quantum information with trapped strontium ionsthe statistical nature of quantum processes is to...

Something wonderful happenswhen small numbers of ions aretrapped in a linear Paul (radio-

frequency, or rf) trap and laser-cooled.The ions become nearly motionlessand line up neatly along the trapaxis—each confined to its own tinyspace of about 100 micrometers orless in any direction. Because the ionsare frozen in place, experimentalphysicists can continually observethem for up to months at a time andgain uncommon insight into the quantum realm.

For example, single ions exhibitquantum-mechanical effects that couldnever be observed in a large ensembleof ions or neutral atoms. A large fieldof study in quantum optics has in factemerged with the development of iontraps (Thompson et al. 1997). In addi-tion, the internal transitions of a nearlymotionless ion are only slightly affected by Doppler shifts, and the ion can be superbly isolated fromunwanted electric fields and noisymagnetic fields. This characteristicmakes a trapped ion a useful testingground for many physical theories that

predict very small shifts of the atomicenergy levels (Berkeland et al. 1999).Finally, a focused laser beam can inter-act first with one specific ion, then adifferent one—a capability that meanswe can control complicated interactionsbetween states of a particular ion andbetween different ions. For this reason,the ion trap has shown considerablepromise as the basis for a quantumcomputer. (See the article “Ion-TrapQuantum Computation” on page 264.)

In this article, I discuss some ofour activities with trapped and laser-cooled ions. I focus on an experimentthat provides a fundamental test ofquantum-mechanical randomness butalso mention a spectroscopy experi-ment that is a prerequisite to thedevelopment of a quantum logic gate.For background material, see the pre-viously mentioned article, “Ion-TrapQuantum Computation,” which dis-cusses the operational principles of alinear Paul trap and laser cooling.

We conduct our experiments usingsingly ionized strontium atoms. Figure 1(a) is an illustration of our linear Paul trap (Berkeland 2002).

Most of the trap has been created withoff-the-shelf components and requiresno precise or otherwise demandingmachining to assemble. This feature issignificant because it shows that iontrapping with linear traps can be anaccessible technology for groups withlimited resources.

Figure 1(b) shows the transitionswe use in the strontium ion 88Sr+. Weuse the 422-nanometer transition toDoppler-cool the ions. We also collectthe 422-nanometer fluorescent lightfrom the decay of the P1/2 state andfocus it onto a detector to image theions. Light at 1092 nanometers drivesthe D3/2 ↔ P1/2 transition to preventthe atoms from pooling in the long-lived D3/2 state, in which they wouldnot scatter any 422-nanometer light. A 674-nanometer diode laser drivestransitions between the S1/2 groundstate and the D5/2 state, which lives anaverage of 0.35 seconds. This transi-tion can be used to couple the S1/2 andD5/2 states of the ion with its motionalstates, any of which may be used asqubits in a quantum computer. TheS1/2 ↔ D5/2 transition is also driven

178 Los Alamos Science Number 27 2002

Quantum Information with

Trapped Strontium IonsDana J. Berkeland

so that quantum jumps can beobserved in the experiments discussednext.

Quantum Randomness

In the article “A New Face forCryptography” on page 68, the

authors describe the quantum cryp-tography project at Los Alamos.Cryptography applications, whetherclassical or quantum, require stringsof numbers (typically 1s and 0s) thatare as random as possible. Generatingrandom numbers, however, is not atrivial matter. In fact, the randomnumber generators found in various

computer programs are do not yieldvery random numbers because theyare based on algebraic processes thatare intrinsically deterministic.

It is generally accepted that pro-ducing strings of truly random num-bers requires measuring the randomoutcome of a quantum-mechanicalprocess. One example of a random

Number 27 2002 Los Alamos Science 179

(a) A schematic of the linear trap depicts five 88Sr+ ions alongits axis (not to scale). The ions in this trap are confined radial-ly in a time-averaged potential that is created by applying 100 V at a frequency of 7 MHz to the two electrodes shown.The other two electrodes are held at a constant potential.The tubular electrodes (labeled “sleeves”) are held at constantpotentials up to 100 V, relative to the other electrodes, to stopthe ions from leaking out of the ends of the trap. The pictureof five Sr+ ions was made by focusing the 422-nm light scat-

tered from the ions onto an intensified charge-coupled-devicecamera. The ions are spaced about 20 µm from each other.(b) The diagram shows the relevant energy levels of Sr+ andthe corresponding transitions (not to scale). We use 422-nmlight from a frequency-doubled diode laser to Doppler-cool theions and collect the scattered 422-nm light to detect the ions.A fiber laser generates 1092-nm light that keeps the ions frombecoming stuck in the long-lived D3/2 state. A very stablediode laser at 674 nm drives the narrow S1/2 ↔ D5/2 transition.

About forty strontium ions lined up in our linear Paul trap are visible because they scatter laserlight. The apparent gaps are due to other ions that do not scatter the light.

Figure 1. Strontium Ion Linear rf Paul Trap

outcome is a photon hitting a beamsplitter (Jennewein et al. 2000). Thephoton has a probability to either passthrough the optic or reflect off it, andonly a measurement determines itsfate. Another example is the decay ofradioactive nuclei, which emit, say,

alpha particles at unpredictable times(Silverman et al. 2000). Although boththose processes are believed to be ran-dom, they suffer from one major draw-back in a test of their statistics: As inany experimental setup, all the detec-tors have physical limitations.

Therefore, we cannot be sure that wewould detect every photon or alphaparticle. It is possible that some non-random processes might be overlookedin analyzing the incomplete data set.

In contrast, a very clean way to testthe statistical nature of quantumprocesses is to analyze the behavior ofan atom undergoing quantum jumps(Erber 1995). Quantum jumps are thesudden transitions from one quantumstate to another. As Figure 2 shows, astrontium ion in the S1/2 ground statewill absorb a photon from a laser tunedto 422 nanometers and “jump” to theP1/2 excited state. Because the P1/2state is short-lived, the ion quicklyreturns to the S1/2 state by emitting a422-nanometer photon in a randomdirection. Once it returns to the S1/2state, the ion can absorb and emitanother photon, and because the life-time of the P1/2 excited state is soshort, the ion will scatter millions ofphotons per second. We can detectenough of the scattered light with anoptical system to observe the ion butnot enough to determine every time theion jumps to and from the P1/2 state.

To directly observe quantumjumps, we simultaneously illuminatethe ion with a 422- and a 674-nanometer laser light. In addition tojumping to the P1/2 state, now the ioncan also jump to the D5/2 state. Assoon as that transition occurs, the ionwill stop scattering 422-nanometerlight. The scattered light will returnthe moment the ion has left the D5/2state. As Figure 2 shows, we can very easily record every time a single ion makes a transition to theD5/2 state and every time it returns tothe S1/2 state. According to quantumtheory, the exact times of those transi-tions are completely unpredictable.Surprisingly, this prediction has notbeen tested with data sets comprisingmuch more than about a thousand consecutive events. It is important totest very large sets of data because itis harder to make a nonrandom series


Quantum Information with Trapped Strontium Ions

Figure 2. Quantum Jumps in a Single Trapped 88Sr+ Ion (a) When illuminated by 422-nm radiation, a single strontium ion will cycle betweenthe S1/2 and P1/2 states and will scatter millions of photons per second. Some of thescattered light can be collected with a simple optical detector in order to monitorthe state of the ion. (b) If the ion is simultaneously illuminated with 674-nm radia-tion, it will occasionally undergo a transition (“quantum jump”) from the S1/2 stateto the long-lived D5/2 state. The scattered light then disappears. (c) This plot showstypical data from the quantum-jump experiment. When the count rate is over50 counts per 10 ms, the atom is cycling between the S1/2 and P1/2 states. When thecount rate suddenly falls to less than 50 counts per 10 ms, the atom has made atransition into the D5/2 state. We continuously monitor the ion’s scattering rate fornearly an hour to observe tens of thousands of these transitions.

300

200

100

0 1 2 3 4 5

Cou

nts/

10 m

s

422 nm

42P1/2

32D5/2

42S1/2

42P1/2

32D5/2

42S1/2

674 nm

DetectorPhotons

422-nmlaser beam Ion

674-nmlaser beam

(a) An ion excited to the short-lived 42P1/2 state scatters millions of photons per second

(c) Data from a quantum-jump experiment

Time (s)

(b) The scattering stops when the ion jumps to the long-lived 32D5/2 state

Ton Toff

of numbers appear random if theseries is very long.

Many tests can be used to deter-mine the degree of randomness in astring of data. Figure 3(a) shows theresult of one such test applied to ourquantum-jump data (Itano et al. 1990).A single atom was continuously moni-tored until it had made over 34,000transitions in and out of the D5/2 state.We record the length of each timeperiod Ton,i, during which the atomcontinually scatters 422-nanometerphotons, and the length of each subse-quent time period Toff,i, during whichthe ion scattered no photons because itwas in the D5/2 state. For example, inthe figure, the values of Toff areToff,1 = 0.23 second, Toff,2 = 0.1 sec-ond, Toff,3 = 0.61 second, andToff,4 = 0.17 second.

We then sift through the data todetermine the number of times a par-ticular pair of values (Toff,i, Toff,i+1)occurs and make the color-coded plotshown in Figure 3(a). The symmetryand shapes of these graphs reflect several important characteristics ofthe data. For example, a pair of val-ues, say (Toff,i, Toff,i+1) = (0.23 sec-ond, 0.1 second), is just as likely tooccur as the pair (0.1 second, 0.23second)—a long period of fluores-cence is no more likely to be fol-lowed by a short one than a shortperiod is likely to be followed by along one. Essentially, plots like theseindicate that the ion has no memoryof what it was doing just the briefestmoment before it fluoresces. Thisfundamental feature of quantumprocesses has not previously beentested precisely. It is also exactlywhat one would like to see in a ran-dom number generator.

We can easily convert the quan-tum-jump data into a string of 1s and0s. If Ton,i is more than a set amountof time, we assign to that event thevalue 0. Likewise, if Ton,i is less thanthis time, we will assign the value 1to the event. Figure 3(b) gives an



Figure 3. Analyzing Quantum-Jump DataThe scatter plots show consecutive periods that the ion spends (a) scattering422-nm photons (Ton,i, Ton,i+1) and (b) not scattering 422-nm photons (Toff,i, Toff,i+1).Because these graphs are symmetric about their diagonal axis, we can tell that theion is just as likely to spend a long time scattering photons followed by a short timescattering photons as it is to spend a short time followed by a long time scatteringphotons. This is one of many indications that the ion has no memory of when it hasmade a transition between the S1/2 and D5/2 states. (c) The quantum-jump data canalso be converted to digital data. The first set of numbers shows a string of consecu-tive times spent in the D5/2 state (Toff,i). If the ion spends 30 ms or more in the D5/2state, the event is assigned a value of 0. Otherwise, the event is assigned a value of1. These assignments are shown in set 2. With strings of tens of thousands of thesedigital numbers, we can use established protocols to test the randomness of ourquantum-jump data.

example of this conversion for a typical set of data.

Digitizing our data lets us usesome of the established protocolsthat test the randomness of digitaldata. (One such standard is outlinedin the U.S. Federal InformationProcessing Standards publication140-2). An example of such a test isthe following: In a string of 1s and0s, we count how many times thetwo-digit patterns (0,0), (0,1), (1,0)and (1,1) appear. We then comparethese numbers with the valuesexpected for an ideal, randomsequence. It is easy to calculate howlikely it is that the measured sets ofvalues differ from the expected ones,so that we can decide whether or notour quantum-jump data are randomaccording to the given protocol. Weare collecting continuous sequencesof data, tens of thousands of eventslong, that can be used for these tests.

Quantum Computing

We are also beginning some ofthe tasks that are prerequisites tomaking a quantum logic gate with atrapped ion. Perhaps the most criticalstep is coherently driving transitionsbetween specific qubit states. In theexperiments we are considering, thestrontium S1/2 ground state corre-sponds to the qubit state |�⟩, whereasthe D5/2 excited state corresponds tothe |�⟩ qubit state. The stable 674-nanometer diode laser couples thequbit states to each other and tostates of the ion’s quantized externalmotion that would also be qubitstates (Monroe et al. 1995).

The stability of the laser is oneof several parameters that can limitthe performance of a quantum com-puter. If the laser frequency and phasewere constant, we could almostalways complete quantum logic oper-

ations perfectly. For example, startingwith the ion in the S1/2 state, wecould reliably create a specific super-position of the S1/2 and D5/2 states:

(1)

However, if the phase or frequencyof the laser is not perfectly stablewhile this operation is taking place,the result of the operation may be, forexample,

(2)

In this case, the new wave functionhas a small phase error. If this opera-tion is repeated many times, the accu-mulations of these small errors couldinvalidate the results of a quantumcomputation. Because every laser hasa nonzero linewidth (proportional tothe laser’s frequency), such errors areinevitable. One way to reduce thelikelihood of introducing the errors isto perform the logic operation quickly,that is, faster than the typical timescales of the frequency fluctuations of the laser, although it is easier toperform a quantum-gate operationslowly. Thus, it is critical that thelaser be very stable with its linewidthas small as possible.

We have measured our laserlinewidth using a procedure related to the quantum-jump experimentdescribed earlier. First, we turn offthe 422-nanometer light, letting the ion decay to the S1/2 state. Thenwe illuminate the ion with a pulse of 674-nanometer laser light. (The422-nanometer light remains off dur-ing this step, because that light willperturb the S1/2 state and broaden theS1/2 ↔ D5/2 transition.) We thendetermine whether or not the laserhas driven the atom from the S1/2to the D5/2 state by shining the

SS D

1/21/2 5/2=

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i

SS D

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20

25

10

5

0–10 –5 0

Laser frequency (kHz)

Num

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of S

→ D

tran

sitio

ns

5 10 15 20

15

Figure 4. Measurement of the Laser Linewidth The plot shows data taken from the narrow sideband of the S1/2 ↔ D5/2 transition ina single trapped 88Sr+ ion. The solid line is a Lorentzian line shape that is fitted tothe data. In one laser probe cycle, the atom starts in the S1/2 state. Next, the coolinglight is turned off while the 674-nm light is pulsed on for 0.001s. Then the coolinglight is turned on again, and we see if any 422-nm light is scattered into the detec-tor. If not, then the 674-nm laser has successfully transferred the ion to the D5/2state. This process is repeated 100 times for each laser frequency.

422-nanometer light on the ion. We detect light scattered by the ion ifit is not in the D5/2 state, but onlybackground light (the small amountof light scattered off the trap and vacuum chamber) if the ion is in the D5/2 state. Figure 4 shows the numberof times the 674-nanometer lasertransfers the ion to the D5/2 state asthe laser frequency is scanned overone of the motional sidebands of the S1/2 ↔ D5/2 transition. The figure also shows the result of fitting aLorentzian-shaped curve to thesedata. From the shape of the fitted curve and from a few key experimentalparameters, we can determine that thelaser linewidth is about 4 kilohertz orless, which is about one percent ofone billionth of the absolute frequen-cy of the laser light (445 terahertz).

This laser linewidth is sufficientlynarrow so that we can perform specific, coherent operations on qubitstates. However, to perform the oper-ations needed for a quantum logicgate, the ions must be cooled muchmore than they are at present, so thatthe quantum state of the ion can beinitialized to the ground state of itsmotion. We are currently workingtoward this goal and on further narrowing the linewidth of the 674-nanometer laser. In addition,we are working on or anticipate per-forming several other quantum-opticsexperiments. The apparatus presentedhere, along with ion traps in general,can facilitate significant contributionsto the field of quantum informationand quantum computation. �

Acknowledgments

I would like to thank RichardHughes for suggesting the study ofthe randomness of quantum jumpsand for his indispensable role inbringing ion-trap technology to thefield of quantum information andquantum computation at Los Alamos.In addition, I am grateful to DaisyRaymondson for her work on some of the laser systems used in thisexperiment and to the Los AlamosSummer School for initially bringingher to Los Alamos.

Further Reading

Berkeland, D. J., A Linear Paul Trap forStrontium Ions. (To be published in Rev. Sci. Instrum.)

Berkeland, D. J., J. D. Miller, F. C. Cruz,B. C. Young, R. J. Rafac, X.-P. Huang et al.1999. High-Resolution, High-AccuracySpectroscopy of Trapped Ions. In AtomicPhysics 16. Proceedings of the InternationalConference. Edited by W. E. Baylis and G. W. F. Drake, 29. New York: AIP Press.

Erber, T. 1995. Testing the Randomness ofQuantum-Mechanic: Natures UltimateCryptogram. Ann. N. Y. Acad. Sci. 755: 748.

Itano, W. M., J. C. Bergquist, F. Diedrich, andD. J. Wineland. 1990. Quantum Optics ofSingle, Trapped Ions. In Coherence andQuantum Optics VI, 539. Edited by J. H. Eberly, et al. New York: Plenum Press.

Jennewein, T., U. Achleitner, G. Weihs,H. Weinfurter, and A. Zeilinger. 2000. AFast and Compact Quantum RandomNumber Generator. Rev. Sci. Instrum. 71 (4): 1675.

Monroe, C., D. M. Meekhof, B. E. King,,W. M. Itano, and D. J. Wineland. 1995.Demonstration of a Fundamental QuantumLogic Gate. Phys. Rev. Lett. 75: 4714.

Silverman, M. P., W. Strange, C. Silverman,and T. C. Lipscombe. 2000. Tests forRandomness of Spontaneous QuantumDecay. Phys. Rev. A 61: 042106.

Thompson, R. C., K. Dholakia,J.-L. Hernandez-Pozos, G. Zs. K. Horvath,J. Rink, and D. M. Segal. 1997.Spectroscopy and Quantum Optics with IonTraps. Physica Scr. T72: 24.



Dana J. Berkeland has performed a widevariety of experiments in atomic and opticalphysics. She received her Ph.D. from YaleUniversity in 1995 after precisely measuringStark shifts in lithium and cesiumand measuring theground-state Lambshift in hydrogen to an accuracy of 6 parts per million.This measurement,performed withMalcolm Boshier,was at the time themost precise meas-urement of the quan-tity. Dana spent three years as a NationalResearch Council postdoctoral fellow at theNational Institute of Standards andTechnology in Boulder, Colorado, where sheevaluated a trapped-mercury-ion microwavefrequency standard to a fractional accuracy of3 × 10–15 (making it one of the most accuratefrequency standards in the world). In 1998,she came to Los Alamos National Laboratoryas a J. Robert Oppenheimer postdoctoral fel-low to build a laboratory for quantum-opticsexperiments with trapped strontium ions andbecame a technical staff member in 2000. She has over a decade of experience in build-ing and developing laser systems and relatedoptics. Dana has over seven years of experi-ence in building and working with ion trapping systems.

quantum information with trapped strontium ionsthe statistical nature of quantum processes is to...

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