An atomic electron is prepared in a state closely analogous to Schrödinger’s coherent superposition of “live cat” and “dead cat.” The electronic state is a coherent superposition of two spatially localized wave packets separated by approximately 0.4 mm at the opposite extremes of a Kepler orbit. State-selective ionization is used to verify that only every other atomic level is populated in the “cat state,” and a Ramsey fringe measurement is used to verify the coherence of the superposition

Noel, Michael W.

Stroud, C. R., Jr.

English

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Bryn Mawr CollegeScholarship, Research, and Creative Work at Bryn MawrCollegePhysics Faculty Research and Scholarship Physics1996Excitation of an Atomic Electron to a CoherentSuperposition of Macroscopically Distinct StatesMichael W. NoelBryn Mawr College, mnoel@brynmawr.eduC. R. Stroud Jr.Let us know how access to this document benefits you.Follow this and additional works at: http://repository.brynmawr.edu/physics_pubsPart of the Physics CommonsThis paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. http://repository.brynmawr.edu/physics_pubs/77For more information, please contact repository@brynmawr.edu.Custom CitationMichael W. Noel and C. R. Stroud, Jr., "Excitation of an Atomic Electron to a Coherent Superposition of Macroscopically DistinctStates," Phys. Rev. Lett. 77, 1913 (1996).PHYSICAL REVIEWLETTERSVOLUME 77 2 SEPTEMBER 1996 NUMBER 10Excitation of an Atomic Electron to a Coherent Superposition of Macroscopically Distinct StatesMichael W. Noel and C. R. Stroud, Jr.The Institute of Optics, University of Rochester, Rochester, New York 14627(Received 15 November 1995)An atomic electron is prepared in a state closely analogous to Schrödinger’s coherent superpositionof “live cat” and “dead cat.” The electronic state is a coherent superposition of two spatiallylocalized wave packets separated by approximately 0.4 mm at the opposite extremes of a Keplerorbit. State-selective ionization is used to verify that only every other atomic level is populated inthe “cat state,” and a Ramsey fringe measurement is used to verify the coherence of the superposition.[S0031-9007(96)01071-X]PACS numbers: 03.65.Bz, 32.80.Rm, 42.50.MdBecause the description of reality appears so differentin classical and quantum physics, the boundary betweenthe macroscopic and microscopic worlds has long beena fascinating area of research. Schrödinger brought thedistinction between the two descriptions into sharp focusby introducing a cat that was placed in a quantumsuperposition of “live cat” and “dead cat” [1].Schrödinger’s cat has been the subject of an enormousnumber of papers and books [2], and has even become fa-miliar to the layman. With recent developments in tech-nology in the mesoscopic regime (where dimensions areof the order of a micron) that lies at the boundary be-tween the microscopic and macroscopic regimes, therehave been proposals of practical systems that are realiza-tions of Schrödinger’s cat state. The systems consideredvary greatly, from nonlinear optics [3,4] and micromasers[5,6] to molecules [7,8] and trapped ions [9].In these realizations a system with macroscopic (ornearly macroscopic) dimensions that is reasonably wellmodeled as a harmonic oscillator is excited into a su-perposition of two coherent states. This simple act ofsuperposing two quasiclassical states results in some re-markably nonclassical features such as sub-Poissonianand oscillatory number state distributions, squeezing, andinterference [10–12].A superposition of two harmonic oscillator coherentstates forms a Schrödinger cat state when the two are cho-sen to be localized at macroscopically separated points. Astate of this type can be writtenjcstdl ≠ Nfjastdl 1 eifj 2 astdlg , (1)where N is a normalization constant. The coherent statescan be expanded out in terms of number statesjastdl ≠ e2jaj2y2X`n≠0jas0djnpn!e2isn11y2dvtjnl . (2)In this case each of the coherent states forms a mini-mum uncertainty wave packet oscillating back and forthperiodically between its turning points in a parabolic po-tential. There are two phases associated with this oscilla-tion. The phase of the complex number as0d determinesthe locations of the two wave packets at t ≠ 0. For realas0d, the two wave packets start out at the opposite ex-tremes of the oscillatory motion separated by the distance2as0d. During a single period of the oscillation the twowave packets will pass through each other as each goesthrough the same motion. The constant f is the phasedifference between the de Broglie waves associated withthe two packets. When the two wave packets overlapthis phase will determine the nature of the interferencebetween them. As we will see later, it also determinesthe population distribution in the various eigenstates. Atsome particular time in the motion the system will in gen-eral be in a superposition which is simultaneously locali-zed in two different locations. If the two positions are0031-9007y96y77(10)y1913(4)$10.00 © 1996 The American Physical Society 1913VOLUME 77, NUMBER 10 P HY S I CA L REV I EW LE T T ER S 2 SEPTEMBER 1996macroscopically distinguishable then this state is similarto Schrödinger’s cat state in that it represents a quantummechanical superposition of two classically distinguish-able physical states which are localized in one or the otherof the two classical positions only by the act of a measure-ment which destroys the superposition.We have used two phase-coherent laser pulses to excitea single atomic electron into a state well approximated byEq. (1), in which the electron is simultaneously localizedin two wave packets each oscillating in and out about thenucleus in a single Kepler orbit. The diameter of the orbitis approximately 0.4 mm, which is not quite macroscopic,but mesoscopic, of the order of size of a transistor on acomputer chip. We characterize the wave packet cat statein two ways. First, the quantum eigenstate distribution ismeasured for various phase differences between the twowave packets. Next, we probe the interference betweenthe two wave packets to demonstrate the coherent, ratherthan statistical, nature of the superposition.A radial wave packet is formed when an atom inter-acts with a short laser pulse tuned to the Rydberg se-ries where the atomic energy levels are closely spaced[13–16]. The result of this interaction is the excitationof some of the population from the ground state to a co-herent superposition of Rydberg states. The number ofstates in the superposition depends on the bandwidth ofthe transform limited laser pulse and, in the weak fieldlimit, is simply the Fourier transform of the pulse enve-lope. Since the superposition is of energy eigenstates allwith the same angular momentum, the localization of theelectron is in the radial coordinate only. The electron (ini-tially localized to a radial shell near the core) oscillates inand out in a breathing motion at the classical Kepler pe-riod. For short times a radial wave packet is very similarto the coherent state of a harmonic oscillator which alsoconsists of a superposition of energy eigenstates [Eq. (2)]and results in a localized probability oscillating back andforth in the harmonic potential. In the harmonic oscil-lator this classical oscillation will continue indefinitelyprovided there is no further interaction after the initial ex-citation. Since the atomic potential is only approximatelyharmonic, the radial wave packet oscillates classically foronly a limited period of time. The complex dynamics ofatomic electron wave packets have been described in de-tail by several authors [17–20].A Schrödinger cat state of the atom is formed byinteracting with a pair of phase-coherent time-delayedlaser pulses. The separation of these pulses is one-halfof the Kepler period. The first pulse excites a portion ofthe population to a radial wave packet state. One-half ofthe Kepler period later the wave packet excited by thefirst pulse has moved to the outer turning point and thesecond laser pulse arrives to excite another wave packetat the inner turning point. The phase between the twolaser pulses is very carefully controlled. The result of thissequence is that a single electron is now in a superpositionof being at two well separated spatial locations (the innerand outer turning points).We set up the experiment as follows. A 25 ps, 10 mJ,nearly transform-limited laser pulse was tuned to couplethe ground state of a potassium atom to the Rydbergseries. This pulse was divided into two time-delayedpulses with a beam splitter and pair of delay lines. Westabilized and controlled the time delay between the twopulses with an active servo system. The pulse pair wasthen directed onto a potassium atomic beam where weexcited the radial wave packet cat state.The quantum state distribution was measured for vari-ous phase differences between the two wave packets. Thepresence of a second wave packet modifies the state dis-tribution compared to that of a single wave packet. Thisis one of the signatures of this type of coherent super-position state. Two particularly dramatic state distribu-tions occur for the so called even and odd coherent states,jal 1 j 2 al and jal 2 j 2 al, respectively. The evencoherent state contains only even eigenstates jnl, and theodd state is a superposition of only the odd eigenstates.We use state-selective field ionization to measure thequantum state distribution of our atomic cat state. Thistechnique consists of ramping on a dc electric field whichionizes the population from different Rydberg states atdifferent times as each state’s ionization potential isreached during the ramp. The ions are collected withan electron multiplier and binned according to theirarrival time. The resulting time-resolved ion signal haspeaks which correspond directly to different Rydbergenergy eigenstates in the superposition. The time bins inour experiment are 50 ns wide and the peak separationvaries from about 200 ns to 1 ms. These peaks arenot equally spaced in time since the ionization potentialgoes as 1y16n4. Each peak also has a width due tofield inhomogeneities and geometrical averaging in theinteraction region. The population in a given Rydbergstate is the integrated signal in the peak associated withthat state.Figure 1(a) shows a reference trace taken with onedelay arm blocked. This is the state distribution for asingle wave packet. There are about five states in thesuperposition centered near n ≠ 66. Next, we measuredthe distribution for excitation with the pulse pair. InFig. 1(b) the phase between pulses was set to eliminatepopulation from the odd Rydberg states. This mimics thestate distribution for the even harmonic oscillator cat state.The phase can also be set to eliminate population from theeven Rydberg states [Fig. 1(c)]. This is similar to the oddharmonic oscillator cat state.The modification of the population distribution is a di-rect indication of the interference between the two qua-siclassical wave packets. Another characteristic featureof Schrödinger’s cat state is the interference between the1914VOLUME 77, NUMBER 10 P HY S I CA L REV I EW LE T T ER S 2 SEPTEMBER 1996FIG. 1. Quantum state distribution of the radial wave packetsmeasured using state-selective field ionization. Wave packetsexcited with (a) a single laser pulse, (b) a pulse pair of phasedifference p , and (c) a pulse pair of phase difference 0. Eachof the peaks in these time-resolved ion signals corresponddirectly to a given Rydberg state in the superposition. Theodd Rydberg states are shaded light gray, and the even statesare shaded dark gray.two wave packets in phase space. We have looked atthis phase space interference at a time when the twowave packets spatially overlap to demonstrate the coher-ent rather than statistical nature of the superposition. In astatistical mixture the wave packets behave like ensemblesof classical particles and simply add incoherently. How-ever, if the superposition is coherent, the wave packets caninterfere, and their wave nature is revealed. The interfer-ence between a pair of coherently excited wave packetswas recently described as a type of Young’s double slitinterferometer [21]. Here we measure the interference ina similar way by probing with a third short laser pulse.The location of a radial electron wave packet can bedetermined by interacting with a time-delayed probe laserpulse. If the wave packet is near the core when the probepulse arrives, there is a strong interaction between the two.On the other hand, if the wave packet is away from thecore when the probe pulse arrives, the interaction betweenthe two is very weak. If the probe pulse is identical tothe pulse which originally excited the wave packet, thetwo can interact coherently [22]. In this case a type ofRamsey fringe is observed (rapid oscillation in the excitedstate population at the optical period). The amplitude ofthe Ramsey fringes is large when the wave packet is nearthe core where it can interact with the probe pulse, andsmall when the wave packet is away from the core.To measure the interference between the wave packetsin our atomic Schrödinger cat state we must look at a timewhen the two wave packets spatially overlap and are nearthe core, where the probe interacts strongly. Initially thetwo wave packets are well separated and do not overlap atall near the core. However, since the atomic potentialis not harmonic, the wave packets do not remain welllocalized, but spread as they oscillate. We probe at a fixedtime in the evolution when the wave packets have spreadenough so that their overlap near the core is very good.The Ramsey fringe amplitude is plotted as a function ofthe phase difference between the two wave packets in thecat state in Fig. 2. The excellent fringe visibility indicatestwo things. First, the wave packets must be in a coherentsuperposition to interfere at all, and second, the probe wasin fact fixed at a time when the wave packets were welloverlapped so they could strongly interfere.The Schrödinger cat state which we have excited inthe atom displays many of the nonclassical featureswhich result from the superposition of two quasiclassicalstates. Although the extent of our wave packet’s orbit isenormous compared to the size of a ground state atomand its behavior very nearly that of a classical planetarysystem (at least for short times) it is still far from being aliving, breathing macroscopic cat.FIG. 2. Schrödinger cat state probed with a third time-delayedlaser pulse. The time delay of the probe pulse is fixed at tenKepler periods after the initial excitation. At this time the twowave packets have spread enough that their spatial overlap isvery good. The Ramsey fringe amplitude is measured as afunction of the phase difference between the two wave packetsin the cat state revealing strong interference between the two.1915VOLUME 77, NUMBER 10 P HY S I CA L REV I EW LE T T ER S 2 SEPTEMBER 1996A cat has other macroscopic properties in additionto size. In particular, it has many degrees of freedomwhereas the linear harmonic oscillator, no matter howlarge, has only one degree of freedom. One might arguethat the number of degrees of freedom is the essential fea-ture distinguishing the microscopic from the macroscopic.However, we note that a protein molecule has many de-grees of freedom and yet it remains fundamentally a quan-tum object. Many definitions have been offered of theclassical limit, and it is not clear that any of them sufficesfor all cases. Until we can actually achieve a quantumsuperposition involving indisputably macroscopic objectslike cats, it is perhaps interesting to investigate, as wehave, quantum superpositions of states of objects whichare at least of dimensions approaching the macroscopic.In a perfectly isolated quantum system the size of ourcat state could be increased to truly macroscopic dimen-sions, all the while maintaining its quantum coherence.However, in any real experimental situation this completeisolation is of course impossible. The ways in which theseenvironmental interactions lead to the decoherence of amacroscopic superposition vary depending on the quan-tum system of interest are some of the motivations forstudying such states [23,24]. One way in which the envi-ronment couples to our atomic electron cat state is throughthe interaction with blackbody radiation. The time scalefor mixing due to blackbody transitions goes as n2 whilethe Kepler period scales as n3 [25]. So, as the wavepackets are excited higher in the Rydberg series creatingmore macroscopic superpositions there is a point at whichthe coherence between them will be lost before the wavepackets can complete even a single orbit.An atomic electron wave packet which was localized attwo well-separated spatial positions was observed severalyears ago by Yeazell and Stroud [19]. This is the formthat results when a single atomic wave packet is allowedto evolve freely to what is referred to as the one-halffractional revival [26]. In this paper we have shownthat the one-half fractional revival is a Schrödinger catstate, and by exciting it directly with a pair of laserpulses we were able to control the phase between the twowave packets and more fully characterize the peculiar,nonclassical properties of the Schrödinger cat state.We would like to acknowledge helpful discussions withI. A. Walmsley, J. Bromage, M. F. Van Leeuwen, John D.Corless, and James A. West. This work was supported inpart by the Army Research Office.Note added.—A Schrödinger cat state has recentlybeen excited with a single ion in a trap [27].[1] E. Schrödinger, Naturwissenschaften 23, 807 (1935); 23,823 (1935); 23, 844 (1935); J. D. Trimmer, Proc. Am.Philos. Soc. 124, 323 (1980).[2] John R. Gribbin, In Search of Schrödinger’s Cat: Quan-tum Physics and Reality (Bantam Books, Toronto, 1984).[3] B. Yurke and D. Stoler, Phys. Rev. Lett. 57, 13 (1986).[4] Shang Song, Carlton M. Caves, and Bernard Yurke, Phys.Rev. A 41, 5261 (1990).[5] Pierre Meystre, John J. Slosser, and Martin Wilkens, Opt.Commun. 79, 300 (1990).[6] M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, andN. Zagury, Phys. Rev. A 45, 5193 (1992).[7] J. Janszky, An.V. Vinogradov, and T. Kobayashi, Phys.Rev. A 50, 1777 (1994).[8] I. A. Walmsley and M.G. Raymer, Phys. Rev. A 52, 681(1995).[9] R. L. de Matos Filho and W. Vogel, Phys. Rev. Lett. 76,608 (1996).[10] U.M. Titulaer and R. J. Glauber, Phys. Rev. 145, 1041(1966).[11] W. Schleich, M. Pernigo, and Fam Le Kien, Phys. Rev. A44, 2172 (1991).[12] V. Buzˇek, A. Vidiella-Barranco, and P. L. Knight, Phys.Rev. A 45, 6570 (1992).[13] Jonathan Parker and C.R. Stroud, Jr., Phys. Rev. Lett. 56,716 (1986).[14] G. Alber, H. Ritsch, and P. Zoller, Phys. Rev. A 34, 1058(1986).[15] A. ten Wolde, L.D. Noordam, A. Lagendijk, and H. B.van Linden van den Heuvell, Phys. Rev. Lett. 61, 2099(1988).[16] John A. Yeazell, Mark Mallalieu, Jonathan Parker, andC. R. Stroud, Jr., Phys. Rev. A 40, 5040 (1989).[17] I. Sh. Averbukh and N. F. Perelman, Phys. Lett. 139A, 449(1989).[18] Z. D. Gaeta and C.R. Stroud, Jr., Phys. Rev. A 42, 6308(1990).[19] John A. Yeazell and C.R. Stroud, Jr., Phys. Rev. A 43,5153 (1991).[20] D. R. Meacher, P. E. Meyler, I. G. Hughes, and P. Ewart,J. Phys. B 24, L63 (1991).[21] Michael W. Noel and C. R. Stroud, Jr., Phys. Rev. Lett.75, 1252 (1995).[22] L. D. Noordam, D. I. Duncan, and T. F. Gallagher, Phys.Rev. A 45, 4734 (1992).[23] Juan Pablo Paz, Salman Habib, and Wojciech H. Zurek,Phys. Rev. D 47, 488 (1993).[24] Wojciech H. Zurek, Salman Habib, and Juan Pablo Paz,Phys. Rev. Lett. 70, 1187 (1993).[25] Thomas F. Gallagher, Rydberg Atoms (Cambridge Univer-sity Press, New York, 1994).[26] This is similar to the way a cat state is formed in anonlinear optical system [3]; by injection of a coherentstate into a nonlinear medium which causes the wavepacket to disperse and reform as two or more separatedwave packets.[27] C. Monroe, D.M. Meekhof, B. E. 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Excitation of an Atomic Electron to a Coherent Superposition of Macroscopically Distinct States

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Excitation of an Atomic Electron to a Coherent Superposition of Macroscopically Distinct States

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