††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: These authors contributed equally.

# Molecular assembly of ground state cooled single atoms

L. R. Liu  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.  J. D. Hood
Department of Chemistry and Chemical Biology, Harvard University, Cambridge,
Massachusetts, 02138, USA.  Department of Physics, Harvard University,
Cambridge, Massachusetts, 02138, USA.  Harvard-MIT Center for Ultracold Atoms,
Cambridge, Massachusetts, 02138, USA.  Y. Yu  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.
J. T. Zhang  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.  K. Wang
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.  Y.-W. Lin  Department of Chemistry and
Chemical Biology, Harvard University, Cambridge, Massachusetts, 02138, USA.
Department of Physics, Harvard University, Cambridge, Massachusetts, 02138,
USA.  Harvard-MIT Center for Ultracold Atoms, Cambridge, Massachusetts, 02138,
USA.  T. Rosenband  Department of Physics, Harvard University, Cambridge,
Massachusetts, 02138, USA.  K.-K. Ni<EMAIL_ADDRESS>Department of
Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts,
02138, USA.  Department of Physics, Harvard University, Cambridge,
Massachusetts, 02138, USA.  Harvard-MIT Center for Ultracold Atoms, Cambridge,
Massachusetts, 02138, USA.

###### Abstract

We demonstrate full quantum state control of two species of single atoms using
optical tweezers and assemble the atoms into a molecule. Our demonstration
includes 3D ground-state cooling of a single atom (Cs) in an optical tweezer,
transport by several microns with minimal heating, and merging with a single
Na atom. Subsequently, both atoms occupy the simultaneous motional ground
state with 61(4)% probability. This realizes a sample of exactly two co-
trapped atoms near the phase-space-density limit of one, and allows for
efficient stimulated-Raman transfer of a pair of atoms into a molecular bound
state of the triplet electronic ground potential $a^{3}\Sigma^{+}$. The
results are key steps toward coherent creation of single ultracold molecules,
for future exploration of quantum simulation and quantum information
processing.

††preprint: APS/123-QED

##  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].

These prospects hinge on the precise generation and control of ultracold
molecules with well defined internal and motional quantum states. Many
approaches for trapping and cooling molecules to ultracold temperatures are
being pursued [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. Recent highlights
include rapid progress made with laser cooling of molecules [17, 18, 19, 20],
the creation of quantum degenerate gases of fermionic KRb [21], assembling
single molecules [22] and loading single molecules into an optical tweezer
array [23]. Bulk samples of ultracold molecules have already proven a
versatile platform, enabling the study of ultracold chemistry [24] and quantum
spin models [25]. In addition, strongly interacting phenomena can be explored
with a lower entropy gas and with single molecule addressability [26].
Inspired by techniques in optical single atom manipulation [27, 28, 29, 30,
31, 32], we proposed a realization through single particle control of
molecules without relying on collisions for cooling [33].

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 $a^{3}\Sigma^{+}$.

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.

Figure 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 $\mathbf{k_{R}}$
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 $\mu$m.

##  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 quasi-independent 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 [35, 29, 30]. 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 ground-state 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.

Figure 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 $\Delta n=+1(-1)$ sidebands for the
axial and two radial directions, respectively. Above: Spectra after sub-
optimal RSC reveals the $\Delta n=-1$ sidebands, and hence the motional
frequencies. The 3D ground state population is $P_{0}^{3D}=44(5)\%$. Below:
Spectra after cooling with optimized motional frequencies, yielding
$P_{0}^{3D}\geq 96(3)\%$.

In our scheme (Fig. 2), the Raman transition occurs between Cs ground-state
hyperfine levels $|F=4,m_{F}=-4;n\rangle$ and $|3,-3;n-1\rangle$, 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
$\Delta=2\pi\times 44$ GHz from the Cs $D_{2}$ line at 852 nm, and with Rabi
rates $\Omega_{F3}$ and $\Omega_{F4}$, respectively. The tweezer has a power
of 14.3 mW and beam waist of 0.84 $\mu$m. To achieve motional coupling, the
laser beams are arranged as shown in the inset of Fig. 2(a). This
configuration yields substantial two-photon momentum transfer,
$\Delta\vec{k}=\vec{k}_{F4(i)}-\vec{k}_{F3}$, while the energy difference
associated with the hyperfine level and motional state change is supplied by
their relative detuning, $\delta$. This resonance condition is maintained for
all relevant motional states, $n$.

The atom is initially prepared in $|4,-4\rangle$ by OP (independently of the
motional state, $n$). For this, we use $\sigma^{-}$-polarized beams resonant
with $|4,-3\rangle\rightarrow|4^{\prime},-4\rangle$ and
$|3,-3\rangle\rightarrow|4^{\prime},-4\rangle$ transitions, where the primed
levels denote sub levels of the $6P_{3/2}$ manifold of Cs. During the first
step of RSC, a Raman $\pi$-pulse drives the transition
$|4,-4;n\rangle\rightarrow|3,-3;n-1\rangle$. Subsequently, OP pumps the atom
to $|4,-4;n-1\rangle$. OP preserves the motional state with high probability.
Thus, in each RSC cycle, $n$ decreases on average. The process repeats until
the atom reaches the dark state $|4,-4;0\rangle$, thereby deterministically
preparing the internal and the motional quantum state of the atom.

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 $\pi$-pulse, which is inversely related to the peak effective
Raman Rabi rate $\Omega_{F3}\Omega_{F4}/2\Delta=2\pi\times 33~{}$kHz
($2\pi\times 7\,$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 $9.2~{}\mu$K, corresponding to a
mean axial motional quantum number $\bar{n}_{a}=9$, leads to non-negligible
occupation of levels up to $n_{a}\approx 40$.

Due to the $\sqrt{n}$ scaling of sideband transition strengths [36], it was
necessary to “sweep” the Raman pulse durations in descending order starting
from $n_{a}^{init}$=41. Furthermore, to overcome decoherence, which reduces
the transfer fidelity of each pulse, we repeat the sweep, but each time with a
smaller $n_{a}^{init}=\left\\{41,31,16,11,6\right\\}$. The entire process
takes $\approx 100$ ms.

We characterize two cooling experiments in Fig. 2(b): (1) sub-optimal cooling
was used with slightly off-resonant $\delta\neq\omega_{trap}$ to reveal the
location of the $\Delta n=-1$ sidebands. (2) optimal cooling is obtained by
setting $\delta=\omega_{trap}$, 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 $\Delta n=-1$ and $\Delta n=+1$ Raman
sideband transition heights. A successful transition changes the state from
$|4,-4\rangle$ to $|3,-3\rangle$ and is revealed by state selective imaging:
light that is resonant with the cycling
$|4,-4\rangle\rightarrow|5^{\prime},-5\rangle$ transition ejects only
$|4,-4\rangle$ atoms. The remaining atoms in $|3,-3\rangle$ are then imaged.
We obtain the average occupation number $\bar{n}$ from the ratio of sideband
heights via $I_{-1}/I_{+1}=\frac{\bar{n}}{\bar{n}+1}$. 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 $P_{0}^{3D}$.

This procedure yields
$\\{\bar{n}_{a},\bar{n}_{r1},\bar{n}_{r2}\\}=\\{0.03(3),0.00(1),0.01(1)\\}$,
corresponding to $P_{0}^{3D}\geq 96(3)\%$ for optimal cooling.

The signal contrast in Fig 2B does not reach unity due to the $\approx
300~{}\mu$s 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 $\dot{\Delta n_{a}}\approx 0.3$ 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 $\mu$m 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].

Figure 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 $2.95~{}\mu$m 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 $\mu$s 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 $\Delta n=-1$ 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.

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.

After running this sequence (in the absence of Na) followed by its time
reverse for detection, Raman sideband thermometry on the separated tweezer
shows minimal motional excitation of Cs
($\\{\Delta\bar{n}_{a},\Delta\bar{n}_{r1}\\}=\\{0.01(5),0.00(2)\\}$) (Fig.
3(b)).

We further explore different trap powers for merging of Cs and Na atoms into
one tweezer. To prevent spin-changing collisions [22], we first prepare Na in
$|2,2\rangle$ and Cs in $|4,4\rangle$. Then, we merge the atoms and measure
the joint axial ground state fraction $P_{n_{a}=0}^{Na}\times
P_{n_{a}=0}^{Cs}$ 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. 1.

The 976 nm beam can make Na spill from the 700 nm tweezer and gain kinetic
energy. This limits the ratio $P_{\mathrm{700nm}}/P_{\mathrm{976nm}}$ 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. 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 $P_{\mathrm{700nm}}/P_{\mathrm{976nm}}$ 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. 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 $\sim 1.5\%$ with the acoustic drive
frequency.

We choose powers of $P_{\mathrm{976nm}}=14.3$ mW and $P_{\mathrm{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 $P_{0}^{Na}\times
P_{0}^{Cs}=0.80(3)\times 0.76(4)=0.61(4)$. 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

Figure 4: (a) Level diagram for two-photon Raman transfer from an atom pair to
a weakly-bound molecule. Two lasers $L_{1}$ and $L_{2}$ with a frequency
difference $\delta$ derived from an AOM are phase coherent and drive the atoms
from the tweezer motional ground state ($v^{\prime\prime}=25$) to the weakly-
bound molecular state $a^{3}\Sigma^{+}(v^{\prime\prime}=24$ or
$v^{\prime\prime}=-1)$. A large detuning $\Delta$ from the excited state
$c^{3}\Sigma(v^{\prime}=0)$, which decays at a rate $\Gamma_{e}$, reduces
spontaneous emission during molecular transfer, which occurs when the two-
photon frequency difference $\delta$ is resonant with the binding energy. (b)
Photoassociation spectroscopy of the intermediate excited state. With the
laser $L_{2}$ off, the $L_{1}$ 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. The
$c^{3}\Sigma^{+}_{1}(v^{\prime}=0,J^{\prime}=2,F^{\prime}=7)$ state is
observed at 288,698.64(6) GHz. (c) Two-photon Raman resonance for transferring
single atoms to a molecule. With a detuning $\Delta=2\pi\times 3.2$ GHz, the
frequency difference $\delta$ of the $L_{1}$ and $L_{2}$ beams is scanned
around the binding energy of the $a^{3}\Sigma^{+}(v^{\prime\prime}=24)$ 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.

After the merge, both Na $|F=2,m_{F}=2\rangle$ and Cs $|4,4\rangle$ atoms
occupy the motional ground state of the same tweezer, which corresponds to the
vibrational level $v^{\prime\prime}=25$ of the combined tweezer potential and
electronic ground molecular potential $a^{3}\Sigma^{+}$ [33]. As shown in Fig.
4(a), we transfer the atom pair into the least-bound molecular state
$v^{\prime\prime}\\!=\\!-1$ (or $v^{\prime\prime}=24$) 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.

The creation of a molecule in the weakly-bound state is an important step
toward subsequent transfer to deeply-bound molecular states via Stimulated
Raman Adiabatic Passage (STIRAP) [8, 7, 9, 10, 11, 12, 13, 14, 15, 16]. In
previous work, weakly-bound molecules were produced using Fano-Feshbach
resonances. In this work, we instead use an all-optical technique [38] to
generalize the weakly-bound molecule production to atoms without suitable
Fano-Feshbach resonances.

We choose $v^{\prime}\\!=\\!0$ of the molecular potential $c^{3}\Sigma^{+}$ 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 near-threshold and trap states minimizes their contribution to
spontaneous emission [33]. We choose the weakly-bound ground state
$v^{\prime\prime}\\!=\\!-1$ as our target state because its FCF is the most
similar to that of the motional ground state ($v^{\prime\prime}\\!=\\!25$).
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 $c^{3}\Sigma^{+}$ 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
$\sigma^{+}$ polarized light and a beam radius of $w\approx 15$ $\mu$m, the
atom merge sequence was immediately reversed to separate the surviving atoms
for detection. The two-body 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
$c^{3}\Sigma_{\Omega=1}^{+}(v^{\prime}=0,J^{\prime}=2,F^{\prime}=7)$ state,
where $J^{\prime}$ is total angular momentum excluding nuclear spin, and
$F^{\prime}$ is the total angular momentum including nuclear spin. The
uncertainty is dominated by the wavemeter inaccuracy of 60 MHz. We also
observe the $J^{\prime}=1$ and $J^{\prime}=3$ rotational lines and fit them to
$v_{J^{\prime}}=v_{0}+BJ^{\prime}(J^{\prime}+1)$ to obtain a rotational
constant of $B=1.1$ GHz The lack of a $J^{\prime}=0$ state confirms
$\Omega=1$.

To achieve two-photon Raman transfer to the ground molecular state, we first
located the least-bound state
$a^{3}\Sigma^{+}(v^{\prime\prime}=24,N^{\prime\prime}=0,F^{\prime\prime}=6)$
via dark-resonance spectroscopy and calibrated the single-photon Rabi rates of
the two individual beams (Appendix E). We then increased the detuning $\Delta$
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
$\Delta=2\pi\times 3.2~{}$GHz. The two beams $L_{1}$ and $L_{2}$ propagate
along $\mathbf{k_{R}}$ as shown in Fig. 1, with beam radii
$\\{w_{0}^{x},w_{0}^{y}\\}=\\{10,23\\}~{}\mu$m 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
$v^{\prime\prime}=25$ 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
$\Gamma_{\text{Raman}}\approx 149~{}$Hz, larger than the Raman transfer rate
$\Omega_{R}=2\pi\times 50~{}$Hz. Although increasing the detuning $\Delta$
improves the ratio of Raman transfer to scattering rate, the fixed scattering
rate of $\Gamma_{\text{tweezer}}=30~{}$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 $\Omega_{1}\Omega_{2}$ can be
more than 200 times higher for the same beam power, thereby allowing $\Delta$
to increase to reduce off-resonant scattering, while maintaining the same
$\Omega_{R}$.

##  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 $a^{3}\Sigma^{+}$. In the future,
deriving the molecular transfer and tweezer beams from a single laser may
reduce off-resonant scattering 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 $B^{1}\Pi$ and $c^{3}\Sigma$.

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].

###### Acknowledgements.

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.

##  Appendix A Trajectory for merging two atoms into one tweezer

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

$x(t)=x_{\text{min
jerk}}(t,d,T)=d\left(10(\frac{t}{T})^{3}-15(\frac{t}{T})^{4}+6(\frac{t}{T})^{5}\right)$

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

$x(t)=\left\\{\begin{array}[]{lr}x_{\text{min jerk}}(t,2\Delta f,2\Delta
t)&{\text{for }}0\leq t\leq\Delta t\\\ \frac{15}{4}\frac{\Delta f}{2\Delta
t}&{\text{for }}\Delta t<t<T-\Delta t\\\ x_{\text{min jerk}}(t-T+2\Delta
t,2\Delta f,2\Delta t)+\alpha T\frac{15}{4}\frac{\Delta f}{2\Delta
t}&{\text{for }}T-\Delta t<t\leq T\end{array}\right\\}$

where $\Delta f=d/(2+\frac{15}{4}\frac{\alpha}{1-\alpha})$ and $\Delta
t=\frac{1}{2}T(1-\alpha)$ are the distance covered and time elapsed,
respectively, of the minimum jerk trajectory portion, and $\alpha$ 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 $d=2.5~{}\mu$m, $T=7.6$ ms, and $\alpha=0$. For the
data in Fig 3C, we use $d=2.95~{}\mu$m and $\alpha=0.95$. We find the hybrid
trajectory is more robust against parametric heating.

##  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 $\Delta t$ and position grid spacing $\Delta x$.
The accuracy of the split operator method is then set by $[T,V]\Delta t^{2}$,
where $T$ and $V$ are the kinetic and potential energy operators,
respectively. For the simulation data presented here, we use a time step of
$\Delta t=0.1~{}\mu$s and spatial grid spacing $\Delta x$ = 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 $\mu$m and 4.875 $\mu$m ($z_{R}=1.006~{}\mu$m)
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.

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

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
$10\times$ faster (i.e. 2.95 $\mu$m 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-563AHF3.11 which 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 $FSR=97.5$ kHz. This is consistent with $FSR=v/2L$ where the
length of the acousto-optic crystal $L\approx 2$cm and the speed of sound is
$v=3.63~{}mm/\mu$s.

Scanning the RF drive frequency by 9.44 MHz moves the 976 nm trap 2.95 $\mu$m
in the focal plane.

Therefore, the acoustic cavity causes the trap depth to oscillate at a
frequency $v_{move}~{}9.44$ MHz $/(FSR\times 2.95~{}\mu$m), 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.

##  Appendix D Simulated tweezer power 2D scan

Figure 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.

We perform a numerical simulation of the dynamics of merging atoms into one
tweezer with different 700 nm and 976 nm tweezer powers. This yields the plots
in Fig 6A and B for Na and Cs, respectively. We find that heating regions
arising from double-well for Na and anti-trapping of Cs are qualitatively
reproduced (discrepancy in the exact size of the heating regions are
attributed to aberrations of the tweezers which cause the actual trap depth to
be different than expected).

We observe more overall heating in the experimental data compared to
simulation, even in the regions that have no specific heating mechanism. This
is likely caused by axial misalignment, which we estimate to be about 1.5
$\mu$m in this experiment.

##  Appendix E Calibration of $\Omega_{1}$ and $\Omega_{2}$

We use the vector light shift of the 1038 nm beam on the Cs atom to calibrate
the intensity. When the beam power $P_{PA}=20~{}$mW, the vector light shift
was $\Delta_{VLS}=2\pi\times 35.7$ kHz. We did PA to the
$c^{3}\Sigma^{+}(v^{\prime}=0)$ and measured a PA rate of $K_{PA}=1/0.35$ ms.

The excited state lifetime is assumed to be $\Gamma_{e}=1/30.4~{}$ns, the same
as that of Cs $6^{2}P_{3/2}$.

Therefore $\Omega_{1}=\sqrt{\Gamma_{e}K_{PA}}=2\pi\times 49~{}$kHz.

For arbitrary vector lightshift $\Delta_{VLS}$,
$\Omega_{1}=\sqrt{\frac{\Delta_{VLS}}{2\pi\times 35.7~{}kHz}}2\pi\times
49~{}kHz$.

Assuming no change in PA beam alignment, for arbitrary PA beam power $P_{PA}$,
$\Delta_{VLS}$, $\Omega_{1}=\sqrt{\frac{P_{PA}}{20~{}mW}}2\pi\times
49~{}kHz=2\pi\times 11~{}kHz/\sqrt{mW}$.

For the Raman transfer, we had a $\Delta_{VLS}=2\pi\times 16.3~{}$kHz, so
$\Omega_{1}=2\pi\times 33~{}$kHz.

The theoretical FCF’s yield $\Omega_{2}=292\times\Omega_{1}$.

##  Appendix F Using equal Raman beam powers

In the limits $d_{2}>>d_{1}$ (the bound-bound transition is much stronger than
the free-bound transition) and $\Delta>>E_{B}/\hbar$, the differential AC
stark shift and scattering rate are both proportional to
$(P_{1}+P_{2})d_{2}^{2}$.

1\. Minimizing the number of photons scattered per $\pi-$time. The time
required for a coherent transfer to occur is the so-called $\pi$-time
$t_{\pi}$, where $\Omega_{R}t_{\pi}=\pi$. This is given by

$t_{\pi}=(2\pi\Delta)/(\sqrt{P_{1}P_{2}}d_{1}d_{2})$

where $d_{i}=\Omega_{i}/\sqrt{P_{i}}$ is proportional to the matrix element
for the transition addressed by $\Omega_{i}$.

Therefore, we want to minimize

$(P_{1}+P_{2})d_{i}^{2}/(\sqrt{P_{1}P_{2}}d_{1}d_{2})$

which occurs when $P_{1}=P_{2}$.

2\. Minimizing the decoherence due to fluctuating differential AC Stark shift.
In the experiment, we assume that the total Raman beam power stability is some
fixed fraction of the total power, $dP_{tot}\propto P_{tot}=P_{1}+P_{2}$.
Differential fluctuations of the Raman resonance must be small compared to the
spectral width of a coherent Raman transition, proportional to the Raman Rabi
rate, $(\sqrt{P_{1}P_{2}}d_{1}d_{2})/(2\Delta)$. Therefore, we also want to
minimize

$(P_{1}+P_{2})/(\sqrt{P_{1}P_{2}})$

As before, this occurs when $P_{1}=P_{2}$.

##  Appendix G Scattering rate of molecules due to tweezer

We calculate the scattering rate from the least-bound state
$v^{\prime\prime}=24$ of the tweezer as a function of tweezer frequency in
Fig. 7. The calculation assumes a tweezer beam power of 15 mW, and a beam
waist of $0.8~{}\mu$m, and transmission through the objective and glass cell
of 0.22. The calculation includes vibrational level $v^{\prime\prime}=24$ of
the $a^{3}\Sigma^{+}$ ground state and a complete basis of vibrational
eigenstates derived from the $c^{3}\Sigma^{+}$ excited state molecular
potentials which are embedded in an isotropic harmonic well with a trap
frequency of 80 kHz, which is the geometric mean of the experiment axial and
radial trapping frequencies. The dipole matrix elements are assumed to be
$3~{}e~{}a_{0}$ times the relevant wavefunction overlap.

Figure 7: Scattering rate due to the 15 mW 976 nm (307 THz) tweezer from
$a^{3}\Sigma(v^{\prime\prime}=24)$. The calculation is performed using the
$c^{3}\Sigma$ potential from Ref. [39] and $a^{3}\Sigma$ potential from Ref.
[47]. The peaks correspond to different vibrational states of the
$c^{3}\Sigma$ potential.

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