when 6 is tuned to /) the blue sideband[) transition of and the effects (radiative linewidth = 19.4 MHz at A = 313 nm; qubits that allow construction of the CN gate. As pointed by step (b), y/2' ofeitherthe twosideband,Ramsey ~/2the pulsesstimulatedcancel. RamanOn the othertransitionshand, entanglesee Fig. 1). Since this radiation does not appreciably out by many authors [6,9,10], if the qubits are not suffi- anylS) componentwith ln), ofa crucialthe quantumpart registerof the intrapped-ionthe )1)l T) statequantumcoupleCNto the I T) state (relative excitation probability: ciently isolated from outside infIuences, decoherences can acquiresgate. a sign change in step (b), and the two Ramsey =5 X 1() s), the fluorescence reading is proportional to add constructively, "Dipping" the =j, For = we collect on =1 photon destroy the quantum interferences that form the computa- pulsesWe realize the controlled-NOTeffectively gate bytargetsequentiallyP(S ). S $, average qubit by ~ radians. The truth table of the CN operation per measurement cycle [16]. Once S is measured, we tion. Several proposed experimental schemes for quantum applying three pulses of the Raman beams to the ion 4715 4714 VOLUME 75, NUMBER 25 PH YS ICAL REVIEW LETTERS 18 DECEMBER 1995

Demonstration of a Fundamental Quantum Logic Gate

C. Monroe, D. M. Meekhof, B.E. King, W. M. Itano, and D. J. Wineland National Institute of Standards and Technology, Boulder, Colorado 80303 (Received 14 July 1995) We demonstrate the operation of a two-bit "controlled-NOT" quantum logic gate, which, in conjunction with simple single-bit operations, forms a universal quantum logic gate for quantum computation. The two quantum bits are stored in the internal and external degrees of freedom of a single trapped atom, which is first laser cooled to the zero-point energy. Decoherence effects are identified for the operation, and the possibility of extending the system to more qubits appears promising.

PACS numbers: 89.80.+h, 03.65.—w, 32.80.Pj We report the first demonstration of a fundamental computers and CN gates involving a dipole-dipole inter- quantum logic gate that operates on prepared quantum action between quantum dots or atomic nuclei [6,7, 11,12] states. Following the scheme proposed by Cirac and may suffer from decoherence efforts. The light shifts on Zoller [1], we demonstrate a controlled-NOT gate on a atoms located inside electromagnetic cavities have been pair of quantum bits (qubits). The two qubits comprise shown to be large enough [13,14] that one could construct two internal (hyperfine) states and two external (quantized a quantum gate where a single photon prepared in the motional harmonic oscillator) states of a single trapped cavity acts as the control qubit [7,15] for the atomic state. atom. Although this minimal system consists of only two However, extension to large quantum registers may be dif- qubits, it illustrates the basic operations necessary for, and ficult. Cirac and Zoller [1] have proposed a very attrac- the problems associated with, constructing a large scale tive quantum computer architecture based on laser-cooled quantum computer. trapped ions in which the qubits are associated with in- The distinctive feature of a quantum computer is its ternal states of the ions, and information is transferred ability to store and process superpositions of numbers between qubits through a shared motional degree of free- [2]. This potential for parallel computing has led to dom. The highlights of their proposal are that (i) deco- the discovery that certain problems are more efficiently VOLUMEherence can75, beNUMBERsmall, 25(ii) extension PHto largeYS ICALregisters REVIEWis LETTERS 18 DECEMBER 1995 solved on a quantum computer than on a classical relatively straightforward, and (iii) the qubit readout can computer [3]. The most dramatic example is an algorithm have nearly unit efficiency. 2 is as follows: presented by Shor [4] showing that a quantum computer In our implementation of a quantum CN P3alogic gate, the 2 Input state Output state should be able to factor large numbers very efficiently. target qubit is spanned by two S&I2 hyperfine ground ~ lS) )l This to be of considerable interest, since the 9Be = = appears states of a single 5C ion (the 2,mF and I lf 2) lo) l) - lo) I l) security of many data encryption schemes [5] relies on the lF = 1, mF = 1) states, abbreviated by the equivalent Io)l Io)l inability of classical computers to factor large numbers. in T) — T) (2) spin-1/2 states l J) and l [)) separated frequency by Ra man A quantum computer hosts a register qubits, each of = of coo/2~ 1.250 6Hz.transitionThe control qubit ln) is spanned I 1) I 1) —I 1)I T) which behaves as quantum mechanical two-level systems the first two harmonic oscillatorDetection states by quantized (o') and and Il)l T) Il)l l). can store arbitrary superposition states of 0 1. of the trapped in (lo) and l 1)), separated in frequency It has been shown that any computation on a register by the vibrational frequency to, /2' = 11 MHz of the The experiment apparatus is described elsewhere of qubits can be broken up into a series of two-bit harmonically trapped ion. Figure 1 displays the relevant [16,17]. A single 9Be+ ion is stored in a coaxial- operations [6], for example, a series of two-bit Be levels. Manipulation between the four basis resonator rf-ion trap [17],which provides pseudopotential 2 energy = "controlled-NOT*' (CN) quantum logic gates, accompa- eigenstates1/2 spanning the two-qubit register (ln)lS) = oscillation frequencies of (cu„au~,cu, )/2~ (11.2, 18.2, MHz the axes the nied by simple rotations on single qubits [7,8]. The CN lO)l g), lO)l y), ll)l g), ll)l T)) is achieved by applying a 29.8) along principal of trap. We cool 10& I S& = gate transforms the state of two qubits e] and e2 from pair of off-resonant laser beams to the ion, which drives the ion so that the n 0 vibrational ground state is occu- )0&)aux& let)le2) to le&)le& s E2), where the s operation is addi- stimulated Raman transitions between basis states. When pied =95% of the time by employing resolved-sideband tion modulo 2. Reminiscent of the classical exclusive-OR FIG.the difference1. Be+ frequencyenergy levels.6 of theThebeamslevels is indicatedset near 6with= stimulated Raman cooling in the x dimension, exactly VOLUME NUMBER 25 18 DECEMBER thick lines form the basis75, of the quantum PHinternalYS ICALas REVIEWin Ref. [16].LETTERSThe two Raman beams each contain 1995 (XOR) gate, the CN gate represents a computation at the coo (the carrier),= transitions are coherently= register:driven =between levels are I 1& and I 2, mF and =1 most fundamental level: the "target" qubit le2) is fiipped internal statesIS) lS) while preservingT) ( Siyq)Fln). Likewise, for2) 6 = mW of power at =313 nm and are detuned =50 6Hz S,y2)F = 1, mF = 1) levels, respectively, separated by red of the excited state. The Raman beams are depending on the state of the "control" qubit le~). con —co,= (the red sideband), transitions= are= coherently= P~y2 coo/2n 1.250 GHz), and )aux) DemonstrationS~y2)F 2, mF of0) a Fundamentalapplied to the ionQuantumin directionsLogicsuchGatethat their wave-vector Experimental realization of a quantum computer re- t5 = driven l and and con (separatedbetweenfrom I 1)lby1)=2.5 MHz);lo)l f),externalforvibrational +levelsco, 1) difference Bk points nearly along the x axis of the quires isolated quantum systems that act as the qubits, and are(the ln)blue= )0)sideband),and )1) transitions(separated byare~,coherently/2 r7= 11.2 drivenMHz). VOLUME 75, NUMBER 25 PH YS ICAL REVIEWC. Monroe, D. M. LETTERSMeekhof, trap;B.E.thusKing,theW.RamanM. Itano,transitionsand D.18J. WinelandareDECEMBERhighly insensitive1995 to the presence of controlled unitary interactions between the Stimulatedbetween lo)lRaman[) andtransitionsll)l 1'). Notebetweenthat S]/2whenhyperfine6 is tunedstatesto are driven through the virtual levelNational= 50Institute ofwithStandardsmotionand Technology,in the otherBoulder,two Coloradodimensions.80303 The Lamb-Dicke qubits that allow construction of the CN gate. As pointed either sideband, the stimulated P~pRaman transitions(5 GHz)entangle a pair of =313 nm laser beams. Measurement of S is accom- (Received 14 July 1995)= authors the suffi- parameter is g 6k xo = 0.2, where xo = 7 nm is out by many [6,9,10], if qubits are not lS) with ln), a crucialthe2 part of the trapped-ion =quantum CN plished by driving cyclingWe I demonstrate1) ~ P3y2)Ftheis asoperation3, mFfollows:3)of a two-bitthe spread"controlled-NOT"of the n =quantum0 wavelogicfunction.gate, which,The incarrier ciently isolated from outside infIuences, decoherences can transitiongate. with o.+-polarizedP3a conjunctionlight and withdetectingsimplethesingle-bitresulting operations, forms a universal quantum logic gate for quantum 2 (ln)) j) )n)) T)) Rabi frequency is Ao2vr = 140 kHz, destroy the quantum interferences that form the computa- ionWefluorescence.realize the controlled-NOTcomputation. gateThe twoby quantumsequentiallybits are stored in the stateinternal and externalVOLUMEdegrees75, NUMBERof freedom 25of a single PH YS ICAL REVIEW LETTERS 18 DECEMBER 1995 th«edInput ()1)l ~~ Output)0)l T)) andstateblue ()0)l g) ~ )1)l T)) tion. Several proposed experimental schemes for quantum three pulses oftrappedthe Ramanatom, whichbeamsis firstto laserthe ioncooled to the zero-point l)energy. Decoherence effects are identified )l applying sideband Rabi frequencies are Ao/2~ = 30 kHz, and for the operation, and the possibility of extending the system to more qubits appearsg, promising. 2 5C according to the following format: lo) I lo) I is as follows: 4714 the auxiliary transitionl) - ()1)ll)T) ~ )0)l $)) Rabi frequency P3a A pulse is PACS onnumbers:the carrier80.+h,transition.65.—w, 32.80. 2 (a) vr/2 applied 89. 03. Pj is g, A, /27r = 12 kHz. The difference frequency of Input state ~ Output state V'~ „,Io)l T) —Io)l T) (2) is The effectWe describedreport the byfirstthe demonstrationoperator (vr/2)of a fundamentalthe Ramancomputersbeams isandtunableCN gatesfrominvolving1200)l toa dipole-dipole1300 MHz inter- Ra man in the notation Ref. 5C lo) I I quantum oflogic gateI 1].that operates on prepared quantumwith the useactionofbetweena doublequantumpass acousto-opticdots or atomic nucleimodulator[6,7, 11,12] l) - lo) l) I 1) I 1) —I 1)I T) transition states. Following the scheme proposed Cirac(AOM),and andmay thesufferRamanfrom decoherencepulse durationsefforts.are Thecontrolledlight shifts on (b) A 2~ pulse isDetectionapplied on the blue sideband by Io)l T) —Io)l T) (2) Zoller [1], we'T)demonstrate a controlled-NOT gatewithon aadditionalatoms switchinglocated insideAOMs.electromagneticSince the Ramancavitiesbeamshave been transition between(o')I and an auxiliary atomic (1) Il)l T) Il)l l). Ra man pair of quantum bits (qubits). The two qubits compriseare generatedshownfromto bea singlelarge enoughlaser and[13,an14]transitionAOM,that onebroadeningcould construct I 1) I 1) —I 1)I T) level )aux) (see Fig. 1). Detection two internal (hyperfine) states and two external (quantizedof the Ramana quantumtransitionsgate duewhereto a finitesingle laserphotonlinewidthpreparedis in the The experiment apparatus is described elsewhere (o') T) Il)l (c) A zr/2 motionalpulse is appliedharmonicon oscillator)the carrier statestransition,of a single trapped cavity acts as the control qubit [7,15] for the atomic state. Il)l l). [16,17]. A singlenegligible9Be+[18]. ion is stored in a coaxial- with a matom.phaseAlthoughshift relativethis minimalto (a), systemleading consiststo the of only twoFollowingHowever,Ramanextensioncooling to largethe )0)lquantum[) state,registersbut beforemay be dif- The experiment apparatus is described elsewhere V' —it illustrates the basic operationsresonatornecessaryrf-ionfor, and ficult. whichCirac and Zoller have a very attrac- qubits, I trap provides pseudopotentialproposed 2 operator ( zr/2) of Ref. 1]. application[17],of the CN operation, [1]we apply appropriately [16,17]. A single 9Be+ ion is stored in a coaxial- the problems associated with, constructing a large scale tive quantum computer architecture based on laser-cooled 1/2 The ~/2 pulses in steps (a) and (c) causeoscillationthe spin IS)frequenciestuned and oftimed(cu„au~,Raman cu,pulses)/2~to =the(11.ion,2,which18.2,can resonator rf-ion trap [17],which provides pseudopotential quantum computer.— trapped ions in2 which the qubits are associated with in- to undergo +1/4 and 1/4 of a complete Rabi cycle, prepare an arbitrary state 1/2of the two-qubit register. For oscillation frequencies of (cu„au~,cu, )/2~ = (11.2, 18.2, The distinctive feature of a quantum29.8) MHzcomputeralongis itsthe principalternal states axesof theofions,the andtrap.informationWe coolis transferred respectively, while leaving10& I S& In) unchanged. The auxiliary instance, to prepare a )1)l J) eigenstate, we apply a 7r 29.8) MHz along the principal axes of the trap. We cool ability to store and process superpositionsthe ion so ofthatnumbersthe n =between0 vibrationalqubits throughgrounda sharedstatemotionalis occu-degree of free- transition in step simply reverses the sign of any pulse on the blue sideband followed a pulse on 10& I S& the ion so that the n = vibrational state is occu- )0&)aux& [2]. This(b) potential for parallel computing has led to dom. The highlights of their proposalby ~ are that (i) deco- 0 ground the in the state pied =95% ofthethe time by employing resolved-sideband)0&)aux& two component ofthe discoveryregister that certain)1)l T) problemsby inducingare morea efficientlycarrier herence()0)l $) can be)1)lsmall,T) (ii))1)lextension$)). We toperformlarge registers is pied =95% of the time by employing resolved-sideband complete Rabi cycle from 'T) )0))aux)stimulated— T). Ramanmeasurementscooling to indetectthe thex dimension,population of exactlythe register FIG. 1. Be+ energy levels. The levelssolved onindicateda )1)lquantumwithcomputer= than )1)l=on a classical relatively FIG.straightforward,1. Be+ energyand (iii) levels.the qubit Thereadoutlevelscan indicated with stimulated Raman cooling in the x dimension, exactly thick lines form the basis ofThetheauxiliaryquantumcomputerlevel register:)aux)[3]. isThetheinternalmostSiyzdramaticIF asexample2,inmF Ref.is 0)an [16].algorithmafter Thean havearbitrarytwonearlyRamanthicksequenceunit linesefficiency.beamsformof operations.theeachbasis containofFirst,the quantumwe register: internal as in Ref. [16]. The two Raman beams each contain

ground state, split from the I state virtue of a measure the probability =that the target qubit = = = presented= by Shor= [4]J) showingbythat a quantum computer In our implementationlevels P(Sare IS)=))of a Iquantum1& and CNI T) logic( Siyq)Fgate, the2, mF 2) and =1 mW of power at =313 nm and are detuned =50 6Hz levels are IS) I 1& and I T) Siyq)F 2, mF 2) and =1 mW of power at =313 nm and are detuned =50 ( = = levels, 6Hz Zeeman shift shouldof =2.be5 ableMHzto resultingfactor largefromnumbersa 0.18verymT efficiently.IS) is in thetargetI J)qubitstateS,lS)y2)Fbyis collectingspanned1, mF bythetwo1) ionS&I2lluorescencehyperfinerespectively,ground separated by red of the excited state. The Raman beams are S,y2)F = 1, mF = 1) levels, respectively, separated by = = P~y2 applied magneticThis fieldappears(seetoFig.be of1). considerableAny componentred ofinterest,theof sinceP~y2whentheexcitedo.+-polarizedstates state.of acoo/2nlasersingleThe=radiation9Be1.Raman250 ionGHz),is appliedbeamsand = resonant)aux)are = S~y2)Fand 2, mF = 0) = 250 and = = = (the lf 2,mF 2) applied to the ion in directions such that their wave-vector coo/2n 1. GHz), )aux) S~y2)F 2, mF = 0) (separated from=I 1) by =2.= 5 MHz); external vibrational levels the quantum registersecurity ofin manythe In)data encryption)0) state appliedschemesis unaffected[5]to reliesthe ionwithon theinthedirectionscyclinglF = 1, ImF/) such= 1)P3gz)Fthatstates,theirabbreviated3, mFwave-vector3)bytransitionthe equivalent difference Bk points nearly along the x axis of the from vibrational are ln) = and (separated by r7= 11.2 MHz). (separated I 1) by =2.5 MHz);the blueexternalsidebandinability transitionof classicaloflevelscomputers andto factorthe effectslarge numbers.(radiative linewidth = )0) 4 MHz)1)at A =in ~,/2 by step (b), spin-1/2 statesy/2' l J) and19. l [)) separated 313frequencynm; by = difference Bk points nearly Stimulatedalong theRamanx transitionsaxis ofbetweenthe S]/2 hyperfine states trap; thus the Raman transitions are highly insensitive to are ln) )0) and )1) (separatedof the twobyRamsey~,A/2quantumr7=pulses11.computer2 cancel.MHz).hostsOnatheregisterother ofhand,qubits, eachsee ofFig. 1).coo/2~Since= 1.this250 radiation6Hz. Thedoescontrolnot qubitappreciablyln) is spanned ~/2 are driven through the virtual P~p level (5 = 50 GHz) with motion in the other two dimensions. The Lamb-Dicke Stimulated Raman transitions between which behaveshyperfineas states mechanicaltrap; thustwo-levelthe Raman transitions are highly insensitive to component S]/2of the quantum quantumregister in the state systemscouple to thetheI firststatetwo(relativequantizedexcitationharmonicprobability:oscillator states any )1)l T) by T) a pair of =313 nm laser beams. Measurement of S is accom- parameter is = 6k xo = 0.2, where xo = 7 nm is and can= store arbitrary superposition states of and 1. g are driven through the virtual acquiresP~p levela sign (5change 50in GHz)step withand the motiontwo Ramseyin the0 =5otherX 1()twoofs),thethedimensions.trappedfluorescencein (lo) readingTheand l 1)),Lamb-Dicketheis separatedproportionalin frequencyto = (b), plished by driving cycling I 1) ~ P3y2)F 3, mF 3) the spread of the n = 0 wave function. The carrier a pair of =313 nm laser beams.pulses Measurementadd constructively,It has beenof Sshowneffectivelyis accom-that "Dipping"any computationthe targeton a register=j,= byFortheS vibrational=transitionwe collectfrequencywith o.on+-polarizedaverageto, /2' ==1light11photonMHzand detectingof the the resulting parameter is P(Sg ).6k xo = 0.$, 2, where xo = 7 nm is Rabi frequency is Ao2vr = 140 kHz, of qubits can be broken up into a series of two-bit harmonicallyion trappedfluorescence.ion. Figure 1 displays the relevant (ln)) j) )n)) T)) plished by driving the cyclingqubit I by1) ~ radians.P3y2)FThe= 3,truthmFtable 3)of the CN operation per measurement cycle [16]. Once S is measured, we the spread oftwo-bitthe n = 0 wave function. The carrier th«ed ()1)l l) ~ )0)l T)) and blue ()0)l g) ~ )1)l T)) transition with o.+-polarized light and detectingoperations the[6],resultingfor example, a series of Be energy levels. Manipulation between the four basis "controlled-NOT*' (CN) quantum logic gates, accompa- Rabi frequency is Ao2vr = 140 kHz,4715 = sideband Rabi frequencies are Ao/2~ = 30 kHz, and ion fluorescence. (ln)) j) )n)) T)) eigenstates accordingspanning tothethe two-qubitfollowing registerformat: (ln)lS) g, nied simple rotations on single qubits 8]. The CN the auxiliary transition ()1)l T) ~ )0)l Rabi frequency by th«ed [7,()1)l l) ~ )0)llO)l T))g), lO)landy), bluell)l g), ()0)lll)l T))g)is ~achieved)1)l byT))applying a $)) gate transforms the state of two qubits e] and e2 from pair of off-resonant(a) A vr/2laser pulsebeamsistoappliedthe ion,onwhichthe carrierdrives transition. is g, A, /27r = 12 kHz. The difference frequency of sideband Rabi frequencies are Ao/2~ = 30 kHz, and V'~ „, according to the following format: let)le2) to le&)le& s E2), where the s operation is addi- stimulated Ramang, Thetransitionseffect isbetweendescribedbasisbystates.the operatorWhen (vr/2) the Raman beams is tunable from 1200 to 1300 MHz tion modulo Reminiscent the classical exclusive-OR in 2. of the auxiliary transition the()1)ldifferenceT) ~frequency)0)lthe$))notation6Rabiof theoffrequencybeamsRef. isI 1].set near 6 = with the use of a double pass acousto-optic modulator (a) A vr/2 pulse is applied on the carrier(XOR) gate,transition.the CN gate representsis a computation/27r at=the12 kHz.coo (the carrier),The differencetransitions arefrequencycoherently drivenof between (AOM), and the Raman pulse durations are controlled "target"g, A, (b) A 2~ pulse is applied on the blue sideband most fundamentalV'~ level: the qubit „,le2) is fiipped internal states lS) while preserving ln). Likewise, for 6 = with The effect is described by the operator (vr/2) REVIEWS OF , VOLUME 75,'T) JANUARY 2003 additional switching AOMs. Since the Raman beams depending on the state of the "control"the Ramanqubit beams is tunable— fromtransition1200 betweento 1300I MHzand an auxiliary atomic (1) le~). con co, (the red sideband), transitions are coherently are generated from a single laser and an AOM, broadening in the notation of Ref. I 1]. Experimental realization of a quantum computer re- level )aux) (see Fig. 1).t5 = with the use of a doubledriven betweenpass acousto-opticl and modulatorand for con + co, 1)l 1) lo)l f), of the Raman transitions due to a finite laser linewidth is quires isolated quantum systems that act as the qubits, and (the blue sideband), transitions are coherently driven (AOM), and the Raman pulse(c)durationsA zr/2 pulseareis appliedcontrolledon the carrier transition, negligible [18]. (b) A 2~ pulse is applied on the bluethe presencesidebandof controlled unitary interactions between the between lo)l and ll)l 1'). Note that when 6 is tuned to Quantum dynamics of single trapped ions[) with a m phase shift relative to (a), leading to the 'T) qubits that allow construction ofwiththe CNadditionalgate. As pointedswitchingeitherAOMs.sideband, Sincethe stimulatedthe RamanRaman transitionsbeams entangle Following Raman cooling to the )0)l [) state, but before transition between I and an auxiliary atomic (1) V' — suffi- operator ( zr/2) of Ref. I 1]. application of the CN operation, we level )aux) (see Fig. 1). out by many authors [6,9,10], ifarethe generatedqubits are notfrom a singlelS) withlaserln), aandcrucialanpartAOM,of the broadeningtrapped-ion quantum CN apply appropriately D.ciently Leibfriedisolated from outside infIuences,of the decoherencesRaman transitionscan gate.due Theto a~/2finitepulseslaserin linewidthsteps (a) andis (c) cause the spin IS) tuned and timed Raman pulses to the ion, which can destroy the quantum interferences that form the computa- We realizeto undergothe controlled-NOT+1/4 and —gate1/4byofsequentiallya complete Rabi cycle, prepare an arbitrary state of the two-qubit register. For (c) A zr/2 pulse is applied on the carrier transition, negligible [18]. Universitytion. Several proposed of Coloradoexperimental andschemes Nationalfor quantum Instituteapplying of Standardsthreerespectively,pulses andofwhile Technology,the Ramanleaving beamsIn) Boulder,unchanged.to the ion The auxiliary instance, to prepare a )1)l J) eigenstate, we apply a 7r with a m phase shift relative to (a), leading to the Following Raman cooling to the )0)l state, but before V' — Colorado4714 80305-3328 transition in[)step (b) simply reverses the sign of any pulse on the blue sideband followed by a ~ pulse on operator ( zr/2) of Ref. I 1]. application of the CN operation,componentwe of the registerappropriatelyin the )1)l T) state by inducing a the carrier ()0)l $) )1)l T) )1)l $)). We perform two apply — The ~/2 pulses in steps (a) and (c) cause the spin IS) tuned and timed Raman pulsescompleteto theRabi ion,cycle whichfrom )1)lcan'T) )0))aux) )1)l T). measurements to detect the population of the register R. Blatt The auxiliary level )aux) is the IF = 2, mF = 0) after an arbitrary sequence of operations. First, we to undergo +1/4 and —1/4 of a complete Rabi cycle, prepare an arbitrary state of the two-qubit register. For Siyz ground state, split from the I J) state by virtue of a measure the probability P(S =)) that the target qubit respectively, while leaving In) unchanged.InstitutThe fu¨r Experimentalphysik,auxiliary instance, Universitato prepare¨t Innsbruck,a )1)l J) eigenstate, A-6020 Innsbruck,we apply Austriaa 7r Zeeman shift of =2.5 MHz resulting from a 0.18 mT IS) is in the I J) state by collecting the ion lluorescence transition in step (b) simply reverses the sign of any pulse on the blue sideband followedapplied magneticby a field~ pulse(see Fig.on1). Any component of when o.+-polarized laser radiation is applied resonant = = = the quantum register in the state is unaffected with the I transition component of the register in the )1)l T)C.state Monroeby inducing a the carrier ()0)l $) )1)l T) )1)l $)). We perform In)two )0) cycling /) P3gz)F 3, mF 3) by the blue sideband transition of step (b), and the effects (radiative linewidth = 19.4 MHz at A = 313 nm; complete Rabi cycle from )1)l 'T) )0))aux) —)1)l T). measurements to detect the population of the register y/2' FOCUS Center and Department of Physics, Universityof the oftwo Michigan,Ramsey ~/2 Annpulses Arbor,cancel. On the other hand, see Fig. 1). Since this radiation does not appreciably The auxiliary level )aux) is the Siyz IF = 2, mF = 0) after an arbitrary sequence of operations. First, we Michigan 48109-1120 any component of the quantum register in the )1)l T) state couple to the I T) state (relative excitation probability: ground state, split from the I J) state by virtue of a measure the probability P(Sacquires=)) thata signthechangetargetin stepqubit(b), and the two Ramsey =5 X 1() s), the fluorescence reading is proportional to pulses add constructively, effectively "Dipping" the target P(S =j,). For S = $, we collect on average =1 photon Zeeman shift of =2.5 MHz resulting from a 0.18 mT IS) is in the I J) state by collecting the ion lluorescence applied magnetic field (see Fig. 1). D.Any Winelandcomponent of when o.+-polarized laser radiationqubit by ~is radians.applied Theresonanttruth table of the CN operation per measurement cycle [16]. Once S is measured, we = = = the quantum register in the In) )0)Nationalstate is Instituteunaffected of Standardswith the andcycling Technology,I /) P3gz)F Boulder, Colorado3, mF 3) 80305-3328transition 4715 by the blue sideband transition of step (b), and the effects (radiative linewidth y/2' = 19.4 MHz at A = 313 nm; of the two Ramsey ~/2 pulses cancel.(PublishedOn the other 10hand, March 2003)see Fig. 1). Since this radiation does not appreciably any component of the quantum register in the )1)l T) state couple to the I T) state (relative excitation probability: acquires a sign change in step (b), andSinglethe trappedtwo Ramsey ions represent=5 X 1() elementarys), the fluorescence quantum systemsreading thatis proportional are well isolatedto from the pulses add constructively, effectively environment."Dipping" the Theytarget can beP(S brought=j,). nearlyFor S to= rest$, we by lasercollect cooling,on average and both=1 theirphoton internal electronic qubit by ~ radians. The truth table statesof the andCN externaloperation motionper canmeasurement be coupled to andcycle manipulated[16]. Once by is fields.measured, This makeswe them ideally suited for quantum-optical and quantum-dynamical studies under well-controlled4715 conditions. Theoretical and experimental work on these topics is reviewed in the paper, with a focus on ions trapped in radio-frequency (Paul) traps.

CONTENTS C. Electromagnetically induced transparency cooling 300 1. Cooling in the Lamb-Dicke regime 301 I. Introduction 282 2. Scattering rates in EIT cooling 302 II. Radio-Frequency Traps for Single Charged Particles 283 3. Experimental results 303 A. Classical motion of charged particles in rf traps 283 V. Resonance Fluorescence of Single Ions 303 1. Classical equations of motion 283 A. Excitation spectroscopy, line shapes 303 2. Lowest-order approximation 284 B. Nonclassical statistics, antibunching, and 3. Typical realizations 285 squeezing 304 B. Quantum-mechanical motion of charged C. Spectrum of resonance fluorescence, homodyne particles in rf traps 285 detection of fluorescence 305 1. Quantum-mechanical equations of motion 286 VI. Engineering and Reconstruction of Quantum States 2. Lowest-order quantum approximation 287 of Motion 307 C. Special quantum states of motion in ion traps 287 A. Creation of special states of motion and 1. The number operator and its eigenstates 287 internal-state/motional-state entanglement 307 2. Coherent states 288 3. Squeezed vacuum states 288 1. Creation of number states 307 4. Thermal distribution 289 2. Creation of coherent states 308 III. Trapped Two-Level Atoms Coupled to Light Fields 289 3. Creation of squeezed states 310 A. The two-level approximation 290 4. ‘‘Schro¨ dinger-cat’’ states of motion 310 B. Theoretical description of the coupling 290 5. Arbitrary states of motion 312 1. Total Hamiltonian and interaction B. Full determination of the quantum state of Hamiltonian 290 motion 312 2. Rabi frequencies 291 1. Reconstruction of the number-state density 3. Lamb-Dicke regime 292 matrix 313 4. Resolved sidebands 292 2. Reconstruction of s-parametrized 5. Unresolved sidebands 293 quasiprobability distributions 313 6. Spectrum of resonance fluorescence 293 3. Experimental state reconstruction 314 C. Detection of internal states 294 VII. in the Motion of a Single 1. The electron shelving method 294 Atom 315 2. Experimental observations of quantum A. Decoherence background 316 jumps 295 B. Decoherence reservoirs 316 D. Detection of motional-state populations 295 1. High-temperature amplitude reservoir 316 IV. Laser Cooling of Ions 296 2. Zero-temperature amplitude reservoir 318 A. Doppler cooling 296 3. High-temperature phase reservoir 319 B. Resolved-sideband cooling 298 C. Ambient decoherence in ion traps 320 1. Theory 298 VIII. Conclusions 320 2. Experimental results 299 Acknowledgments 320

0034-6861/2003/75(1)/281(44)/$35.00 281 ©2003 The American Physical Society Leibfried et al.:Quantumdynamicsofsingletrappedions 311

FIG. 15. Steps for creation of a Schro¨ dinger-cat state (Monroe et al., 1996). For detailed explanation, see text.

the components of a superposition of internal states to be associated to different motional states. The evolving state of the system during the sequence is summarized in Fig. 15. Following (a) laser cooling to the ͉g͉͘n ϭ0͘ state, the Schro¨ dinger-cat state was cre- this common motion, jiggling back and forth Quantum on the quantumated computer by applying chip. Electric several sequentialforces pulses of the Raman like two pendulum weights connected by a can move the ionbeams: strings (b)without A ␲/2 disturbing pulse on their the carrier transition split the spring. Researchers can excite the common information internal states, wavehence functionpreserving into the andata equal they superposition of states motion by applying photon pressure from a laser carry. And researchers͉g͉͘0͘ couldand ͉e entangle͉͘0͘. (c) one The string selective dipole force of the science offers coherent displacement beams excited the motion corre- beam modulated at the natural oscillation fre- with another to transfer data and perform pro- an opportunity lated with the ͉e͘ component to a state ͉␣͘. (d) A ␲ pulse quency of the trap [see box on opposite page]. cessing tasks thaton require the carrier the action transition of many then log swapped- the internal states More important, the laser beam can be made to radically ic gates. The resultingof the architecture superposition. would (e) some Next,- the displacement beams to affect the ion only if its magnetic orientation is change what resembleexcited the familiar the motion charge-coupled correlated with the new ͉e͘ compo- up, which here corresponds to a qubit value of 1. device (CCD) usednent in to digital a second cameras; coherent just as state a ͉␣ei␾͘. (f) A final ␲/2 What is more, these microscopic bar magnets computing. CCD can move electricpulse on charge the carrieracross an combined array of the two coherent states. rotate their orientation while they are oscillating capacitors, a quantumThe relative chip could phases propel [␾ and strings the phases of steps (b), (d), Scientists may and (f)] of the steps aboveLeibfried et wereal.:Quantumdynamicsofsingletrappedions controlled by phase 311 in space, and the amount of rotation depends on finally realize of individual ionslocking through the a rfgrid sources of linear that traps. created the frequency split- whether one or both of the ions are in the 1 state. Many of the trapped-ionting of the Ramanexperiments or displacement at NIST beams, respectively. The net result is that if we apply a specific laser their dream have involved shuttlingThe state ions created through after a multi step- (e) is a superposition of force to the ions for a carefully adjusted duration, zone linear trap.two Extending independent this idea coherent to much states, each correlated with FIG. 16. Phase signal of a Schro¨ dinger-cat state for different of creating an internal state of the ion (for ␾ϭ␲), we can create a CNOT gate. When the qubits are larger systems, however, will require more magnitudes of the coherent displacement (Monroe et al., initialized in superposition states, the action of a quantum sophisticated structures with͉␣͘ a͉e multitude͘ϩ͉Ϫ␣͉͘ gof͘ elec- 1996). this gate entangles the ions, making it a funda- trodes that couldFIG. guide 15.͉⌿ Steps͘ theϭ for ions creation in of any a Schro direction.¨ dinger-cat. state (Monroe (145) machine that Leibfried et al.:Quantumdynamicsofsingletrappedionset al.Leibfried, 1996). Foret detailed al.:Quantumdynamicsofsingletrappedions explanation,& see text. 311 311 mental operation for the construction of an arbi- The electrodes would have to be very small—in or decoherence. The experiment was continuously the components of a superposition of internal states to trary quantum computation among many ions. can tackle tasks the range of 10 betoIn associated100 this millionths state, to different the of motional widely a meter states. separated—to coherent states re- repeated—cooling, state preparation, detection—while Researchers at several laboratories—includ- confine and controlplaceThe the evolving the ion-shuttling state classical of the system notionsprocedure during ofthe sequence ‘‘dead’’ and ‘‘alive’’ in slowly sweeping the relative motional phase ␾ of the once thought isSchro summarized¨ dinger’s in Fig. original 15. Following thought (a) laser cooling experiment. to The coher- ing groups at the University of Innsbruck, the precisely. Fortunately,the ͉g͉͘n ϭ 0the͘ state, builders the Schro of¨ dinger-cat trapped- state was cre- coherent states. atedence by applying of this several mesoscopic sequential pulses superposition of the Raman was verified by Figure 16 shows the measured P (␾) for a few differ- University of Michigan at Ann Arbor, the impossible. ion quantum computers can take advantage of g beams:recombining (b) A ␲/2 pulse the on coherent the carrier transition wave-packet split the components in ent values of the coherent-state amplitude ␣, which is set National Institute of Standards and Technology microfabricationwave techniques, function into such an equal as microelec superposition- of states ͉theg͉͘0͘ finaland ͉e͉͘ step0͘. (c) (f). The selective This resulted dipole force in of different the degrees of by changing the duration of application of the displace- (NIST) and the University of Oxford—have dem- tromechanical coherentsystemsinterference displacement (MEMS) of beams the and excited two semicon wavethe motion- packets corre- as the relative e ment beams [steps (c) and (e) from above]. The unit FIG. 15. Steps for creation of a Schro¨ dinger-cat state (Monroe latedphase with␾ the of͉ ͘ component the displacement to a state ͉␣͘. (d) forces A ␲ pulse [steps (c) and (e)] onstrated working CNOT gates. Of course, et al., 1996). For detailed explanation, see text.ductor lithography,on the that carrier are transition already then used swapped to the con internal- states visibility (CӍ1) of the interference feature near ␾ϭ0 none of the gates works perfectly, because they struct conventionalofwas the computer superposition.varied. The chips. (e) nature Next, the of displacement the interference beams depended on verifies that superposition states were produced instead FIG. 15. Steps for creation of a Schroexcited¨ dinger-cat the motion state correlated (Monroe with the new ͉e͘ compo- the phases of steps (b), (d),i␾ and (f) and was set to cause are limited by such things as laser-intensity fluc- the components of a superposition of internalOver states the to pastnent year to a secondseveral coherent research state ͉␣ groupse ͘. (f) A final ␲/2 of statistical mixtures, and the feature clearly narrows as be associated to differentet al., motional 1996). For states. detailed explanation,pulse see on the text. carrier combined the two coherent states. this common motion, jiggling back and forth destructive interferenceon the quantum of thecomputer wave chip. packetsElectric forces in the ͉g͘ ␣ increases. The amplitude of the Schro¨ dinger-cat state tuations and noisy ambient electric fields, which have demonstratedQuantumThe relative the first phases integrated [␾ and the phasesion traps. of steps (b), (d), The evolvinglike state two of pendulum the system weights during connected the sequence by a can move the ion strings without disturbing their andstate. (f)] ofThe the interference steps above were was controlled directly by phase measured by detect- was extracted by fitting the interference data to the ex- compromise the integrity of the ions’ laser-excit- is summarized inspring. Fig. Researchers 15. Following can (a)excite laserScientists the coolingcommon at to informationthe University of Michiganinternal states,and hencethe preserving the data they the components¨ of a superpositionlockinging theof the internal rf probability sources thatstates createdP to(␾ the) frequency that the split- ion was in the ͉g͘ the ͉g͉͘n ϭ0͘ state,motion the by applying Schrodinger-cat photon pressure state from was a laser cre- science offers carry. Andg researchers could entangle one string pected form of the interference fringe. The extracted ed motions. Currently researchers can make a ated by applying severalbe associated sequential pulses toLaboratory different of the Raman motional for tingPhysicalinternal of states. the Raman Sciences state or for displacement at a the given Universi beams, value respectively.- of ␾. The signal for par- beam modulated at the natural oscillation fre- an opportunityThe state created afterwith step another (e) is ato superposition transfer data and of perform pro- values of ␣ agreed with an independent calibration of beams: (b) A ␲/2quency pulse ofThe onthe trap the evolving [ carriersee box on transition opposite state page of split]. the the systemtwoticular independent during choices the coherent sequence ofcessing the states,phases tasks each that correlated require in (b), the withaction (d), of and many log (f)- is two-qubit gate that operates with a “fidelity” of ty of Maryland employed a gallium arsenide FIG. 16. Phase signal of a Schrothe¨ dinger-cat displacement state for different forces. Coherent-state amplitudes as wave function intoMore an important, equal the superposition laser beam can be of made states to radicallyan internal state of the ionic (forgates.␾ Theϭ␲ resulting), architecture would some- is summarized in Fig. 15. Following (a) laser cooling to magnitudes of the coherent displacement (Monroe et al., slightly above 99 percent, meaning that the ͉g͉͘0͘ and ͉e͉͘0to͘. affect (c) the The ion selective only if its magnetic dipolesemiconductor orientation force of is the structure for their1 quantumwhat resemble chip. the familiar charge-coupled high as ␣Ӎ2.97(6) were measured, corresponding to an the ͉g͉͘n ϭ0͘ state, the Schrochange¨ dinger-cat ͉␣͉͘e state͘ϩ͉Ϫ␣͘ was͉g͘ cre-Ϫ␣2(1Ϫcos ␾) 2 1996). probability of the gate operating in error is less coherent displacementup, which beamshere corresponds excited to the a qubitInvestigators motion value corre-of 1. at NIST͉⌿P͘ϭ developedg͑␾͒ϭ ͓1 aϪ devicenew.Ce (CCD) ion-trap used in digitalcos (145) cameras;͑␣ sin just␾ as͒ ͔a , (146) average of n Ӎ9 vibrational quanta in the state of mo- lated with the ͉eWhat͘ component atedis more, by these to applying amicroscopic state ͉␣͘. several (d)bar Amagnets␲ pulse sequential computing. pulses of&2 the RamanCCD can move electric charge across an array of or decoherence. The experimenttion. This was continuously indicates a maximum spatial separation of than 1 percent. But a useful quantum computer rotate their orientation while theygeometry are oscillating in which the ions float abovecapacitors, a achip’s quantum chip could propel strings on the carrier transitionbeams: then swapped (b) A ␲ the/2 internal pulse on states theScientists carrierIn this state, transition may the widely split separated the coherent states re- repeated—cooling, state preparation, detection—while in space, and the amount of rotation depends on where ␣ is the magnitudeof individual ions of through the coherenta grid of linear statestraps. and C 4␣x ϭ83(3) nm, which was significantly larger than the may need to achieve a fidelity of about 99.99 per- of the superposition.wave (e) Next, function the displacement intosurface. an beams Groups equalplace superposition at theAlcatel-Lucent classical notions of states and of ‘‘dead’’ Sandia and ‘‘alive’’ in slowly sweeping the relative motional0 phase ␾ of the whether one or both of the ions are in the 1 state. finally realize Many of the trapped-ion experiments at NIST excited the motion correlated with the new ͉e͘ compo- Schroϭ1¨ dinger’s is the original expected thought visibility experiment. of The the coher- fringescoherent in the states. absence single wave-packet size characterized by x0ϭ7.1(1) nm cent for error-correction techniques to work The net͉g result͉͘0͘ is thatand if we͉e iapply͘␾͉0͘National a. specific (c) The laser Laboratories selectivetheirence dream of dipole this have mesoscopic forcefabricated superpositionhave of theinvolved even shuttling wasfan- verified ions through by a multi- nent to a second coherent state ͉␣e ͘. (f) A final ␲/2 of any mechanisms decreasing the contrast,Figure such 16 shows as, for the measuredas wellPg(␾) as for a few typical differ- atomic dimension (Ӎ0.1 nm). The pulse on the carrierforce tocoherent combined the ions for a thecarefully displacement two adjusted coherent duration, beamsstates. of excitedcreatingrecombining the the motion coherentzone corre- wave-packet linear trap. components Extending this in ideaent to much values of the coherent-state amplitude ␣, which is set properly. One of the main tasks of all trapped- we can create a CNOT gate. Whencier the qubitsion traps are on thesilicon final stepchips. (f). Much This resultedlarger work systems, in remains different however, degrees will of require more individual wave packets were thus clearly spatially sepa- The relative phaseslated [␾ and with the phases the e of stepscomponent (b), (d), to aexample, state ␣͘. imperfect (d) A ␲ pulse state preparation, motionalby changing heating, the duration of application of the displace- initialized in superposition states,͉ ͘ the action of a quantuminterference͉ of the two wavesophisticated packets structures as the with relative a multitude of elec- ion research groups is to reduce the background and (f)] of the steps above were controlledto be by done phase on these chip traps. The atomic noise ment beams [steps (c) and (e) from above]. The unit this gateon entangles the carrier the ions, transitionmaking it a funda then- swappedphase ␾ of the the internal displacementtrodes states forces that could [steps guide (c) the and ions (e)] in any visibilitydirection. (C 1) of the interference feature near ␾ϭ0 locking the rf sources that created the frequency split-machine that Ӎ noise enough to reach these goals, and although mental operation for the constructionemanating of an arbi- fromwas nearby varied. surfaces The nature must ofThe the electrodesbe interference reduced, would depended have to be on very smallverifies—in that superposition states were produced instead ting of the Raman orof displacement the superposition. beams, respectively. (e) Next, theRev. displacementMod. Phys., Vol. 75, beams No. 1, January 2003 trary quantum computation among many ions. canthe tackle phases of tasks steps (b), (d),the and range (f) of and 10 to was 100 set millionths to cause of a meterof statistical—to mixtures, and the feature clearly narrows as this effort will be daunting, nothing fundamen- The state createdexcited after step the (e) ismotion a superpositionperhaps correlated by of cooling withdestructive thethe interference newelectrodese compo- of thewith wave liquid packets in the g Researchers at several laboratories—includ- ͉ ͘ confine and control the ion-shuttling͉ ͘ procedure␣ increases. The amplitude of the Schro¨ dinger-cat state two independent coherent states, each correlated withoncestate. thought Thei␾ interference was directly measured by detect- tal stands in the way of its achievement. ing groupsnent at the to University a second of nitrogenInnsbruck, coherent the or stateliquid͉ ␣helium.e ͘. (f) And A researchers finalprecisely.␲/2¨ Fortunately, must the builders ofwas trapped- extracted by fitting the interference data to the ex- ingFIG. the 16. probability Phase signalP ( of␾) a that Schro thedinger-cat ion was state in the for͉g different͘ an internal stateUniversity of the ion of (forMichigan␾ϭ␲ at), Ann Arbor, the impossible. g ion quantum computers can take advantagepected of form of the interference fringe. The extracted pulse on the carrierskillfully combined choreographinternalmagnitudes the two state coherentthe of for the amovement given coherent value states. of displacement ␾of. The ions signal (Monroe for par- etvalues al., of ␣ agreed with an independent calibration of National Institute of Standards and Technology microfabrication techniques, such as microelec- ͉␣͉͘e͘ϩ͉ϪThe␣͉͘g͘ relative phases [␾ and theticular1996). phases choices of of steps the phases (b), in (d), (b), (d), and (f) is the displacement forces. 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This indicates a maximum spatial separation of are limited by such things as laser-intensity fluc- repeated—cooling, stateOver preparation, the past year detection—while several research groups quantum computer out of trapped ions? Unfor- placeLEVITATED the classical STRINGting notionsof eight of of the calcium ‘‘dead’’ Raman ions and or are ‘‘alive’’ displacement confined in where in a vacuum beams,␣ is the magnitude chamber respectively. of and the coherent laser- states and C 4␣x0ϭ83(3) nm, which was significantly larger than the tuations and noisy ambient electric fields, which ϭslowly1 is the sweeping expected visibility the relativehave of demonstrated the motionalfringes in the the phase first absence integrated␾ of singletheion traps. wave-packet size characterized by x ϭ7.1(1) nm tunately, it appears that longer strings of ions— Schrocooled¨ dinger’s to be original nearly thought at rest. experiment. 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The I of any mechanisms decreasing the contrast, such as, for Ӎ ence of this mesoscopic superposition was verified by those containing more than about 20 qubits— ed motions.two Currently independent researchers coherent can make a states,example,Figure each imperfect 16 correlated shows state the preparation,Laboratory measured with forP motional gPhysical(␾) for Sciences heating, a few at differ- theindividual Universi- wave packets were thus clearly spatially sepa- recombining thetwo-qubit coherent gate wave-packetthat operates with components a “fidelity” of in ent values of the coherent-statety of Maryland amplitude FIG.employed␣ 16., whicha gallium Phase is setarsenide signal of a Schro¨ dinger-cat state for different the final step (f). Thisan resulted internal in state different of thedegrees ion of (for ␾ϭ␲), would be nearly impossible to control because slightly above 99 percent, meaning that the Rev.by changingMod. 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www.SciAm.com © 2008 SCIENTIFIC AMERICAN, INC. SCIENTIFIC AMERICAN 69 progress Architecture for a large-scale ion-trap quantum computer

D. Kielpinski*, C. Monroe† & D. J. Wineland‡

* Research Laboratory of Electronics and Center for Ultracold Atoms, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA † FOCUS Center and Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1120, USA ‡ Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA ......

Among the numerous types of architecture being explored for quantum computers are systems utilizing ion traps, in which quantum bits (qubits) are formed from the electronic states of trapped ions and coupled through the Coulomb interaction. Although the elementary requirements for quantum computation have been demonstrated in this system, there exist theoretical and technical obstacles to scaling up the approach to large numbers of qubits. Therefore, recent efforts have been concentrated on using quantum communication to link a number of small ion-trap quantum systems. Developing the array-based approach, we show how to achieve massively parallel gate operation in a large-scale quantum computer, based on techniques already demonstrated for manipulating small quantum registers. The use of decoherence-free subspaces significantly reduces decoherence during ion transport, and removes the requirement of clock synchronization between the interaction regions.

quantum computer is a device that prepares and the Coulomb coupling necessary for entangling gates3,21. Lasers are manipulates quantum states in a controlled way, offering focused through the interaction region to drive gates. We then move significant advantages over classical computers in tasks the ions again to prepare for the next operation. such as factoring large numbers1 and searching large We can realize the trapping and transport potentials needed for databases2. The power of quantum computing derives the QCCD using a combination of radio-frequency (r.f.) and Afrom its scaling properties: as the size of these problems grows, the quasistatic electric fields. Figure 1 shows only the electrodes that resources required to solve them grow in a manageable way. Hence a support the quasistatic fields. By varying the voltages on these useful quantum computing technology must allow control of large electrodes, we confine the ions in a particular region or transport quantum systems, composed of thousands or millions of qubits. them along the local trap axis, which lies along the thin arrows in Fig. The first proposal for ion-trap quantum computation involved 1. Two more layers of electrodes lie above and below the static confining a string of ions in a single trap, using their electronic states electrodes, as shown in Fig. 2. Applying r.f. voltage to the outer layers as qubit logic levels, and transferring between creates a quadrupole field that confines the ions transverse to the ions through their mutual Coulomb interaction3. All the elementary local trap axis by means of the ponderomotive force22. This geometry requirements for quantum computation4—including efficient quan- allows stable transport of the ions around ‘T’ and ‘X’ junctions, so we tum state preparation5–7, manipulation7–10 and read-out7,11,12—haveprogresscan build complex, multiply connected trap structures. been demonstratedArchitecture in this system. for But a large-scale manipulating a large ion-trap number of ions in a single trap presents immense technical difficulties, and scaling argumentsquantum suggest computer that this scheme is limited to compu- 13–15 tations on tensD. Kielpinski of ions*, C. Monroe† &. D. One J. Wineland way‡ to escape this limitation involves

* Research Laboratory of Electronics and Center for Ultracold Atoms, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA quantum communication† FOCUS Center and Department of Physics, between University of Michigan, a Ann number Arbor, Michigan 48109-1120, of small USA ion-trap ‡ Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado 80305, USA quantum registers...... Recent proposals along these...... lines that use 16–18 19 photon couplingAmong the numerousand types spin-dependent of architecture being explored for Coulomb quantum computers interactions are systems utilizing ion traps, in which quantum bits (qubits) are formed from the electronic states of trapped ions and coupled through the Coulomb interaction. Although have not yetthe been elementary tested requirements in for the quantum laboratory. computation have been The demonstrated scheme in thispresented system, there exist theoretical and technical obstacles to scaling up the approach to large numbers of qubits. Therefore, recent efforts have been concentrated on here, however,using quantum uses communication only quantum to link a number ofmanipulation small ion-trap quantum systems. techniques Developing the array-based that approach, we show how to achieve massively parallel gate operation in a large-scale quantum computer, based on techniques already have alreadydemonstrated been individually for manipulating small experimentally quantum registers. Theprogress use demonstrated. of decoherence-free subspaces significantly reduces decoherence during ion transport, and removes the requirement of clock synchronization between the interaction regions.

ArchitectureThe quantum forCCDquantum a large-scale computer is a device that ion-trap prepares and the Coulomb coupling necessary for entangling gates3,21. Lasers are manipulates quantum states in a controlled way, offering focused through the interaction region to drive gates. We then move significant advantages over classical computers in tasks the ions again to prepare for the next operation. quantumTo build up computer a large-scalesuch as factoring quantum large numbers1 andcomputer, searching large We we can realize have the trapping proposed and transport potentials a needed for databases2. The power of quantum computing derives the QCCD using a combination of radio-frequency (r.f.) and D.‘quantum Kielpinski*, C. Monroe† & charge-coupled D. J.Afrom Wineland its scaling‡ properties: as the device’ size of these problems (QCCD) grows, the architecturequasistatic electric fields. Figure consisting 1 shows only the electrodes that * Research Laboratory of Electronics andresources Center for Ultracold required Atoms, to Massachusettssolve them Institute growof in Technology, a manageable Cambridge, way. Massachusetts Hence a 02139,support USA the quasistatic fields. By varying the voltages on these †ofFOCUSa Center large and Department numberof Physics,useful University quantum of Michigan, computingof Ann interconnected Arbor, technology Michigan 48109-1120, mustallow USA control ion of large traps.electrodes, we By confine changing the ions in a particular the region or transport ‡ Time and Frequency Division, Nationalquantum Institute of systems, Standards and composed Technology, Boulder, of thousands Colorado 80305, or millions USA of qubits. them along the local trap axis, which lies along the thin arrows in Fig...... The first proposal for ion-trap quantum...... computation involved 1. Two more layers of electrodes lie above and below the static operating voltagesconfining a string of of theseions in a single traps, trap, using theirwe electronic can states confineelectrodes, a as shown few in Fig.ions 2. Applying in r.f. each voltage to the outer layers Among the numerous types ofas architecture qubit logic being levels, explored and transferring for quantum quantum computers information are systems between utilizing ioncreates traps, a in quadrupole which field that confines the ions transverse to the quantumtrap bits or (qubits) shuttle are formedions from through ions the electronic their from mutual states of Coulomb trapped trap ions interactionto and coupled trap.3. All through the In elementary the Coulomb any interaction.local particular trap Although axis by means trap, of the ponderomotive we can force22. This geometry the elementary requirements for quantum computation have been demonstrated4 in this system, there exist theoretical and technical obstacles to scalingrequirements up the approach for to quantum large numbers computation of qubits. Therefore,—including recent efficient efforts quan- have beenallows concentrated stable transport on of the ions around ‘T’ and ‘X’ junctions, so we usingmanipulate quantum communicationtum a to linkfew state a number preparation ions of small5–7, ion-trap manipulationusing quantum7–10 the systems.and read-out methods Developing7,11,12 the—have array-based alreadycan build approach, complex, we demonstrated, multiply connected trap structures. show how to achieve massivelybeen parallel demonstrated gate operation in this in system. a large-scale But manipulating quantum computer, a large number based on techniques already demonstrated for manipulatingof ions small in quantum a single trapregisters. presents The immense use of decoherence-free technical difficulties, subspaces and significantly reduces decoherencewhile during the ion connections transport, and removes the requirement between of clock synchronization traps allow between the interactioncommunication regions. between scaling arguments suggest that this scheme is limited to compu- 13tations on tens of ions13–15. One way to escape this limitation involves 20 sets ofquantum ions computerquantum. isBecause a device communication that prepares both and betweenthe the Coulomb a number speed coupling of necessary small of ion-trap for quantumentangling gates3,21. Lasers logic are gates and manipulates quantumquantum states in a controlled registers. way, Recent offering proposalsfocused through along the these interaction lines region that to use drive gates. We then move significant advantages over classical computers16–18 in tasks the ions again to prepare for the next operation.19 the shuttlingsuch as factoring largephotonspeed numbers coupling1 and searching areand largelimited spin-dependentWe can realize by Coulomb the thetrapping interactions and trap transport strength, potentials needed for shuttling ions databases2. The powerhave of quantum not yet computing been tested derives in thethe laboratory. QCCD using The ascheme combination presented of radio-frequency (r.f.) and Afrombetween its scaling properties: memory as thehere, size of however, these problems anduses grows, only the quantum interactionquasistatic manipulation electric fields. techniques Figure regions 1 shows that only the should electrodes that consume an resources required to solve them growhave in already a manageable been way. individually Hence a experimentallysupport the quasistatic demonstrated. fields. By varying the voltages on these useful quantum computing technology must allow control of large electrodes, we confine the ions in a particular region or transport quantumacceptably systems, composedsmall of thousandsThe quantum or fraction millions CCD of qubits. ofthem a along clock the local trap cycle.axis, which lies along the thin arrows in Fig. The first proposal for ion-trap quantum computation involved 1. Two more layers of electrodes lie above and below the static confining a string of ions in a singleTo trap, build using up their a large-scale electronic states quantumelectrodes, computer, as shown we in Fig.have 2. Applyingproposed r.f. avoltage to the outer layers as qubitFigure logic levels, and transferring 1 shows‘quantum quantum charge-coupled information a diagram between device’creates (QCCD) of a quadrupole the architecture field proposed that consisting confines the ions transverse device. to the Trapped ions ions through their mutual Coulombof interactiona large number3. All the elementary of interconnectedlocal trap ionaxis by traps. means By of the changing ponderomotive the force22. This geometry requirements for quantum computationoperating4—including voltages efficient of these quan- traps,allows we stablecan confine transport a of few the ions around in each ‘T’ and ‘X’ junctions, so we storing quantum5–7 7–10 information7,11,12 are held in the memory region. To tum state preparation , manipulationtrap or shuttleand read-out ions from—have trap tocan trap. build In complex, any particular multiply trap, connected we can trap structures. been demonstrated in this system. But manipulating a large number Figure 1 Diagram of the quantum charge-coupled device (QCCD). Ions are stored in ofperform ions in a single trap presents a logic immensemanipulate technical gate, a few difficulties, ions we using and move the methods the already relevant demonstrated, ions into an interaction scaling arguments suggest that thiswhile scheme the connections is limited to compu-between traps allow communication between the memory region and moved to the interaction region for logic operations. Thin tations on tens of ions13–15. One waysets to escapeof ions this13. limitation Because involves both the speed of quantum logic gates20 and quantumregion communication by applying between a number ofappropriate small ion-trap voltages to the electrode segments. the shuttling speed are limited by the trap strength, shuttling ions arrows show transport and confinement along the local trap axis. quantum registers. Recent proposalsbetween along memory these lines and that interaction use regions should consume an photon coupling16–18 and spin-dependent Coulomb interactions19 haveIn not the yet been interaction tested in the laboratory.acceptably The small region, scheme fraction presented of athe clock cycle. ions are held close together, enabling here, however, uses only quantumFigure manipulation 1 shows techniques a diagram that of the proposed device. Trapped ions have already been individually experimentallystoring quantum demonstrated. information are held in the memory region. To Figure 1 Diagram of the quantum charge-coupled device (QCCD). Ions are stored in The quantum CCD perform a logic gate, we move the relevant ions into an interaction the memory region and moved to the interaction region for logic operations. Thin ToNATURE|VOL build up a large-scale quantum 417|13region computer, by applyingJUNE we have proposed appropriate 2002|www.nature.com/nature a voltages to the electrode segments. arrows show transport and© confinement 2002 along Nature the local trap axis.Publishing Group 709 ‘quantum charge-coupled device’In (QCCD) the interaction architecture region, consisting the ions are held close together, enabling of a large number of interconnected ion traps. By changing the operating voltages of these traps,NATURE|VOL we can confine 417|13 a few JUNE ions 2002|www.nature.com/nature in each © 2002 Nature Publishing Group 709 trap or shuttle ions from trap to trap. In any particular trap, we can manipulate a few ions using the methods already demonstrated, while the connections between traps allow communication between sets of ions13. Because both the speed of quantum logic gates20 and the shuttling speed are limited by the trap strength, shuttling ions between memory and interaction regions should consume an acceptably small fraction of a clock cycle. Figure 1 shows a diagram of the proposed device. Trapped ions storing quantum information are held in the memory region. To Figure 1 Diagram of the quantum charge-coupled device (QCCD). Ions are stored in perform a logic gate, we move the relevant ions into an interaction the memory region and moved to the interaction region for logic operations. Thin region by applying appropriate voltages to the electrode segments. arrows show transport and confinement along the local trap axis. In the interaction region, the ions are held close together, enabling

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