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Entanglement of Atomic Qubits using an Optical Frequency Comb D. Hayes,∗ D. N. Matsukevich, P. Maunz, D. Hucul, Q. Quraishi, S. Olmschenk, W. Campbell, J. Mizrahi, C. Senko, and C. Monroe Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA (Dated: January 21, 2010) We demonstrate the use of an optical frequency comb to coherently control and entangle atomic qubits. A train of off-resonant ultrafast laser pulses is used to efficiently and coherently transfer population between electronic and vibrational states of trapped atomic ions and implement an entangling quantum logic gate with high fidelity. This technique can be extended to the high field regime where operations can be performed faster than the trap frequency. This general approach can be applied to more complex quantum systems, such as large collections of interacting atoms or molecules. 0 1 PACSnumbers: 03.67.-a,32.80.Qk,37.10.Vz,37.10.Rs 0 2 The optical frequency comb generated from an ultra- maticlasersarephase-lockedorasinglecwlaserismodu- n fastlaserpulsetrainhasrevolutionizedopticalfrequency lated by an acousto-optic (AO) or an electro-optic (EO) a J metrology [1–4] and is now playing an important role in modulator. However, the technical demands of phase- 1 high resolution spectroscopy [5]. The spectral purity yet locked lasers and the limited bandwidths of the modu- 2 largebandwidthofopticalfrequencycombsalsoprovides lators hinder their application to experiments. Here we a means for the precise control of generic quantum sys- exploit the large bandwidth of ultrafast laser pulses in ] h tems,withexamplessuchasthequantumcontrolofmul- a simple top-down approach toward bridging large fre- p tilevelatomicsystems[6,7],lasercoolingofmoleculesor quencygapsandcontrollingcomplexatomicsystems. By - exoticatomicspecies[8,9], andquantumstateengineer- starting with the broad bandwidth of an ultrafast laser t n ing of particular rovibronic states in molecules [10, 11]. pulse, a spectral landscape can be sculpted by interfer- a The optical frequency comb may become a crucial com- encefromsequentialpulses, pulseshapingandfrequency u ponentinthefieldofquantuminformationscience,where shifting. In this paper, we start with a picosecond pulse q [ complex multilevel quantum systems must be controlled and, through the application of many pulses, generate a with great precision [12]. frequency comb that drives Raman transitions by stim- 2 ulating absorption from one comb tooth and stimulating v InthisLetter,wereporttheuseofanopticalfrequency 7 comb generated from an ultrafast mode-locked laser to emission into another comb tooth as depicted in Fig. 1. 2 Because this process only relies on the frequency differ- efficiently control and faithfully entangle two trapped 1 ence between comb teeth, their absolute position is ir- atomic ion qubits. The optical pulse train drives stimu- 2 relevant and the carrier-envelope phase does not need to . latedRamantransitionsbetweenhyperfinelevels[13,14], 1 be locked [2]. As an example of how this new technique accompanied by qubit state dependent momentum kicks 0 promises to ease experimental complexities, the control 0 [15]. Thecoherentaccumulationofthesepulsesgenerates of metastable-state qubits separated by a terahertz was 1 particular quantum gate operations that are controlled recently achieved using cw lasers that are phase-locked : throughthephaserelationshipbetweensuccessivepulses. v through a frequency comb [18], but might be controlled i Thisprecisespectralcontroloftheprocessalongwiththe X directly with a 100 fs Ti:sapph pulsed laser. large optical bandwidth required for bridging the qubit At a fixed point in space, an idealized train of laser r frequency splitting forms a simple method for control- a pulses has a time-dependent electric field that can be ling both the internal electronic and external motional written as states of trapped ion qubits, and may be extended to mostatomicspecies. Thissameapproachcanbeapplied N (cid:88) tocontrollargertrappedioncrystalswithmoreadvanced E(t)= f(t−nT)eiωct, (1) pulse-shaping techniques, and can also be extended to a n=1 strong pulse regime where only a few high power pulses where f(t) is the pulse envelope, T is the time between are needed for fast quantum gate operations in trapped successive pulses (repetition rate ν = 1/T), N is the R ions [15–17]. number of pulses in the train and ω is the carrier fre- c High fidelity qubit operations through Raman transi- quency of the pulse. For simplicity, any pulse-to-pulse tions are typically achieved by phase-locking frequency optical phase shift is ignored since the offset frequency components separated by the energy difference of the in the comb is unimportant. The Fourier transform qubit states. This is traditionally accomplished in a of Eq. (1) defines a frequency comb characterized by bottom-uptypeofapproachwhereeithertwomonochro- an envelope f˜(ω) ≡ F[f(t)] centered around the opti- 2 FIG. 2: Schematic of the experimental setup showing the paths of the pulse trains emitted by a mode-locked Ti:Sapphire(Ti:Sapph)laser,wheretheopticalpulsesarefre- quency shifted by AOs. Single qubit rotations only require a singlepulsetrain,buttoaddressthemotionalmodesthepulse trainissplitintotwoandsentthroughAOstotunetherela- tiveoffsetofthetwocombs. Welocktherepetitionrate(ν ) R by first detecting ν with a photodetector (PD). The output R of the PD is an RF frequency comb spaced by ν . We band- R passfilter(BP)theRFcombat12.685GHzandthenmixthe signalwithalocaloscillator(LO).Theoutputofthemixeris sentintoafeedbackloop(PID)whichstabilizesν bymeans R FIG.1: TheStokesRamanprocessdrivenbyfrequencycombs of a piezo mounted on one of the laser cavity mirrors. When isshownhereschematically. Anatomstartinginthe|↓(cid:105)state locked, ν is stable to within 1 Hz for more than an hour. R can be excited to a virtual level by absorbing a photon from Asanalternative,insteadoflockingtherepetitionrateofthe the blue comb and then driven to the |↑(cid:105) state by emitting a pulsed laser, an error signal could be sent to one of the AOs photon into the red comb. Although drawn here as two dif- to use the relative offset of the two combs to compensate for ferentcombs,ifthepulsedlaser’srepetitionrateoroneofits a change in the comb spacing. harmonics is in resonance with the hyperfine frequency, the absorptionandemissioncanbothbestimulatedbythesame frequencycomb. Becauseoftheevenspacingofthefrequency comb, all of the comb teeth contribute through different vir- of q. As shown in Fig. 3, when n = 2 (q = 313 and tual states which result in indistinguishable paths and add ν = 40.39 MHz), application of the pulse train drives R constructively. oscillationsbetweenthequbitstatesofasingleion. How- ever, when n = 3 (q = 469.5 and ν = 26.93 MHz), the R qubit does not evolve. cal frequency ω and teeth separated by ν whose in- The Rabi frequency of these oscillations can be esti- c R dividual widths scale like ∼ ν /N. The Raman res- mated by considering the Hamiltonian resulting from an R onance condition will be satisfied when a harmonic of infinite train of pulses. After adiabatically eliminating the repetition rate is equal to the hyperfine qubit splitting ω , implying that the parameter q ≡ ω /2πν , 0 0 R is an integer. To demonstrate coherent control with a pulse train, 171Yb+ ions confined in a linear Paul trap are used to encode qubits in the 2S hyperfine clock 1/2 states |F =0,m =0(cid:105)≡|↓(cid:105) and |F =1,m =0(cid:105)≡|↑(cid:105), F F having hyperfine splitting ω /2π = 12.6428 GHz. For 0 statepreparationanddetectionweusestandardDoppler cooling, optical pumping, and state-dependent fluores- cence methods on the 811 THz 2S ↔2P electronic 1/2 1/2 transition [19]. The frequency comb is produced by a frequency-doubled mode-locked Ti:Sapphire laser at a carrierfrequencyof802THz,detunedby∆/2π =9THz from the electronic transition. The repetition rate of the laser is ν = 80.78 MHz, with each pulse having a du- R FIG.3: AfterDopplercoolingandopticalpumpingtothe|↓(cid:105) ration of τ ≈1 psec. The repetition rate is phase-locked state, a single pulse train is directed onto the ion. When the to a stable microwave oscillator as shown in Fig. 2, pro- ratioofqubitsplittingtopulserepetitionrate,q,isaninteger, viding a ratio of hyperfine splitting to comb spacing of pairsofcombteethcandriveRamantransitionsasshownby q = 156.5. An EO pulse picker is used to allow the pas- the blue circular data points. However, if the q parameter is sage of one out of every n pulses, decreasing the comb a half integer, the qubit remains in the initial state as shown spacing by a factor of n and permitting integral values by the red square data points. 3 theexcited2P stateandperformingtherotating-wave For I =0.15 W/cm2, the data shown in Fig. 3 is con- 1/2 sat approximation, the resonant Rabi frequency of Raman sistent with an average intensity I¯≈500 W/cm2. transitions between the qubit states is given by a sum over all spectral components of the comb teeth as indi- cated in Fig. 1 ((cid:126)=1): In order to entangle multiple ions, we first address the motion of the ion by resolving motional sideband tran- |µ|2(cid:80) E E (cid:18) ω τ (cid:19) sitions. As depicted in Fig. 2, the pulse train is split Ω= l l l−q ≈Ω 0 , (2) ∆ 0 eω0τ/2−e−ω0τ/2 intotwoperpendicularbeamswithwavevectordifference k along the x−direction of motion. Their polarizations whereµisthedipolematrixelementbetweentheground are mutually orthogonal to each other and to a weak and excited electronic states, E ≡ ν f˜(2πkν ), and magnetic field that defines the quantization axis [20]. k R R q is an integer. In the approximate expression above, We control the spectral beatnotes between the combs by the sum is replaced by an integral and each pulse is de- sending both beams through AO modulators (driven at (cid:112) scribedbyf(t)= π/2E sech(πt/τ)withτ (cid:28)T,where frequencies ν and ν ), imparting a net offset frequency 0 1 2 Ω =(ν τ)|µE |2/∆=sγ2/2∆isthetime-averagedres- of ∆ω/2π = ν −ν between the combs. For instance, 0 R 0 1 2 onantRabifrequencyofthepulsetrainands=I¯/I is in order to drive the first upper/lower sideband transi- sat theaverageintensityI¯=ν c(cid:15) /2(cid:82) dt|f(t)|2scaledtothe tion we set |2πjν +∆ω| = ω ±ω , with j an integer R 0 R 0 t 2S ↔ 2P saturation intensity. Note the net transi- and ω the trap frequency. In order to see how the side- 1/2 1/2 t tionrateissuppressedunlessthesingle-pulsebandwidth bands are spectrally resolved, we consider the following is large compared to the hyperfine frequency (ω τ (cid:28)1), Hamiltonian of a single ion and single mode of harmonic 0 in which case Ω ≈ Ω . In our experiments, ω τ ≈ 0.08. motion interacting with the Raman pulse train: 0 0 H =ω a†a+ ω0σ + θp (cid:88)δ(t−nT)(cid:16)σ ei(kxˆ+∆ωt)+σ e−i(kxˆ+∆ωt)(cid:17), (3) eff t 2 z 2 + − n where θ = ΩT is the change in the Bloch angle due to to satisfy the resonance condition for the red sideband, p a single pulse, σ is the Pauli-z operator, σ are raising ϑ ≡(ω +∆ω−ω )T =2πj, where j is an integer, then z ± r 0 t and lowering operators, xˆ is the x−position operator of the sum in Eq. (5) is approximately given by, the trapped ion, a† and a are the raising and lowering N−1 operators of the x−mode of harmonic motion and the q (cid:88) Q ≈iηsinNϑr/2eiϑr(N−1)/2σ a+h.c. (7) parameterhasbeenassumedtonotbeanintegerorhalf- n sinϑ /2 + r n=0 integer. Intheinteractionpicture,theevolutionoperator after N pulses is given by VN, where The coefficient in Eq. (7) is the same as the field amplitude created by a diffraction grating of N slits, whose (cid:20) (cid:21) narrow peaks have an amplitude equal to N. In the V =exp[−iH T]exp −iθp (cid:0)σ eikxˆ+σ e−ikxˆ(cid:1) (4) limit ω T (cid:28)1, the other terms in Eq. (5) that drive the 0 2 + − t carrier and other sideband transitions can be neglected when N (cid:29) (ω Tη)−1. This is analogous to the de- and H = ω a†a+1/2(ω +∆ω)σ . The time evolution t 0 t 0 z structiveinterferenceofamplitudesawayfromthebright operator is given by, peaks in a diffraction grating. For ω /2π = 1.64 MHz, t N−1 T =12.4 ns and η =0.1, the sidebands are well-resolved VN =exp[−iH0NT](Iˆ−iθ2p (cid:88) Qn+O(θp2)) (5) when N (cid:29)80. n=0 For many applications in quantum information, the Q ≡σ ei(ω0+∆ω)nTD(iηeiωtnT)+h.c. , (6) motional modes of the ion must be cooled and initial- n + ized to a nearly pure state. Fig. 4 shows that the pulsed where D(α) = exp[αa† − α∗a] is the harmonic oscil- laser can also be used to carry out the standard tech- lator displacement operator in phase space, and η = niques of sideband cooling [20] to prepare the ion in the k(cid:112)(cid:126)/2mωt is the Lamb-Dicke parameter. In the motionalgroundstatewithnearunitfidelity. Theset-up (cid:112) Lamb-Dicke regime, η (cid:104)a†a(cid:105)+1 (cid:28) 1, we can write alsoeasilylendsitselftoimplementingatwo-qubitentan- D(iηeiωtnT) ≈ 1+iη(eiωtnTa† +e−iωtnTa) turning the gling gate by applying two fields whose frequencies are sum in Eq. (5) into a geometric series. If, for exam- symmetrically detuned from the red and blue sidebands ple, the offset frequency between the combs ∆ω is tuned [21,22]. Bysimultaneouslyapplyingtwomodulationfre- 4 FIG. 5: The parity oscillation that is used to calculate the fidelity of thespinstateoftwoionswithrespecttothemax- imally entangled state |χ(cid:105) after performing the entangling gate. Thephaseφoftheanalyzingpulseisscannedbychang- ing the relative phase of the rotation pulses. The offset and lack of full contrast in the parity signal can be attributed to state detection errors. tionontheBlochspherewithphaseφ[25]. Thecontrast of the parity signal, Π , is used to calculate the fidelity FIG. 4: (a) Using a Raman probe duration of 80µs, (N ∼ C 6500),afrequencyscanofAO1showstheresolvedcarrierand F = (ρ↑↑,↑↑+ρ↓↓,↓↓)/2+ΠC/4. The measured popula- motional sideband transitions of a single trapped ion. The tions of |↓↓(cid:105) and |↑↑(cid:105) together with the data shown in transitionsarelabeled,(∆n ,∆n ),toindicatethechangein Fig. 5 yield a fidelity F =0.86±0.03, thereby signaling x y the number of phonons in the two transverse modes that ac- that the two ions are entangled after the application of companyaspinflip. Thexandymodesplittingiscontrolled the pulse train. by applying biasing voltages to the trap electrodes. Unla- Wehavedemonstratedfullcontrolandentanglementof beledpeaksshowhigherordersidebandtransitionsandtran- sitionstootherZeemanlevelsduetoimperfectpolarizationof twoatomicqubitsusinganopticalfrequencycomb. This the Raman beams. (b) Ground state cooling of the motional work represents a significant simplification over current modes via a train of phase-coherent ultra-fast pulses. The methods for optical control of trapped ion qubits, and red open-circle data points show that after Doppler cooling also points the way toward future advances with higher and optical pumping, both the red and blue sidebands are power laser pulses. For example, such pulses allow much easily driven. The blue filled-circle data points show that af- largerdetuningsfromresonanceandasuppressionofde- ter sideband cooling, the ion is close to the motional ground coherence from spontaneous emission [26] while main- state, (n¯ ≤ 0.03), as evidenced by the suppression of the x,y red-sideband transition. taining gate speed. When only a few high-power pulses are used in ways similar to the experiment, it also be- comes possible to suppress sources of motional decoher- ence through the use of fast entanglement schemes [16]. quencies to one of the comb AO frequency shifters, we createtwocombsinoneofthebeams. Whenthesecombs This work is supported by the Army Research Office aretunedtodrivetheredandbluesidebands(inconjunc- (ARO)withfundsfromtheDARPAOpticalLatticeEm- tion with the third frequency comb in the other beam), ulator(OLE)Program,IARPAunderAROcontract,the the ion experiences a spin-dependent force in a rotated NSF Physics at the Information Frontier Program, and basis as described in Ref. [23]. Ideally, when the fields the NSF Physics Frontier Center at JQI. aredetunedfromthesidebandsbyanequalandopposite amountδ =2ηΩ,adecouplingofthemotionandspinoc- curs at gate time t =2π/δ, and the spin state evolve to g the maximally entangled state |χ(cid:105) = |↓↓(cid:105)+eiϕ|↑↑(cid:105). In ∗ [email protected] the experiment, t =108µs (N ∼8700 pulses). g [1] Th. Udem, R. Holzwarth, and T. W. Ha¨nsch. Nature The entanglement is verified by the measurement of 416, 233 (2002). a fidelity-based entanglement witness [24]. The fidelity [2] S.T.CundiffandJ.Ye. Rev.Mod.Phys.75,325(2003). [3] J. L. 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