††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Raman Sideband Cooling of Molecules in an Optical Tweezer Array

Yukai Lu  Department of Physics, Princeton University, Princeton, New Jersey
08544 USA  Department of Electrical and Computer Engineering, Princeton
University, Princeton, New Jersey 08544 USA  Samuel J. Li  Department of
Physics, Princeton University, Princeton, New Jersey 08544 USA  Connor M.
Holland  Department of Physics, Princeton University, Princeton, New Jersey
08544 USA  Lawrence W. Cheuk<EMAIL_ADDRESS>Department of Physics,
Princeton University, Princeton, New Jersey 08544 USA

###### Abstract

Ultracold molecules, because of their rich internal structures and
interactions, have been proposed as a promising platform for quantum science
and precision measurement. Direct laser-cooling promises to be a rapid and
efficient way to bring molecules to ultracold temperatures. For trapped
molecules, laser-cooling to the quantum motional ground state remains an
outstanding challenge. A technique capable of reaching the motional ground
state is Raman sideband cooling, first demonstrated in trapped ions and atoms.
In this work, we demonstrate for the first time Raman sideband cooling of
molecules. Specifically, we demonstrate 3D Raman cooling for single CaF
molecules trapped in an optical tweezer array, achieving average radial
(axial) motional occupation as low as $\bar{n}_{r}=0.27(7)$
($\bar{n}_{z}=7.0(10)$). Notably, we measure a 1D ground state fraction as
high as 0.79(4), and a motional entropy per particle of $s=4.9(3)$, the lowest
reported for laser-cooled molecules to date. These lower temperatures could
enable longer coherence times and higher fidelity molecular qubit gates
desirable for quantum information processing and quantum simulation. With
further improvements, Raman cooling could also be a new route towards
molecular quantum degeneracy applicable to many laser-coolable molecular
species including polyatomic ones.

Ultracold molecules have been proposed as a new platform for exploring many
areas in physics ranging from simulation of quantum many-body Hamiltonians, to
quantum information processing, to precision measurements in searches for
physics beyond the Standard Model DeMille (2002); Carr  _et al._ (2009); Bohn
_et al._ (2017); Blackmore  _et al._ (2018). Yet, cooling and fully
controlling molecules have been long-standing experimental challenges. One
route to produce ultracold molecules is via assembly from atoms, for which
cooling techniques are well-developed. This approach has successfully been
used to produce bialkali molecules, enabling explorations in ultracold
chemistry De Miranda  _et al._ (2011); Liu  _et al._ (2021) and the creation
of quantum-degenerate molecular gases well-suited for studying long-ranged
interacting many-body systems De Marco  _et al._ (2019); Schindewolf  _et al._
(2022).

In contrast to assembly from ultracold atoms, methods that directly cool could
be broadly applicable to a large variety of molecular species including
polyatomic ones. In particular, direct laser-cooling of molecules has seen
great advances recently. Starting with molecular magneto-optical traps near
the Doppler limit of $\sim 100\,\mu\text{K}$ Barry  _et al._ (2014); Truppe
_et al._ (2017); Anderegg  _et al._ (2017); Collopy  _et al._ (2018); Vilas
_et al._ (2022), sub-Doppler cooling techniques have allowed experiments to
enter into the $\mu\text{K}$ regime Truppe  _et al._ (2017); Cheuk  _et al._
(2018); Caldwell  _et al._ (2019); Ding  _et al._ (2020); Vilas  _et al._
(2022); Langin  _et al._ (2021); Wu  _et al._ (2021); Hallas  _et al._ (2023).
Importantly, sub-Doppler cooling has enabled optically trapped molecular
samples with record densities Anderegg  _et al._ (2018); Cheuk  _et al._
(2018); Wu  _et al._ (2021) and arrays of single molecules trapped in optical
tweezers Anderegg  _et al._ (2019); Holland  _et al._ (2022a); Bao  _et al._
(2022).

Access to molecular samples at even lower temperatures could enable new
possibilities. For example, molecular tweezer arrays have recently emerged as
a promising platform for quantum science. Notably, recent work has shown high-
fidelity control over molecular positions and internal states, coherent
dipolar interactions, and implementation of an entangling two-qubit gate
Holland  _et al._ (2022a); Bao  _et al._ (2022). Yet, residual thermal motion
limits the achievable coherence times and gate fidelities. These limitations
can be largely eliminated by cooling to the motional ground state.

One technique capable of cooling to the motional ground state is Raman
sideband cooling (RSC) Heinzen and Wineland (1990). First, a Raman process
transfers a molecule initially in internal state $\left|\uparrow\right\rangle$
to $\left|\downarrow\right\rangle$ while removing $\Delta n$ quanta of
motional energy. Subsequently, optical pumping reinitializes the internal
state to $\left|\uparrow\right\rangle$ while largely preserving the motional
state. By iterating over these two steps, cooling is achieved. RSC was first
proposed and demonstrated for trapped ions and atoms in optical lattices
Monroe  _et al._ (1995); Hamann  _et al._ (1998); Kerman  _et al._ (2000), and
has since been used to cool single atoms in tweezer traps to their motional
ground states Kaufman  _et al._ (2012); Thompson  _et al._ (2013), for imaging
in quantum gas microscopes Cheuk  _et al._ (2015); Parsons  _et al._ (2015),
and to produce single molecules via assembly from two RSC-cooled atoms He  _et
al._ (2020); Zhang  _et al._ (2022a); Ruttley  _et al._ (2023). RSC also
provides a rapid and efficient (low-loss) all-optical method to create atomic
Bose-Einstein condensates Hu  _et al._ (2017); Urvoy  _et al._ (2019),
circumventing the need for evaporation.

Figure 1:  Raman Sideband Cooling Scheme. (a) Motional-changing two-photon
Raman transitions between $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$ are driven using laser beams detuned by
$\Delta\approx-2\pi\times 42~{}\text{GHz}$ from the $X^{2}\Sigma(v=0,N=1)\to
B^{2}\Sigma(v=0,N=0)$ transition. $\delta$ denotes the two-photon detuning.
Optical pumping into $\left|\uparrow\right\rangle$ is performed using light
addressing the $X^{2}\Sigma(v=0,N=1)\to A^{2}\Pi_{1/2}(v=0,J=1/2,+)$
transition. (b) A magnetic field $\vec{B}$ is applied in the radial plane, and
is oriented at an angle of $\theta$ relative to the polarization axis of the
tweezer light $(\hat{\varepsilon}\parallel\hat{x})$. Raman laser beams
$R_{1},R_{2}$ ($R_{3},R_{4}$) address the radial (axial) directions. $R_{1}$
and $R_{2}$ are optionally retro-reflected. Optical pumping light (OP) is
applied radially.

Raman sideband cooling of optically trapped and laser-cooled molecules faces
two main challenges that arise from the complex internal structure of
molecules Caldwell and Tarbutt (2020). First, state-dependent optical trapping
leads to inhomogeneous broadening of Raman transitions, preventing resolved
addressing of cooling sidebands and decreasing transfer efficiency. Second,
efficient optical pumping is difficult because of the large number of
molecular states and the degradation of free-space selection rules in deep
optical traps.

In this work, we demonstrate Raman sideband cooling of molecules for the first
time. We overcome the above challenges by devising a RSC scheme for CaF
molecules that provides both narrow Raman transitions and efficient optical
pumping. Our scheme does not require high magnetic fields, in contrast to the
one proposed in Caldwell and Tarbutt (2020). We demonstrate 3D Raman cooling
for CaF molecules trapped in an optical tweezer array and achieve average
motional quanta as low as $\bar{n}_{r}=0.27(7)$ and $\bar{n}_{z}=7.0(10)$ in
the radial and axial directions, respectively.

Our work begins with single CaF molecules that are cooled by
$\Lambda$-enhanced gray molasses Cheuk  _et al._ (2018) and trapped in a 1D
array of linearly polarized optical tweezers Anderegg  _et al._ (2019);
Holland  _et al._ (2022b, a). Raman beams are sent along the radial and axial
directions, and are near-detuned from the $X^{2}\Sigma(v=0,N=1)\rightarrow
B^{2}\Sigma(v=0,N=0)$ transition (Fig. 1). Optical pumping is achieved on the
$X^{2}\Sigma(v=0,N=1)\rightarrow A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ transition.

We identify a suitable pair of internal states
$\\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\\}$ for RSC.
Constrained by optical pumping, we consider optically cyclable states from
$X^{2}\Sigma(v=0,N=1)$. In free space, selection rules enable optical pumping
into the stretched states
$\left|\pm\right\rangle=\left|N=1,J=3/2,F=2,m_{F}=\pm 2\right\rangle$.
Specifically, these states are dark to $\sigma_{\pm}$ and $\pi$ light
addressing the $X^{2}\Sigma(v=0,N=1)\rightarrow A^{2}\Pi_{1/2}(v=0,J=1/2,+)$
transition. In deep tweezer traps, the trapping light can admix in bright
states and modify selection rules, degrading optical pumping. The admixture
can be reduced by providing a well-defined quantization axis with a magnetic
field $\vec{B}$ along the polarization axis of the trapping light Thompson
_et al._ (2013); Caldwell and Tarbutt (2020). However, at certain fields,
because of tensor ac Stark shifts, level crossings can occur, increasing
bright state admixtures. Our calculations indicate that
$\left|\uparrow\right\rangle=\left|-\right\rangle$ is immune from these
crossings at low fields and over a large range of trap depths, therefore
robustly providing low bright state admixtures Sup . In particular, the bright
state population admixture remains below $10^{-4}$ even for traps as deep as
$k_{B}\times 2\,\text{mK}$ at $B=4.4\,\text{G}$.

For $\left|\downarrow\right\rangle$, we seek a state that is connected by a
two-photon Raman transition to $\left|\uparrow\right\rangle$ and has minimal
differential ac Stark shifts with $\left|\uparrow\right\rangle$. Our
calculations indicate that
$\left|\downarrow\right\rangle=\left|N=1,J=3/2,F=1,m_{F}=0\right\rangle$
satisfies these requirements. In particular, while most $N=1$ states
experience fractional differential Stark shifts at the $10^{-1}$ level, those
between $\left|\uparrow\right\rangle$ and $\left|\downarrow\right\rangle$ are
suppressed to $10^{-2}$ even in deep traps with depths $\sim k_{B}\times
1\,\text{mK}$. The shifts can be further reduced by changing the angle
$\theta$ between $\vec{B}$ and the tweezer polarization, with $10^{-3}$
fractional shifts possible at specific “magic” angles. For example, at a trap
depth of $k_{B}\times 0.3\,\text{mK}$ and magnetic field of
$B=5.5~{}\text{G}$, the magic angle is $\theta\approx 57^{\circ}$.

Figure 2:  Raman Linewidths and Optical Pumping Characterization. (a) Raman
linewidths $\Gamma$ versus $\theta$, measured at a tweezer depth of
$k_{B}\times 326(7)\,\mu\text{K}$. $\Gamma$ is smallest at the magic angle
$\theta_{m}=56.3(3)^{\circ}$, indicated by the dashed vertical line.
Representative spectra versus two-photon detuning $\delta$, along with
Lorentzian fits (solid), are shown in the sub-panels. (b)
$\left|\uparrow\right\rangle$ population $P_{\uparrow}$ versus optical pumping
time $t$ at a tweezer depth of $k_{B}\times 930(20)\,\mu\text{K}$. Red
triangles (blue circles) show data for $\theta=0^{\circ}$
($\theta=\theta_{m}$). Solid curves show simultaneous fits to early and late
time dynamics. (c) The figures-of-merit $\kappa$ and $P_{s}$ at $\theta_{m}$
versus trap depth $U/U_{0}$ ($U_{0}=k_{B}\times 930(20)\,\mu\text{K}$) are
shown by blue circles, with solid lines as guides to the eye. The
corresponding values for $\theta=0^{\circ}$ are shown by the red shaded
regions and indicate significantly better optical pumping.

We experimentally verify the properties of
$\\{\left|\uparrow\right\rangle,\left|\downarrow\right\rangle\\}$ through
Raman spectroscopy and optical pumping dynamics. We first probe the
inhomogeneous broadening of Raman transitions arising largely from
differential Stark shifts. At a trap depth $U=k_{B}\times
326(7)\,\mu\text{K}$, we measure the linewidth $\Gamma$ of the carrier
($\Delta n=0$) transition using co-propagating Raman beams. We prepare
molecules in $\left|\uparrow\right\rangle$ and measure the population in
$\left|\downarrow\right\rangle$ ($P_{\downarrow}$) versus the two-photon
detuning $\delta$. At $\theta=0^{\circ}$ and $B=4.4~{}\text{G}$, we measure a
linewidth of $\Gamma=2\pi\times 26.5(3)\,\text{kHz}$. This is below the radial
trapping frequency $\omega_{r}=2\pi\times 117.3(4)\,\text{kHz}$, allowing
radial sidebands ($\Delta n\neq 0$) to be resolved. At $B=5.5~{}\text{G}$, as
a function of $\theta$, we find that $\Gamma$ reaches a minimum of $\approx
2\pi\times 7\,\text{kHz}$ at the magic angle of $\theta_{m}=56.3(3)^{\circ}$,
as predicted (Fig. 2(a)). Notably, at $\theta_{m}$, $\hbar\Gamma/U\approx
10^{-3}$, and the linewidth $\Gamma$ is below the axial trapping frequency
$\omega_{z}\approx 2\pi\times 26\,\text{kHz}$, allowing axial sidebands to be
resolved Sup .

Next, we measure optical pumping dynamics. Starting with molecules initially
distributed over all 12 hyperfine states of $X^{2}\Sigma(v=0,N=1)$, we measure
the $\left|\uparrow\right\rangle$ population ($P_{\uparrow}$) after variable
durations of optical pumping. At short times, $P_{\uparrow}$ increases as
molecules are pumped into $\left|\uparrow\right\rangle$, and subsequently
saturates to $P_{s}$. At long times, $P_{\uparrow}$ decreases due to molecular
loss arising from heating or decay into undetected states due to imperfect
darkness of $\left|\uparrow\right\rangle$ (Fig. 2(b)). The darkness of
$\left|\uparrow\right\rangle$, essential for efficient optical pumping, can
therefore be quantified by $\kappa=\tau_{1}/\tau_{2}$, where $\tau_{1}$
($\tau_{2}$) is the $1/e$ rise (fall) time of $P_{\uparrow}$. $P_{s}$ provides
a complementary measure of optical pumping efficiency. When $\theta=0^{\circ}$
and $B=4.4\,\text{G}$, $\kappa\approx 8\times 10^{3}$ and $P_{s}\approx 0.8$,
even at a deep trap depth of $k_{B}\times 930(20)\,\mu\text{K}$ (Fig. 2(c)).
In comparison, in the magic configuration ($\theta=\theta_{m}$,
$B=5.5~{}\text{G}$), we find that both $\kappa$ and $P_{s}$ decrease with
increasing trap depth, indicating degrading selection rules. For all tweezer
depths explored, $\kappa$ is a factor of 6 to 20 lower compared to that at
$\theta=0^{\circ}$. This shows that optimal optical pumping (at
$\theta=0^{\circ}$) and Raman linewidths (at $\theta=\theta_{m}$) cannot be
simultaneously achieved.

For Raman sideband cooling, deep tweezer depths help preserve the motional
state during optical pumping, which is critical for cooling. Although the
magic configuration at $\theta_{m}$ provides the narrowest Raman transitions,
optical pumping is severely degraded at deep depths. On the other hand, at
$\theta=0^{\circ}$, optical pumping is efficient even in deep tweezers. The
Raman linewidth is slightly broader but sufficient to resolve the radial
sidebands. We therefore choose to perform Raman cooling at $\theta=0^{\circ}$.

Figure 3:  Raman Sideband Spectra at $\theta=0^{\circ}$ and Adiabatic Trap
Lowering Curves. (a) Population transfer $P_{\downarrow}$ using radial Raman
beams versus $\delta$, the two-photon detuning from the carrier ($\Delta
n=0$). The solid blue line shows a fit using a sum of nine Lorentzians with an
offset. The black (red) dashed line marks the carrier ($\Delta n_{r}=-2$
sideband). (b) Population transfer $P_{\downarrow}$ using axial Raman beams
versus $\delta$. Solid curve is a guide to the eye. The red dashed line shows
the two-photon detuning $\delta=2\omega_{r}$ used during cooling. (c) Raman
cooling sequence consists of (i) Raman transfers at a tweezer depth of
$U_{R}=k_{B}\times 326(7)\,\mu\text{K}$, and (ii) optical pumping at a higher
depth of $U_{0}=k_{B}\times 930(20)\,\mu\text{K}$. (d) Probing temperature via
adiabatic trap lowering. The tweezer depth is lowered to $U_{\text{spill}}$
over $1\,\text{ms}$, held $10\,\text{ms}$ to allow hot molecules to escape,
and increased back to full depth for detection. The surviving fraction $f$
versus $U_{\text{spill}}/U_{0}$ is shown for no cooling (blue circles), radial
cooling (RC) (green diamonds), and radial/axial cooling (RAC) (red squares).
Inset: $f$ versus number of cooling cycles $N$ at a fixed lower depth
$U_{\text{spill}}=k_{B}\times 0.72(2)~{}\mu\text{K}$ (dashed line in main
plot).

We next verify Raman motional coupling by driving sideband transitions at
$\theta=0^{\circ}$. The Raman coupling between different motional states is
characterized by the Lamb-Dicke parameter $\eta=|\Delta\vec{k}|l/\sqrt{2}$,
where $\Delta\vec{k}$ is the difference in wave-vectors between the two Raman
beams and $l=\sqrt{\hbar/(m\omega)}$ is the harmonic oscillator length, $m$
being the molecular mass and $\omega$ the trapping frequency. With molecules
initially prepared in $\left|\uparrow\right\rangle$, we probe radial and axial
motional state transfer at $U=k_{B}\times 326(7)\,\mu\text{K}$,
$B=4.4\,\text{G}$. We pulse on Raman beams and measure $P_{\downarrow}$ versus
$\delta$. The radial Lamb-Dicke parameter is $\eta_{r}=0.46$ at this depth,
allowing us to observe resolved radial sidebands at $\delta=(\Delta
n_{r})\omega_{r}$ up to $|\Delta n_{r}|=4$ (Fig. 3(a)), where
$\omega_{r}=2\pi\times 117.3(4)\,\text{kHz}$. Axially, the weaker confinement
leads to a larger Lamb-Dicke parameter of $\eta_{z}=1.34$, allowing
significant motional coupling up to $|\Delta n_{z}|\sim 10$ (Fig. 3(b)). We
observe that the axial spectrum is significantly broader than the measured
carrier Raman linewidth, indicating that motion-changing Raman transfers are
indeed occurring.

Having demonstrated motional state-changing Raman transfer, we next construct
a radial cooling sequence consisting of two discrete steps: optical pumping
and Raman transfer on a cooling $(\Delta n<0)$ sideband. We optically pump at
a deep tweezer depth of $U_{0}=k_{B}\times 930(20)\,\mu\text{K}$ to minimize
increase in motional quanta. We estimate that $\sim 19$ photons are required
for optical pumping, increasing the energy by an equivalent of $\Delta
n_{r}^{\text{op}}\sim 1.3$ radial quanta. To attain net cooling, we therefore
address the $\Delta n_{r}=-2$ sideband. To obtain sufficient motional coupling
and reduce inhomogeneous broadening, we perform Raman transfer at a reduced
tweezer depth $U_{R}=k_{B}\times 326(7)\,\mu\text{K}$, the same depth where we
measured Raman linewidths. Each cooling cycle has a duration of
$0.65\,\text{ms}$ and the trap depths are ramped adiabatically over
$0.2\,\text{ms}$ between $U_{0}$ and $U_{R}$ (Fig. 3(c)).

To verify cooling, we first indirectly probe the temperature via adiabatic
reduction of the tweezer depth Cooper  _et al._ (2018); Tuchendler  _et al._
(2008). Hot molecules are spilled progressively as the trap is lowered. At a
fixed final trap depth $U_{\text{spill}}$, the surviving fraction $f$
increases with lower temperatures. After 90 cycles of radial cooling (RC), we
indeed observe that $f$ increases, indicating radial cooling (Fig. 3(d)). We
next add axial cooling by switching on additional axial Raman beams with the
same two-photon detuning. This radial/axial cooling sequence (RAC)
simultaneously addresses the $\Delta n_{r}=-2$ radial and the $\Delta
n_{z}\approx-9$ axial sidebands. With 90 cycles of RAC, $f$ increases further
compared to radial cooling alone (Fig. 3(d)), indicating successful cooling in
all directions.

The cooling rate of RAC can be probed by measuring $f$ versus the number of
cooling cycles $N$. Fixing $U_{\text{spill}}=k_{B}\times
0.72(2)~{}\mu\text{K}$, we find a $1/e$ cooling timescale of $N_{c}=51(14)$
cycles. We also quantify loss during cooling. With Raman and optical pumping
beams off but keeping the tweezer depth ramps, we observe a $1/e$ lifetime of
$1.28(15)\times 10^{3}$ cycles. Remarkably, with Raman cooling on, the
lifetime increases to $2.7(4)\times 10^{3}$ cycles, corresponding to a
fractional loss of $3.66(6)\times 10^{-4}$ per cycle. This difference could
arise from Raman cooling compensating for technical heating, or from Raman
beams repumping molecules that decay into $X^{2}\Sigma(v=0,N=3)$ due to off-
resonant scattering of tweezer light Holland  _et al._ (2022b).

Although adiabatic trap reduction provides qualitative evidence of 3D cooling,
it does not directly provide a quantitative temperature. For thermometry, we
rely on Raman spectroscopy. We first cool at $\theta=0^{\circ}$, and then
perform spectroscopy in the magic configuration ($\theta=\theta_{m}$ and
$B=5.5\,\text{G}$) to minimize Raman broadening. Along the radial direction,
the $\Delta n_{r}=\pm 1$ sidebands are well-resolved. Given the radial Lamb-
Dicke parameter and the temperature regime, the Raman motional coupling
depends weakly on the motional state $n_{r}$ Sup . This allows us to observe
coherent $\Delta n_{r}=\pm 1$ sideband transfer and also simplifies
interpretation of spectra Sup . We apply $\pi$-pulses and measure the
resulting transfer around each sideband (Fig. 4(a)). The ratio $\mathcal{A}$
between the peak transfer fractions of the heating versus cooling sidebands
allows us to extract 1D ground state fractions and temperatures. Assuming
either uniform motional coupling or thermal occupation, we find a 1D radial
ground state fraction of $P_{0}=1-1/\mathcal{A}=0.60(6)$ after radial/axial
cooling (RAC). Assuming only a thermal distribution, the mean radial motional
occupation $\bar{n}_{r}$ is given by $\bar{n}_{r}=1/(\mathcal{A}-1)$. We find
$\bar{n}_{r}=0.66\,(16)$ ($\tilde{T}_{r}=k_{B}T_{r}/(\hbar\omega_{r})=1.1(2)$)
compared to $\bar{n}_{r}=1.4\,(4)$ ($\tilde{T}_{r}=1.8(5)$) before cooling.

We probe the axial temperature similarly. Although we can observe resolved
axial sidebands spaced up to $|\Delta n_{z}|\sim 10$, extracting a temperature
is difficult due to the complex lineshape arising from high axial temperatures
and a large Lamb-Dicke parameter Sup . Nevertheless, robust thermometry is
possible by probing high-order sidebands in the unresolved regime Thompson
_et al._ (2013); Zhang  _et al._ (2022b). The wings of the spectra become
Gaussian, and the spectra can be understood as Doppler-sensitive two-photon
transfer Sup . By fitting the wings, one can robustly extract a temperature.
Experimentally, we increase the Raman Rabi coupling such that the wings of the
spectra appear smooth (Fig. 4(b)). Fitting the spectra gives an axial
temperature of
$\tilde{T}_{z}=k_{B}T_{z}/(\hbar\omega_{z})=7.5\begin{subarray}{c}+1.0\\\
-0.7\end{subarray}$ ($\bar{n}_{z}=7.0\begin{subarray}{c}+1.0\\\
-0.7\end{subarray}$) after RAC, compared to
$\tilde{T}_{z}=26.5\begin{subarray}{c}+4.0\\\ -3.1\end{subarray}$
($\bar{n}_{z}=26\begin{subarray}{c}+4\\\ -3\end{subarray}$) before cooling.

Figure 4:  Raman Thermometry. (a,c) Radial Raman spectra showing the carrier
along with $\Delta n_{r}=\pm 1$ sidebands. Grey squares and light gray circles
show the carrier after and before cooling, respectively. Red (blue) squares
show the $\Delta n_{r}=1$ ($\Delta n_{r}=-1$) sideband after cooling; light
red (light blue) circles show the $\Delta n_{r}=1$ ($\Delta n_{r}=-1$)
sideband before cooling. (b,d) Unresolved axial Raman spectra. Blue squares
(light blue circles) show spectra after (before) cooling. (a,b) Spectra after
90 cycles of RAC (darker colors) compared to spectra before Raman cooling
(lighter colors). (c,d) Spectra after 90 cycles of RAC and 30 cycles of IRC
(darker colors) and spectra before Raman cooling (lighter colors). For radial
(axial) data, solid lines show fits to Lorentzian (Gaussian) distributions
with a vertical offset. For all panels, the dashed line and the shaded region
($\pm 1$ standard deviation) show the independently measured offset without
Raman beams.

Lastly, we demonstrate a way to reach even lower temperatures. In our RAC
scheme, since $\Delta n_{r}=-2$ sidebands are addressed, molecules can
accumulate in $n_{r}=1$, limiting the radial ground state fraction. To
circumvent this while maintaining net cooling, we apply an additional radial
cooling sequence, where we interlace radial cooling cycles that separately
address the $\Delta n_{r}=-2$ and $\Delta n_{r}=-1$ sidebands (Fig. 4(c)).
With an additional 30 cycles of interlaced radial cooling (IRC), we observe
that the 1D radial ground state fraction increases to $P_{0}=0.79(4)$,
corresponding to an average radial occupation of $\bar{n}_{r}=0.27(7)$
($\tilde{T}_{r}=0.65(9)$). Because the axial direction is not cooled during
IRC, the axial temperature increases to $\tilde{T}_{z}=13.5(14)$
($\bar{n}_{z}=13.0(14)$), still below the initial temperature.

When the motion is highly quantized, as in our case, a useful figure-of-merit
in addition to temperature is the motional entropy. This quantifies how many
motional states are populated and indicates the level of control over the
initial motional state. With RAC, we obtain a motional entropy per particle of
$s=5.2\begin{subarray}{c}+0.5\\\ -0.4\end{subarray}$ compared to
$s=7.5\begin{subarray}{c}+0.8\\\ -0.7\end{subarray}$ before cooling. With IRC,
we further reduce the motional entropy per particle to $s=4.9(3)$, the lowest
reported to date for laser-cooled molecules. Since most of the entropy is in
the axial motion, lower entropies and temperatures could be reached with
increased axial confinement and further optimized Raman pulse sequences Yu
_et al._ (2018); Zhang  _et al._ (2022b); Spence  _et al._ (2022).

The motional entropy also allows us to quantify the efficiency of our Raman
cooling scheme. In evaporative cooling of atomic and molecular gases, a common
efficiency metric is $\gamma=-d\ln(\text{PSD})/d\ln(N)$ where $N$ is the
particle number, and PSD is the phase space density Ketterle and Van Druten
(1996); Li  _et al._ (2021); Bigagli  _et al._ (2023); Lin  _et al._ (2023).
One can generalize this metric to $\gamma_{q}=ds/d\ln(N)$, where $s$ is the
motional entropy per particle Sup . $\gamma_{q}$ coincides with $\gamma$ in
the classical regime and is convenient when in the highly quantized regime.
Using our loss measurements, we estimate that $\gamma_{q}=70(28)$ for RAC,
indicating highly efficient cooling with little loss.

Our demonstration of Raman sideband cooling of molecules in this work opens up
several new possibilities. In the near term, the lower temperatures achieved
could significantly improve coherence times and provide more coherent dipolar
interactions between molecules, enabling high-fidelity quantum gates and
quantum simulation with long evolution times. Longer term, our work opens the
door to direct laser-cooling of trapped molecules to their 3D motional ground
states. This would be a key step towards full quantum control of molecules,
and could enable efficient production of ensembles with low motional entropy
suited for quantum simulation of itinerant many-body systems. Potentially,
this could provide an all-optical route towards quantum degeneracy Hu  _et
al._ (2017); Urvoy  _et al._ (2019) that may be broadly applicable to other
laser-coolable molecules including polyatomic ones.

##  I Acknowledgements

We thank Jeff Thompson, Waseem Bakr, and the Bakr group for fruitful
discussions. This work is supported by the National Science Foundation under
Grant No. 2207518. L.W.C. acknowledges support from the Sloan Foundation.
S.J.L. acknowledges support from the Princeton Quantum Initiative.

##  II Methods

###  II.1 Preparation of Molecules

CaF molecules in the $X^{2}\Sigma(v=0,N=1)$ manifold are created in a single-
stage cryogenic buffer gas cell Hutzler  _et al._ (2012), slowed via chirped
slowing, and loaded into a DC magneto-optical trap (MOT). The MOT is
subsequently switched off, $\Lambda$-cooling is applied, and the molecules are
loaded into an optical dipole trap (ODT) with the aid of a repulsive ring trap
in the presence of $\Lambda$-cooling Lu  _et al._ (2022). The molecules are
optically transported and loaded into a reconfigurable array of 37 optical
tweezers, which are created with focused laser beams of 781 nm light projected
vertically through a microscope objective Holland  _et al._ (2022b). For
normalization, after tweezer loading, the occupation of the tweezers are
detected non-destructively with $\Lambda$-imaging Holland  _et al._ (2022a).
After non-destructive detection, the molecules are spread out over all 12
hyperfine states in the $X^{2}\Sigma(v=0,N=1)$ manifold.

###  II.2 State-Resolved Detection

To probe the population in $\left|\downarrow\right\rangle$, we first lower the
trap depth to $U_{\text{MW}}=k_{B}\times 130(3)\,\mu\text{K}$ and rotate the
magnetic field ($B=4.4\,\text{G}$) to $\theta=53^{\circ}$. Microwaves at $\sim
20.5$ GHz are used to transfer the population from
$\left|\downarrow\right\rangle=\left|N=1,J=3/2,F=1,m_{F}=0\right\rangle$ to
$\left|N=0,J=1/2,F=1,m_{F}=-1\right\rangle$ via a Landau-Zener sweep. A pulse
of light resonant with the $X^{2}\Sigma(v=0,N=1)\rightarrow
A^{2}\Pi_{1/2}(v=0,J=1/2,+)$ transition removes any molecules remaining in
$X^{2}\Sigma(v=0,N=1)$ Holland  _et al._ (2022b, a). A second Landau-Zener
microwave sweep transfers molecules into
$\left|N=1,J=1/2,F=0,m_{F}=0\right\rangle$. Finally, the population is
measured via $\Lambda$-imaging Cheuk  _et al._ (2018), which detects all
molecules in $X^{2}\Sigma(v=0,N=1)$. An analogous approach is used to measure
population in $\left|\uparrow\right\rangle$.

###  II.3 Optical Pumping

For optical pumping at $\theta=0^{\circ}$, we use a single beam (OP) in the
radial (horizontal) plane. The beam makes an angle of $42^{\circ}$ relative to
the tweezer polarization axis $\hat{\varepsilon}$ ($\hat{x}$). It is optimized
to have minimal $\sigma_{+}$ component along the quantization axis set by the
magnetic field (along $\hat{x}$ at $\theta=0^{\circ}$). For the data exploring
optical pumping at the magic angle $\theta=\theta_{m}$ (Fig. 2(b,c)), we use a
second optical pumping beam (OP2) propagating along $\hat{y}$. This beam is
optimized to have minimal $\sigma_{+}$ component along the quantization axis
set by the magnetic field.

###  II.4 Raman Coupling

To achieve Raman coupling between $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$, we use light with two frequency components
$\omega_{1}$ and $\omega_{2}$ detuned near the
$X^{2}\Sigma(v=0,N=1)-B^{2}\Sigma(v=0,N=0)$ transition (single-photon detuning
of $\Delta=-2\pi\times 42\,\text{GHz}$). $\omega_{1}$($\omega_{2}$) nominally
couples to $\left|\uparrow\right\rangle$($\left|\downarrow\right\rangle$).
These components are generated by acousto-optical modulators (AOMs), allowing
the two-photon detuning $\delta$ to be varied.

To achieve motional coupling in the radial (horizontal) $xy$-plane, we send
beams $R_{1}$ and $R_{2}$ along the $\hat{x}+\hat{y}$ and $\hat{x}-\hat{y}$
directions, respectively. $R_{1}$($R_{2}$) carries a single frequency
component $\omega_{1}$($\omega_{2}$) and is linearly polarized vertically
along $\hat{z}$. The two beams are optionally retro-reflected. The retro-
reflections are controlled by shutters that can be controlled mid-sequence. To
address axial motion, we send beams $R_{3}$ and $R_{4}$ along $-\hat{z}$ and
$\hat{z}$, respectively. The beam $R_{3}$ is linearly polarized along
$\hat{y}$, while $R_{4}$ is linearly polarized perpendicular to
$\theta=\theta_{m}$.

###  II.5 Spectroscopy Sequences

We use the following sequence for the data presented in Fig. 2(a). First,
molecules are pumped into $\left|\uparrow\right\rangle$ by turning on OP for
$2\,\text{ms}$ in a bias magnetic field of $B=4.4\,\text{G}$ at
$\theta=0^{\circ}$. Next, the tweezer depth is decreased to $U_{R}$ and the
magnetic field is rotated to the desired angle $\theta$. Both $\omega_{1}$ and
$\omega_{2}$ are delivered via a single vertical beam $R_{4}$ along the
tweezer axis. A Raman pulse is applied for $170\,\mu\text{s}$ with an
estimated two-photon Rabi coupling of $\Omega_{R}\approx 2\pi\times 3$ kHz.
Finally, the population in $\left|\downarrow\right\rangle$ is measured.

For all spectra in Figs. 3 and 4, the radial Raman beams $R_{1}$ and $R_{2}$
are not retro-reflected. All spectra in Fig. 3 are taken in a bias field of
$B=4.4\,\text{G}$ at $\theta=0^{\circ}$. For the radial spectrum in Fig. 3(a),
radial beams $R_{1}$ and $R_{2}$ are applied for 1 ms, probing the radial
direction $\hat{y}$. For the axial spectrum in Fig. 3(b), $\omega_{1}$ is
delivered via $R_{3}$ and $\omega_{2}$ via $R_{4}$, and the beams are applied
for 1 ms.

All spectra in Fig. 4 are taken in a magnetic field of $B=5.5\,\text{G}$ at
$\theta=\theta_{m}$. For radial spectra in Fig. 4(a,c), $R_{1}$ and $R_{2}$
are used. The beams are applied for $30\,\mu\text{s}$ (carrier) or
$50\,\mu\text{s}$ ($\Delta n_{r}=\pm 1$ sidebands), with $\Omega_{R}\approx
2\pi\times 20$ kHz. For axial spectra in Fig. 4(b,d), $\omega_{1}$ is
delivered via $R_{3}$ and $\omega_{2}$ via $R_{4}$. The beams are applied for
$100\,\mu\text{s}$, with an estimated Rabi frequency of $\Omega_{R}\approx
2\pi\times 20$ kHz.

###  II.6 Cooling Sequences

Prior to Raman sideband cooling, we optically pump molecules into
$\left|\uparrow\right\rangle$ by applying beam OP for 2 ms.

For the radial/axial cooling sequence (RAC), both radial beams $R_{1}$ and
$R_{2}$ are on and retro-reflected. Axially, we send $\omega_{1}$ into
$R_{3}$, and $\omega_{2}$ into $R_{4}$. We perform Raman transfer on the
radial $\Delta n_{r}=-2$ sideband for $150\,\mu\text{s}$, and optical pumping
with OP for $150\,\mu\text{s}$. The Raman transfer occurs at tweezer depth
$U_{R}$, and the optical pumping occurs at depth $U_{0}$. The tweezer depth
ramps occur over $200\,\mu\text{s}$ between each of these steps.

For the interlaced radial cooling sequence (IRC), we only turn on radial beams
$R_{1}$ and $R_{2}$, with $R_{2}$ retro-reflected. Raman transfer on the
$\Delta n_{r}=-2$ ($\Delta n_{r}=-1$) sideband occurs for $150\,\mu\text{s}$
($300\,\mu\text{s}$).

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