0 A pr 2 01 9 Molecular assembly of ground state cooled single atoms

L. R. Liu, 2, 3, ∗ J. D. Hood, 1, 3, ∗ Y. Yu, 2, 3, ∗ J. T. Zhang, 2, 3 K. Wang, 2, 3 Y.-W. Lin, 1, 3 T. Rosenband, and K.-K. Ni 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. (Dated: May 1, 2019)


I. INTRODUCTION
Building up complex many-body systems from simpler, well-understood constituents is a promising approach toward understanding and controlling quantum mechanical behavior.Using ultracold molecules as building blocks would allow new explorations of quantum chemical dynamics [1], novel quantum many-body phases [2], and quantum computation [3][4][5].
In this paper, we experimentally demonstrate key steps toward such an "ultracold molecular assembler" [33].We obtain full quantum state control including cooling, transport, and merging of two different single atoms.We perform two-photon dark resonance spectroscopy to locate the least-bound NaCs molecular state of the electronic triplet ground potential a 3 Σ + .We then transfer two single ground state-cooled atoms in the same tweezer to the weakly-bound molecular state using a two-photon Raman pulse.In the following sections, we detail each experimental assembly step.

II. CONTROLLING THE QUANTIZED MOTION OF ATOMS
A schematic of the apparatus is shown in Fig. 1.As described in our previously work [22], we generate two tweezer traps at different wavelengths for quasiindependent manipulation of single Na and Cs atoms.One of the beams is steerable, so that initially separate tweezer traps can be merged.Single-atom fluorescence images confirm simultaneous trapping of single Na and Cs atoms side-by-side as shown in Fig. 1.
Using standard polarization gradient cooling (PGC), it is possible to cool the motion of single Cs or Na atoms to an average of tens of quanta in a tight tweezer trap.To further cool the atoms into the lowest motional state, we use 3-dimensional Raman Sideband Cooling (3D RSC), first demonstrated with single ions [34] and more recently with single neutral atoms [29,30,35].We operate in the resolved sideband regime where the linewidth of the cooling transition is less than the trap frequency (10-100's of kHz).We have previously demonstrated groundstate cooling of single Na [32].Here, we demonstrate 3D RSC of a single Cs atom in an optical tweezer.To our knowledge, we report the highest 3D ground-state probability for single atoms in tweezers to date.
The RSC sequence consists of two steps: a coherent two-photon Raman transition that connects two internal states while removing a motional quantum, and an optical pumping (OP) step that re-initializes the internal state of the atom.The two steps are repeated until the atom reaches the motional ground state.
In our scheme (Fig. 2), the Raman transition occurs between Cs ground-state hyperfine levels |F = 4, m F = −4; n and |3, −3; n − 1 , which are about 9.2 GHz apart.Here, n is the motional quantum number.The transition is driven by two phase-locked diode lasers, "F3" and "F4", both red-detuned by ∆ = 2π × 44 GHz from the Cs D 2 line at 852 nm, and with Rabi rates Ω F 3 and Ω F 4 , respectively.The tweezer has a power of 14.3 mW and beam waist of 0.84 µm.To achieve motional coupling, the laser beams are arranged as shown in the inset of Fig. 2(a).This configuration yields substantial twophoton momentum transfer, ∆ k = k F 4(i) − k F 3 , while the energy difference associated with the hyperfine level and motional state change is supplied by their relative detuning, δ.This resonance condition is maintained for all relevant motional states, n.
We switch between the three Raman F4(i) directions in the sequence i = 3, 1, 2, 1 to cool the atomic motion along all three axes of the tweezer.The tweezer potential has a cigar shape with two near-degenerate, tightly confined.radial directions and a loosely confined axial (along the tweezer beam propagation) direction.
The linewidth of the Raman transition is Fourier broadened due to the finite duration of a π-pulse, which is inversely related to the peak effective Raman Rabi rate Ω F 3 Ω F 4 /2∆ = 2π × 33 kHz (2π × 7 kHz) for radial (axial) trap axes.The smaller energy splitting of the axial motion necessitates a smaller Raman coupling along that direction.An 8.6 G magnetic field is applied throughout RSC along the OP propagation direction to define the quantization axis.
All Raman pulses in this experiment for cooling and spectroscopy use a Blackman window temporal intensity profile to reduce off-resonant excitation of the carrier.The starting temperature of 9.2 µK, corresponding to a mean axial motional quantum number na = 9, leads to non-negligible occupation of levels up to n a ≈ 40.
Due to the √ n scaling of sideband transition strengths [36], it was necessary to "sweep" the Raman pulse durations in descending order starting from n init a =41.Furthermore, to overcome decoherence, which reduces the transfer fidelity of each pulse, we repeat the sweep, but each time with a smaller n init a = {41, 31, 16, 11, 6}.The entire process takes ≈ 100 ms.
We characterize two cooling experiments in Fig. 2(b): (1) sub-optimal cooling was used with slightly offresonant δ = ω trap to reveal the location of the ∆n = −1 sidebands.(2) optimal cooling is obtained by setting δ = ω trap , as determined by the sideband locations in (1).
To characterize the cooling performance, we use sideband thermometry [34].Following RSC, we measure the ratio of ∆n = −1 and ∆n = +1 Raman sideband transition heights.A successful transition changes the state from |4, −4 to |3, −3 and is revealed by state selective imaging: light that is resonant with the cycling |4, −4 → |5 ′ , −5 transition ejects only |4, −4 atoms.The remaining atoms in |3, −3 are then imaged.We obtain the average occupation number n from the ratio of sideband heights via I −1 /I +1 = n n+1 .By assuming a thermal distribution, we extract a temperature and a ground state probability along each axis.The product of the ground state probabilities in all three dimensions gives the 3D ground state probability P 3D 0 .This procedure yields {n a , nr1 , nr2 } = {0.03(3),0.00(1), 0.01(1)}, corresponding to P 3D 0 ≥ 96(3)% for optimal cooling.The signal contrast in Fig 2B does not reach unity due to the ≈ 300 µs coherence time for driving motional sideband transitions.Furthermore, different pulse durations were used on the two radial axes, leading to a further difference in contrast.However, the sideband ratios, used to extract the final ground state population, are unaffected.
A final consideration is that any wait time between the end of RSC and molecule formation needs to be minimized because the atoms can be heated by off-resonantly scattering photons from their respective tweezers.This occurs at a rate of ∆ n a ≈ 0.3 Hz.To avoid unnecessary waiting, we perform the Na and Cs RSC sequences concurrently so that they end at the same time.We have verified experimentally that RSC of one species does not affect the atom of the other species.

III. PREPARING BOTH NA AND CS IN THE GROUND STATE OF THE SAME TWEEZER
As shown in Fig. 1, two optical tweezers trap a single Cs and Na atom approximately 3 µm apart.Both tweezer beams are combined on a dichroic mirror and focused by a NA= 0.55 objective.The position of the Cs tweezer can be moved by changing the drive frequency of an upstream acousto-optic deflector (AOD).While merging two separately confined identical ground-state atoms into one potential well requires delicate quantum tunneling [37], merging different atomic species is more straightforward.Due to their different atomic polarizability as a function of wavelength, two different color optical tweezers allow the two atoms to be manipulated quasi-independently [22].
One tweezer beam at a wavelength of 700 nm confines Na at the intensity maximum while repelling Cs.A second tweezer beam at a wavelength of 976 nm strongly confines Cs while weakly attracting Na.As shown in Fig. 3(a), translation of the 976 nm beam to overlap the 700 nm beam, followed by gradual turn-off of the 700 nm beam leaves the two atoms confined in the same tweezer trap, all within 10 ms.The exact trajectory is detailed in Appendix A.
We further explore different trap powers for merging of Cs and Na atoms into one tweezer.To prevent spinchanging collisions [22], we first prepare Na in |2, 2 and Cs in |4, 4 .Then, we merge the atoms and measure the joint axial ground state fraction P N a na=0 × P Cs na=0 as a function of beam powers (Fig. 3(c)).We identify three issues that can cause excess heating during the merge and require careful beam-power selection to overcome: 1.The 976 nm beam can make Na spill from the 700 nm tweezer and gain kinetic energy.This limits the ratio P 700nm /P 976nm to be above 0.37, indicated by the red triangle in Fig. 3(c), and the left panel at 5.7 ms in Fig. 3(a).
2. The 700 nm beam can dominate the 976 nm beam and repel Cs from the trap.This limits the power ratio of the beams P 700nm /P 976nm to be below 2.7, indicated by the left purple shaded triangle in Fig. 3(c), and the right panel at 8.55 ms in Fig. 3a.
3. Modulation in the tweezer power during trap movement causes parametric heating of the atoms.A weak acoustic standing wave in the AOD crystal results in an overall efficiency that modulates ∼ 1.5% with the acoustic drive frequency.
We choose powers of P 976nm = 14.3 mW and P 700nm =7.1 mW (also used in Fig. 3(b)) for all subsequent experiments.These powers yield the trap potentials depicted by the solid lines in Fig. 3(a)), i.e., approximately 2 mK for Cs and 1 mK for Na, respectively.We characterize with 3D Raman sideband thermometry that we have prepared two atoms in the same tweezer with a phase space density (PSD) of P N a 0 ×P Cs 0 = 0.80(3) × 0.76(4) = 0.61 (4).In this experiment, lower optical pumping fidelity resulted in a higher initial Cs temperature as compared to Sec. 2.

IV. TWO-PHOTON RAMAN TRANSFER TO THE LEAST-BOUND MOLECULAR GROUND STATE
After the merge, both Na |F = 2, m F = 2 and Cs |4, 4 atoms occupy the motional ground state of the same tweezer, which corresponds to the vibrational level v ′′ = 25 of the combined tweezer potential and electronic ground molecular potential a 3 Σ + [33].As shown in Fig. 4(a), we transfer the atom pair into the least-bound molecular state v ′′ = −1 (or v ′′ = 24) via a two-photon Raman pulse using two beams L 1 and L 2 that couple the initial and final states to a single intermediate electronic excited state.
We choose v ′ = 0 of the molecular potential c 3 Σ + as the excited intermediate state because it has suitable Franck-Condon factors (FCF's) with both the initial and final states, and because its large detuning from the nearthreshold and trap states minimizes their contribution to spontaneous emission [33].We choose the weakly-bound ground state v ′′ = −1 as our target state because its FCF is the most similar to that of the motional ground state (v ′′ = 25).The spontaneous emission from the excited state during Raman transfer is proportional to the ratio of the FCF's of the two ground states to the excited state.While a STIRAP pulse sequence could also potentially be used for this transfer, we have found from simulations that large Stark shifts of the two-photon detuning and the longer required duration result in poor efficiency.
Initial search for the intermediate excited state relied on photoassociation (PA) spectroscopy of the two atoms.Guided by the c 3 Σ + potential curve from Ref. [39], we scanned the frequency of a tunable diode laser around 1038 nm wavelength (L 1 ) until the laser was resonant with the excited state, and molecule formation was indicated by simultaneous atom loss.
Specifically, after illuminating the atoms for 75 ms with 15 mW of σ + polarized light and a beam radius of w ≈ 15 µm, the atom merge sequence was immediately reversed to separate the surviving atoms for detection.The twobody loss spectrum is shown in Fig. 4(b).The Lorentzian fit gives a transition frequency of 288,698.54(6)GHz, which we identify as the c 3 Σ + Ω=1 (v ′ = 0, J ′ = 2, F ′ = 7) state, where J ′ is total angular momentum excluding nuclear spin, and F ′ is the total angular momentum including nuclear spin.The uncertainty is dominated by the wavemeter inaccuracy of 60 MHz.We also observe the J ′ = 1 and J ′ = 3 rotational lines and fit them to v J ′ = v 0 + BJ ′ (J ′ + 1) to obtain a rotational constant of B = 1.1 GHz The lack of a J ′ = 0 state confirms Ω = 1.
To achieve two-photon Raman transfer to the ground molecular state, we first located the least-bound state a 3 Σ + (v ′′ = 24, N ′′ = 0, F ′′ = 6) via dark-resonance spectroscopy and calibrated the single-photon Rabi rates of the two individual beams (Appendix E).We then increased the detuning ∆ in order to reduce population of the excited state, which decays rapidly.
Figure 4c shows the Raman resonance for a pulse length of 100 ms and ∆ = 2π × 3.2 GHz.The two beams L 1 and L 2 propagate along k R as shown in Fig. 1, with beam radii {w x 0 , w y 0 } = {10, 23} µm and identical beam powers of 15 mW to minimize scattering (Appendix F).The resonance is fit to a Lorentzian centered at 298.0795 (6) MHz with a FWHM of 8(2) kHz.The 70(10)% transfer efficiency matches closely to the relative ground-state fraction of the Na+Cs atom pair, while the 21(2)% background level can be explained by spontaneous Raman scattering of the tweezer light from the v ′′ = 25 state, followed by a spin-changing collision [22].
We have not yet observed coherent atom-molecule oscillations between the initial and final state and believe the main source of decoherence is off-resonant scattering of the Raman light from the least-bound molecular state.For the above conditions, this scattering rate is Γ Raman ≈ 149 Hz, larger than the Raman transfer rate Ω R = 2π × 50 Hz.Although increasing the detuning ∆ improves the ratio of Raman transfer to scattering rate, the fixed scattering rate of Γ tweezer = 30 Hz due to the tweezer (Appendix G) provided a further constraint.
A potential solution for future work is replacing the 976 nm tweezer with a 1038 nm tweezer that can also serve as the molecular transfer beam.Due to the tight focusing of the tweezer, the product Ω 1 Ω 2 can be more than 200 times higher for the same beam power, thereby allowing ∆ to increase to reduce off-resonant scattering, while maintaining the same Ω R .

V. SUMMARY AND OUTLOOK
We have described experimental steps towards coherent assembly of single molecules from individual atoms.Starting with side-by-side trapping of the constituent atoms (Cs and Na) in optical tweezers, we have demonstrated ground-state cooling of Cs to its 3D ground state (96 %) and merging single Cs and Na atoms into the same tweezer while maintaining both atoms in the motional ground state with 61 % probability.These tools of dual-species single-atom manipulation can be extended to other species and tweezer wavelengths, providing a valuable resource to investigate interactions, collisions, and coherent spectroscopy and creation of molecules.
With two atoms in a tweezer, we have probed their electronic ground and excited molecular potentials.The resulting information enabled two-photon Raman transfer of 70 % of the atom pairs into the least bound molecular state of the triplet ground electronic potential a 3 Σ + .In the future, deriving the molecular transfer and tweezer beams from a single laser may reduce off-resonant scat-tering of the transferred molecule, which is otherwise long-lived.The transfer from the weakly-bound state to the ro-vbirational ground state could then be achieved by performing STIRAP with an excited state from the mixed potentials B 1 Π and c 3 Σ.
For studies of ultracold chemistry, quantum information, and many-body physics, the number of atom pairs could be scaled up by employing an array of single atom tweezer traps as a starting point [40,41].

ACKNOWLEDGMENTS
This work is supported by the Arnold and Mabel Beckman Foundation, as well as the NSF (PHYS-1806595), the AFOSR Young Investigator Program, the Camille and Henry Dreyfus Foundation, and the ARO DURIP (W911NF1810194).J.T.Z acknowledges support from the NDSEG fellowship.The speed at which we choose to transport a Cs atom in the 976 nm tweezer and subsequently merge it with the 700 nm tweezer is constrained by two main factors: (1) minimizing heating due to jerk (time-derivative of acceleration) at the endpoints, and (2) avoiding trap depth oscillations at a frequency that could cause parametric heating [42].
To address (1), we use the so-called "minimum-jerk trajectory" [43] to transport Cs.It is designed to translate the equilibrium point of a classical harmonic oscillator with minimal motional excitation.The displacement x as a function of time t is given by where d is the total distance traveled and T is the total move time.However, the minimum jerk trajectory has a variable moving speed that is problematic for constraint (2).Because the tweezer is transported by sweeping the RF frequency that drives the AOD in Fig. 1, the trap depth oscillations arising from imperfections of the AOD (see Section C) would sweep through a band of frequencies and be more likely to excite a parametric heating resonance.
Therefore, we devise a hybrid trajectory which uses constant velocity in the middle and minimum jerk at the endpoints.Thus, the oscillation frequency is constant for the middle part and the parameters can be more easily chosen to avoid parametric resonances.The displacement as a function of time for the hybrid trajectory is given by

Appendix B: Simulating merging of two tweezers
To find the fastest speed at which we can merge single Na and Cs atoms tweezers into the same tweezer, we simulate their time evolution using the split operator method [44].
The atomic polarizabilities are taken from Table 2 of Ref. [45].The initial and final trap eigenfunctions are calculated with the Fourier Grid method [46].The ground state population at the end of the sequence is given by the squared overlap of the wavefunction following the time evolution with the ground state of the final trap.The accuracy of these simulations is determined by the time step ∆t and position grid spacing ∆x.The accuracy of the split operator method is then set by [T, V ]∆t 2 , where T and V are the kinetic and potential energy operators, respectively.For the simulation data presented here, we use a time step of ∆t = 0.1 µs and spatial grid spacing ∆x = 1 nm, and have checked that the results of the simulation converge at these values.
The tweezer waist is estimated from scalar Gaussian beam propagation simulation of the input beam (whose waist we can measure), including the effect of the beam clipping on the objective aperture.The simulated electric field intensities along the radial and axial directions are fitted independently to those of a Gaussian beam.We find that doing so gives an input beam that is Gaussian except that the Rayleigh range is scaled by 1.39, to account for aberrations.
For the 976 nm tweezer, for 15 mW measured before a final beam expanding telescope, 9 mm waist input before the objective, the radial and axial waists at the tweezer are 0.844 µm and 4.875 µm (z R = 1.006 µm) respectively.We match the calculated and measured radial and axial trapping frequencies of 125.7 kHz and 24.1 kHz respectively, by inserting a transmission coefficient T = 0.27 by hand.This includes transmission through many optical elements: dichroics, objective, glass cell, and electrode plate surfaces.Similarly, for the 700 nm tweezer, 6.6 mm input waist, 48 mW power before a final beam expanding telescope, T= 0.36 gives 530.5 kHz and 92.7 kHz radial and axial trap frequencies, in good agreement with measurements.
By scanning the merge time and calculating the final wavefunction overlap with the motional ground state wavefunction, we find that we can scan more than 10× faster (i.e.2.95 µm in < 1 ms) using a minimum jerk trajectory and still remain in the ground state with > 99.9% probability (see Fig 5), provided there are no technical imperfections.

Appendix C: Derivation of trap depth oscillation frequency
We use an IntraAction A2D-563AHF3.11which can deflect the beam in two dimensions.The electro-optic medium is not angle-cut, and forms an acoustic cavity.The amplitude of the intracavity field affects the AOD diffraction efficiency and depends on RF drive frequency.Therefore, as the RF drive frequency is scanned to move the tweezer, the trap depth oscillates, in this case by 1%.
By scanning the tweezer position along the merge axis and measuring the period of the intensity fringes, we measure the free spectral range of the acoustic cavity to be F SR = 97.5 kHz.This is consistent with F SR = v/2L where the length of the acousto-optic crystal L ≈ 2cm and the speed of sound is v = 3.63 mm/µs.
Scanning the RF drive frequency by 9.44 MHz moves the 976 nm trap 2.95 µm in the focal plane.
Therefore, the acoustic cavity causes the trap depth to oscillate at a frequency v move 9.44 MHz /(F SR × 2.95 µm), where v move is the speed at which the trap moves.For our hybrid trajectory in Section III, the trap depth oscillation during the linear part is therefore 9.9kHz.).The calculation is performed using the c 3 Σ potential from Ref. [39] and a 3 Σ potential from Ref. [47].The peaks correspond to different vibrational states of the c 3 Σ potential.

FIG. 1 .
FIG. 1. Single atom trapping and transport for molecular assembly.(a) Schematic of apparatus.Two neighboring optical tweezers at 976 nm and 700 nm trap a single Cs (blue) and Na (orange) atom in the vacuum chamber.Both tweezer beams are combined on a dichroic mirror and focused by an objective.The 976 nm tweezer can be moved in the focal plane by changing the drive frequency of an upstream acousto-optic deflector (AOD).Once atoms are cooled and merged into the same tweezer, a laser propagating along kR transfers them into a bound molecular state in the presence of a quantization B field.(b) Experimental assembly steps of ultracold molecules demonstrated in this paper.A single Na and Cs atom are cooled, merged into the same trap, and transferred to a weakly-bound molecule.(c) Single-shot fluorescence image of single Na and Cs atoms in adjacent tweezers separated by 3 µm.

FIG. 2 .
FIG.2.3D motional control of a single Cs atom.(a) Level scheme for Cs RSC.F3 and F4 Raman beams coherently couple adjacent motional states to reduce motional energy, while optical pumping provides the dissipation needed for cooling.[Inset]Directions of laser beams.Switching Raman F4 beam directions allows addressing motion in all 3 dimensions.(b) 3D sideband thermometry for Cs after RSC.Black, blue, and red spectral peaks in the unshaded (shaded) region correspond to ∆n = +1(−1) sidebands for the axial and two radial directions, respectively.Above: Spectra after sub-optimal RSC reveals the ∆n = −1 sidebands, and hence the motional frequencies.The 3D ground state population is P 3D 0 = 44(5)%.Below: Spectra after cooling with optimized motional frequencies, yielding P 3D 0 ≥ 96(3)%.

FIG. 3 .
FIG.3.Merging atoms in two tweezers while maintaining quantum motional states.(a) Radial cuts of optical potential experienced by Na and Cs during the merge time sequence.Blue and orange lines show paths of the 976 nm and 700 nm tweezers, respectively.The 976 nm tweezer containing Cs is translated by 2.95 µm in 7.6 ms until it overlaps with the 700 nm tweezer.Then, the 700 nm tweezer power is linearly ramped from 48 mW to 0 mW in 1.5 ms, followed by a 50 µs wait.Dashed potential in the left 5.7 ms panel (marked by red square) shows the conditions for non-ideal tweezer powers, leading to spilling of Na.Dashed potential in the 8.55 ms panel (marked by red circle) shows the conditions for a different set of non-ideal tweezer powers giving rise to anti-trapping for Cs.(b) Raman sideband spectroscopy to characterize heating associated with atom transport.Top: A control experiment holding the atoms stationary for 18 ms.Bottom: After the round-trip merge sequence (the sequence shown in (a) followed by its time reverse) Dashed blue lines indicate expected position of ∆n = −1 sidebands.The round-trip sequence causes minimal heating.Inset: Coordinates of the transport direction vs a thermometry axis.Blue and orange circle represent 976 nm and 700 nm tweezers respectively.(c) Na+Cs joint axial ground state fraction after round-trip merge sequence as a function of 700 nm and 976 nm tweezer powers.The lower triangle in red corresponds to spilling of Na.Red square is an exemplary point in this regime, whose radial potential is plotted with a dashed line in the correspondingly marked panel in (a).Upper triangle in purple indicates anti-trapping of Cs.Red circle is an exemplary point, whose potential is plotted with a dashed line in the correspondingly marked panel in (a).Dark purple stripe shows parametric heating resonance (due to technical imperfection) during transport of Cs.Our usual operating point is indicated by the star.

FIG. 4 .
FIG. 4. (a)Level diagram for two-photon Raman transfer from an atom pair to a weakly-bound molecule.Two lasers L1 and L2 with a frequency difference δ derived from an AOM are phase coherent and drive the atoms from the tweezer motional ground state (v ′′ = 25) to the weakly-bound molecular state a 3 Σ + (v ′′ = 24 or v ′′ = −1).A large detuning ∆ from the excited state c 3 Σ(v ′ = 0), which decays at a rate Γe, reduces spontaneous emission during molecular transfer, which occurs when the two-photon frequency difference δ is resonant with the binding energy.(b) Photoassociation spectroscopy of the intermediate excited state.With the laser L2 off, the L1 laser drives the atoms to the excited molecular vibrational states when resonant.The Na + Cs two-body loss probability is measured as a function of the PA frequency for a 75 ms pulse duration.Thec 3 Σ + 1 (v ′ = 0, J ′ = 2, F ′ = 7)state is observed at 288,698.64(6)GHz.(c) Two-photon Raman resonance for transferring single atoms to a molecule.With a detuning ∆ = 2π × 3.2 GHz, the frequency difference δ of the L1 and L2 beams is scanned around the binding energy of the a 3 Σ + (v ′′ = 24) state.The Raman resonance is observed at 298.0795(6) MHz with a FWHM of 8(2) kHz, indicating transfer of the atom pair to the weakly-bound molecular state.
Appendix A: Trajectory for merging two atoms into one tweezer ∆t < t < T − ∆t x min jerk (t − T + 2∆t, 2∆f, 2∆t) + αT15   4   ∆f 2∆t for T − ∆t < t ≤ T ) and ∆t = 1 2 T (1 − α) are the distance covered and time elapsed, respectively, of the minimum jerk trajectory portion, and α is the fraction of the trajectory that is linear motion and can range from 0 (fully minimum jerk) to 1 (fully linear).For data in Fig 3B, we use d = 2.5 µm, T = 7.6 ms, and α = 0.For the data in Fig 3C, we use d = 2.95 µm and α = 0.95.We find the hybrid trajectory is more robust against parametric heating.

FIG. 5 .
FIG.5.Minimum merge time.We numerically simulate the motional excitation as a function of merge time with fixed trap depth.

FIG. 6 .
FIG. 6. Simulated 2D tweezer power scan.Numerical simulation of the axial ground state population for Na and Cs following the merge sequence described in Sec.III.All the fundamental heating mechanisms (delineated by purple lines) are qualitatively reproduced.