Rabi flopping between ground and Rydberg states with dipole-dipole atomic interactions
T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, M. Saffman
aa r X i v : . [ qu a n t - ph ] N ov Rabi flopping between ground and Rydberg states with dipole-dipole atomicinteractions
T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman
Department of Physics, University of Wisconsin, 1150 University Avenue, Madison, WI 53706 (Dated: November 1, 2018)We demonstrate Rabi flopping of small numbers of Rb atoms between ground and Rydbergstates with n ≤
43. Coherent population oscillations are observed for single atom flopping, whilethe presence of two or more atoms decoheres the oscillations. We show that these observations areconsistent with van der Waals interactions of Rydberg atoms.
PACS numbers: 32.80.-t, 32.80.Rm, 03.67.-a
Atoms in highly excited Rydberg states with principalquantum number n >> d ∼ qa n , with q the electron chargeand a the Bohr radius. Two such Rydberg atoms canbe strongly coupled via a dipole-dipole interaction. Itwas recognized in recent years that the large interactionstrength can potentially be used for fast quantum gatesbetween qubits stored in stable ground states of neutralatoms[1]. When several atoms are sufficiently close to-gether the presence of a single excited atom can causea shift in the energy of all other atoms which is largeenough to prevent resonant excitation of more than oneatom in a sample. This “dipole blockade” mechanism hasthe potential for creating strongly coupled ensembles con-taining moderate numbers of atoms. Such ensembles canbe used for gates[2], as well as several other quantum in-formation tasks including state preparation[3], fast mea-surement protocols[4], and collective encoding of multi-qubit registers[5].A number of recent experiments have revealed signa-tures of “dipole blockade” by showing that the proba-bility of multiple excitation of Rydberg atoms is sup-pressed at high n [6]. However, none of the experimentsto date have demonstrated blockade at the level of a sin-gle atomic excitation which is crucial for applications toquantum information processing. In order to be usefulfor quantum gates it is also necessary to be able to co-herently excite and de-excite a Rydberg state so that theatom is available for further processing. In this letter wedemonstrate important steps towards the goal of a fastneutral atom Rydberg gate. We start by preparing sin-gle atom states in micron sized optical traps and observecoherent Rabi oscillations between ground and Rydbergstates with n ≤
43 at rates as high as Ω R / π = 0 . . We then show that the presence of two or more atoms inthe trap causes dephasing of the Rabi oscillations. Com-parison with theoretical calculations of the strength ofthe Rydberg van der Waals interactions[7], confirms thatour observations are consistent with the presence of Ry-dberg interactions.The experiment starts by loading a far-off-resonanceoptical trap (FORT) from a Rb vapor cell magneto-
FIG. 1: (color online) Experimental arrangement, see text fordetails. optical trap (MOT) as described in our recent letter[8].For the experiments reported here, between 1 and 10atoms are loaded into a 10 mK deep FORT (570 mW of1030 nm light focused to a 1 /e intensity radius waist of w = 2 . µ m). The radial and axial oscillation frequenciesare 130 and 12 kHz. The average number of atoms iscontrolled by varying the amount of time for which theMOT and FORT lasers are simultaneously on from 25 -400 ms. Atom temperatures in the FORT are measuredby performing a drop and recapture measurement, andcomparing the probability of recapture with numericalcalculations. We consistently find temperatures from 5 −
10 % of the FORT depth, which corresponds to 0 . − Rb,which we attribute to degradation of the laser coolingby large FORT induced differential Stark shifts of the5 s / and 5 p / levels. With a conservative estimate of1 mK for the temperature, the spatial distribution of theatoms is quasi one-dimensional with standard deviationsof σ z = 0 . µ m and σ x = 3 . µ m . We prepare single atom states in the FORT despitePoissonian loading statistics by using a two-measurementsequence. After the MOT to FORT loading period, weconduct a first measurement of the number of atoms inthe FORT by scattering MOT light (3 pairs of counter-propagating beams) detuned by − FIG. 2: (color online) Correlation between first and secondmeasurement distributions, without Rydberg excitation be-tween measurements. The dashed lines show cuts for selectingsingle atom states. the cycling transition while chopping the FORT on andoff at rates between 0 . − . × s − . The probing MOTlight is chopped out of phase with the FORT, eliminat-ing the need for tuning to the Stark shifted atomic res-onance. Scattered photons are collected with a fast lens(NA=0.4) and focused onto a cooled electron-multiplyingCCD camera. We estimate our detection efficiency in-cluding finite solid angle, optical losses, and camera quan-tum efficiency to be about 2.7%. We observe single atomphotoelectron rates of about 10 s − (here time is theprobing time with the MOT beams on, with the totalmeasurement time about 2 . /e FORT lifetime of about 3 s . We verify the reliabilityof preselecting single atom states by performing a sec-ond measurement, shown on the ordinate. We see thatdespite some loss of atoms during the first measurement,single atom states can be prepared with about 85% prob-ability, with a 15% admixture of zero atom states. Notethat the reliability of selecting states with two or moreatoms is much worse. We believe that this is due to lightassisted collisions causing rapid loss out of the FORTduring the first measurement [9].We excite Rydberg states using two-photon transitionswith 780 and 480 nm lasers as shown in Fig. 1. The Ryd-berg beams E , E are focused to waists of w ≃ µ mand spatially overlapped with the FORT. These beamsare generated by locking a 780 nm laser and a 960 nmlaser to different longitudinal modes of the same stablereference cavity with finesse F ∼ ,
000 and linewidth
FIG. 3: (color online) Rabi flopping experiment to 43 d / with P = 1 . µ W , P = 10 . / π = − . . Each data point is the average of 40 pre-selected single atomexperiments, with the bars showing ± ∼ < µ saveraging time. The 960 nm light is then amplified andfrequency doubled to create the 480 nm Rydberg excita-tion light. Acousto-optic modulator (AOM) based noiseeaters are used as necessary to reduce amplitude fluctua-tions to a few percent. The frequencies of both lasers arethen shifted with AOM’s to match the desired Rydberglevel.After preparing a single atom state the probabilityof transition to a Rydberg level is measured as a func-tion of the pulse length of E , E . If the atoms arenot prepared in a single ground state Zeeman level theywill be coupled to a superposition of different RydbergZeeman levels. In the presence of background magneticfields these levels have different Zeeman shifts which de-cohere the coherent population oscillations we are inter-ested in. We therefore start by optically pumping intothe | f = 2 , m f = 2 i Zeeman state using σ + polarizedlight ( E p in Fig. 1) near resonant to the | s / f = 2 i ↔| p / f ′ = 3 i transition with a B = 10 − T bias field.The ˆ z polarized excitation light only couples this stateto the | nd / , m j = 1 / i Zeeman state. We have verifiedthe expected shift of − . T ( µ s) µ s T/2 T/2 t
780 nm480 nm1030 nm µ s
100 ns P ( g r ound ) FIG. 4: (color online) Rabi flopping experiment with singleatoms to 28 d / using a double pulse sequence. The excitationparameters were the same as in Fig. 3 except that P =3 . µ W , P = 9 . / π = − . . The solidcurve is a theoretical calculation assuming Ω R / π = 0 . µ s gap of 0 .
53 MHz . The inset showsthe timing sequence. spectroscopy with and without the field on.In order to measure the probability of Rydberg exci-tation we use the fact that the calculated photoioniza-tion rate of the d states due to the FORT light is largecompared to the radiative decay rate back to the groundstate[10]. The timing sequence for Rabi flopping is shownin Fig. 3. We turn off the FORT light for a fixed lengthof time which is long enough to perform the Rydberg ex-citation yet short enough that we do not lose the atom inthe absence of a Rydberg pulse. We then perform a Ry-dberg pulse of variable length T , after which we restorethe FORT light. Photoionization by the FORT light thusperforms a projective measurement of the atomic state,and after 100 ms with the FORT on, the ground statepopulation is measured using MOT light. The result-ing data points for the ground state probability are thennormalized to 1 . T = 0 to correct for a 10-20% lossrate due to the single atom selection measurement, mo-tional losses during the FORT drop period, and FORTloss due to background collisions. A curve fit to thedata of Fig. 3 with the function (1 − a ) + ae − tτ cos(Ω R t )gives a Rabi frequency of Ω R = 2 π × .
49 MHz, whereasour theoretical value with no adjustable parameters is[11]Ω R = Ω Ω /
2∆ = 2 π × .
55 MHz. We attribute theapproximately 11% lower experimental value to spatialmisalignment, and a small fraction of the Rydberg lightbeing present in servo sidebands from the laser locks.The fit also gives a decay time of τ = 8 µ s which isconsistent with Doppler averaged numerical calculations.The data show that the atom is returned to the groundstate with better than 90 % probability and that Ryd-berg state excitation is achieved with 70 −
80% proba-bility. The lack of perfect Rydberg excitation is due toseveral factors which we estimate as Doppler broadeningof the excitation( ∼ − FIG. 5: (color online) Rabi flopping experiment to 43 d / with different atom numbers. Each data point is the averageof 150 measurements, with the bars showing ± ( ∼ ∼ π pulse time of ∼ µ s this requiresstaying in the Rydberg level for at least 2 µ s. We haveverified that coherence can be maintained over this timeby performing a double pulse experiment as shown inFig. 4. For this experiment the maximum probability ofRydberg excitation is only about 50%. This is becauseduring the 2 µ s free evolution between the pulses the780 nm Rydberg laser induced light shifts are no longerpresent which results in a free evolution for 2 µ s at finitedetuning. We have fit the measured data by assuming adetuning of 0 .
53 MHz. This is consistent with the 780nm beam induced light shift of the ground state which isΩ /
4∆ = 2 π × .
58 MHz for our estimated experimentalparameters.The above Rabi flopping curves change dramaticallywhen more than one atom is present in the FORT. Fig-ure 5 shows the results with average atom numbers of¯ N = 0 . , . , and 8. These data were obtained by simplyadjusting the loading time for a desired ¯ N (the actualdistribution is close to Poissonian), and then applyingvarying length Rydberg pulses, without first using thesingle atom selection procedure of Fig. 2. We see thatat ¯ N = 1 . N = 8 there is essentially no oscillationleft. We emphasize that Rabi flopping with high visi-bility and good coherence between ground state levels isobserved with as many as 10 atoms in the FORT[8].In order to explain these observations we must ac-count for Rydberg interactions. In the absence of anexternal electric field Rb atoms excited to the 43 d / state experience a F¨orster interaction due to the nearresonance of the process 43 d / + 43 d / → p / +41 f / , / . The corresponding energy defect is ¯ hδ = FIG. 6: (color online) Eigenvalues (triangles) and overlap fac-tors (squares) for the two-atom van der Waals eigenstates with − ≤ M x ≤ U (45 p / ) + U (41 f / , / ) − U (43 d / ) . Using recentmeasurements of the Rb quantum defects[13], we find δ/ π = − . , − . f / and f / states re-spectively. This small energy defect naively implies along range 1 /R interaction, with R the atomic separa-tion. However, due to Zeeman degeneracy of the Ryd-berg levels, there are linear superpositions of two-atom d / states which are excited by the Rydberg lasers, yethave small dipole-dipole interactions[14]. We have devel-oped a full theory of this situation which will be pub-lished elsewhere[7]. For the present discussion we notethat in the limit where the atoms have a relatively largeseparation we obtain a van der Waals interaction of theform V dd = C D φ /R , with C ∼ d / ¯ hδ , and D φ aneigenvalue. The eigenvalues depend on M , the projec-tion of the two-atom angular momentum on the inter-atomic axis, and for the | d / i states very small valuesof D φ occur for M = 0 . In our experimental geometrywe excite states with M z = 1 along ˆ z, but in a quasione-dimensional FORT large values of R occur for atompairs aligned along ˆ x. States with M z = 1 are linear su-perpositions of states with − ≤ M x ≤
5, and the stateswith M x = 0 overlap with eigenstates for which D φ isvery small.Figure 6 shows the overlap coefficients κ φ = h φ | M z =1 i , together with the eigenvalues D φ for the (2 j +1) = 36eigenvectors. We see that there is a strong overlap withthe state with the smallest eigenvalue, D φ, min = 0 . . For this interaction | d / i has C = 450 GHz µ m and at a characteristic two-atom separation in the FORTof p h R i = √ σ x = 7 . µ m the interaction strengthdue to the smallest eigenvalue is only 4 . C . At the same time there isstrong coupling to states with D φ = 0 .
81 which give acharacteristic interaction strength of 1 . R , but is further accentuated bycoupling to both small and large eigenvalues. In the lim-its of very weak interactions, or very strong interactionsgiving an effective blockade, we do not expect rapid de-phasing in the multiple atom regime, which is a signatureof a range of coupling strengths.In conclusion we have observed coherent Rabi oscil-lations between ground and Rydberg states which is animportant step towards demonstration of a neutral atomRydberg gate. We have shown how van der Waals inter-actions lead to dephasing of the oscillations when severalatoms are present, and elucidated the role of F¨orster zerostates in the dephasing. Future work will explore alter-native interaction geometries that do not couple to smalleigenvalues as a means of demonstrating dipole blockade.This work was supported by the NSF and the Disrup-tive Technology Office. We are grateful to Pasad Ku-latunga and Marie Delaney for experimental contribu-tions at an early stage of this work. [1] D. Jaksch, et al, Phys. Rev. Lett , 2208 (2000).[2] M. D. Lukin, et al, Phys. Rev. Lett. , 037901 (2001).[3] M. Saffman and T. G. Walker, Phys. Rev. A , 065403(2002).[4] M. Saffman and T. G. Walker, Phys. Rev. A , 042302(2005).[5] E. Brion, K. Mølmer, and M. Saffman, Phys. Rev. Lett.to appear (2007).[6] D. Tong, et al, Phys. Rev. Lett. , 063001 (2004);K. Singer, et al, ibid, , 163001 (2004); T. C. Liebisch,A. Reinhard, P. R. Berman, and G. Raithel, ibid, ,253002 (2005); erratum , 109903 (2007); T. Vogt, et al,ibid, , 083003 (2006); T. Vogt, et al, ibid, , 073002(2007); P. Bohlouli-Zanjani, J. A. Petrus, and J. D. D.Martin, ibid, , 203005 (2007); R. Heidemann, et al,ibid, , 163601 (2007).[7] T. G. Walker and M. Saffman, unpublished.[8] D. D. Yavuz, et al, Phys. Rev. Lett. , 063001 (2006).[9] D. Sesko, et al, Phys. Rev. Lett. , 961 (1989); K. D.Nelson, X. Li, and D. S. Weiss, Nature Phys. , 556(2007).[10] M. Saffman and T. G. Walker, Phys. Rev. A , 022347(2005).[11] For ˆ z polarized light Ω = √ q E ¯ h R p / s / and Ω = √ q E ¯ h R d / p / . Using quantum defect wavefunctions wefind R p / s / = 5 . a , R d / p / = 0 . a . [12] The ratio of room temperature radiative transitions tophotoionization rates is about (1 . × s − ) / (3 . × s − ) which implies a 4% reduction in our measuredexcitation probability relative to the true value.[13] W. Li, I. Mourachko, M. W. Noel, and T. F. Gallagher, Phys. Rev. A , 52502 (2003); J. Han, et al, Phys. Rev.A , 054502 (2006).[14] T. G. Walker and M. Saffman, J. Phys. B: At. Mol. Opt.Phys.38