d. tong et al- local blockade of rydberg excitation in an ultracold gas

8/3/2019 D. Tong et al- Local Blockade of Rydberg Excitation in an Ultracold Gas

1/4

Local Blockade of Rydberg Excitation in an Ultracold Gas

D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Cote, E. E. Eyler, and P. L. Gould

Physics Department, University of Connecticut, Storrs, Connecticut 06269, USA(Received 15 Februar y 2004; published 3 August 2004)

In the laser excitation of ultracold atoms to Rydberg states, we observe a dramatic suppression causedby van der Waals interactions. This behavior is interpreted as a local excitation blockade: Rydberg atoms

strongly inhibit excitation of their neighbors. We measure suppression, relative to isolated atomexcitation, by up to a factor of 6.4. The dependences of this suppression on both laser irradiance andatomic density are in good agreement with a mean-field model. These results are an important steptowards using ultracold Rydberg atoms in quantum information processing.

DOI: 10.1103/PhysRevLett.93. 063001 PACS numbers: 32.80. Rm , 03.67. Lx, 34. 20.Cf

The possibility of a computer that operates accordingto the principles of quantum mechanics has attractedgrowing interest from a variety of research fields [1,2].A number of possible implementations are being inves-tigated, including solid-state systems, nuclear magnetic

resonance, cavity quantum electrodynamics, t rapped di-polar molecules, trapped ions, and trapped neutral atoms.A key element to any successful system is the ability tocontrol the coherent interactions between the fundamen-tal building blocks (qubits). Highly excited Rydbergatoms with principal quantum numbers n * 30 have theadvantage that they interact quite strongly with eachother, allowing information to be exchanged quickly [3].Here we report an i mportant advance towards using ultra-cold Rydberg atoms in quantum computing. We observethat the laser excitation of a macroscopic sample of ultra-cold atoms t o high-lying Rydberg states can be dramati-cally suppressed by their strong long-range interactions.

This leads to a local blockade effect, where the excitationof one atom prevents excitation of its neighbors. Ourobservations agree well with a model based on mean-fieldinteractions.

In a high-n Rydberg state, the electron spends most ofits time quite far from the nucleus [4]. As a result, theenergy of this highly excited state is very sensitive toexternal perturbations, including those caused by neigh-boring Rydberg atoms. A system of two ultracold Rydbergatoms, subject to these long-range interactions, has beenproposed as a possible realization of a quantum logic gate[3,5]. Rydberg states combine the advantages of longradiative lifetimes and strong long-range interactions,allowing information to be exchanged before decoher-ence sets in, even when the atoms are sufficiently sepa-rated to allow individual addressing. If the atoms areultracold, they can be highly localized, e.g., in the sitesof an optical lattice, allowing control of their interactionsand efficient detection of their quantum state. An out-standing challenge is t he assurance t hat at most, a singleRydberg atom is produced at a given site. Towards thisend, the concept of an excitation blockade has been pro-

posed [6,7]. With multiple atoms o ccupying a sufficientlylocalized site, the strong Rydberg-Rydberg interactionsallow at most one Rydberg excitation. Further excitationsare blocked by t he large energy level shif ts that push theresonant frequencies outside the laser bandwidth. Strong

dipole coupling and entanglement between a pair ofclosely spaced molecules in a crystal has recently beenobserved [8], making this system another candidatefor quantum logic gates operating by the blockademechanism.

For two atoms in np states, separated by a distance R,the C6=R

6 van der Waals (vdW) interaction dominates atlong ra nge [9,10]. T he rapid n11 scaling of the C6 coeffi-cient indicates the advantage of using high-n Rydbergstates. T he original proposals for quantum gates [3] andan excitation blockade [6] with Rydberg atoms involveddc electric fields that mix the angular momentum states,giving rise to a C3=R

3 dipole-dipole interaction. The vdW

interaction, which we have employed, has the i mportantadvantage of providing nonzero energy level shifts forevery molecular state regardless of the relative orientationof the atoms. By contrast, dipole interactions vanish forcertain orientations, reducing the efficiency of the dipoleblockade mechanism. In addition, the faster falloff of thevdW interaction may provide g reater nearest-neighborsite selectivity in a lattice.

In the experiment we start with a macroscopic sampleof about 107 ultracold 85Rb atoms in the 5s ground stateand illuminate them with a UV pulse resonant with atransition to an np Rydberg state [11]. Once a Rydbergatom is produced, the strong vdW interaction shifts theenergy levels of neighboring atoms, suppressing theirexcitation. It is this suppression and its dependence onlaser irradiance and atomic density that we observe in theexperiment.

The ultracold sample, with a peak density up to1011 cm3 and a temperature of100 K, is providedby a diode-laser-based vapor-cell magneto-optical trap(MOT). Trapped atoms in a specific hyperfine level of theground state are excited to np3=2 states with n 3080

VOLUME 93, NUMBER 6P H Y S I C A L R E V I E W L E T T E R S week ending

6 AUGUST 2004

063001-1 0031-9007=04=93(6)=063001(4)$22.50 2004 The American Physical Society 063001-1


2/4

by a narrowband 297 nm laser pulse of 8.6 ns duration(FWHM), generated by frequency doubling of a pulse-amplified cw laser as in our earlier work [11]. The band-width of about 100 MHz, measured by scanni ng over the30p resonance, is about twice the Fourier transform limitdue to frequency chirping in the pulsed amplifier [12].This h igh spectral resolution is necessar y to observe t heeffects of the small vdW shifts. In order to excite the

highest density region, the UV light is focused into theMOT cloud, yielding a cylindrical excitation volume500 m long and 220 m in diameter (FWHM).Within 60 ns after t he laser pulse, a1500 V=cm electricfield is applied, ionizing the Rydberg atoms and acceler-ating the ions towards a microchannel plate (MCP) de-tector. The trapping light is turned off when the UV pulsear rives in order to prevent di rect photoionization fromthe 5p level.

The MCP is calibrated using two methods. The first isbased on the signal from near-threshold photoionizationof the 5s ground state. The measured density distributionin the MOT, obtained f rom the trapped -atom fluorescenceprofile, is combined with the measured UV beam pa-rameters and the known photoionization cross section[13,14] to calculate the number of photoions per laserpulse. The second method is based on the n 30Rydberg signal, which behaves as isolated-atom excita-tion at all intensities used, because for n 30 the vdWinteraction is relatively weak. A linear fit is combinedwith calculations utilizing the 5s ! 30p oscillatorstrength of Ref. [15], including the effects of a linearlaser frequency chirp corresponding to the observedbandwidth. The chirp reduces the isolated-atom excita-tion efficiency by a factor of 2, the ratio of the band-

width to the Fourier t ransform l imit. The two calibrationsagree within 2%, fortuitously even better than t he domi-nant uncertainties in this comparison, 1520% in thephotoionization cross sections [13,14] and 10 20% i nthe laser linewidth. This confirms that we can makeaccurate quantitative predictions of excitation probabil-ities in the absence of a blockade.

The dependence of the Rydberg signal on the peak UVirradiance is shown in Fig. 1 for n 30, 70, and 80. Thesignal plotted is the fraction of the entire MOT samplethat is excited. For each n, the irradiance values are scaledto n 30 by t he factor 30=n3 in order to account forthe decrease in transition strength with increasing n. Here

n n , and 2:6415 is the quantum defectfor p3=2 states [16]. Note that the n 30 saturation in-

tensity for isolated atoms, defined as that required for anunchirped -pulse in the center of the beam, is0:36 MW=cm2. With this irradiance scaling, the variousns would fall on a universal isolated-atom excitationcurve if the Rydberg levels were unshifted by atomicinteractions. Th is is seen to be the case for the very lowestintensities, at which the Rydberg atoms are sufficientlysparse that interactions between them are negligible.

The sal ient feature of Fig. 1 is the d ramatic suppressionof Rydberg excitation for n 70 and 80 relative to theisolated-atom (n 30) excitation curve. As expected, the

suppression is larger for n 80 due to its stronger vdWinteraction, reaching a factor of 6.4 at the highest inten-sities shown.

We model the suppression of Rydberg excitation bysolving the Bloch equations for the ground-state andexcited-state amplitudes, cg and ce, of a given atom.

The key point is to include the energy level shift " dueto interactions with nearby Rydberg atoms. If we consideran np3=2 Rydberg atom located at ri, labeled jpii, the first-

order shift due to its interaction Vint with jpki is "ik

hpipkjVintri rkjpipki, so its total shi ft is t he sum overall atoms, "i ki "ik. At large separations, "ik is domi-

nated by the vdW interaction corresponding to a pair ofmolecular states 1g and3u , labeled 1 and 2,

respectively [911]. For n 70, C6 2:64 1022 a:u:

for both states.For simplicity, we consider a spherical domain of ra-

dius Rd and volume Vd which contains several atoms, butby definition, only a single Rydberg atom jpii. Hence, Rdis determined using t he condition

Z

Vd

d3rjcer; tj2 1; (1)

FIG. 1. Comparison of Rydberg excitation for the unblock-aded (isolated-atom) 30p state and the blockaded 70p and 80pstates at a peak density of 6:5 1010 cm3. Irradiances arescaled by n=303 to account for the n dependence of the

electric dipole transition probability. Insets show the regionnear the origin with an expanded scale. The dashed line forn 30 is a least-squares fit to the data, while the solid curvesfor n 70 and n 80 are theoretical predictions, using asingle adjustable scaling parameter (see text). We measurethe MOT density profile for each run and correct for its effecton the isolated-atom signals by multiplying the n 30 resultsby 1.29 (n 70 inset), 1.19 (n 80 inset) and 1.24 (mainfigure, using the average for n 70; 80).


6 AUGUST 2004

063001-2 063001-2


3/4

where , the local density of atoms, is assumed uni-form within the domain. From the definition of a domain,Rdt and the density of excited atoms et (assumedlocally uniform in and around the domain) are relatedself-consistently via eVd 1; initially, Rd ! 1 (ase ! 0), but reaches a finite value as the laser pulse ends.

The amplitude cer; t of an atom located at r dependson its shift "r; t. In our mean-field model, we calculate

this shift by replacing the discrete sum by an integral overall other excited atoms, which by construction resideoutside the domain V0 V Vd, and by consideringtheir density et to be locally uniform:

"r; t etZ

V0d3r0

C6jr r0j6

X21

jhprpr0 jij2: (2)

Using y r= Rd, the shift can be rewritten as

"y; t ~C6 gy=R6dt; (3)

where the effective vdW coefficient, ~C6, and the spatialvariation of the shift, gy (with g0 1), are obtained

by numerical integration of Eq. (2). Substituting

R3d t Zjyj1

d3yjcey; tj2 (4)

into Eq. (3) leads to nonlinear Bloch-like equations forthe time-dependent amplitudes cgy; t and cey; t,

id

dtcg

2eit

2

ce; (5a)

id

dtce

2 ~C6g

Zjyj1

d3yjcej2

2

ce

2eit

2

cg:

(5b)

Here, characterizes the chirp of the laser pulse and tis the Rabi frequency. As shown in Fig. 2, both " and ce,evaluated just after the laser pulse, depend upon thelocation r of the atom within the domain. At the center,the distance from any other excited atom is maximized,and the shift is therefore mi nimized. At the periphery ofthe domain, the proximity of external Rydberg atomsincreases the shift, leading to a stronger suppression ofexcitation.

Using the adapted zeroth order wave functions for themolecular states 1 and 2 [10], and averaging overprojections mj of the excited states jn pj3=2 mji foratoms outside the domain and integrating over all pos-

sible angles of molecular a xes, we find ~C6 7=60 C6. Bysolving numerically Eq. (5), we find the local density ofexcited atoms e; 0 after the Gaussian laser pulse ofpeak Rabi frequency 0 has ended. To compare with theexperimental measurements, this function is averagedover the density and laser intensity profiles in the trappedsample to provide the overall fraction of atoms which isexcited. Since they appear together, uncertainties in both

and ~C6 are taken into account by a single scaling factor

defined via ~C1=2

6; ~C 1=26 . We note that laser

frequency chirping causes Rydberg excitation to be lesssuppressed; for example, the predicted n 70 signal atthe highest irradiances shown in Fig. 1 is larger by afactor of 2 than t he chir p-free prediction.

The data and the predictions of the model describedabove are compared in Fig. 1 for both n 70 and n 80.We lim it t he scaled ir radiance to less than 0:35 MW=cm2

because the validity of the model is questionable when the

isolated-atom excitation approaches saturation. For thecurves shown, we use a scaling factor 2:4. Thisreflects a possible underestimate of the absolute atomicdensity (known only to within a factor of 2) and/or theeffective C6 coefficient. With this scaling factor, theagreement between the model and the data is quitegood. For n 80, we observe a maximum overall sup-pression by a factor of 6.4 relative to isolated-atom ex-citation, and the theory indicates the suppression reachesa factor of 19 in the center of the MOT.

We have also measured the excitation fraction as afunction of atomic density at a fixed laser irradiance.The density is varied by transiently transferring thepopulation between the two hyp erfine levels of the g roundstate, F 2 and F 3, which are separated by 3.0 GHz.Since our UV laser bandwidth is very narrow, it excitesselectively from only one of these levels. Therefore,changing their relative population immediately preced-ing t he UV pulse, while keeping other parameters of t heMOT fixed, varies the effective atomic density availablefor excitation. This population transfer is due to the slow( 100 s) optical pumping into F 2 which occurswhen the repumping laser is turned off before the

FIG. 2 (color online). Mean-field model of excitation sup-pression. Within a given domain, any one of the numerousground-state atoms (small spheres) may be excited into aRydberg state (light sphere). Because of strong Rydberg-Rydberg vdW interactions, the mean-field energy shift " de-pends on the particular location, as does the excitation proba-bility jcej

2. Atoms near the domain center are less shifted and

their excitation is more probable than those near the periphery.The graphs correspond to n 80 atoms at 6:5 1010 cm3 and scaled irradiance I 0:187 MW=cm2.


6 AUGUST 2004

063001-3 063001-3


4/4

trapping laser. The extent of the t ransfer is controlledby the ti me interval between switching off these twolasers. The effective density is measured by comparingthe isolated-atom signals from F 2 and F 3 to then 30 state.

The measured excitation fraction for n 80 as a func-tion of atomic density is shown in Fig. 3 along with theprediction of the model. Note that the excitation fraction

is with respect to t he number of atoms in t he appropriatehyperfine level. Again, we choose the scaling factor to be 2:4. Just as for the irradiance dependence in Fig. 1,the data and the model agree quite well. Note that in theabsence of a blockade, one would expect a constant ex-citation fraction (here 6.4%) instead of the rapid decreaseobserved.

We have verified that the suppression of excitation wehave observed is not due to Stark shifts from chargescreated during the UV laser pulse. At sufficiently highintensities, we do observe significant numbers of freeelectrons within 30 ns of a resonant excitation pulse.However, at the maximum irradiance used in the presentwork, 5:1 MW=cm2 for n 80, we observe less than 100such electrons. In t he worst case, if these electrons wereall produced during the laser pulse, the resulting averagemicrofield would be less than 20 mV/cm. I f all were toleave the excitation volume, the space charge field at itsedge would be50 mV=cm. The resulting Stark shifts for80p are 0.9 MHz and 5.6 MHz, respectively, negligiblecompared to the laser bandwidth of100 MHz.

The prospects of achieving a complete blockade in amicroscopic sample, such as at a single site in an optical

lattice, look very promising. For example, with a densityof 1012 cm3 and a laser bandwidth of 1 MHz, we con-servatively estimate t hat in a spherical sample of 50atoms, the probability of exciting more than one atom ton 80 is less than 1%. In addition, quantum gates [3,6]using adjacent sites of an optical lattice can be realizedvia the vdW blockade and vdW interactions between sites.We estimate, using a worst-case analysis, that in a CO2

laser lattice this is possible for n * 85 if the Rydbergexcitation laser bandwidth is & 1 MHz.

In conclusion, we have observed a local blockade ofRydberg state excitation in a macroscopic sample due tostrong vdW interactions. The dependence of this dramaticsuppression on laser irradiance, atomic density, and prin-cipal quantum number are in good agreement with amodel based on a mean-field t reatment of the atomicinteractions. We have observed a sample-averaged sup-pression by up to a factor of 6.4 relative to isolated-atomexcitation. Increasing the atomic density should result insignificantly la rger suppressions. A full blockade in amicroscopic sample would represent an important nextstep toward quantum computing with ultracold Rydbergatoms.

This work was supported by the University ofConnecticut Research Foundation, the ResearchCorporation, and Grants No. PHY-998776 and No.ITR-0082913 from the National Science Foundation.

[1] Quantum Computation and Quantum InformationTheory, edited by C. Macchiavello, G. M. Palma, andA. Zeilinger (World Scientific, Singapore, 2000).

[2] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information (Cambridge University,Cambridge, England, 2000).

[3] D. Jaksch, et al., Phys. Rev. Lett. 85, 2208 (2000).[4] T. F. Gallagher, Rydberg Atoms (Cambridge University,

Cambridge, England, 1994).[5] I. E. Protsenko, G. Reymond, N. Schlosser, and

P. Grangier, Phys. Rev. A 65, 052301 (2002).[6] M. D. Lukin, et al., Phys. Rev. Lett. 87, 037901 (2001).[7] M. Saffman a nd T. G. Walker, Phys. Rev. A 66, 065403

(2002).[8] C. Hettich, et al., Science 298, 385 (2002).[9] C. Boisseau, I. Simboten, and R. Cote, Phys. Rev. Lett.

88, 133004 (2002).

[10] M. Marinescu, Phys. Rev. A 56, 4764 (1997).[11] S. M. Farooqi, et al., Phys. Rev. Lett. 91, 183002 (2003).[12] N. Melikechi , S. Gangopadhyay, and E. E. Eyler, J. Opt.

Soc. Am. B 11, 2402 (1994).[13] G.V. Mar r and D. M. Creek, P roc. R. Soc. London A 304,

233 (1968).[14] D. Ciampini, et al., Phys. Rev. A 66, 043409 (2002).[15] L. N. Shabanova a nd A. N. K hlyustalov, Opt. Spectrosc.

(USSR) 56, 128 (1984).[16] C. J. Lorenzen and K. Niemax, Phys. Scr. 27, 30 0

(1983).

FIG. 3. Density dependence of the n 80 Rydberg signal.Relative densities a re accurate to 5%10%, but the absolutepeak density, nominally 6:5 1010 cm3 at a relative densityof 1.0, is uncertain by as much as a factor of 2. MOT fractionsare expressed in terms of the available ground-state population

in the selected hyperfine level (see text). The solid curve showsthe mean-field theoretical prediction, using the same scalingparameter as for Fig. 1. The inset shows the low-densitybehavior.


6 AUGUST 2004

063001-4 063001-4

d. tong et al- local blockade of rydberg excitation in an ultracold gas

Documents