Algorithmic Ground-state Cooling of Weakly-Coupled Oscillators using Quantum Logic
Steven A. King, Lukas J. Spie?, Peter Micke, Alexander Wilzewski, Tobias Leopold, José R. Crespo López-Urrutia, Piet O. Schmidt
AAlgorithmic Ground-state Cooling of Weakly-Coupled Oscillators using QuantumLogic
Steven A. King, ∗ Lukas J. Spieß, Peter Micke,
1, 2
Alexander Wilzewski, Tobias Leopold, Jos´e R. Crespo L´opez-Urrutia, and Piet O. Schmidt
1, 3 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, Welfengarten 1, 30167 Hannover, Germany (Dated: February 26, 2021)Most ions lack the fast, cycling transitions that are necessary for direct laser cooling. In mostcases, they can still be cooled sympathetically through their Coulomb interaction with a second,coolable ion species confined in the same potential. If the charge-to-mass ratios of the two ion typesare too mismatched, the cooling of certain motional degrees of freedom becomes difficult. This limitsboth the achievable fidelity of quantum gates and the spectroscopic accuracy. Here we introduce anovel algorithmic cooling protocol for transferring phonons from poorly- to efficiently-cooled modes.We demonstrate it experimentally by simultaneously bringing two motional modes of a Be + -Ar mixed Coulomb crystal close to their zero-point energies, despite the weak coupling between theions. We reach the lowest temperature reported for a highly charged ion, with a residual temperatureof only T . µ K in each of the two modes, corresponding to a residual mean motional phononnumber of h n i . .
4. Combined with the lowest observed electric field noise in a radiofrequency iontrap, these values enable an optical clock based on a highly charged ion with fractional systematicuncertainty below the 10 − level. Our scheme is also applicable to (anti-)protons, molecular ions,macroscopic charged particles, and other highly charged ion species, enabling reliable preparationof their motional quantum ground states in traps. I. INTRODUCTION
Laser cooling has ushered in a new era of spectroscopicprecision and accuracy. Atoms and ions can be broughtpractically to rest, greatly suppressing Doppler broad-ening and shifts. The required strong, cycling electronictransitions in the laser-accessible range only exist in a fewspecies, however. This limitation can be overcome usingso-called sympathetic cooling, whereby a second ion, re-ferred to as the ‘cooling ion’ (or ‘logic ion’, dependingon the application), is confined in the same trap togetherwith the ion of interest, which we will refer to as the ‘spec-troscopy ion’. Their mutual Coulomb interaction leads tocoupled motion of the ions, which can be damped usingthe cooling ion [1]. Applications to-date include quantuminformation processing [2, 3], ultra-high-accuracy opticalatomic clocks [4], molecular ions [5–9], multiply chargedions [10], highly charged ions (HCI) [11, 12], and heliumions [13].The motional coupling between the ions depends onhow well the charge-to-mass ratios of the two speciesmatch. At large mismatches, the ions can move almostindependently in the trap. This reduces the efficiency ofsympathetic cooling in modes of motion where the cool-ing ion is almost stationary [14–16], thereby lengtheningthe cooling time, and raising the equilibrium tempera-ture where heating mechanisms are present. Further-more, the control and manipulation of these modes viathe cooling ion (e.g., for sideband cooling or thermome-try) is challenging. As such, these modes were expected ∗ Contact: [email protected] to pose limitations to quantum protocols [16] and achiev-able spectroscopic accuracy [17].Several attempts were made to address this problemby coupling the weakly-cooled modes to others that aremore effectively cooled. For example, in a linear radiofre-quency (rf) Paul trap, a tilt was applied to a two-ioncrystal previously aligned along the trap symmetry axisby means of a transverse electric field [18]. An alter-native approach was to mix the modes using additionalradiofrequency fields [19]. These techniques work well ifthe coupling is not too weak, but below a certain limitthe induced mixing still cannot afford effective cooling.In this work, we demonstrate how to remove phononsfrom weakly coupled modes using algorithmic cooling.It consists of a quantum protocol where coherent opera-tions transfer entropy (or heat) from one part of a systemto another, from which it can be removed by coupling,e. g., to a bath [20, 21]. Originally, algorithmic cool-ing was developed to improve the polarization of a solidstate spin sample without cooling the environment in nu-clear magnetic resonance experiments [22]. Later, it wasutilized to remove entropy from a quantum gas in anoptical lattice [23, 24]; demon-like algorithmic quantumcooling was realized in a photonic quantum optical net-work [25], and a partner-pairing algorithm was proposedto cool the motion of a single-species trapped ion quan-tum computer [26]. Here, we demonstrate the techniqueby cooling weakly-coupled motional modes of a trapped,sympathetically-cooled highly charged ion. By using theinternal spin of the HCI, excitation of the weakly-coupledmode is coherently mapped onto a different mode thatcan be cooled efficiently by the cooling ion. The tech-nique we present is widely applicable and could be used to a r X i v : . [ phy s i c s . a t o m - ph ] F e b D opp l e r c oo li ng & R epu m p i ng bea m s HC I a x i a l bea m P au l t r ap z - a x i s R a m an bea m Q uan t i z a t i on a x i s R a m an bea m H C I r a d i a l b e a m x y r f gnd r f gnd x y Figure 1. Simplified depiction (not to scale) of the two-ion Coulomb crystal of a single Be + ion (red, left) and a single HCI(purple, right) confined in a linear Paul trap. The laser beams needed for their manipulation are shown. The ions are locatedon the trap axis ( z ); radial motion evolves on the ( x , y ) coordinates defined by the trap electrodes which are at 45 ◦ to theplane on which all the laser beams lie. The Doppler cooling, repumper, and Raman laser beams all intersect the z -axis at anangle of 30 ◦ . The beam for addressing axial motion of the HCI is delivered along the z -axis, and the beam for addressing itsradial motion is delivered perpendicular to it. Inset: radial cross section of the ion trap, showing the orientation of the radialtrap axes as defined by the radiofrequency (rf) and ground (gnd) electrodes. improve the performance of many sympathetically-cooledsystems with mismatched charge-to-mass ratios betweenthe particles. II. EXPERIMENTAL SETUP
Detailed descriptions of our setup can be found in ref-erences [12, 27–30]. In short: a two-ion crystal composedof a single Be + ion and a single Ar ion is confinedin a cryogenic linear rf Paul trap, which is driven at afrequency of Ω = 2 π × . ◦ to the horizontal plane in a four-fold symmetric pattern.We define the z direction to be along the axial directionof the trap, with the two radial directions x and y ly-ing perpendicular to it and along the axes of the bladeelectrodes, as illustrated in Fig. 1.Choosing the axial confinement to be weaker than theradial confinement and careful compensation of stray dcelectric fields forces the ions to arrange themselves alongthe z axis, where the rf field has a node. The ions canoscillate along the three axes either in-phase with oneanother as part of center-of-mass motion, or out-of-phasein so-called ‘stretching’ or ‘rocking’ modes, leading to sixnormal modes of motion in total [2, 14, 31]. Under our typical trapping conditions, the eigenfrequencies ω/ π ofthese modes lie in the range of 1 − + having the highest charge-to-mass ratio ofany singly-charged ion with a suitable laser-cooling tran-sition, its mismatch to Ar is so great that the ampli-tudes of motion for the Be + ion in the radial in-phasemodes are two orders of magnitude smaller than in theother four modes [17, 30]. This slows cooling since thelasers addressing the Be + ion cannot efficiently removeenergy from these weakly-coupled radial (WCR) modes.Furthermore, operations on the WCR modes by resolvedsideband techniques suffer from the extremely weak cou-pling strength on the Be + ion. The decoupling is muchless severe in the axial direction of the trap, where theconfinement is provided by a static dc potential. Hence,this direction is generally preferred for quantum logic op-erations because they can be performed using either ion.The laser beams for manipulation of the two-ion crys-tal are shown in Fig. 1. They are delivered in the hor-izontal plane, and therefore each beam projects equallyonto the x and y axes of the trap. We use a 441 nmlaser for performing operations on the Ar by drivingthe P / → P / fine-structure transition. The lasercan enter from two possible directions in order to coupleto either radial or axial motion of the ion crystal as re-quired. The Doppler cooling and repumping lasers have 𝜔 𝜔 𝜔 ↑ ↓↓ (a) (b) (c) (d) (e) r z HCI Be } } } + r Be + ↓ HCI ↓ Be + ↓ HCI r zz z r z Figure 2. Upper: Scheme of the quantum logic sideband cooling sequence representing the state of the ion crystal before eachof the depicted laser pulses is applied. Pulses (a) and (b) on the HCI (purple symbols) swap a phonon from one of the WCRmodes ( r ) into the axial out-of-phase mode ( z ) through the excited electronic state of the HCI. Pulse (c) on the Be + ion (redsymbols) removes the added phonon from the axial out-of-phase mode and converts it into an electronic excitation of the Be + ion. Pulse (d) dissipatively resets the Be + to its initial electronic state through spontaneous decay, completing the sequenceand ensuring unidirectionality and cooling. Solid arrows depict driven transitions, undulating arrows represent spontaneousdecay. |↓i and |↑i represent the ground and excited electronic states of the relevant ion, respectively. Lower: Representation ofthe sequence as a quantum circuit composed of a series of SWAP quantum gates. | i i and | i i represent the presence and lackof a phonon in mode i , respectively. wavelengths near 313 nm for driving the S / → P / and S / → P / transitions in the Be + ion, respec-tively. They intersect the z axis at an angle of 30 ◦ , andare precisely aligned with the quantization axis, definedby an applied magnetic field with a flux density of ap-proximately 20 µ T (200 mG).Operations on the axial motional modes using the Be + ion employ stimulated Raman transitions between the F = 2 and F = 1 hyperfine sublevels of the S / state.We drive them with two beams derived from a third313 nm laser; each beam intersects the trap z axis atan angle of 30 ◦ , with the effective wavevector of the twobeams lying along the z -axis. III. EXPERIMENTAL SEQUENCE
The main experimental sequence runs as follows. Wefirst apply 200 ms of Doppler cooling to ensure effectivecooling of the WCR modes (see section IV). The two ax-ial modes are then cooled close to their ground states us-ing stimulated Raman transitions driven on the Be + ion.After that, we optically pump the HCI into the desiredground state using quantum-logic-assisted state prepara-tion [12]. The WCR modes are then optionally cooledusing the quantum algorithm shown in Fig. 2, which re-lies on driving resolved sidebands of the transitions in each of the ions that are either lower in energy than thecarrier (red-sideband transition, RSB) or higher in en-ergy (blue-sideband transition, BSB). This is achieved bydetuning the involved laser away from the carrier in theappropriate direction by the respective motional modefrequency. (a) The excitation of the WCR mode is co-herently mapped onto the electronic state of the HCI bydriving the P / → P / magnetic-dipole transition inthe Ar ion using a laser beam with projection purelyonto the radial trap axes. The laser is tuned to the RSBof the WCR mode to be cooled, and results in the re-moval of one phonon from this mode if the excitationis successful. (b) A second pulse from the same lasermaps the electronic excitation of the Ar ion onto thestrongly-coupled axial out-of-phase motional mode. Itis implemented by applying a beam with a projectionpurely onto the axial direction, with the laser frequencytuned to the RSB of the axial out-of-phase motion. Thisadds a phonon to the axial out-of-phase mode only ifpulse (a) had removed a phonon from the WCR mode.(c) This axial motional excitation is then mapped ontoan electronic excitation in the Be + ion by driving theappropriate RSB of the stimulated Raman transition onthe Be + ion. (d) Dissipation and irreversibility as re-quired for cooling is provided by resetting the Be + ionto its initial electronic state through spontaneous decayfrom the P / level after excitation by the repumping ( [ F L W D W L R Q S U R E D E L O L W \ ' H W X Q L Q J I U R P F D U U L H U 0 + ] x 5 6 % y 5 6 % y % 6 % x % 6 % Figure 3. WCR motional sidebands of the two-ion crystalobserved on the HCI before (black) and after (red) resolvedsideband cooling. The error bars are statistical only, governedby quantum projection noise. The observed difference in thefull-width-at-half-maximum values between the two cases iscaused by the different lengths of the interrogation pulse used,which were adjusted in each case to produce the highest pos-sible contrast. The scans have been re-centered for presenta-tion, correcting for small drifts in the secular frequencies. laser, completing the cycle. We then repeat steps (a)-(d)for the second WCR mode, and this whole sequence sev-eral times until (e) a temperature close to the motionalground state is reached. In practice, we cycle throughsteps (c) and (d) several times to neutralize the recoilof the ion crystal due to spontaneous photon scattering[8, 28, 32, 33], and a short optical pumping step for theHCI is implemented before each cycle to ensure that thecorrect electronic state has been prepared beforehand.An advantage of the weak coupling is that, unlike theother four modes of the crystal, the WCR modes suffervery little heating from spontaneous decay of the Be + ion, as we elaborate upon in section IV.The first two pulses (a) and (b) on the Ar ion couldalso in principle be joined to implement a direct SWAPgate between motional excitation of the WCR mode andthe axial mode by employing a Raman coupling. Thiswould be advantageous in systems with excited electronicstate lifetimes shorter than the gate time, or where pop-ulation of the excited state needs to be avoided.Fig. 3 shows a sideband spectrum measured on the HCIafter eight cooling pulses on each WCR mode, shown to-gether with the Doppler-cooled case for comparison. Thefinal state of the HCI after the interrogation pulse is de-termined using quantum logic [4], which resembles a sim-plified version of the algorithmic cooling sequence usingonly pulses (b) and (c) to map the electronic excitation ofthe HCI onto the electronic state of the Be + , followed bydetection of the electronic state of the Be + using state-dependent fluorescence [12]. The mean phonon occupa-tion number h n i can be obtained from the asymmetrybetween the peak RSB and BSB excitation probabilities[34], with a small correction applied owing to the finitebackground signal caused by imperfect ground-state cool- ing of the axial mode used for the quantum logic opera-tion. The slight asymmetry after Doppler cooling impliesmean occupation numbers of 3.1(13) and 3.3(13) phononsin the x and y modes respectively, where the number inbrackets represents the 1 σ uncertainty on the last sig-nificant figure. These values are close to the calculatedvalue [30] of h n i ≈
3. The strong suppression of the redsidebands after ground-state cooling indicates prepara-tion of a nearly pure motional ground state, with meanoccupation numbers in the x and y modes of 0.39(15) and0.21(8) phonons. If a thermal distribution of states is as-sumed [35], this is equivalent to probabilities of approx-imately 72% and 83% of occupying the ground states ofthe two modes, or residual temperatures of T ≈ µ Kand 121 µ K above the zero-point energy. These valuesare close to what would be expected from our experi-mental parameters, and could be further improved withmodifications to the cooling sequence [30].
IV. DEMONSTRATION OF THE WEAKCOUPLING
We experimentally demonstrate the weak coupling ofthe Be + cooling laser to the WCR modes by measuringthe time constant for Doppler cooling of these modes.Precise measurements of values of h n i & + ion, and the amplitude ofeither the red or blue sideband of the WCR mode undertest is then measured as described in section II. We in-terleave the measurements of the red and blue sidebandson a cycle-by-cycle basis to reduce sensitivity to drifts.Mode temperatures reach a steady-state value for longDoppler cooling pulses. We fit the data by means of asemiclassical model [36, 37] with our approximate exper-imental parameters, but with the Lamb-Dicke parameterand initial value for h n i as fit parameters, with the re-sults shown in Fig. 4. The fitted Lamb-Dicke parameteris 3 . × − · cos ( θ cool ), where θ cool ≈ ◦ is the in-tersection angle between the cooling laser and the radial ' R S S O H U F R R O L Q J W L P H V 0 H D Q S K R Q R Q Q X P E H U n Figure 4. Heating of the y WCR mode out of the groundstate during the Doppler cooling pulse applied to the Be + ion(black points), with the expected background from anoma-lous heating removed. The data is fitted (red line) using asemiclassical model. modes [30]. The quoted uncertainty is only from the fitand does not include additional uncertainties in experi-mental parameters. This is in good agreement with thecalculated value [30] of 3 . × − · cos ( θ cool ). The 1 /e time constant for the heating is 70 ms, and equilibriumis only reached for cooling times in the range of hundredsof milliseconds. V. LIMITATIONS TO THE COOLING PROCESS
The main limitation to the rate at which energy can beremoved by the algorithmic cooling scheme is the dutycycle resulting from dead time and overhead such as axialground state cooling and optical pumping steps amongothers. Each step is limited by the shortest possible pulsedurations for each part of the process, and the requirednumber of repetitions of each step. The minimum pulselength for both ions is limited by off-resonant couplingto other nearby transitions that are not part of the cool-ing process, such as the much stronger carrier transi-tions that change only the electronic state [34]. Temporalshaping of the laser pulse can mitigate this to some de-gree by limiting the high-frequency content of the laserspectrum [38]. Ultimately, when the Rabi frequency ap-proaches the splitting between the lines, the resulting linebroadening will cause the features to merge. In our ex-periment we employ laser pulse durations on the orderof 20 µ s for the Raman operations for the Be + ion, and180 µ s for operations on the HCI, resulting in respectivefull-width-at-half-maximum (FWHM) values of 40 kHzand 4.4 kHz for the observed motional sidebands on thetwo ions. The FWHM values for the carrier transitionsare approximately one order of magnitude larger in bothcases, but they are still well-resolved from the motionalsidebands.The number of repetitions of each step of the algorith- Mode ω/ π Γ h S E ( ω )(MHz) (quanta/s) (V m − Hz − )WCR x . × − WCR y . × − Table I. Properties of the WCR modes of the two-ion crystal:frequency ω/ π , heating rate Γ h , and calculated electric fieldnoise power spectral density at this frequency S E ( ω ). mic cooling process depends on experimental imperfec-tions in implementing the gates, including inefficienciesin optical pumping. Additional time is also needed forthe frequent reprogramming of frequency generators usedto tune the laser frequencies by means of acousto-opticmodulators. In our current implementation, a single al-gorithmic cooling cycle has a duration of 4.5 ms. If bothWCR modes are cooled in parallel by interleaving cyclesthat individually address the x and y modes, a maximumcooling rate Γ c of approximately 111 phonons per secondcould be achieved for both modes. This assumes that theappropriate electronic state of the HCI and the groundstates of the axial modes of the crystal have been pre-pared in advance. Despite the significant dead time inour algorithmic cooling cycle, the achievable cooling rateis still an order of magnitude higher than the Dopplercooling rate near to equilibrium (see section IV), and al-lows much lower temperatures to be reached.A key parameter governing the equilibrium tempera-ture after ground state cooling of the WCR modes is theanomalous heating rate Γ h caused by fluctuating electricfields at the position of the ions. The heating rates of the( x, y ) modes of a single Be + ion in this trap were previ-ously measured to be (1.9(3), 0.7(2)) phonons per secondat mode frequencies of (2.5, 2.2) MHz [28]. Significantlyhigher heating rates will be observed for modes where theHCI motion is dominant, however, since the high chargestate of the HCI leads to a much stronger coupling toelectric field noise (for a mode dominated by a single ionwith charge Z , Γ h ∝ Z [39, 40]).For measuring the anomalous heating rates of theWCR modes, we first cool them both to the motionalground state as described in section III. They are thenallowed to heat freely for periods of up to 0.5 seconds. Af-ter the chosen delay, we determine the mean number ofphonons per mode by measuring the excitation probabil-ities on the red and blue sidebands using quantum logic[34]. The heating rates, summarized in Table I, can beconverted to a single-sided power spectral density for theelectric-field noise that would cause an equivalent heat-ing, assuming that it arises from noise at ω and not fromcoupling to micromotion sidebands at frequencies Ω andΩ ± ω [40]. The value of 3 . × − V m − Hz − mea-sured for the WCR y mode is to our knowledge the lowestvalue ever reported for a radiofrequency ion trap [40, 41].The reduction in noise compared to the older single-iondata given earlier is likely due to subsequent improve-ments in the rf and dc circuitry used to drive the trap. ( [ S H U L P H Q W D O F \ F O H 0 H D Q S K R Q R Q Q X P E H U n Figure 5. Suppression of the heating of the x WCR mode.Without additional algorithmic cooling, the ion heats up dueto the anomalous electric field noise (black circles). A lin-ear fit (black dashed line) implies a heating rate of 8.9(16)phonons/second, consistent with the previously measuredvalue for this mode (see Table I). With one algorithmic cool-ing cycle applied on each WCR mode per experimental cycle,this heating is suppressed (red triangles), with a fitted heatingrate of 0.18(15) phonons/second (red dashed line).
For comparison against lowly-charged systems, this levelof noise would lead to a heating rate of 0.012 quanta/s fora single Ca + ion at this motional frequency. Identifyingthe origin for the difference in noise spectral density forthe two radial directions will be subject of future work. VI. KEEPING THE HCI IN THE GROUNDSTATE
Under normal operation, it is desirable to minimizedead time by keeping the Doppler cooling pulses sig-nificantly shorter than the 200 ms value used for themeasurements above. The disadvantage of this is thatanomalous heating of the WCR modes is not suppressedby the short cooling pulses. Therefore, the WCR modesgradually heat up until the increased Doppler coolingrate caused by the higher ion temperature can balancethe anomalous heating [36, 37]. We demonstrate this bypreparing the two modes in the ground state and thenperforming a typical ‘clock’ experiment. The clock cycleresembles the experiment detailed in section IV, but withtwo main modifications: (1) omitting the additional pulsefrom the cooling laser that was used to cause heating outof the ground state, and (2) rather than measuring thered and blue sidebands after a single cycle, we perform avariable number of cycles (each of duration 32 ms) beforereading out the ion temperature. Each cycle contains atotal Doppler cooling time of less than 1 ms. Withoutalgorithmic cooling, it can be seen that the mode tem-perature slowly increases between cycles in line with theexpected anomalous heating rate. This is shown in Fig. 5,where the temperature of the x WCR mode is displayedas a ‘worst-case’ (since we observe a higher heating rate for this mode than for the y WCR mode). The anoma-lous heating rates of only a few quanta per second canbe suppressed by only occasionally applying red sidebandpulses: we add a single algorithmic cooling cycle for eachof the two WCR modes to the end of our normal op-tical pumping routine for the HCI [12] while repeatingthe above measurement. As discussed in section V, thisadds only 9 ms of overhead to the experimental cycle, butnevertheless the effect of anomalous heating of the WCRmodes can be completely suppressed as the heating ratesare lower than the maximum cooling rate of 24 phononsper second under these conditions.
VII. CONCLUSIONS
To conclude, we have achieved algorithmic ground-state cooling of the radial in-phase motional modes ofa two-ion Coulomb crystal containing a HCI, despite theexceptionally weak coupling between the ions in thesemodes. In conjunction with the ground-state coolingof the axial modes of the crystal, this is the coldestHCI prepared in a laboratory thus far. The techniquedemonstrated here is very general, and could be ap-plied to a plethora of ions that cannot be directly laser-cooled and would have an unavoidably large charge-to-mass ratio mismatch with their cooling ion, as isthe case for (anti-)protons [42, 43], highly charged ions[11, 12, 17] and trapped charged macroscopic particles,such as nanospheres [44–46], graphene [47, 48] or nanodi-amonds [49, 50].For the discussed Be + -Ar system, if the meanphonon numbers in the WCR modes could be kept belowthe conservative target of h n x,y i = 0 .
5, time dilation fromthe residual ion velocity in these modes would lead to atotal fractional systematic shift of only − × − on theAr transition resonance frequency. This eliminatesthe final obstacle for the development of an optical fre-quency standard based on highly charged ions [17] withan accuracy that could surpass that of the best opticalfrequency standards available today [51–54]. ACKNOWLEDGMENTS
The authors would like to thank Erik Benkler andThomas Legero for their contributions to the frequencystabilization of the HCI spectroscopy laser, GiorgioZarantonello for fruitful discussions, and Ludwig Krin-ner for helpful comments on the manuscript. Theproject was supported by the Physikalisch-TechnischeBundesanstalt, the Max-Planck Society, the Max-Planck–Riken–PTB–Center for Time, Constants andFundamental Symmetries, and the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation)through SCHM2678/5-1, the collaborative research cen-tres SFB 1225 ISOQUANT and SFB 1227 DQ-mat, andunder Germany’s Excellence Strategy – EXC-2123 Quan-tumFrontiers – 390837967. This project 17FUN07 CC4Chas received funding from the EMPIR programme co-financed by the Participating States and from the Euro- pean Union’s Horizon 2020 research and innovation pro-gramme. S.A.K. acknowledges financial support from theAlexander von Humboldt Foundation. [1] D. J. Larson, J. C. Bergquist, J. J. 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Oelker, J. M. Robinson, S. L.Bromley, W. L. Tew, J. Ye, and C. J. Kennedy, JILASrI optical lattice clock with uncertainty of 2 × -18 ,Metrologia , 065004 (2019). upplemental Material for Algorithmic Ground-state Cooling of Weakly-CoupledOscillators using Quantum Logic Steven A. King, ∗ Lukas J. Spieß, Peter Micke,
1, 2
Alexander Wilzewski, Tobias Leopold, Jos´e R. Crespo L´opez-Urrutia, and Piet O. Schmidt
1, 3 Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Institut f¨ur Quantenoptik, Leibniz Universit¨at Hannover, Welfengarten 1, 30167 Hannover, Germany (Dated: February 26, 2021)
In this supplemental material, we describe in more de-tail the experimental setup and elaborate upon some ofthe techniques and calculations used in the main paper.
I. EXPERIMENTAL SETUP
This section is dedicated to the coherent laser manip-ulation of Be + and the major changes to the apparatussince publication of Ref. [1]. A. Laser systems for Be + The relevant level structures of the Be + and Ar ionsare shown in Fig. 1. Sideband operations on the Be + ionare performed using stimulated Raman transitions driven GHz (a) (b) (c) F = 1 F = 2 m , M C oo li ng m , E R epu m p m , E R a m an R a m an GHz
GHz ↑ ↓ PP P P S Figure 1. Partial term schemes of Be + and Ar (energiesnot to scale), showing the transitions used for manipulationof the two-ion crystal. (a) Doppler cooling and repumpingusing the Be + ion, (b) driving stimulated Raman transitionson the Be + ion, and (c) driving sideband transitions using theAr ion. Hyperfine sublevels of the P / and P / statesof Be + are omitted for clarity. ∗ Contact: [email protected] between the F = 2 ( m F = 2) and F = 1 ( m F = 1) sub-levels of the S / state. The Raman beams are derivedfrom the same 313 nm source, which has a detuning ofapproximately −
103 GHz from the S / → P / tran-sition. Operating at this detuning allows cancellation ofthe induced ac Stark shifts from the individual beams toa high degree, as they have a roughly equal and oppo-site detuning from the S / → P / transition. How-ever, this leads to a reduction of the Rabi frequenciescaused by destructive interference between the two path-ways [2, 3]. The beams have a relative detuning near1.25 GHz, which matches the hyperfine splitting in the S / ground state. Variation of this parameter allowsthe addressing of other spectral features. The first Ra-man beam (‘Raman 1’ in Fig. 1) and the second Ramanbeam (‘Raman 2’) respectively address the F = 2 and F = 1 hyperfine sublevels. The second Raman beamcounter-propagates with respect to the cooling laser. Ithas σ + /σ − polarization in order to minimize the ac Starkshift induced by this beam [2]. The first Raman beamcan be delivered from one of two possible directions, en-suring that in combination with the second Raman beamthe effective projection of the two beams is either axial orradial, as required. Due to the geometry of the vacuumchamber, the two possible directions for the first Ramanbeam are also both oriented at an angle of 30 ◦ to z , suchthat that they can not have pure π -polarization. Theirassociated Stark shifts can still be reduced to the kHz-level by careful tuning of the laser polarization, which isonly a few percent of the 40 kHz transition linewidthsproduced under our typical operating conditions.After the driving of a stimulated Raman transition, anindependent 313 nm laser source [4] is used to recycle (re-pump) the Be + ion to the S / , F = 2 state by excitationto the P / level [1, 5] from which it can spontaneouslydecay to the desired state. B. Stabilization and monitoring of laser powers
Relatively large shifts in laser frequency of up to10 MHz are required to probe the motional sidebandsof the atomic transitions. This leads to changes inthe diffraction efficiency of the acousto-optic modulators(AOM) used to tune the lasers. To ensure identical laserpowers and hence stabilize the Rabi frequencies and laser- a r X i v : . [ phy s i c s . a t o m - ph ] F e b induced systematic shifts of the resonances, we stabilizethem on a pulse-by-pulse basis using signals from pho-todiodes located near to the ion trap. Feedback is thenapplied to the drive power of the appropriate AOM us-ing a STEMlab (formerly Red Pitaya)-based system [6]in a sample-and-hold configuration. The status of thesesample-and-hold systems along with various other sys-tems such as laser locks are monitored on a cycle-by-cyclebasis by our experiment-control system. Cycles whereone or more parameters were outside tolerable ranges arediscarded and repeated. C. Temporal shaping of laser pulses
As a consequence of the small applied magnetic field(20 µ T) and resulting first-order Zeeman shifts, impurelaser polarizations, low motional frequencies ( ∼ + ion observed inthis direction have many closely separated lines includ-ing pronounced intermodulation peaks. To reduce theprobability of off-resonant driving of unwanted nearbytransitions whilst maintaining high Rabi frequencies, al-most all laser pulses coupling to first-order motional side-bands in this direction are temporally shaped such thatthe Rabi frequency evolves with a profile that closely re-sembles a Blackman waveform, thereby suppressing theproblematic ‘wings’ of the Rabi line profile [7]. D. Stabilization of rf trapping voltage
As the trap depth must be reduced in order to reloada new HCI after a charge-exchange collision [5, 8], theradial motional frequencies are prone to drift after re-turning to the typical values used during the presentedexperiments. This is likely attributable to the increasedrf power dissipation causing changes in the temperatureand hence the electrical conductivity of both the trapand rf resonator which are mounted within the cryostat[1]. To avoid this, the rf trap depth is actively stabi-lized by monitoring the pickup on an antenna next tothe ion trap used for manipulation of the Be + ion usingmicrowave fields. Feedback is applied to the amplitude ofthe rf synthesizer used to drive the trap in an approachsimilar to that of reference [9]. The resulting radial secu-lar frequencies are typically stable at the fractional levelof 10 − . II. MODE TEMPERATURES AFTER DOPPLERCOOLING
The cooling laser is delivered in the horizontal plane atan angle of 30 ◦ to the trap z axis, and therefore has a sig-nificantly weaker projection onto the radial modes of the Coulomb crystal than onto the axial modes, with an in-tersection angle of θ cool = arccos(sin 30 ◦ · cos 45 ◦ ) ≈ ◦ .This leads to an increase in the equilibrium temperatureof the radial modes after Doppler cooling compared tothe ideal case. Using the notation of reference [10], inthe absence of anomalous heating, the total energy afterDoppler cooling for mode i can be expressed as: h E i i = 2 h E kin ,i i = (cid:18) f si f i (cid:19) ~ Γ4 , (1)where h E kin ,i i is the kinetic energy in mode i , f si isa factor determined by the angular distribution of thespontaneously emitted photons, f i is a factor determinedby the projection of the cooling laser onto the directionof motion, ~ is the reduced Planck constant, and Γ is thelinewidth of the transition used for laser cooling. Thiscan be equated with the energy of the quantum harmonicoscillator in terms of the mean number of phonons h n i i : h E i i = (cid:18) h n i i + 12 (cid:19) ~ ω i . (2)For typical frequencies of ω = 2 π × . f x,y ≈ . σ + -polarized photons be-ing scattered during Doppler cooling ( f sx,sy ≈ . π ×
18 MHz linewidth for the cooling transitionin Be + lead to a mean vibrational quantum number of h n x,y i ≈ h n x,y i ≈ . III. NORMAL MODE AMPLITUDES
In this section, we present the calculated extents ofthe ground-state wavefunctions for the two ions in eachof the six normal modes of the Coulomb crystal, includ-ing the amplitude scaling factors arising from the un-equal distribution of energy in the modes between theions. These were calculated for our typical experimen-tal conditions by extending the approaches of references[11–13] to include the different charges of the two ions.The Lagrangian for the system (including the trappingpotential and Coulomb interaction) is solved to yield thenormal modes of the ion crystal, in the usual coordinatesystem where the motion of the highly charged ion (HCI)is scaled by a factor ( m Ar /m Be ) − / . The amplitudes ofmotion z i for the two ions (with i ∈ Be + , Ar ) can bedetermined by treating them as quantum harmonic oscil-lators in the ground state of a given mode, yielding therelation: z i = | b i | r ~ m i ω , (3) Axis Mode ω/ π Be + amplitude Ar amplitude(MHz) (nm) (nm) x IP (WCR) 4.62 0.15 5.2 x OP (SCR) 1.39 20 0.13 y IP (WCR) 4.42 0.17 5.4 y OP (SCR) 1.15 22 0.15 z IP 1.15 18 6.3 z OP 1.56 11 7.2Table I. Calculated frequencies of the axial ( z ) and ra-dial ( x, y ) normal modes for the two ions in a Be + - Ar Coulomb crystal under our approximate experimental trap-ping conditions, along with the extents of the ground-statewavefunctions for the individual ions in these modes. IPand OP refer to in- and out-of-phase motion, respectively,and SCR and WCR refer to the strongly-coupled and weakly-coupled radial modes, respectively. The Be + ion has a muchsmaller motional amplitude in the x and y IP modes, leadingto greatly inhibited laser cooling of these modes. where b i is the appropriate component of the normalizedeigenvector for the mode for ion i , ~ is the reduced Planckconstant, m i is the ion mass, and ω/ π is the secularfrequency of the mode.The results are summarized in Table I. The amplitudeof only 0.15 nm for the Be + ion in the x in-phase (IP)mode, referred to as the x weakly-coupled radial (WCR)mode, corresponds in our setup to a Lamb-Dicke param-eter of only 3 . × − · cos ( θ cool ), more than two ordersof magnitude smaller than the value of 0 . · cos ( θ cool ) forthe x out-of-phase (OP) mode. In the axial ( z ) direction,the ions do not display the same level of decoupling, andthe laser cooling efficiency using the Be + remains high[13]. IV. DETERMINING h n i FROM MOTIONALSIDEBAND SPECTRA
We derive the ground state population and heatingrates from the mean phonon number h n i i for mode i ,which has been determined using the sideband ratio tech-nique [15]. The value of h n i can be calculated from theobserved contrasts A on the red (RSB) and blue side-bands (BSB): h n i = A RSB /A BSB − A RSB /A BSB . (4)This ratio is independent of the employed Rabi probetime, as long as it is the same for RSB and BSB. A. Quantum logic background correction
A finite background appears on the quantum logic sig-nal even when the laser interrogating the HCI is far off-resonant or physically blocked. This arises from imper- fect ground-state cooling of the axial out-of-phase mode,leading to a weak but non-zero red sideband on the Be + Raman transition. This is taken into account in all mea-surements, where a systematic correction of -1.0(5)% isapplied to all measured excitation probabilities to ac-count for its value and typical instability. This correctionis applied after the averaging of multiple datasets to avoidinadvertent and invalid averaging of this correction. Thevalue of the background should be identical for the blueand red sideband measurements, as they are performed inan interleaved fashion. The uncertainty in the extractedvalue of h n i is therefore likely slightly overestimated asthere is correlation in the background correction that isnot accounted for when propagating the uncertainties. B. Scanning over the motional sidebands
For Fig. 3 in the main text, the frequency of the laseraddressing the HCI was scanned over the red and bluesidebands of the two WCR modes, and the line profileswere fitted. In order to minimize sensitivity to drifts inthe motional frequencies, each red sideband and its bluecounterpart were scanned in a narrow range around eachfeature. The individual probe frequencies were cycledthrough in a pseudo-random sequence, further reducingthe correlation between successive measurements.For the Doppler cooled-case, a simple squared-sincfunction was used to fit the data described by a ther-mal superposition of lines with n -dependent Rabi fre-quencies. For the ground-state-cooled case, a Rabi lineprofile assuming a pure motional quantum state was fit-ted to the data. Line profile distortions, most noticeablein the wings of the excitation profile, arise from insta-bility ( ∼ − ) of the motional mode frequencies on thetimescale of the experiments, dephasing from the residualpopulation in higher Fock states, and noise on the exci-tation laser. To avoid the resulting underestimation ofthe fitted excitation amplitude and to be consistent withall other h n i measurements in this manuscript, we choseto use the offset-corrected (see subsection IV A) experi-mental data point closest to the fitted line center for thecalculation of h n i , at the expense of increased statisticaluncertainty. C. Measurements of peak excitation
In order to improve the efficiency of the measurements,full scans over each line were not performed during mea-surements of anomalous heating rates (such as those dis-played in Fig. 4) or during measurements of the Doppler-cooling rate of the WCR modes, but measurements wererestricted to the points of peak excitation. To limit anypotential systematic error arising from drifts in the WCRmode frequencies during the measurements, the durationof the waiting time to allow the ion to heat was pseudo-randomized, along with whether the measurement was
Figure 2. Simplified depiction of the two main experimental sequences used in this work, with the cooling phases (yellow),clock operation (pink) and experimental (blue) phases indicated. GSC = ground-state cooling. The ions involved in each stepare also indicated. Sequence (a) was used to measure anomalous heating rates for the WCR modes, Rabi flopping, and forsideband scans. Sequence (b) was used to demonstrate suppression of the heating of the WCR modes during simulated clockoperation, as displayed in Fig. 5 in the main manuscript, with the central clock operation section repeated a variable numberof times in order to measure the evolution of the WCR mode temperature over an increasing number of cycles. made on the blue or red sideband.
V. GROUND-STATE COOLING OF THE WCRMODESA. Summary of main approaches to algorithmiccooling
Two main approaches are taken for ground-state cool-ing of the WCR modes. The first is that a separate exper-imental phase is dedicated to cooling the WCR modes,as shown in Fig. 2(a). This directly follows the op-tical pumping phase for the HCI, and is composed ofeight pulses on the first-order red sideband of each of thetwo WCR modes, applied in an interleaved fashion to re-duce the impact of anomalous heating [1]. Owing to theca. 10 ms lifetime of the excited electronic state of theHCI [5, 16], after each cooling pulse the HCI must bereturned to the ground state using a simplified versionof the optical pumping routine presented in reference [5].As the temperatures of the WCR modes decrease duringthe ground-state cooling process, the optimal pulse dura-tion on these motional sidebands that leads to the high-est probability for transfer to the excited state increasesgradually [14]. This is addressed by gradually increasingthe length of the red sideband pulse of the WCR modebetween algorithmic cooling cycles. Under typical con-ditions, an initial value of 90 µ s is used, which is slowly increased to 180 µ s. These values are a compromise be-tween the optimum values for the x and y WCR modes,which are visible in Fig. 3. This cooling approach wasfollowed when scanning over the motional sidebands, ob-serving Rabi oscillations (see section VI), or measuringanomalous heating rates.The second approach to cooling has been employed toproduce Fig. 5 of the main text. After initial cooling ofthe two WCR modes close to their ground states in aprocedure similar to the cooling phase in Fig. 2(a), a se-quence resembling optical clock operation is started (seeFig. 2(b)). During this sequence, only a single algorith-mic cooling cycle on each of the two WCR sidebands isadded after the HCI electronic state preparation [5] dur-ing each repetition of the clock sequence. In this case, thepulse time is fixed at 180 µ s. While lacking the raw cool-ing power of the first approach, this allows an already-cold HCI to be held in the ground state whilst addingminimal experimental overhead. B. Steady-state temperature after algorithmiccooling
As mentioned in subsection V A, the optimum pulselength to achieve the maximum population transfer tothe excited state depends on the Fock state n . Underour experimental conditions, the Rabi frequencies forthe n = 5 → n = 4 transitions of the WCR modes 3 X O V H G X U D W L R Q V P (a) 3 X O V H G X U D W L R Q V P (b) Figure 3. Probability of exciting the HCI to the excited state P ↑ ( t ) by driving the first blue sideband of the (a) x , and (b) y WCR modes, observed on the HCI as a function of interrogation time before (black) and after (red) algorithmic cooling to theground state was applied. The clean Rabi flopping behaviour visible after cooling is a clear indication of a high population inthe vibrational ground state of both radial in-phase modes. Details of the fitting function is given in the text. are approximately double that of the n = 1 → n = 0transitions. This means that there will be no popula-tion transfer on the former transition by a pulse whoselength matches the so-called π -time for the latter, lead-ing to difficulty in cooling Fock states higher than n = 5to the ground state. Our use of varying pulse lengthsduring the cooling process mitigates this to some degree,and rough simulations (not including anomalous heating)using our chosen experimental parameters indicate thatapproximately 80% of the population reaches the groundstate in a given mode, which matches our experimentalobservations. Further fine-tuning of the pulse lengths,the order in which they are applied, the total numberof pulses, and the use of higher-order sidebands to re-move multiple phonons per pulse as is done for the axialmodes [1], could all potentially increase the ground statepopulations from the values reached here. VI. COHERENT OPERATIONS ON WCRMODE SIDEBANDS
The low temperature of the two weakly-coupled radial(WCR) modes after algorithmic cooling can also be ob-served by the ability to drive coherent oscillations (Rabiflopping) on the blue sidebands, as shown in Fig. 3. Be-fore cooling, no coherent oscillations can be observed dueto dephasing from the different Debye-Waller factors [14]of the various Fock states in the initial thermal distribu-tion. After ground-state cooling, the dominant contribu-tor to the signal is the n = 0 → n = 1 transition, leadingto a clean oscillation. The maximum observed excitationof approximately 65% is limited by a combination of im-perfect state preparation of the HCI, imperfect quantumlogic pulses, and dephasing from the residual Fock statedistribution after cooling. The fit to the BSB oscillationson the x WCR mode displayed in Fig. 3(a) takes the form: P ↑ ( t ) = A X m P m,n sin (Ω m → m +1 ,n t ) (5)where P ↑ ( t ) is the probability of finding the ion in theexcited state |↑i at time t , A is the maximum contrast, P ( m, n ) is the probability of occupation of Fock states m and n in the x and y WCR modes, respectively, andΩ m → m +1 ,n is the Rabi frequency for driving state m, n to state m + 1 , n [14]. For this purpose, an infinite co-herence time was assumed. To reduce the number of freeparameters in the fit, identical temperatures and ther-mal distributions in the two modes after cooling wereassumed, and any higher-order couplings were neglected.Fock states up to m, n = 50 were included in the fit. Asimilar fit was carried out for the BSB oscillations on the y WCR mode, displayed in Fig. 3(b), but with the ap-propriate Rabi frequency Ω m,n → n +1 and the sum beingover n . The fitted values of h m i = 0 . h n i = 0 . h m i = 6 . h n i = 7(1) extracted for the Doppler-cooled data are higher than those determined from themeasurements of the peak contrast, which is likely dueto correlations between the fitting parameters in such acomplex fit. VII. ANOMALOUS HEATING
We consider heating of a single ion by electric fieldnoise of a single mode with frequency ω . The heatingrate Γ h (measured in phonons per second) is related to 7 L P H Z L W K R X W F R R O L Q J V 0 H D Q S K R Q R Q Q X P E H U n x : &