Assembly of a rovibrational ground state molecule in an optical tweezer
William B. Cairncross, Jessie T. Zhang, Lewis R. B. Picard, Yichao Yu, Kenneth Wang, Kang-Kuen Ni
AAssembly of a rovibrational ground state molecule in an optical tweezer
William B. Cairncross,
1, 2, 3, ∗ Jessie T. Zhang,
1, 2, 3, ∗ Lewis R. B. Picard,
1, 2, 3
Yichao Yu,
1, 2, 3
Kenneth Wang,
1, 2, 3 and Kang-Kuen Ni
2, 1, 3, † Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Chemistry and Chemical Biology,Harvard University, Cambridge, Massachusetts 02138, USA Harvard-MIT Center for Ultracold Atoms, Cambridge, Massachusetts 02138, USA
We demonstrate the coherent creation of a single NaCs molecule in its rotational, vibrational, andelectronic (rovibronic) ground state in an optical tweezer. Starting with a weakly bound Feshbachmolecule, we locate a two-photon transition via the | c Σ , v = 26 i excited state and drive coherentRabi oscillations between the Feshbach state and a single hyperfine level of the NaCs rovibronicground state | X Σ , v = 0 , N = 0 i with a binding energy of D = h × . . ± . Trapped arrays of individually controlled interactingatoms have enabled a range of studies in quantum infor-mation and quantum many-body physics that are nowreaching beyond what can be computed on classical ma-chines [1–6]. Substituting atoms with polar molecules,which feature rich internal states with tunable long-rangedipolar interactions, further expands the opportunitiesfor quantum simulation of novel phases of matter [7–11], high fidelity quantum information processing [12–15], precision measurements [16–18], and studies of coldchemistry [19–21]. However, this molecular complexityalso presents challenges in obtaining the same level ofcontrol in molecules as in atoms, motivating the devel-opment of new approaches [22–30]. Recent experimentshave now realized coherent control of molecules at thelevel of individual particles and single internal quantumstates for ions through cooling and readout of co-trappedatoms [31, 32] and with neutrals in optical tweezers [33–38]. In select cases, motional control of single moleculeshas also been attained.Associating single molecules from individual atomscleanly maps the full quantum state control that is at-tainable for the constituent atoms onto the molecules. Bycontrolling the motional states of the atoms prior to as-sociation, both the motional and the internal state of theresultant molecule can be controlled. Previously, full con-trol has been demonstrated for single molecules in weaklybound states [35, 37, 38]. In order to realize tunabledipolar interactions—the key ingredient for simulatingstrongly correlated, novel phases of matter—these singlemolecules must be transferred to a low-lying vibrationalstate where they possess an appreciable molecule-framedipole moment.In this Letter, we demonstrate the coherent associ-ation of a single rotational, vibrational, and electronic(rovibronic) ground-state molecule in an optical tweezer.Specifically, we perform “molecular assembly” [Fig. 1(a)]to form a single ground-state NaCs molecule starting from individually trapped single atoms that are first mag-netoassociated into a weakly bound Feshbach molecularstate. We transfer the Feshbach molecules to the rovi-bronic ground state using a coherent two-photon detunedRaman pulse, demonstrating internal state transfer ofultracold molecules in a new parameter regime. The re-sulting molecule possesses a large molecule-frame dipolemoment of 4.6 Debye [39, 40], and we measure a trappedlifetime of 3 . ± . Na and
Cs atoms are loaded from a dual-species magneto-optical trap into tweezers at 700 nm and1064 nm, respectively [Fig. 1(a), first panel]. The atomsare loaded stochastically, and an initial non-destructiveimaging step allows for postselection of the ∼
30% ofexperimental sequences where both atoms are initiallypresent. Both atoms are then cooled to their respec-tive 3D motional ground states using polarization gra-dient cooling followed by Raman sideband cooling. Af-ter cooling, we bring the atoms together into a singletrap, and then perform Feshbach magnetoassociation toconvert the atom pair into a weakly bound molecule[35] [Fig. 1(a), center panel]. The Feshbach moleculeis formed in a single internal state with a binding en-ergy of ∼ ∼
300 ms.Next, we proceed to transfer the molecule fromthe Feshbach state to the rovibronic ground state a r X i v : . [ phy s i c s . a t o m - ph ] J a n Trap & cool Merge Molecule association Internal statetransfer (a)
Internuclear distance R (Å) E ne r g y ( c m –1 ) X a c B Na(3s)+Cs(6p ) P
922 nm S
635 nm (b) v''=0 v'=26
Feshbach state
FIG. 1. Schematic overview of assembling rovibronic groundstate molecules in optical tweezers. (a) Sequence of molecu-lar assembly from individually trapped atoms to ground statemolecule. This work focuses on the outlined final step of inter-nal state transfer. (b) Selected potential curves of the NaCsmolecule, showing the Raman transfer scheme from the Fesh-bach state to the rovibronic ground state via | c Σ , v = 26 i .Pump and Stokes laser Rabi frequencies are labeled Ω P andΩ S , respectively. Inset: Geometry of the optical tweezer,magnetic bias field, and Raman transfer lasers. | X Σ , v = 0 , N = 0 i [Fig. 1(a), last panel], where v and N are the vibrational and rotational quantum num-bers, respectively, of X Σ. As in earlier work that usedensembles of molecules, we employ a two-photon opticaltransfer via an electronically excited state [Fig. 1(b)] [22–24, 42–46]. Earlier works have used stimulated Ramanadiabatic passage (STIRAP) to transfer population fromthe Feshbach state. In STIRAP, modulation of pumpand Stokes laser intensities as a function of time causes acoherent dark state to adiabatically evolve from the ini-tial to final states while the intermediate state remainsnearly unpopulated, thus minimizing scattering. In thiswork, we instead implement a detuned Raman transfer,and by doing so we demonstrate that high-fidelity cre-ation of ground state molecules is possible in a differentparameter regime than previously explored. The relevantparameters for a coherent transfer are the excited statelinewidth Γ, the pump and Stokes Rabi frequencies Ω
P,S ,and the ratio Ω R /R sc , where Ω R is the Raman Rabi fre-quency and R sc is the (detuning-dependent) scatteringrate. After locating the pump and Stokes transitions, we investigate these properties to find suitable parametersto perform Raman transfer.A two-photon transfer to the ground state has notbeen previously performed in NaCs, necessitating first asearch for and characterization of intermediate states aswell as locating the ground state resonance. We choose | c Σ , v = 26 i as an intermediate state due to severalfactors: (1) it has relatively high Franck-Condon over-lap with the Feshbach state, (2) it is expected to havestrong transition dipole moments to both the Feshbachand rovibronic ground states due to the large spin-orbitcoupling constant of the Cs atom, and (3) it is accessiblefrom these states with convenient laser wavelengths of922 nm and 635 nm, respectively [Fig. 1(b)]. Followinga prediction based on the potential curves of Ref. [47],we located | c Σ , v = 26 i using photoassociation spec-troscopy. Figure 2(a) shows a high-resolution spectrumof the transition from the Feshbach state to the J = 1and J = 2 manifolds along with a model that incor-porates the excited state structure, where J is the ro-tational plus electronic angular momentum in the c Σ state. We then located the | X Σ , v = 0 i state usingAutler-Townes spectroscopy. The details of the model-ing and quantum number assignment are reported else-where [48].The linewidth of the intermediate state has a signif-icant influence on our state transfer scheme. In ear-lier molecular association experiments, the molecular ex-cited state used as an intermediate had a width com-parable to that of the atomic transition to which it isasymptotically connected. In our system, that wouldbe the Cs 6s →
6p transition, with a natural linewidthof order Γ atom / π ≈ | c Σ , v = 26 , J = 1 , M J = 1 i to beΓ / π = 120(30) MHz—more than an order of magni-tude larger than the atomic linewidth [Fig. 2(b)], where M J is the projection of J onto the laboratory mag-netic field. We have not been able to determine theorigin of the increased linewidth, and further investi-gation is warranted. We also characterized the scat-tering arising from all | c Σ , v = 26 i lines when red-detuned from resonance, as shown in Fig. 2(c). We findscattering rates consistent with our independent mea-surements of the | c Σ , v = 26 i linewidth and transi-tion strength, with the addition of a background scatter-ing rate of [200(100) µ s] − that may arise from further-detuned states.While the observed linewidth of the c Σ state is notideal for state transfer, we find that the strength of thepump transition offsets this issue. We characterize thestrength of the 922 nm pump transition by measuringthe depletion time on the J = 1 , M J = 1 resonance at325129.64(2) GHz at low laser power, where the lifetime τ is long compared to 1 / Γ. In this limit, τ ≈ Γ / Ω P .We independently calibrate the pump laser intensity us-ing a measurement of the vector AC Stark shift of the N a + C ss u r v i v a l (a) Pump frequency (GHz) – 325130 GHz0 1 2 3 4–1–2–3–4 J' = 1 J' = 2 M' ≈ –1 0 1 M' ≈ –1 0 1 2 (b) Pump frequency (GHz) – 325130 GHz–0.5 –0.4 –0.3 –0.2 ( s –1 ) (c) Pump frequency (GHz) – 325130 GHz–20 –10–30–40–50 R sc FIG. 2. Characterization of the | c Σ , v = 26 i state of NaCs by depletion of Feshbach molecules. (a) Depletion spectrumin the vicinity of 325130 GHz using σ + + σ − polarization, showing the J = 1 , c Σ states[48]. (b) Linewidth characterization of the J = 1 , M J = 1 level of | c Σ , v = 26 i used for Raman transfer when probed with σ + polarization. The curve is a fit to a Lorentzian lineshape, with fit linewidth Γ = 2 π × R sc due to the pump laser when red-detuned from | c Σ , v = 26 i . The curve is a fit to thescattering rate derived from the same model as in part (a). Inset: the linear dependence of the scattering rate on the pumplaser power at the −
21 GHz detuning is consistent with single-photon scattering.
Cs hyperfine ground state, allowing us to extract thetransition dipole moment. We find a Rabi frequencyΩ P / π = 6 . × p P P / (1 mW) where P P is thepump laser power, corresponding to a transition dipolemoment of µ P = 0 . e a . We attribute this largetransition strength (when compared to other species ofFeshbach molecules, cf . [23]) to the closed-channel domi-nated character of the Feshbach state. The tight confine-ment provided by the diffraction-limited optical tweezerallows us to use a small ∼ µ m waist for the Ramanlasers, allowing Rabi frequencies up to ∼
60 MHz for thistransition with P P ≈
100 mW.Having characterized the excited state properties, wecan then identify the specific c Σ levels whose quantumnumbers allow a two-photon transition to the rovibronicground state. In the presence of a large magnetic biasfield, the initial (Feshbach), intermediate ( c Σ ), and fi-nal ( X Σ) states of our transfer scheme are in very dif-ferent angular momentum coupling regimes. As a result,both initial and final states can couple to many excitedstates. Only a few intermediate levels couple to both and can thus give rise to Raman transitions, while oth-ers contribute only to loss via one-photon scattering. Wefind that choosing both lasers to have σ + polarization[Fig. 1(b) inset] gives the simplest spectrum within thecurrent geometrical constraints of our apparatus. In thiscase, the Raman transition amplitude is dominated bythe | J = 1 , M J = 1 i state of | c Σ , v = 26 i , connect-ing the Feshbach state to the | M I Na = 3 / , M I Cs = 5 / i hyperfine component of the | X Σ , N = 0 i rotationalstate, where M I is the projection of the nuclear spin I onto the laboratory magnetic field. With the pumplaser fixed on the J = 1 , M J = 1 resonance, we locatedthe Stokes transition at 472166.04(2) GHz using Autler-Townes spectroscopy. This measurement provides a valueof the NaCs binding energy D = 147038 . | c Σ , J = 1 , M J = 1 i , we scan the pump beamfrequency to calibrate the Stokes laser Rabi fre-quency Ω S and the X Σ → c Σ transition dipole -1 -0.5 0 0.5 1 Stokes detuning (MHz) N a + C s s u r v i v a l (a) Pulse duration (µs) N a + C s s u r v i v a l (b) FIG. 3. (a) Raman resonance and (b) Raman Rabi oscillation between Feshbach state and the rovibronic ground state of NaCsvia | c Σ , v = 26 i . Dashed lines and gray bars show the Feshbach molecule contrast, which was collected simultaneously withthe data in (b) and is shown there by the first blue circle and orange triangle. The data are simultaneously fit to a Rabilineshape including loss and decoherence. moment. As a function of pump detuning, we observea characteristic Autler-Townes doublet, and we findΩ S / π = 158(8) MHz × p P S / (1 mW), where P S isthe Stokes laser power [49]. Using the estimated peakintensity of the Stokes beam, we obtain a transitiondipole moment of µ S = 0 . e a . We find that thesinglet fraction of the | c Σ , v = 26 i state is 32(3)%; asubstantial admixture which we attribute to the largespin-orbit coupling in the Cs 6p state and the closeproximity of B Π vibrational states.To perform Raman transfer, we detune both pump andStokes lasers by 21 GHz to the red of the | c Σ , v = 26 i manifold. With a 3 µ s pulse, we locate the Ra-man resonance shown in Fig. 3(a) by scanning theStokes laser detuning. We observe only a single reso-nance, consistent with our expectation from calculationsbased on Ref. [50] that we populate predominantly the | M I Na = 3 / , M I Cs = 5 / i hyperfine component of therovibronic ground state for our choice of laser polariza-tions. Fixing the Stokes laser frequency on resonanceand varying the pulse duration, we observe coherent Rabioscillations [Fig. 3(b)]. We measure a Raman Rabi fre-quency of Ω R / π = 187(2) kHz, consistent with the the-oretical value of 210(30) kHz with pump Rabi frequencyΩ P / π ≈ P P = 50 mW, and Stokes Rabifrequency Ω S / π ≈ P S = 2 . R /R sc = 27(7), indicating that coherenttransfer dominates over loss.We find a one-way transfer efficiency of 82(10)% fromFeshbach molecules to rovibronic ground state molecules.Incorporating the present Feshbach molecule creation fi-delity of 38(1)%, the overall efficiency for creation ofground-state molecules from individual atoms is 31(4)%,and the round-trip efficiency from atoms to ground statemolecules and back is 25(4)%. The dominant factor lim-iting the overall molecule creation fidelity from atoms is that of Feshbach molecule creation, which is currentlylimited by heating of the atoms during the trap mergestep.Figure 3(b) shows that another significant factor limit-ing ground-state molecule formation is decoherence. Wefit a dephasing time of γ − = 17(5) µ s, while the scat-tering time is R − = 23(6) µ s. The observed decoher-ence can be accounted for by fluctuating AC Stark shiftsarising from drifts in the pump and Stokes laser intensi-ties. At present, the optical power in each of these beamsdrifts by ∼
5% due to thermal variation in the labora-tory environment. For the data shown here, these powerswere not actively stabilized. In planned improvements toour apparatus, we will actively stabilize the Raman laserpowers so that the transfer efficiency will be limited byoff-resonant scattering from c Σ .Since our experiment involves only a single molecule ina deep optical trap, we expect the ground state lifetimeto be primarily limited by scattering of the trap lightor collisions with background gas. At our typical trapintensity of 80 kW / cm , we find a ground-state lifetimeof 3 . ± . × and 27 × (Fig. 4). We find that the rovibronic groundstate lifetime can be reduced to 0.5(1) s and 130(40) msat trap intensities of 0.8 MW/cm and 2.2 MW/cm ,respectively, consistent with a linear scaling.Assuming a linear dependence on trap intensity,we find a loss rate for ground state molecules of2 . − (MW / cm ) − (Fig. 4 inset). The precisionof our measurement is limited by a maximum cycletime of 1.5 s due to thermal fluctuations of the appa-ratus. Using the theoretical ground state polarizabilityof NaCs from Ref. [51], we expect a scattering rate of54 s − (MW / cm ) − , suggesting either an overestimateof the theoretical polarizability or a high proportion of t (s) g . s . m o l e c u l e pop .
80 kW/cm Intensity (MW/cm ) l o ss r a t e ( s - ) FIG. 4. Characterization of the lifetime of a rovibronic groundstate NaCs molecule in an optical tweezer at different values ofthe trap intensity. Atomic background has been subtracted.Inset: the increase in molecule loss rate with laser intensityis consistent with a linear trend corresponding to one-photonscattering.
Rayleigh over Raman scattering at this wavelength.The rovibronic ground state molecule primarily inher-its the motional quantum state of the Feshbach molecule,which arises from the individually laser cooled atoms. Weestimate that the rovibronic ground state molecule occu-pies the motional ground state with 65(5)% probability,compared to 75(5)% for the Feshbach molecule [49]. Thisexcitation of the motion of the rovibronic ground statemolecule arises from two sources: First, the absorptionof a pump photon and emission of a Stokes photon im-parts a coherent momentum kick to the molecule, re-moving it from the ground state with ∼
8% probabil-ity. Second, the Feshbach and rovibronic ground statesexperience different optical trapping frequencies due totheir differential polarizability, leading to a wavefunctionmismatch that projects population onto excited motionalstates with ∼
6% probability. These effects can be mit-igated by performing the Raman transfer more slowlywith a larger detuning and at a higher trap frequency.However, loss of Feshbach molecules and finite laser co-herence will limit the maximum transfer time. In thelimit of resolved motional sidebands during Raman trans-fer, both effects can be eliminated. It is also possible toapply the Raman lasers perpendicular to the tweezer axisin order to take advantage of the higher trap frequenciesin the radial direction.In summary, we have demonstrated the coherent cre-ation of a single rovibronic ground state molecule pre-dominantly in the motional ground state of an opti-cal tweezer. The 31(4)% overall fidelity of ground-statemolecule production demonstrated here is not fundamen-tally limited at any step, and can be improved with opti-mization of the atom ground-state cooling and merging. This work completes the final step of molecular as-sembly in our work towards arrays of molecules in themotional ground state of optical tweezers. With a sin-gle quantum-state-controlled rovibrational ground statemolecule in hand, recent demonstrations of scaling tolarger arrays with atoms [5, 6] then serve as a startingpoint for parallel molecular assembly in the near future,providing a platform for engineering controlled long-range entangling interactions between molecules. Withthe addition of established techniques for microwave andelectric field control [52, 53], molecular qubits for quan-tum computing applications [13, 15] and simulations thatfurther our understanding of quantum phases of matter[7–11, 54] are all within experimental reach.We thank Lee R. Liu and Constantin Arnscheidt forearly experimental assistance. 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Ye, Nature ,521 (2013). upplemental Material:Assembly of a rovibrational ground state molecule in an optical tweezer William B. Cairncross,
1, 2, 3, ∗ Jessie T. Zhang,
1, 2, 3, ∗ Lewis R. B. Picard,
1, 2, 3
Yichao Yu,
1, 2, 3
Kenneth Wang,
1, 2, 3 and Kang-Kuen Ni
2, 1, 3, † Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Department of Chemistry and Chemical Biology,Harvard University, Cambridge, Massachusetts 02138, USA Harvard-MIT Center for Ultracold Atoms, Cambridge, Massachusetts 02138, USA
PUMP AND STOKES RABI FREQUENCY CALIBRATION
We calibrate the pump Rabi frequency by measuring the Feshbach (FB) molecules loss rate as a function of laserpower when the laser is resonant with the | J = 1 , M J = 1 i state. In the regime where Ω P (cid:28) Γ, the depletion rateis 1 /τ ≈ Ω P / Γ. We fit the inverse FB molecule lifetime versus pump power as shown in Fig. S1, and extract a Rabifrequency of Ω P / π = 6 . × q P P . Pump power (mW) / ( s - ) R sc = 2.0(5) 10 s -1 mW -1P = 6.2(8) MHz/mW FIG. S1. Depletion rate of Feshbach molecules as a function of pump laser power, for calibration of pump laser Rabi frequency.
We calibrate the Stokes laser Rabi frequency using Autler-Townes spectroscopy. After locating the rovibrationalground state, we tune the Stokes laser onto resonance with the | J = 1 , M = 1 i state, and measure the depletion ofFeshbach molecules as a function of pump laser frequency. We observe an Autler-Townes doublet feature as shown inFig. S2, and fit it to a model function for the 2-body survival probability P Na+Cs after a pulse duration t [S1] P Na+Cs ( t ) = P atom + P mol exp (cid:18) − Ω P t P − ∆ S ) | Ω S + 2 i (∆ P − ∆ S )(Γ + 2 i ∆ P ) | (cid:19) , (S.1)where P atom is an atomic background population, P mol is the Feshbach molecule creation fidelity, and ∆ P,S arethe one-photon detunings of pump and Stokes lasers, respectively. We obtain a value of Ω S / π = 283(9) MHz at P S = 3 . S / π = 158(8) MHz × p P S / (1 mW) for the Stokes laser Rabi frequency. MOTIONAL GROUND STATE FRACTION
In Ref. [S2], we estimated that the center of mass (COM) ground state fraction of Feshbach molecules was 77(5)%using Raman sideband thermometry. In that work, we also observed a Feshbach molecule creation fidelity of 47(1)%,consistent with a relative motional ground state population of 58(4)%. Presently, we observe a slightly reducedFeshbach molecule creation fidelity of 38(3)%, while the atomic ground state cooling conditions have not changed a r X i v : . [ phy s i c s . a t o m - ph ] J a n -200 -100 0 100 200 300 Pump detuning (MHz) N a + C s s u r v i v a l S = 283(9) MHz, P S = 3.2 mW FIG. S2. Autler-Townes splitting of | c Σ , v = 26 , J = 1 , M J = 1 i used for calibration of Stokes laser Rabi frequency. from Ref. [S2]. The reduced fidelity of Feshbach molecule creation arises from an axial misalignment of the dualspecies optical tweezers, which primarily leads to heating of the Na atom during the merge process. Because Nais much lighter than Cs, heating of the Na atom primarily contributes to excitation of the relative motional degreeof freedom, as opposed to the COM. Using the Feshbach molecule creation fidelity as a proxy for thermometry, weestimate that the COM ground state fraction of Feshbach molecules under present conditions is 75(5)%. In thefollowing, we neglect state changes of the ∼
25% initially motionally excited molecules.Two main effects cause COM motional excitation of the molecule during Raman transfer. The first arises from thedifferential wavenumber of the pump and Stokes lasers, ∆ k . In the Raman transfer process, motional sidebands ofthe COM degree of freedom are not resolved: The Raman Rabi frequency is of order Ω R ≈ π ×
200 kHz, while theaxial trap frequency during transfer is ω z ≈ π × ~ ∆ k . The mean occupation number resulting from this kick is ¯ n ≈ . α Na+Cs (1064 nm)] = 1397 a from the sum of atomic polarizabilities, while the X Σ polarizabilityis Re[ α X Σ (1064 nm)] = 936 a from [S3]. The ratio of trap depths is then 1397 / ≈ .
5, so that ratio of trapfrequencies is √ . f below, and the ratio of harmonic oscillator lengths is (1 . / . The probability ofremaining in the motional ground state is P ← = | h n x = 0 , n y = 0 , n z = 0 | e i ∆ kz | n x = 0 , n y = 0 , n z = 0 i | . (S.2)The initial Feshbach molecule COM spatial wavefunction is essentially projected onto the rovibrational ground stateCOM wavefunction. Because the Raman transfer lasers propagate along the z axis, we can separate the integrals andevaluate them straightforwardly, finding P ← = 8 f / ( f + 1) exp (cid:18) − f η f + 2 (cid:19) , (S.3)where η = ∆ k z ho ≈ .
413 is the Lamb-Dicke parameter. We find P ← ≈ . × ∗ W. B. C. and J. T. Z. contributed equally to this work. † [email protected][S1] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. , 633 (2005).[S2] J. T. Zhang, Y. Yu, W. B. Cairncross, K. Wang, L. R. B. Picard, J. D. Hood, Y.-W. Lin, J. M. Hutson, and K.-K. Ni,Phys. Rev. Lett. , 253401 (2020).[S3] R. Vexiau, D. Borsalino, M. Lepers, A. Orban, M. Aymar, O. Dulieu, and N. Bouloufa-Maafa, Int. Rev. Phys. Chem.36