High-Fidelity Entanglement and Detection of Alkaline-Earth Rydberg Atoms
Ivaylo S. Madjarov, Jacob P. Covey, Adam L. Shaw, Joonhee Choi, Anant Kale, Alexandre Cooper, Hannes Pichler, Vladimir Schkolnik, Jason R. Williams, Manuel Endres
HHigh-fidelity entanglement and detection of alkaline-earth Rydberg atoms
Ivaylo S. Madjarov, ∗ Jacob P. Covey, ∗ Adam L. Shaw, Joonhee Choi, Anant Kale, AlexandreCooper, † Hannes Pichler, Vladimir Schkolnik, Jason R. Williams, and Manuel Endres ‡ Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
Trapped neutral atoms have become a prominent platform for quantum science, where entan-glement fidelity records have been set using highly-excited Rydberg states. However, controlledtwo-qubit entanglement generation has so far been limited to alkali species, leaving the exploitationof more complex electronic structures as an open frontier that could lead to improved fidelities andfundamentally different applications such as quantum-enhanced optical clocks. Here we demonstratea novel approach utilizing the two-valence electron structure of individual alkaline-earth Rydbergatoms. We find fidelities for Rydberg state detection, single-atom Rabi operations, and two-atomentanglement surpassing previously published values. Our results pave the way for novel applica-tions, including programmable quantum metrology and hybrid atom-ion systems, and set the stagefor alkaline-earth based quantum computing architectures.
Recent years have seen remarkable advances in gener-ating strong, coherent interactions in arrays of neutralatoms through excitation to Rydberg states, character-ized by large electronic orbits . This has led to profoundresults in quantum science applications, such as quantumsimulation and quantum computing , includinga record for two-atom entanglement for neutral atoms .Furthermore, up to 20-qubit entangled states have beengenerated in Rydberg arrays , competitive with resultsin trapped ions and superconducting circuits . Manyof these developments were fueled by novel techniques forgenerating reconfigurable atomic arrays and mitiga-tion of noise sources . While previous Rydberg-atom-array experiments have utilized alkali species, atoms witha more complex level structure, such as alkaline-earthatoms (AEAs) commonly used in optical latticeclocks , provide new opportunities for increasing fideli-ties and accessing fundamentally different applications,including Rydberg-based quantum metrology , quan-tum clock networks , and quantum computing schemeswith optical and nuclear qubits .Here we demonstrate such a novel Rydberg array archi-tecture based on AEAs, where we utilize the two-valenceelectron structure for single-photon Rydberg excitationfrom a meta-stable clock state as well as auto-ionizationdetection of Rydberg atoms (Fig. 1). We find leading fi-delities for Rydberg state detection, ground- to Rydberg-state coherent operations, and Rydberg-based two-atomentanglement (Table I). More generally, our results con-stitute the highest reported two-atom entanglement fi-delities for neutral atoms as well as a proof-of-principle for controlled two-atom entanglement betweenAEAs. We further demonstrate a high-fidelity entangle-ment operation with optical traps kept on, an important ∗ These authors contributed equally to this work † Permanent address: Institute for Quantum Computing, Univer-sity of Waterloo, 200 University Ave West, Waterloo, Ontario,Canada ‡ [email protected] a clockqubitdetection P S S e R B (i)(ii) b
408 nm317 nm698 nm
Fig. 1 | Population and detection of Rydberg statesin non-interacting and interacting configurations. a ,The relevant level structure (left), and electronicconfiguration (right) for strontium −
88. The Rydberg-groundstate qubit is defined by a metastable ‘clock’ state | g (cid:105) andthe 5s61s S m J = 0 Rydberg state | r (cid:105) (highlighted with apurple box), which we detect by driving to an auto-ionizing5p61s state | r ∗ (cid:105) . The clock state | g (cid:105) is initialized from theabsolute ground state | a (cid:105) . b , We use atom-by-atom assemblyin optical tweezers to prepare an effectively non-interactingconfiguration ((i), blue box and data-points throughout) anda strongly Rydberg-blockaded pair configuration ((ii), redbox and data-points throughout) . The blockade radius R B ,where two-atom excitation is suppressed, is indicated by adashed circle. Throughout, purple and black circles indicate | r (cid:105) and | g (cid:105) atoms, respectively. The Rydberg,auto-ionization, and clock beams all propagate along theaxis of the atom array and address all atoms simultaneously.Averaged fluorescence images of atoms in configurations (i)and (ii) are shown. See Methods and SupplementaryInformation for further detail. step for gate-based quantum computing . As de-tailed in the outlook section, our results open up a hostof new opportunities for quantum metrology and com-puting as well as for optical trapping of ions.Our experimental system combines various novelkey elements: First, we implement atom-by-atom as- a r X i v : . [ phy s i c s . a t o m - ph ] N ov Table I | Uncorrected and SPAM-corrected fidelitiesfor single-atom and Rydberg-blockaded pulses.
The‘T’ indicates settings where the tweezers are on duringRydberg excitation.Quantity Uncorrected SPAM-correctedSingle-atom π -pulse 0 . . π -pulse 0 . . π -pulse 0 . . π -pulse 0 . . π -pulse, T 0 . . π -pulse, T 0 . . ≥ . ≥ . ≥ . ≥ . sembly in reconfigurable tweezer arrays for AEAs(Fig. 1b). Second, we sidestep the typical protocol fortwo-photon excitation to S-series Rydberg states, whichrequires significantly higher laser power to suppress in-termediate state scattering, by transferring atoms to thelong-lived P clock state | g (cid:105) . We treat | g (cid:105) as aneffective ground state from which we apply single-photonexcitation to a S Rydberg state | r (cid:105) . Third, insteadof relying on loss through tweezer anti-trapping as in al-kali systems, we employ a rapid auto-ionization schemefor Rydberg state detection. In contrast to earlier imple-mentations of auto-ionization detection in bulk gases ,we image remaining neutral atoms instead of detectingcharged particles.More generally, our findings improve the outlook forRydberg-based quantum computing , optimiza-tion , and simulation . These applications all relyon high fidelities for preparation, detection, single-atomoperations, and entanglement generation for which webriefly summarize our results: we obtain a state prepara-tion fidelity of 0 . . The new auto-ionizationscheme markedly improves the Rydberg state detectionfidelity to 0 . − . . We also push the lim-its of single and two-qubit operations in ground- toRydberg-state transitions . For example, we find π -pulse fidelities of 0 . . . Finally, using a conserva-tive lower-bound procedure, we observe a two-qubit en-tangled Bell state fidelity of ≥ . ≥ . | g (cid:105) and | r (cid:105) (Fig. 2a) and the auto-ionization de-tection scheme (Fig. 2b) in an essentially non-interactingatomic configuration ((i) in Fig. 1b). To detect atomsin | r (cid:105) we excite the core valence-electron from a 5s to e Time (ns) P r o b a b ili t y ( P ) b Time (ns) P r o b a b ili t y ( P ) a Fig. 2 | Rabi oscillations and auto-ionization. a ,Array-averaged probability P of detecting an atom after aresonant Rydberg pulse and subsequent auto-ionization as afunction of Rydberg pulse time, showing high-contrast Rabioscillations with frequency Ω R = 2 π × . µ s. b , P as afunction of auto-ionization pulse time at a fixed Rydbergpulse time of 70 ns corresponding to a π -pulse (followed by asecond π -pulse). The solid line is a fit to a Gaussian,phenomenologically chosen to capture the finite switch-ontime of the auto-ionization beam . Inset: illustration of theauto-ionization process. In both a and b , data is uncorrectedand averaged over ≈ −
100 experimental cycles pertimestep and over an array of approximately 14 atoms.Error bars indicate a 1 σ binomial confidence interval. a 5p level, which then rapidly auto-ionizes the Rydbergelectron (inset of Fig. 2b) . The ionized atoms are darkto subsequent detection of atoms in | g (cid:105) , providing themeans to distinguish ground and Rydberg atoms.We use a | g (cid:105) ↔ | r (cid:105) Rabi frequency of Ω R ≈ π × − π -pulse on | g (cid:105) ↔ | r (cid:105) , then apply an auto-ionization pulse for a variable duration (Fig. 2b), andthen perform a second π -pulse on | g (cid:105) ↔ | r (cid:105) before mea-surement. The detected population decreases to zero witha 1 /e time of τ A = 35(1) ns. We can compare τ A to thelifetime of | r (cid:105) , which is estimated to be τ | r (cid:105) ≈ µ s ,placing an upper bound on the | r (cid:105) -state detection effi-ciency of 0 . π -pulse fidelity of 0 . .To probe our longer-time coherence, we drive the Ry-dberg transition for as long as 7 µ s (Fig. 3a). The decayof the contrast on longer timescales is well modeled by aGaussian profile of the form C ( t ) = C exp( − t /τ C ). We b Time (µs) P r o b a b ili t y ( - P { } ) a P r o b a b ili t y ( P ) Fig. 3 | Long-time Rabi oscillations for single and blockaded atoms. a , Array-averaged Rabi oscillations for thenon-interacting configuration (i), depicted by the inset. We operate with Ω R = 2 π × . /e coherence of ≈
42 cycles. b , Same as in a but for the blockaded configuration (ii), depicted by the inset.We plot 1 − P { } , where P { } is the array-averaged symmetrized probability of detecting one atom of an initial pair (and notboth). We observe a blockade-enhanced Rabi frequency of ˜Ω R = 2 π × . /e coherence of ≈
60 cycles. Inboth a and b , data is uncorrected and averaged over ≈
10 experimental cycles per timestep and over an array ofapproximately 14 atoms in a or 10 pairs in b . Error bars indicate a 1 σ binomial confidence interval. find that τ C ≈ µ s is consistent with our data, and corre-sponds to a 1 /e coherence of ≈
42 cycles. To our knowl-edge, this is the largest number of coherent ground-to-Rydberg cycles that has been published to date . Lim-itations to short- and long-term coherence are discussedand modeled in detail in Ref. . The main contributingfactors are laser intensity and phase noise (which bothcan be improved upon with technical upgrades, such ascavity filtering of phase noise ), and finite Rydberg statelifetime.We now turn to the pair-interacting configuration ((ii)in Fig. 1b) to study blockaded Rabi oscillations . Foran array spacing of 3 . µ m, we anticipate an interactionshift of V B ≈ π ×
130 MHz for the n = 61 Rydbergstate in the S series . In this configuration, simulta-neous Rydberg excitation of closely-spaced neighbors isstrongly suppressed, and an oscillation between | gg (cid:105) andthe entangled W -Bell-state | W (cid:105) = ( | gr (cid:105) + e iφ | rg (cid:105) ) / √ √ , as observed in our data. We show our resultsfor long-term coherent oscillations in Fig. 3b and find a1 /e coherence time corresponding to ≈
60 cycles. Resultsfor short-term oscillations are shown in Fig. 4a and thefidelity values are summarized in Table I.We now estimate the Bell state fidelity associated witha two-atom (blockaded) π -pulse. While parity oscillationsprovide a standard metric for entanglement fidelity , theyrequire site-resolved laser addressing. We leave this tech-nique for future work, and instead provide a lower boundfor the Bell state fidelity based on measured populationsat the (blockaded) π -time and a lower bound on the pu-rity of the two-atom state. The latter is obtained by mea- suring the atomic populations at the (blockaded) 2 π time,under the assumption that the purity does not increasebetween the π and the 2 π time. For a detailed discus-sion and analysis of this bound and the validity of theunderlying assumptions, see Ref. . With this approach,we find uncorrected and SPAM-corrected lower boundson the Bell state fidelity of 0 . . off during Rydberg excitation. Thepotential application of Rydberg gates to large circuitdepth quantum computers motivates the study of block-ade oscillations with the tweezers on . In particular, weforesee challenges for sequential gate-based platforms iftweezers must be turned off during each operation toachieve high fidelity. In systems implementing gates be-tween the absolute ground and clock states for example,blinking traps on and off will eventually lead to heatingand loss, ultimately limiting the number of possible oper-ations. Furthermore, while individual tweezer blinking ispossible in one dimension, the prospects for blinking in-dividual tweezers in a two dimensional array are unclear:a two-dimensional array generated by crossed acousto-optic deflectors cannot be blinked on the level of a singletweezer, and one generated by a spatial light modulatorcannot be blinked fast enough to avoid loss. Repulsivetraps such as interferometrically-generated bottles orrepulsive lattices have been developed in lieu of stan-dard optical tweezer arrays in part to help maintainhigh-fidelity operations while keeping traps on.Despite finding that our Rydberg state is anti-trapped(with a magnitude roughly equal to that of the ground Time (ns) P r o b a b ili t y ( - P { } ) P r o b a b ili t y ( - P { } ) ab Fig. 4 | Short-time Rydberg-blockaded Rabioscillations with tweezers off and on . a , Short-timeRabi-oscillations for the blockade configuration (ii) with thetraps off , depicted by the inset. b , Same as in a but withtweezers on during Rydberg interrogation with a | g (cid:105) -statedepth of U /h ≈ .
94 MHz. The blockade-enhanced Rabifrequency is ˜Ω R = 2 π × . a and b , data isuncorrected and averaged over ≈ −
100 experimentalcycles per timestep and over an array of approximately 10pairs. Error bars indicate a 1 σ binomial confidence interval. state trapping) at our clock-magic wavelength of λ T =813 . , we observe high-fidelity entanglement evenwhen the tweezers remain on during Rydberg interro-gation. Certain factors make this situation favorable foralkaline-earth atoms. One is the ability to reach lowertemperatures using narrow-line cooling, which suppressesthermal dephasing due to trap light shifts. Furthermore,a lower temperature allows for ramping down of tweezersto shallower depths before atoms are lost, further allevi-ating dephasing. Finally, access to higher Rabi frequen-cies provides faster and less light-shift-sensitive entan-gling operations.We study short-time blockaded Rabi oscillations bothwith the tweezers switched off (Fig. 4a) and left on (Fig. 4b). We find similar fidelities for the π - and 2 π -pulses in both cases (Table I). Further, we estimate alower bound for the Bell state fidelity in the tweezer on case, and find uncorrected and corrected values of ≥ . ≥ . , andour observations show promise for Rydberg-based quan-tum computing in a standard tweezer array .Our work bridges the gap between the fields of Ryd-berg atom arrays and optical clocks , opening the doorto Rydberg-based quantum-enhanced metrology ,including the programmable generation of spin-squeezedstates in recently demonstrated tweezer clocks , and quantum clock networks . Further, the demon-strated entangling operations provide a mechanismfor two-qubit gates in AEA-based quantum compu-tation and simulation architectures leveraging opticaland nuclear qubits . More generally, the observedentanglement fidelities could enable gate fidelities forlong-lived ground states approaching fault-toleranterror correction thresholds . In addition, the highRydberg- and ground-state detection-fidelities couldplay an important role in applications based on samplingfrom bit-string probability distributions . Finally, byauto-ionizing the Rydberg electron with high fidelityand noting that we expect the remaining ion to staytrapped, we have found a potential new approach to theoptical trapping of ions in up to three dimensionalarrays . Such a platform has been proposed forion-based quantum computing as well as for hybridatom-ion systems . Note added.—
Recently, we became aware of work inytterbium tweezer arrays demonstrating trapping ofRydberg states . METHODS
We briefly summarize the relevant features of our Srexperiment . We employ a one-dimensional arrayof 43 tweezers spaced by 3.6 µ m. Atoms are cooled closeto the tranverse motional ground state using narrow linecooling , with an average occupation number of ¯ n r ≈ . T r ≈ . µ K), in tweezers of ground-state depthU ≈ k B × µ K ≈ h × . ω r ≈ π ×
78 kHz.For state preparation (Fig. 1a), we drive from the 5s S absolute ground state (labeled | a (cid:105) ) to the 5s5p P clock state (labeled | g (cid:105) ) with a narrow-line laser , reach-ing Rabi frequencies of Ω C ≈ π × . ≈
710 G (otherwise set to ≈
71 G for theentire experiment). We populate | g (cid:105) with a π -pulse reach-ing a loss-corrected fidelity of 0 . F = U /
10) to obtaina clock state population without and with loss correc-tion of 0 . . .We treat the long-lived state | g (cid:105) as a new ground state,from which we drive to the 5s61s S , m J = 0 Rydbergstate (labeled | r (cid:105) ). The | g (cid:105) ↔ | r (cid:105) Rydberg transition oc-curs at a wavelength of λ R = 316 . /e beam radius of 18(1) µ m. We readily achieve a | g (cid:105) ↔ | r (cid:105) Rabi frequency of Ω R ≈ π × − ≈
30 mW, and up to Ω R ≈ π ×
13 MHz with full op-timization of the laser system and beam path. To detectatoms in | r (cid:105) we drive the strong transition to 5p / / ( J = 1 , m J = ± | r ∗ (cid:105) . This transition excitesthe core ion, which then rapidly auto-ionizes the Ryd-berg electron. The ionized atoms are dark to subsequentdetection of atoms in | g (cid:105) with the high-fidelity schemedescribed in Ref. , providing the means to distinguishground and Rydberg atoms. We switch off the ramped-down tweezers during the Rydberg pulse , after whichwe apply an auto-ionization pulse while rapidly increas-ing the depth back to U for subsequent read-out.The Rydberg and clock laser beams are linearly polar-ized along the magnetic field axis, and the auto-ionizationbeam is linearly polarized perpendicular to the magneticfield axis. Accordingly, we excite to auto-ionizing stateswith m J = ±
1. The tweezers are linearly polarized alongthe axis of propagation of the Rydberg, clock, and auto-ionization beams – perpendicular to the magnetic fieldaxis.
ACKNOWLEDGEMENTS
We acknowledge discussions with Chris Greene andHarry Levine as well as funding provided by the In-stitute for Quantum Information and Matter, an NSFPhysics Frontiers Center (NSF Grant PHY-1733907),the NSF CAREER award (1753386), the AFOSR YIP(FA9550-19-1-0044), the Sloan Foundation, and FredBlum. Research was carried out at the Jet PropulsionLaboratory and the California Institute of Technologyunder a contract with the National Aeronautics andSpace Administration and funded through the Presi-dent’s and Director’s Research and Development Fund(PDRDF). JPC acknowledges support from the PMAPrize postdoctoral fellowship, and JC acknowledgessupport from the IQIM postdoctoral fellowship. HPacknowledges support by the Gordon and Betty MooreFoundation. AK acknowledges funding from the LarsonSURF fellowship, Caltech Student-Faculty Programs.
AUTHOR CONTRIBUTIONS
M.E. conceived the idea and initiated the study. I.M.,J.P.C., A.S., J.C., A.C., and V.S. designed and carriedout the experiments. I.M., J.P.C., A.S., J.C., A.K.,and H.P. performed theory and simulation work. I.M.,J.P.C., A.S., J.C. contributed to data analysis. I.M.,J.P.C., A.S., J.C., and M.E. contributed to writingthe manuscript and supplementary information. J.P.C.,J.W., and M.E. supervised and guided this work. I.M.and J.P.C. contributed equally to this work.
DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
COMPETING INTERESTS
The authors declare no competing interests.
Supplementary Information
A. STATE PREPARATION
The ground state | g (cid:105) of our Rydberg qubit is the 5s5p P metastable clock state of Sr. We populate this statein two stages: first, most atoms are transferred via a co-herent π -pulse on the clock transition. Thereafter, anyremaining population is transferred via incoherent pump-ing.In our regime where the Rabi frequency of the clocktransition (Ω C ≈ π × . ω r ≈ π ×
78 kHz), coher-ent driving is preferable to incoherent pumping becauseit preserves the motional state of an atom, i.e., it doesnot cause heating. However, atomic temperature, trapfrequency, trap depth, and beam alignment contribute tothe transfer infidelity of coherent driving. Although wedrive the clock transition on the motional carrier in thesideband resolved regime, thermal dephasing still playsan important role. Particularly, each motional state hasa distinct Rabi frequency, a thermal ensemble of whichleads to dephasing . This thermal dephasing is less se-vere at higher trapping frequencies; however, this canonly be achieved in our system by using deeper traps,which would also eventually limit transfer fidelity becauseof higher rates of Raman scattering out of the clock state.We therefore perform coherent transfer initially in deepertraps ( ≈ µ K), followed immediately by an adiabaticrampdown to one-tenth of that depth. Finally, precisealignment of the clock beam to the tight, transverse axisof the tweezer is important to ensure that no couplingexists to axial motion, which has a much lower trap fre-quency and thus suffers more thermal dephasing than thetransverse direction.The remaining population is transferred by simultane-ous, incoherent driving of the 5s S ↔ P , 5s5p P ↔ S , and 5s5p P ↔ S transitionsfor 1 ms. This pumping scheme has the clock state as aunique dark state via the decay of 5s6s S to the clockstate and is in general more robust than coherent driv-ing. However, due to photon recoil, differential trapping,and an unfavorable branching ratio of 5s6s S to theclock state (requiring many absorption and emission cy-cles), this process causes significant heating, making itunfavorable as compared to coherent driving. Therefore,we only use this method as a secondary step to transferatoms left behind by the coherent drive.We measure the fidelity of our state transfer by apply-ing a 750 µ s pulse of intense light resonant with the S ↔ P transition immediately after state transfer. The largerecoil force of this pulse rapidly pushes out atoms in S with a fidelity of > . prepara-tion . Taking loss into account, as well as the probabilityof the atom Raman scattering out of the clock state inthe finite time between clock transfer and Rydberg exci-tation (see Sec. C), our overall state preparation fidelitywith both coherent driving and incoherent pumping is F SP = 0 . B. AUTO-IONIZATION AND RYDBERG STATEDETECTION FIDELITY
The auto-ionization beam is resonant with the Sr + ionictransition S / ↔ P / at λ A = 407 . /e beam waist radius is w A o = 16(1) µ m with power P A =2 . A ≈ π × /e timescale of τ A = 35(1) ns to theexpected lifetime of | r (cid:105) , which is τ | r (cid:105) ≈ µ s. That is,we compute the probability that an atom in the Rydbergstate is auto-ionized before it decays away from the Ry-dberg state. This estimate places an upper bound on thedetection fidelity of | r (cid:105) to be 0 . ± µ sin τ | r (cid:105) . Note that when the auto-ionization pulse is notapplied, there is still a residual detection fidelity of | r (cid:105) of0 . | r (cid:105) in the tweezer (thisvalue is smaller than the previously reported < .
98 foralkalis in part because the atoms are colder here than inprevious work ). A lower bound on our detection fidelityis given by the measured π -pulse fidelity after correct-ing for errors in preparation and ground state detection,which gives 0 . A ≈ π × Γ A > Ω A of | r ∗ (cid:105) actu-ally inhibits the | r (cid:105) ↔ | r ∗ (cid:105) transition via the continuousquantum Zeno mechanism . In this regime, the effec-tive auto-ionization rate of the transition continues toscale with Ω A and does not saturate until Ω A (cid:29) Γ A .This is in qualitative agreement with the fact that ourmeasured auto-ionization loss rate continues to increasewith beam intensity. Furthermore, the finite rise timeof the acousto-optic modulator (AOM) that we use forswitching the auto-ionization beam is a limiting factorin achieving faster auto-ionization. Therefore, detectionfidelity can be increased further with higher beam inten-sity as well as faster beam switching. Atom imaged Survived imagingLost in imaging 𝑆 Prepared in |𝑔〉
Lost in transfer
Left in 𝑎 Excited to |𝑟〉
Remained in |𝑔〉 𝐾 Preparation
Excitation 𝑎 (𝑃 𝑎 ) 𝑔 (𝑃 𝑔 )𝑟 (𝑃 𝑟 ) Lost (𝑃 𝑙 ) Bright during imagingDark during imaging Detected in second image ( 𝑃 )Not detected in second image( ) 𝐹 𝐹 Measurement
Fig. 5 | Probability tree for single-atom SPAMcorrection.
Atomic states are color-coded as blue for | a (cid:105) (absolute ground state), red for | g (cid:105) (clock state), purplefor | r (cid:105) (Rydberg state), and dark-gray for lost. Quantitiesabove arrows indicate probabilities. The SPAM correctedquantity of interest, P cr , is highlighted in a purple box. C. STATE PREPARATION ANDMEASUREMENT (SPAM) CORRECTION
At the end of a Rydberg excitation and auto-ionizationsequence, we perform state readout by imaging the ab-sence (0) or presence (1) of atoms. We infer the finalstate of the atom by mapping this binary detection valueto the atomic state as 0 → | r (cid:105) and 1 → | g (cid:105) . However,imperfections in state preparation, imaging fidelity, andstate-selective readout produce errors in this mapping.State preparation and measurement (SPAM) correctionattempts to isolate quantities of the pertinent physics(in this case, Rydberg excitation) from such errors. Inparticular, we attempt to answer the following question:Assuming an atom is perfectly initialized in the groundstate | g (cid:105) , what is the probability that it is in | r (cid:105) after acertain Rydberg excitation pulse?
1. Preparation, excitation, and measurementprocesses
We begin by assuming that an atom/pair has been reg-istered as present via imaging at the start of the exper-iment and that it has no detected neighbors within atwo tweezer spacing. If an atom/pair does not fulfill thiscriterion, it is omitted from our data. For the sake ofsimplicity, we will assume that there are no errors in thisinitial detection stage. In particular, the combination ofhigh imaging fidelity and high array rearrangement fi-delity make errors of this kind exceptionally unlikely.Imaging an atom involves a small probability that theatom will be lost, even if it scatters enough photons to bedetected. We denote by S the probability that a detectedatom survives the first image. After this image, surviving atoms are transferred from the absolute ground state | a (cid:105) to the clock state | g (cid:105) (the ground state of our Rydbergqubit) with a probability of successful transfer denotedby K . There is a small probability L that during thistransfer atoms are lost. The rest, which are not lost butnot successfully transferred, remain in | a (cid:105) with a prob-ability 1 − L − K . The possibilities enumerated up tothis point are represented graphically in Extended DataFig. 5 under “Preparation”.At this point, atoms that have been successfully pre-pared in | g (cid:105) undergo Rydberg excitation. In the singleatom case, they end up in the Rydberg state | r (cid:105) witha probability P cr . For the two atom case, assuming thatboth atoms have been successfully prepared, there arefour possible states in the two-qubit space, with prob-abilities given by P crr , P crg , P cgr , P cgg . Our ultimate goalwill be to solve for these values, which we call “SPAM-corrected”, indicated here with a superscript c .In the two atom case, there is the possibility that oneatom is successfully prepared while the other is not. Inthis case, we expect the successfully prepared atom toexecute single-atom dynamics. In the case of Rabi oscil-lations, the Rabi frequency will be reduced by a factor of √
2. We can thus estimate the Rydberg excitation proba-bility of the prepared atom as P cr ∗ ≡ P cr | Ω t = π cos (Ω t/ t isthe pulse length. Of particular interest are the casesΩ t = π/ √ t = 2 π/ √
2, corresponding to the two-atom π and 2 π pulses, respectively.After excitation follows measurement, which involvesmaking Rydberg atoms dark to imaging (i.e., eitherputting them in a state that scatters no photons or ex-pelling them from the trap) and imaging the remainingbright atoms. In our case, we make Rydberg atoms darkvia auto-ionization. We denote by D the probability thata Rydberg atom is successfully made dark to imaging.Furthermore, we denote by F the probability of correctlyimaging the absence of a bright atom (true negative) andby F the probability of correctly imaging the presence ofa bright atom (true positive). 1 − F gives the probabilityof a false positive, and 1 − F gives the probability of afalse negative.Let P be the probability of an atom being detected aspresent (bright) at the end of the experiment, and simi-larly let P , P , P , P be the corresponding probabili-ties for atom pairs (with the sum of these being 1). Theseare the raw, measured values referred to as “uncorrected”in the main text and hereafter.
2. Determining SPAM probabilities
We now discuss the determination of the various prob-abilities discussed. While some of these quantities are di-rectly measurable, some must be estimated from mea-surements that themselves need SPAM correction. Allprobabilities entering into SPAM correction calculationsare summarized in Extended Data Table II.We determine F and F by analyzing the histogramof detected photons from a typical set of images, simi-larly to the method described in Ref. . The histogramshave a zero- and one-atom peak, and we determine falsepositives and false negatives by the area of these peaksthat extends beyond the binary detection threshold. Lossduring imaging that leads to false negatives is also takeninto account in F . Error bars are given by the standarddeviation across the array.We determine S by taking two consecutive images. Wemeasure the value S , defined as the probability of de-tecting an atom in the second image conditional on itsdetection in the first. Obtaining the true value of S from S requires correcting for false positives and false nega-tives in the second image (where we assume false positivesin the first image are negligible). One can write S as thesum of atoms that survived and were correctly positivelyidentified and that did not survive and were incorrectlypositively identified. Solving for S gives: S = S + F − F + F − K from a value K measured by performing state transfer, using aground-state push-out pulse as described in Sec. A, re-pumping to the ground state, and measuring the prob-ability of detecting an atom in a subsequent image. Toobtain the true K , we correct K for imaging errors aswell as survival probability after imaging. We further-more modify K with the probability R that a success-fully transferred atom goes back to | a (cid:105) due to trap Ra-man scattering in the time delay between state transferand Rydberg excitation. We estimate R = 0 . . We obtain: K = K + F − S + F − − R ) . (2)We note that the total clock state preparation fidelity,an important quantity on its own, can be expressed as Table II | SPAM probabilities.
Probability Symbol ValueImaging true negative F . F . S . S . | g (cid:105) transfer K . | a (cid:105) R . | g (cid:105) transfer K . | g (cid:105) transfer L . D . F SP = SK = 0 . L , we perform state transfer without a push-out pulse, then repump atoms to the ground state andmeasure how many were lost (again correcting for imag-ing loss and imaging errors).Finally, we determine D by comparing the measuredauto-ionization timescale to an estimate of the Rydberglifetime, as described in Sec. B. We assume all decayfrom the Rydberg state is into bright states and thereforeleads to detection errors, which is physically motivatedby the large branching ratio of our Rydberg state to the5s5p P J manifold, whose states are repumped into ourimaging cycle. We neglect other processes that may makea Rydberg atom go dark, such as anti-trapping or decayinto dark states, as these are expected to have a muchlonger timescale.
3. Correcting the single-atom excitationprobabilities
We are now ready to solve for P cr in terms of the uncor-rected value P and the various SPAM probabilities. Forclarity, it will be convenient to define variables for thepopulations of the four possible single-atom states thatan atom can be in at the end of Rydberg excitation: lost, | a (cid:105) , | g (cid:105) , and | r (cid:105) . We will call these populations p l , p a , p g and p r , respectively, with their values determined by theprobability tree in Extended Data Fig. 5 and summarizedin Extended Data Table III.We can write P as a sum of true positive identificationsof bright states plus false positive identification of darkstates (see “Measurement” in Extended Data Fig. 5). Interms of the values defined so far, we have: P = ( p a + p g + p r (1 − D )) F + ( p l + p r D )(1 − F ) . (3)Substituting in the full expression for the populationsfrom Extended Data Table III and solving for P cr , weobtain: P cr = SF + (1 − S )(1 − F ) − LS ( F + F − − P KSD ( F + F − . (4)For the single-atom short-time Rabi oscillations re-ported in Table I of the main text, we observe the bare Table III | Possible states for a single atom.
Note thatthe sum of these populations equals unity.State Symbol ValueLost p l (1 − S ) + SL | a (cid:105) (absolute ground state) p a S (1 − L − K ) | g (cid:105) (clock state) p g SK (1 − P cr ) | r (cid:105) (Rydberg state) p r SKP cr values of P ( π ) = 0 . P (2 π ) = 0 . F SPAM ( π ) = P cr ( π ) = 0 . F SPAM (2 π ) = 1 − P cr (2 π ) =0 .
4. Correcting the two-atom excitation probabilities
For the two-atom case, there are 16 possible states foran atom pair. Similarly to Extended Data Table III, wecan write populations of each of these states in termsof the survival and transfer fidelities in Extended DataTable II, as shown in Extended Data Table IV.We now write the experimentally measured quantities P , P , and P in terms of the values in Extended DataTables II and IV. For notational simplicity we define ¯ F ≡ (1 − F ), and similarly for F and D : P = p ll ( ¯ F F )+ p la ( ¯ F ¯ F )+ p al ( F F )+ p lg ( ¯ F ¯ F )+ p gl ( F F )+ p lr ( ¯ F F D + ¯ F ¯ D ¯ F )+ p rl ( ¯ F DF + F ¯ DF )+ p aa ( F ¯ F )+ p ag ( F ¯ F )+ p ga ( F ¯ F )+ p ar ( F DF + F ¯ D ¯ F )+ p ra ( F ¯ D ¯ F + ¯ F D ¯ F )+ p gg ( F ¯ F )+ p gr ( F DF + F ¯ D ¯ F )+ p rg ( F ¯ D ¯ F + ¯ F D ¯ F )+ p rr ( F ¯ DF D + ¯ F D ¯ F ¯ D + ¯ F F D + F ¯ F ¯ D ) , (5) Table IV | Possible states for two atoms.
Note thatthe sum of these populations equals unity. Terms inside {} have an implied symmetric partner, e.g. p al ≡ p la .States Symbol ValueLost, Lost p ll ((1 − S ) + SL ) { Lost, | a (cid:105)} p la ((1 − S ) + SL ) S (1 − L − K ) { Lost, | g (cid:105)} p lg ((1 − S ) + SL ) SK (1 − P cr ∗ ) { Lost, | r (cid:105)} p lr ((1 − S ) + SL ) SKP cr ∗ | aa (cid:105) p aa S (1 − L − K ) {| ag (cid:105)} p ag S (1 − L − K ) SK (1 − P cr ∗ ) {| ar (cid:105)} p ar S (1 − L − K ) SKP cr ∗ | gg (cid:105) p gg S K (1 − P crg − P cgr − P crr ) | gr (cid:105) p gr S K P cgr | rg (cid:105) p rg S K P crg | rr (cid:105) p rr S K P crr P = p ll ( F )+ p la ( F ¯ F )+ p al ( ¯ F F )+ p lg ( F ¯ F )+ p gl ( ¯ F F )+ p lr ( F D + F ¯ F ¯ D )+ p rl ( F D + ¯ F ¯ DF )+ p aa ( ¯ F )+ p ag ( ¯ F )+ p ga ( ¯ F )+ p ar ( ¯ F F D + ¯ F ¯ D )+ p ra ( ¯ F ¯ D + F D ¯ F )+ p gg ( ¯ F )+ p gr ( ¯ F F D + ¯ F ¯ D )+ p rg ( ¯ F ¯ D + F D ¯ F )+ p rr ( ¯ F ¯ D + F D ¯ F ¯ D + F D + ¯ F ¯ DF D ) , (6) P = p ll ( ¯ F )+ p la ( ¯ F F )+ p al ( F ¯ F )+ p lg ( ¯ F F )+ p gl ( F ¯ F )+ p lr ( ¯ F D + ¯ F F ¯ D )+ p rl ( ¯ F D + F ¯ D ¯ F )+ p aa ( F )+ p ag ( F )+ p ga ( F )+ p ar ( F ¯ F D + F ¯ D )+ p ra ( F ¯ D + ¯ F DF )+ p gg ( F )+ p gr ( F ¯ F D + F ¯ D )+ p rg ( F ¯ D + ¯ F DF )+ p rr ( F ¯ D + ¯ F DF ¯ D + ¯ F D ¯ D + F ¯ D ¯ F D ) . (7)Note that P = 1 − P − P − P . Thus, with the threeabove equations, we can solve for P cgg , P crg , P cgr , and P crr .The full expressions for these solutions are cumbersomeand not shown. The experimentally measured values P , P , P and P are reported in Extended Data Table V.0 D. BELL STATE FIDELITY1. Bounding the Bell state fidelity
Characterizing the state of a quantum system is of fun-damental importance in quantum information science.Canonical tomographic methods addressing this task re-quire a measurement of a complete basis set of opera-tors. Such measurements are often expensive or not ac-cessible. More economic approaches can be employed toassess the overlap with a given target state. For exam-ple the overlap of a two-qubit state with a Bell stateis routinely determined by measuring the populations inthe four computational basis states (yielding the diago-nal elements of the density operator), in addition with ameasurement that probes off-diagonal elements via parityoscillations . To access the latter it is however neces-sary to perform individual, local operations on the qubits.Here, we present a bound on the Bell state fidelity thatcan be accessed with only global control and measure-ments in the computational basis and elaborate on theunderlying assumptions.Specifically, we are interested in the overlap F of theexperimentally created state ρ with a Bell state of theform | W φ (cid:105) = √ ( | gr (cid:105) + e iφ | rg (cid:105) ). This is defined as F = max φ (cid:104) W φ | ρ | W φ (cid:105) = 12 (cid:0) ρ gr,gr + ρ rg,rg + 2 | ρ gr,rg | (cid:1) . (8)Here we denote matrix elements of a density opera-tor ρ in the two-atom basis by ρ i,j = (cid:104) i | ρ | j (cid:105) , with i, j ∈ { gg, gr, rg, rr } . Clearly, a measurement of F re-quires access to the populations in the ground and Ry-dberg states ρ i,i as well as some of the coherences ρ i,j Table V | Experimentally measured two-atom values.
Uncorrected values used to calculate P cgg , P crg , P cgr , and P crr at both the π - and 2 π -times. The ‘T’ superscript indicatesthe values for which the traps were on. We report the valuesof P and P in symmetrized and antisymmetrized form,where P { } = P + P and P [10] = P − P Variable Value P { } ( π ) 0.992(2) P [10] ( π ) 0.01(1) P ( π ) 0.0032(7) P (2 π ) 0.989(2) P [10] (2 π ) 0.004(2) P (2 π ) 0.0036(7) P T { } ( π ) 0.992(2) P T [10] ( π ) 0.004(10) P T ( π ) 0.0032(7) P T (2 π ) 0.985(2) P T [10] (2 π ) -0.003(2) P T (2 π ) 0.0030(6) with i (cid:54) = j . While populations are direct observables (inparticular, we identify ρ i,i with our measured values P i or their SPAM corrected counterparts P ci ), coherencesare not. We can however bound the fidelity F from be-low via a bound on | ρ gr,rg | . Namely, it can be shown viaCauchy’s inequality | ρ a,b | ≤ ρ a,a ρ b,b and the normaliza-tion of states (cid:80) i ρ i,i = 1 that | ρ gr,rg | ≥
12 (tr (cid:8) ρ (cid:9) −
1) + ρ gr,gr ρ rg,rg (9)where tr (cid:8) ρ (cid:9) = (cid:80) i,j | ρ i,j | is the purity.Evaluating the bound given by equation Eq. (9) re-quires access to the purity (or a lower bound thereof).One can bound the purity from below by the populationsin the ground and Rydberg states astr (cid:8) ρ (cid:9) ≥ (cid:88) i ( ρ i,i ) . (10)In general Eq. (10) is a very weak bound. In particular, itdoes not distinguish between a pure Bell state | ψ φ (cid:105) andthe incoherent mixture of the two states | gr (cid:105) and | rg (cid:105) .However, if the state ρ is close to one of the four atomicbasis states (as is the case at the 2 π time of the Rabievolution), the bound Eq. (10) becomes tight. This factallows us to obtain a lower bound for the purity of theBell state in the experiment as follows. The Bell statein our protocol is generated by evolving the state | gg (cid:105) for a time T = π/ ˜Ω R in the Rydberg-blockade regime.Note that the same evolution should lead to a return tothe initial state at time 2 T in the ideal case. Under the assumption that a coupling to the environment decreasesthe purity of the quantum system (see further explorationof this assumption in the following subsection), we canbound the purity of the state at time T by the purity ofthe state at time 2 T , which in turn can be bounded bymeasurements of the atomic populations at time 2 T viaEq. (10):tr (cid:8) ρ ( T ) (cid:9) ≥ tr (cid:8) ρ (2 T ) (cid:9) ≥ (cid:88) i ρ i,i (2 T ) . (11)Using this estimated bound on the purity leads to a lowerbound on the Bell state fidelity F at time T solely interms of the populations in the ground and Rydbergstates at times T and 2 T : F ( T ) ≥ (cid:18) ρ gr,gr ( T ) + ρ rg,rg ( T )+2 (cid:113) max (cid:0) , ( (cid:80) i ρ i,i (2 T ) − / ρ gr,gr ( T ) ρ rg,rg ( T ) (cid:1)(cid:19) . (12)
2. Bounding an increase in purity due tospontaneous decay
Although we make the assumption that the purity ofour state does not increase between times T and 2 T ρ = L ρ = − i [ H, ρ ]+ (cid:88) µ γ µ ( c µ ρc † µ − { c † µ c µ , ρ } )+ (cid:88) µ ¯ γ µ ( h µ ρh µ − { h µ h µ , ρ } )(13)Here we explicitly distinguish incoherent terms generatedby Hermitian jump operators ( h µ = h † µ , e.g. dephasing),and non-Hermitian jump operators ( c µ , e.g. spontaneousemission). We find ddt tr (cid:8) ρ (cid:9) = 2tr { ρ ( L ρ ) } ≤ (cid:88) µ γ µ tr (cid:8) ρc µ ρc † µ − c † µ c µ ρ (cid:9) (14)which simply reflects the fact that the purity of thequantum state can not increase due to the coher-ent part of the evolution or due to any incoherentpart of the evolution that is generated by Hermitianjump operators (dephasing). Thus the coherent partof the evolution does not affect the bound we obtainin the end. Eq. (14) can be obtained from Eq. (13)by noting that tr { ρ [ H, ρ ]) } = tr (cid:8) ρHρ − ρ H (cid:9) =0 and tr { ρ [ h µ , [ ρ, h µ ]] } = − tr { [ h µ , ρ ][ ρ, h µ ] } = − tr (cid:8) ([ ρ, h µ ]) † [ ρ, h µ ] (cid:9) ≤
0, which gives Eq. (14).Now let us assume that the non-Hermitian jump op-erators correspond to decay from the Rydberg state | r (cid:105) into some set of states {| f (cid:105)| f = 1 , , . . . n } that also in-clude the ground state | g (cid:105) ≡ | (cid:105) . The following argumentworks for arbitrary n ≥
1. Since we have two atoms wehave 2 n non-Hermitian jump operators c ( a ) f = | f (cid:105) a (cid:104) r | ,where a = 1 , a by ρ ( a ) ): ddt tr (cid:8) ρ (cid:9) ≤ (cid:88) f,a Γ f tr (cid:26) ρc ( a ) f ρc ( a ) f † − c ( a ) f † c ( a ) f ρ (cid:27) = 2 (cid:88) f,a Γ f ( ρ ( a ) f,f ρ ( a ) r,r − ρ ( a ) r,r ρ ( a ) r,r − (cid:88) e (cid:54) = r ρ ( a ) r,e ρ ( a ) e,r ) ≤ (cid:88) f,a Γ f ( ρ ( a ) f,f ρ ( a ) r,r − ρ ( a ) r,r ρ ( a ) r,r ) (15)where Γ f is the single-atom decay rate from | r (cid:105) to | f (cid:105) .Note that ρ ( a ) f,f ρ ( a ) r,r − ρ ( a ) r,r ρ ( a ) r,r ≤ (1 − ρ ( a ) r,r ) ρ ( a ) r,r − ρ ( a ) r,r ρ ( a ) r,r ≤ /
8. This gives the final result ddt tr (cid:8) ρ (cid:9) ≤ (cid:88) f Γ f = 12 Γ (16)That is, the rate at which the purity increases is upperbounded by half the rate at which a single atom in theRydberg state decays into other states by spontaneousemission. Over a time interval of length T the 2-atompurity can thus not increase by more than T Γ / π -time for T and Rydberg statedecay rate for Γ, we evaluate this bound on the purityincrease to be 3 . × − . This would lead to a decreasein our bound on the Bell state fidelity by 1 . × − forboth the cases of tweezers off and tweezers on, whichis significantly smaller than our quoted error for thesevalues. E. RYDBERG LASER SYSTEM
The Rydberg laser system is based on a Toptica laser,in which an extended cavity diode laser (ECDL) at λ IR = 1266 . ≈ ≈ λ Red = 633 . ≈ . λ UV = 316 . λ IR = 1266 . ≈ ≈
110 kHz. The finesse was measuredby performing cavity ringdown spectroscopy . We cur-rently do not filter the fundamental laser with the cav-ity , but we are prepared to implement this approach.Further discussion on the laser frequency stability canbe found in Appendix F.We use a beam power of P R = 28 . /e waist radius of the beam at the position of the atoms is w R = 18(1) µ m. These conditions correspond to the Rabifrequency used throughout the text of Ω R ≈ π × − ≈
110 mW,for which we observe a Rabi frequency of ≈ π × ≈
40 ns. Whendriving with a Rabi frequency whose π -pulse approachesthis timescale (Ω R ≈ π ×
13 MHz), we observe an asym-metric reduction in Rabi signal contrast by ≈ − π , unlike conventional detuned Rabioscillations where the contrast reduction occurs at theodd multiples of π . We attribute to early-time dynam-ics during the AOM switching. We do not use an opticalfiber, so there is limited spatial – and thus spectral –filtering between the AOM and the atoms. Accordingly,we intentionally work with Ω R ≈ π × − π -pulse time is sufficiently slow compared to theAOM rise and fall times. However, when operating atΩ R ≈ π ×
13 MHz we observe long-time coherence sim-ilar to, or slightly better than, the reported values in themain text for Ω R ≈ π × − F. RYDBERG DECOHERENCE MECHANISMS
For a non-interacting case where Rydberg atoms in atweezer array are well separated, the Hamiltonian H driv-ing Rabi oscillations is H = N (cid:88) i =1 Ω R,i S xi + ∆ i S zi , (17)where Ω R,i and ∆ i are the Rabi frequency and the detun-ing for the atom at site i , S µ are the spin-1/2 operatorswith µ = x, y, z , and N is the total number of atoms.Variations in the Rabi frequency and detuning, mani-festing either as non-uniformity across the tweezer array(e.g. from non-uniform beam alignment) or as randomnoise, lead to a decay in the array-averaged Rabi signal.In our system, we measure a 1 /e decay time of ≈ µs at a Rabi frequency of 6 MHz (see Fig. 3a of the maintext). In this section, we present a model of decoherencemechanisms that accounts for our observed decay.As a preliminary, we begin by confirming that the spa-tial variation of Rabi frequency across different tweezersis less than 0.2%, and that no variation of detuning acrossthe array is observed. We conclude that non-uniformity isnot a dominant contributor to our observed Rabi decay.Therefore, we focus here on three factors that inducerandom noise in the Rabi frequency and detuning: atomicmotion, laser phase noise, and laser intensity noise. Weperform Monte Carlo-based simulations that take intoaccount these noise sources as well as the finite lifetime ≈ µ s of the n = 61 Rydberg state due to spontaneousemission. In the following subsections, we discuss relativecontributions from these noise sources. Time (µs) P r o b a b ili t y ( P ) Fig. 6 | Ramsey interferometry.
We use a detuning of 2MHz between the two pulses to show oscillations with acharacteristic 1 /e decay time τ Ramsey ≈ µ s. Asine-modulated Gaussian decay is used for the fit (solid line).Data is uncorrected and averaged over ≈
40 experimentalcycles per timestep and over an array of approximately 14atoms. Error bars indicate a 1 σ binomial confidence interval.
1. Atomic motion
An atom with a nonzero momentum shows a Dopplershift relative to the bare resonance frequency. At the be-ginning of Rabi interrogation, the momentum distribu-tion, and thus the distribution of Doppler shifts, followsthat of an atom in a trap. More specifically, for an atomat temperature T trapped in a harmonic potential withthe radial trap frequency ω r , the Doppler shift distribu-tion can be modeled as a normal distribution with thestandard deviation ∆ T :∆ T = k L m (cid:115) (cid:126) mω r (cid:126) ω r / k B T ) , (18)where m is the mass of Sr and k L is the wavevector ofthe Rydberg excitation light.The radial temperature of our atomic array (along theaxis of propagation of the Rydberg beam) is measuredvia sideband spectroscopy on the clock transition tobe T r ≈ . µ K at a radial trap frequency of ω r ≈ π ×
78 kHz. We adiabatically ramp down the trap by afactor of 10 before Rydberg interrogation, thereby reduc-ing the temperature and the trap frequency by a factorof √
10 (which we also confirm via further sideband spec-troscopy). Using Eq. 18, we estimate the Doppler broad-ening to be ∆ T ≈ π ×
30 kHz. At a Rabi frequency ofΩ R ≈ π × τ ∼ Ω R / ∆ T ≈
10 ms, which is three orders ofmagnitudes longer than the measured value ≈ µ s. Thisimplies that motional effects are negligible in the Rabidecoherence dynamics.
2. Laser phase noise
Phase noise manifests as random temporal fluctuationof the detuning ∆ in the Hamiltonian in Eq. 17. Since thefrequency of the Rydberg laser is stabilized to a ULE ref-erence cavity via the Pound-Drever-Hall (PDH) method,3we use an in-loop PDH error signal derived from thecavity reflection to extract a phase noise spectrum (seeRef. for the detailed procedures of phase noise extrac-tion). The obtained noise power spectral density, pre-dicting a RMS frequency deviation of ≈ ∼ −
10 kHz, phase noise from the servo bumps centeredat ν SB ≈ . R > ν SB , andin fact dominates the RMS.Since the cavity filters phase noise beyond its linewidth,this noise is suppressed on the measured PDH signal ascompared to the actual noise of the laser light that we usefor Rydberg interrogation. We therefore correct our mea-sured phase noise spectrum with a cavity roll-off factor obtained from the cavity linewidth and finesse, whichresults in an increase in noise as compared to the un-corrected measured spectrum. However, we can also usethe uncorrected spectrum to predict the phase noise wewould have if we used the filtered cavity light to generateour Rydberg light via a technique described in Ref. . Theresults in Extended Data Fig. 7 show simulated resultsboth with and without cavity filtering.Our simulations (without cavity filtering, as in our cur-rent setup) predict a Ramsey decay time of ≈ µ s witha Gaussian envelope, which is consistent with our ex-perimental observation. In principle, Doppler broadening∆ T could also lead to dephasing in Ramsey signals; how-ever, the corresponding 1 /e decay time is expected to be τ Ramsey = √ / ∆ T = 7 . µs , longer than the observed2 µ s, suggesting that laser phase noise is dominant overmotional effects in our Ramsey signal.
3. Laser intensity noise
Our intensity noise predominantly originates directlyfrom the Rydberg laser. This intensity noise is com-posed of both high-frequency fluctuations compared tothe pulse length, and lower frequency (effectively shot-to-shot) fluctuations. Using a UV avalanche photodetec-tor (APD130A2, Thorlabs), we measure that the inten-sity pulse areas between different experimental trials arenormally distributed with fractional standard deviation σ RMS ∼ / √ L , where L is the pulse duration, saturatingto 0 .
8% when
L > µ s. Note that the pulses are too fastto stabilize with an AOM during interrogation, and thatwe employ a sample-and-hold method.In the presence of only intensity noise following anormal distribution with fractional standard deviation σ RMS , one can closely approximate the noise in the Rabifrequency to also be normally distributed and derivean analytical expression for a 1 /e Rabi decay time as τ Rabi = 2 √ / (Ω R σ RMS ) where Ω R is the nominal, noise-free Rabi frequency. In the intensity noise limited regime,we thus expect a Rabi lifetime N Rabi (in oscillation cy-cles) to be Rabi frequency-independent (see the line in
Rabi Frequency (MHz) C o h e r e n t C y c l e s Fig. 7 | Simulated and measured /e coherence vsRabi frequency. The star represents the measured datashown in Fig. 3a, and the circle and square points representnumerical modeling with measured laser phase and intensitynoise profiles. The yellow circles show the case when cavityphase noise filtering is not performed (as in this work), andthe green squares show the case where cavity phase noisefiltering is performed. The horizontal gray line shows theupper limit due to measured intensity noise fluctuationswith RMS deviation of 0 .
8% (see Eq. (19)). Error barsindicate a 1 σ confidence interval. Extended Data Fig. 7): N Rabi = Ω R τ Rabi π = √ πσ RMS . (19)
4. Summary
Including all the discussed noise sources (atomic mo-tion, phase noise, intensity noise) as well the finite statelifetime and a Rydberg probe-induced light shift (dis-cussed in a subsequent section), we calculate N Rabi as afunction of drive frequency, as shown in Extended DataFig. 7. We find that the simulated Rabi oscillation agreeswell with the experimental result at a Rabi frequency of6 MHz. While the Rabi lifetime improves with increasingRabi frequency, it becomes saturated to N Rabi ≈
56 athigh Rabi frequencies due to intensity noise fluctuations.Interestingly, we note that there is a crossover betweena phase noise-limited regime at low Rabi frequencies andan intensity noise-limited regime at higher Rabi frequen-cies, which for our phase and intensity noise profiles oc-curs at Ω R ≈ π × can enhance the long-timeRabi coherence. G. RYDBERG STATE SYSTEMATICS1. State identification and quantum defects
The Rydberg state | r (cid:105) we use for this work is the5s61s S m J = 0 state of Sr. To confirm the quan-tum numbers, we measure the transition wavelengths of n = 48 , , ,
61 for the S series and of n = 47 , , U/U ( M H z ) R2 (MHz ) ( k H z ) a b Fig. 8 | Light shifts of | r (cid:105) from the Rydberg laserand the tweezer light. a , The differential shift of the | g (cid:105) ↔ | r (cid:105) resonance between Ω init R = 2 π × R versus Ω R . This set of data was measured withthe two-rail self-comparison technique utilized in Ref. . Thefit line reflects the quadratic scaling ∆ ν = κ UV | r (cid:105) Ω R , with κ UV | r (cid:105) = 5 . . b , The differential shift of the | g (cid:105) ↔ | r (cid:105) resonance between the dark case U = 0 where thetweezers are extinguished during excitation, and the brightcase with variable | g (cid:105) -state depth U up to U ≈ k B × µ K ≈ h × . ν = κ T | r (cid:105) U, where κ T | r (cid:105) = 18 . . Error barsindicate a 1 σ standard error of the mean. for the D series and find nearly perfect agreement withthe values predicted by the quantum defects given inRef. .
2. Rydberg probe-induced light shift
The pulse generation for our Rydberg interrogation isfacilitated by switching on and off an acousto-optic mod-ulator (AOM). However, due to the finite speed of soundin the AOM crystal, the switch-on and switch-off timesare limited to tens of nanoseconds. This timescale beginsto approach the timescale of our π -pulses for Rabi fre-quencies greater than ≈
10 MHz. This poses a potentialproblem if there is also a significant intensity-dependentlight shift of the resonance frequency due to the Ryd-berg interrogation beam. For example, a detuning thatchanges significantly on the timescale of the Rabi fre-quency could lead to non-trivial dynamics on the Blochsphere, causing unfaithful execution of Rabi oscillations.We note that such an effect scales unfavorably with in-creasing Rabi frequency, as both the relevant timescalebecomes shorter and the magnitude of the shift becomesquadratically larger. To measure this effect, we operate at Rabi frequenciessmaller than 6 MHz to isolate the pure Rydberg probe-induced light shift from any undesired AOM-related tran-sient effects. Using the two-rail self-comparison techniquedescribed in Ref. , we measure the light shift inducedby the Rydberg beam and find it to be described by∆ ν = κ UV | r (cid:105) Ω R with κ UV | r (cid:105) = 5 . , as shownin Fig. 8a.
3. Tweezer-induced light shift
We have demonstrated high-fidelity blockaded Rabi os-cillations without extinguishing the tweezer traps. Togain a partial understanding of this observation, we mea-sure the light shift of | r (cid:105) in the tweezers with wavelength λ T = 813 . w T ≈
800 nm. We measurethe differential shift of the | g (cid:105) ↔ | r (cid:105) resonance betweenthe dark case U = 0 where the tweezers are extinguishedduring excitation, and the bright case with variable | g (cid:105) -state depth U up to U ≈ µ K ≈ h × . ν = κ T | r (cid:105) U, where κ T | r (cid:105) = 18 . . We conclude that κ T | r (cid:105) ≈ − κ T | g (cid:105) at this tweezer wavelength and waist. We leave the de-tailed modeling of the polarizability to future work.
4. Diamagnetic shift from magnetic fields
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