# Direct Laser Cooling of Polyatomic Molecules

Benjamin L. Augenbraun<EMAIL_ADDRESS>Loïc Anderegg  Christian
Hallas  Zack D. Lasner  Nathaniel B. Vilas  John M. Doyle
<EMAIL_ADDRESS>Department of Physics, Harvard University, Cambridge, MA
02138, USA  Harvard-MIT Center for Ultracold Atoms, Cambridge, MA 02138, USA

###### Abstract

Over the past decade, tremendous progress has been made to extend the tools of
laser cooling and trapping to molecules. Those same tools have recently been
applied to polyatomic molecules (molecules containing three or more atoms). In
this review, we discuss the scientific drive to bring larger molecules to
ultralow temperatures, the features of molecular structure that provide the
most promising molecules for this pursuit, and some technical aspects of how
lasers can be used to control the motion and quantum states of polyatomic
molecules. We also present opportunities for and challenges to the use of
polyatomic molecules for science and technology.

###### keywords:

ultracold molecules , laser cooling , polyatomic molecules

††journal: Advances in Atomic, Molecular and Optical Physics

DRAFT: February 18, 2023

###### Contents

  1. 1 Introduction
    1. 1.1 Polyatomic molecules for quantum science
      1. 1.1.1 Quantum information processing
      2. 1.1.2 Quantum simulation
    2. 1.2 Precision measurements
    3. 1.3 Collisions and chemistry
    4. 1.4 Experimental approaches besides direct laser cooling
    5. 1.5 Direct laser cooling
      1. 1.5.1 Forming closed cycling transitions
      2. 1.5.2 Effects of multilevel systems on laser cooling
  2. 2 Molecular structure for laser cooling experiments
    1. 2.1 Electronic structure
    2. 2.2 Vibrational structure
      1. 2.2.1 Vibrational state notation
      2. 2.2.2 Anharmonic coupling
    3. 2.3 Rotational structure
      1. 2.3.1 Linear molecules
      2. 2.3.2 Spherical top molecules
      3. 2.3.3 Symmetric top molecules
      4. 2.3.4 Asymmetric top molecules
      5. 2.3.5 Dependence on spin and electronic angular momenta
      6. 2.3.6 Hyperfine effects and nuclear spin statistics
    4. 2.4 Transitions
      1. 2.4.1 Electronic transitions
      2. 2.4.2 Vibrational transitions
      3. 2.4.3 Measuring vibrational branching ratios
      4. 2.4.4 Repumping transition spectroscopy
      5. 2.4.5 Designing an optical cycle
      6. 2.4.6 Rotational transitions
    5. 2.5 Perturbations
  3. 3 Experimental techniques
    1. 3.1 Cryogenic buffer-gas beams
    2. 3.2 Optical cycling
    3. 3.3 Optical forces
    4. 3.4 Transverse cooling
    5. 3.5 Molecular deceleration
      1. 3.5.1 Radiative slowing
      2. 3.5.2 Zeeman-Sisyphus deceleration
    6. 3.6 Magneto-optical trapping
    7. 3.7 Sub-Doppler cooling
      1. 3.7.1 Grey molasses
      2. 3.7.2 $\Lambda$-enhanced grey molasses
      3. 3.7.3 Single-frequency cooling
    8. 3.8 Optical trapping
    9. 3.9 Preparation and coherent control of single quantum states
  4. 4 Outlook and challenges
    1. 4.1 Toward larger molecules
    2. 4.2 Other molecular motifs
    3. 4.3 Challenges and possibilities for other polyatomic molecules
  5. 5 Conclusion
  6. 6 Acknowledgments

##  1 Introduction

Detailed understanding of polyatomic molecules (those containing three or more
atoms) is central to such diverse fields as chemistry, biology, and
interstellar science. Beyond the inherent interest in their structures and
interactions, physicists also hope to fully control polyatomic molecules at
the single quantum state level for next-generation explorations. The diversity
of electronic structures, geometries, and atomic constituents present in
polyatomic molecules may provide powerful building blocks for quantum
information processing and precision tests of fundamental physics, e.g.
searching for dark matter or for new particles that help explain the matter-
antimatter asymmetry of the universe. However, using complex molecules for
these applications requires us to confront one of their most defining—and
daunting—characteristics: their immensely rich and varied internal structures.
Attempts to tame polyatomic molecules, for example by controlling their
internal (e.g. vibrational/rotational) and external (motional) states have a
long history both in atomic, molecular, and optical (AMO) physics and in
physical chemistry. It is the purpose of this review to describe recent
advances that have introduced direct laser cooling as a new element in the
toolkit of polyatomic molecular control.

The already achieved exquisite control over certain quantum systems has been
realized in no small part by using optical photons—this is a hallmark of
modern quantum science and physical chemistry. The most recent wave of
advances with atoms and molecules relies on the ability to cool, control, and
detect molecules efficiently (and, ideally, nondestructively). Optical
cycling, a process in which molecules are made to rapidly and repeatedly
scatter many hundreds or thousands of photons, can be an effective way to
carry out these tasks. Using photon cycling, scientists have exploited the
mechanical effects of light to prepare and interrogate individual atoms and
diatomic molecules in pristine and dynamically controllable traps (Barredo et
al. (2016); Endres et al. (2016)). Optical photons also allow scientists to
probe the fragile quantum effects that form the heart of modern quantum
technologies (Haroche (2013); Wineland (2013)). The creation of quantum gases
of atoms and the production of useful architectures for quantum computing also
rely on these experimental feats. These wide ranging impacts span many
frontiers of quantum science, as well as cold chemistry, and precision
searches for new fundamental (particle) physics.

###  1.1 Polyatomic molecules for quantum science

Ultracold molecules are a promising platform for quantum simulation and
quantum information processing due to their large electric dipole moments and
the intrinsically long coherence times achievable in low-lying rotational
states. Heteronuclear molecules have molecule-frame electric dipole moments,
typically on the order of several debye (D), that can be accessed in the
electronic ground state, eliminating the need for the short-lived excited
electronic levels often employed in atomic systems. The molecules may interact
via the electric dipole-dipole interaction, whose long-range and anisotropic
behavior enables access to a diverse variety of Hamiltonians for quantum
simulation, as well as enabling entangling gates for quantum information
applications. While not always required to achieve interactions, it can be
advantageous to realize a molecular dipole moment in the laboratory frame by
aligning the atoms or molecules with an external electric field (either DC or
microwave).

While polyatomic molecules, compared to diatomic molecules, possess greater
complexity and additional degrees of freedom that need to be controlled, many
of the structures present in polyatomic molecules have no analogue in atoms or
diatomic molecules. One especially appealing feature of polyatomic molecules
is the existence of parity-doubled states in the ground electronic manifold.
These “parity doublets” comprise nearly-degenerate pairs of quantum states
with opposite parity that can be mixed by a small DC electric field (often
$\lesssim 100$ V/cm depending on the molecule), enabling the molecule to be
easily polarized in the laboratory frame. Moreover, because this method of
polarizing the molecule does not mix end-over-end rotational levels (which
requires much larger electric fields, typically $>$1 kV/cm), the polarized
molecules contain states with positive, negative, and near-zero lab-frame
dipole moment (see Fig. 1).

Figure 1:  Molecular polarization as a function of applied laboratory electric
field. Parameters are typical of YbOH molecules. Blue lines show alignment in
a state without parity doubling ($N\leq 2$, $\left\lvert M_{N}\right\rvert\leq
1$ plotted). Pink lines show alignment of $N=1$ sublevels in a vibrational
level that has parity doubling, e.g. the fundamental bending vibrational
state. We indicate two electric‐field regimes: one where mixing occurs within
a single rotational state and another where mixing occurs among many
rotational states.

The existence of parity-doubled states is a general feature of molecules that
have a nonzero projection of the total angular momentum along the molecule-
frame dipole moment. In particular, for a molecule with total angular momentum
$J$, whose projection onto the laboratory $Z$ axis is $M$ and whose projection
onto the molecular symmetry axis is $K$, the states
$2^{1/2}\lvert\pm\rangle=\lvert J,K,M\rangle\pm(-1)^{J-K}\lvert J,-K,M\rangle$
have opposite parity. States with $K\neq 0$ are universally present in
polyatomic molecules: linear polyatomic molecules have angular momentum about
the internuclear axis in vibrational bending modes, while nonlinear polyatomic
molecules have nonzero moments of inertia about the symmetry axis even in the
electronic and vibrational ground state. The degeneracy of these parity
doublets is lifted by a range of mechanisms, including Coriolis interactions,
nuclear spin-rotation interactions, hyperfine interactions, or molecular
asymmetry (Klemperer et al. (1993)). The resulting energy splitting can be as
large as tens of MHz (notably in linear polyatomic molecules) or as small as
tens of Hz or kHz (e.g., in singlet symmetric top molecules).

For the parity doublet states $\lvert\pm\rangle$ described above, the
Hamiltonian under the influence of an external DC electric field $\mathcal{E}$
is

$H=\begin{pmatrix}-\hbar\Delta/2&-d\mathcal{E}\\\
-d\mathcal{E}&\hbar\Delta/2\end{pmatrix}$ (1)

where $\Delta$ is the parity doublet splitting and

$d\mathcal{E}\equiv\langle+|\vec{d}\cdot\vec{\mathcal{E}}|-\rangle=d_{\text{0}}\mathcal{E}\frac{KM}{J(J+1)}$
(2)

where $d_{0}$ is the permanent dipole moment of the molecule. The energy
eigenvalues are therefore

$E_{\pm}=\pm\frac{1}{2}\hbar\Delta\left[1+4\left(\frac{d\mathcal{E}}{\hbar\Delta}\right)^{2}\right]^{1/2}$
(3)

At low electric fields $d\mathcal{E}/\hbar\Delta\ll 1$ the states split apart
quadratically, while at high electric fields $d\mathcal{E}/\hbar\Delta\gg 1$
the molecule becomes aligned in the laboratory frame and follows linear Stark
shifts:

$E\approx-d_{\text{0}}\mathcal{E}\frac{KM}{J(J+1)},$ (4)

where $K$ and $M$ are signed quantities. The structure of the molecule in this
“polarized” regime is amenable to a number of interesting quantum simulation
and quantum information applications.

####  1.1.1 Quantum information processing

Since the seminal proposal by DeMille (2002), ultracold polar molecules have
generated significant interest as a platform for quantum information
applications, see e.g., Yelin et al. (2006); Ni et al. (2018); Sawant et al.
(2019). The key element of these proposals is the fact that polar molecules
have many long-lived rotational and vibrational degrees of freedom which
enable storage of quantum information, while dipole-dipole interactions enable
transfer of information and entanglement between individual molecules. Recent
progress toward this goal has been realized with diatomic molecules by Holland
et al. (2022) and Bao et al. (2022), who demonstrated dipolar coupling and
entanglement between CaF molecules trapped in a tweezer array.

The linear Stark shifts discussed above for polyatomic molecules make them
especially appealing for certain quantum computing and entanglement schemes,
including those proposed by André et al. (2006); Wei et al. (2011); Yu et al.
(2019); Zhang and Liu (2017). For example, Wei et al. (2011) studied
theoretically entanglement generation in polar symmetric top molecules and
identified two sets of polarized states that could be used as qubits in a
quantum computer. They additionally proposed a method for realizing a CNOT
entangling gate between two such molecules. Interestingly, it was also pointed
out that polarized symmetric top molecules share certain similarities with the
NMR quantum computing platform (see, e.g., Cory et al. (2000)). However,
molecuels in optical tweezer arrays offer the possibility of isolating
individual qubits and controlling entanglement on demand more cleanly than has
been done with liquid-phase NMR and with intrinsic scalability.

Yu et al. (2019) proposed a quantum computing scheme harnessing the parity-
doublet structure of symmetric top molecules, wherein the qubit states are
contained within the $M=0$ manifold, which eliminates first-order electric
field sensitivity, thereby improving robustness to external perturbations.
Interactions are switched on using a third state in either the $M=+1$ or
$M=-1$ manifold, and entanglement is generated using an interaction blockade
mechanism. Using the linear Stark shift structure to switch dipole-dipole
interactions on and off in this manner is a key advantage of polyatomic
molecules.

The large number of rotational and vibrational degrees of freedom in
polyatomic molecules also has potential advantages for quantum information
applications. For instance, error-correcting logical qubits could be
constructed from coherent superpositions of rotational states, as proposed by
Albert et al. (2020). The large number of internal states in polyatomic
molecules could also make them uniquely amenable to “qudit” schemes where
multiple bits of information are stored in the same molecule, as described by
Tesch and de Vivie-Riedle (2002); Sawant et al. (2019). Such schemes could
significantly increase the speed and fidelity of molecular quantum computers
by performing the majority of gates using single-molecule operations, and only
coupling molecules via dipole-dipole interactions when necessary. Full cooling
and control of larger polyatomic molecules with many nuclear spins could also
enable platforms where NMR-based quantum computing is performed within a
single molecule, while scalability is achieved by coupling individual
molecules with dipole-dipole interactions. More theoretical work is required
to explore the feasibility of such schemes.

####  1.1.2 Quantum simulation

The long range and anisotropic dipolar interactions between polar molecules
lend themselves to quantum simulation applications, in particular to
simulation of quantum magnetism models where molecules pinned in a lattice act
as effective spins (Carr et al. (2009); Bohn et al. (2017)). Wall et al.
(2015a) provide a review of this topic. To date, much of this work has been
focused on diatomic molecules, which are naturally suited to simulation of
Heisenberg XY models, as first demonstrated experimentally by Yan et al.
(2013) using KRb molecules. In theoretical work by Micheli et al. (2006), it
was shown that microwave-dressed diatomic molecules with unpaired electron
spins can be used to simulate even more general spin models, though the
technical requirements of this proposal have not yet been realized in present
day experiments.

Compared to diatomic molecules, polyatomic molecules are naturally suited to
simulation of a greater diversity of quantum magnetism models, such as those
described by Wall et al. (2013, 2015b, 2015a). For instance, the unique
rotational structure of molecules with parity doublet structure enables
simulation of XYZ spin models via microwave dressing, as proposed by Wall et
al. (2015b). Critically, the technical requirements of this proposal are
significantly reduced compared to the method of Micheli et al. (2006) for
simulating an XYZ Hamiltonian with diatomic molecules, which requires more
microwave frequencies and significantly smaller lattice spacings to compensate
for the molecules’ sparser internal structure. Another benefit of large
polyatomic molecules may be their small fine and hyperfine splittings, which
can be made comparable to the dipole-dipole interaction energy at larger
molecular spacings. For instance, operating in the regime where the dipolar
interaction energy is comparable to the spin-rotation interaction energy in
molecules with an unpaired spin would enable simulation of a diverse array of
lattice spin models, as described in Micheli et al. (2006). These small
energy-level splittings could, however, present a challenge for achieving full
control of individual quantum states in the molecule, e.g., due to off-
resonant excitations during control pulses.

Wall et al. (2013) have also proposed to use the (nearly) linear Stark shifts
present in many polyatomic molecules as a way to simulate lattice spin models
involving polarized dipoles. Because this proposal involves using electric
dipole moments to simulate magnetic dipoles, the experimental interaction
strengths can be several orders of magnitude larger. Importantly, this analogy
can be extended to molecular states with arbitrarily large $J$, enabling
quantum simulation of magnetic dipoles with integer spin $S=J$. These features
could have interesting applications in simulation of lattice spin models as
well as in the study of bulk dipolar gases (see Lahaye et al. (2009) for a
review on the subject), which have previously been studied using magnetic
atoms with much smaller interaction energies.

###  1.2 Precision measurements

Polyatomic molecules hold further promise for a variety of precision
measurements probing the Standard Model (SM) and beyond-Standard-Model (BSM)
physics (see Hutzler (2020) for a focused review). At least three distinctive
features of polyatomic molecules can be leveraged for improved precision
measurements. The first such feature is the closely-spaced pairs of opposite-
parity states described above, which can be directly mixed by electric fields
to orient molecules in the lab frame, or brought to degeneracy via magnetic
fields.

Mixing opposite-parity states to orient a molecule in the lab frame is useful
for electron electric dipole moment (eEDM) searches, as well as Schiff moment
and magnetic quadrupole moment searches operating in a similar manner. The
proposals by Kozyryev and Hutzler (2017); Maison et al. (2019); Oleynichenko
et al. (2022); Yu and Hutzler (2021); Zakharova and Petrov (2021) describe
experiments using polyatomic molecules in which this capability is useful. The
basic principle of an eEDM measurement using oriented molecules is as follows.
If the electron possesses an electric dipole moment, $\vec{d_{e}}$, then it
must be oriented along or against the electron spin, so that
$|\vec{d_{e}}\cdot\vec{S}|\neq 0$. In a given molecular state, the energy
shift associated with the eEDM is $\langle
H_{d_{e}}\rangle=d_{e}\langle\vec{S}\cdot\mathcal{\vec{E}}_{{\rm
eff}}\rangle$, where the “effective electric field” vector
$\mathcal{\vec{E}}_{{\rm eff}}$ is a state-dependent constant oriented in the
molecular frame, for example along an internuclear axis. In a parity
eigenstate, the electron spin has no average orientation in the molecular
frame, and $\langle\vec{S}\cdot\mathcal{\vec{E}}_{{\rm eff}}\rangle$ vanishes.
The simplest way to obtain a non-vanishing eEDM interaction is to orient the
molecule in the lab frame so that $\vec{\mathcal{E}}_{{\rm eff}}||\hat{Z}$,
where $\hat{Z}$ is the lab $z$-axis, and to simultaneously orient the electron
spin along or against the same axis (e.g., via angular momentum selection
rules on electronic transitions) so that $M_{S}\neq 0$. In this ideal case,
$\langle\vec{S}\cdot\mathcal{\vec{E}}_{{\rm eff}}\rangle$ will have maximal
magnitude and $\langle H_{d_{e}}\rangle$ can be probed. More generally, as
long as the parity of a molecular state is at least partially mixed, then
$\langle\vec{S}\cdot\mathcal{\vec{E}}_{{\rm eff}}\rangle\neq 0$ and the eEDM
can be measured via $\langle H_{d_{e}}\rangle$. Heteronuclear diatomic
molecules in ${}^{2}\Sigma$ electronic states have rotational states of both
positive and negative parity, but they are generally spaced by tens of GHz and
require electric fields on the order of tens of kV/cm to saturate the energy
shifts associated with the eEDM. By contrast, the parity doublets generically
found in polyatomic molecules can be fully mixed at fields at or below 1
kV/cm. Furthermore, in contrast to rotational states, the structure of parity
doublets found in polyatomic molecules, for example $K$-doublets in symmetric
top molecules or $\ell$-doublets in vibrational bending modes of polyatomic
molecules, enables the orientation of molecules to be spectroscopically
reversed in a fixed external electric field. This feature can also be found in
$\Lambda$\- or $\Omega$-doublets of diatomic molecules, and it has already
been a valuable tool for systematic error rejection in ThO and HfF+, such as
the experiments described in ACME Collaboration (2018); Cairncross et al.
(2017). The special feature of polyatomic molecules is that such parity
doublets can be obtained irrespective of the electronic structure of the
selected polyatomic species.

Another case where near-degenerate opposite-parity states are useful is in
probing intrinsic parity-violating Standard Model (SM) effects such as the
vector electron-axial nucleon electroweak current coupling and the nuclear
anapole moment. The total parity-violating interaction in a given electronic
state is characterized by the constant $W_{p}$. By determining the value of
$W_{p}$ in multiple nuclei of the same molecular species, the contribution of
each SM effect could be independently determined. A sensitive method to probe
$W_{p}$ is Stark interference (see DeMille et al. (2008)), where an electric
dipole transition drives a molecule between opposite-parity states separated
by energy $\Delta$. The population transfer contains an interference term
between the driving electric field $E_{0}$ and the parity-violating
interaction $W$ (which is proportional to $W_{p}$ but contains additional
factors from the molecular state). By comparing measurements with opposite
phases of the driving electric field, a measured quantity proportional to
$W/\Delta$ can be obtained. Thus the experimental signal is enhanced when the
states under consideration are brought to near-degeneracy. Whereas rotational
states of diatomic molecules can be brought to near-degeneracy with Tesla-
scale magnetic fields, Norrgard et al. (2019b) showed that in a large class of
linear polyatomic molecules, degeneracy can be achieved with fields of 10 mT
or less, dramatically reducing the experimental complexity of operating an
experiment within the bore of a superconducting magnet. By measuring nuclear-
spin-independent parity-violating effects in light molecules, where
calculations of Standard Model effects are not prohibitively challenging,
parity-violating interactions in the SM may be probed and, perhaps,
distinguished from BSM effects.

A second feature of polyatomic molecules that can be exploited in precision
measurements of BSM physics is the multiplicity of rotational and vibrational
modes. Whereas diatomic molecules have one rotational mode and one vibrational
mode, every polyatomic molecule contains at least three vibrational modes and
up to three rotational modes. Thus accidental near-degeneracies between
rovibronic states can be commonly found at low energies (e.g., below 1000
cm-1), and are nearly guaranteed at higher rovibronic energies where the
density of states increases. As described by Jansen et al. (2014), these near-
degeneracies are useful for probing potential variation of the proton-to-
electron mass ratio, $\mu\equiv m_{p}/m_{e}$, over time: as $\mu$ changes,
each rovibrational level shifts since rotational and vibrational energies
depend directly on the masses of atomic nuclei according to $\delta E=\partial
E/\partial\mu\times\delta\mu$. (Here, changes in physical quantities
associated with changes in $\mu$ are indicated by the prepended symbol
$\delta$.) Pure rotational energies and anharmonic vibrational energies obey
$\delta E=-E\times\delta\mu/\mu$, while pure harmonic vibrational energies
obey $\delta E=-\frac{1}{2}E\times\delta\mu/\mu$. In each case, the _absolute_
energy sensitivity $\delta E$ to $\mu$ variation scales with the overall
energy $E$. In cases of accidental near-degeneracies, it is possible to obtain
$\omega\approx 0$ even when the absolute frequency sensitivity of a transition
$\delta\omega\equiv\delta E_{2}-\delta E_{1}$ does not vanish because $E_{1}$
and $E_{2}$ depend differently on $\mu$. For example, rovibronic transitions
between near-degenerate states can exploit the relatively large absolute
frequency shifts of vibrational energy levels ($E\sim 10$ THz) while being
amenable to the technical convenience of lower-frequency microwave sources
($\omega\sim 10$ GHz) where stable frequency references are readily available
and certain systematic errors like Doppler shifts are suppressed. Transitions
where $\delta\omega/\omega\gg 1$ have been used to set limits on $\mu$
variation in molecules including methanol, ammonia, and KRb (where a
degeneracy between an excited vibrational state and a metastable electronic
state was used by Kobayashi et al. (2019)). Laser-cooled polyatomic molecules
possess convenient rovibronic near-degeneracies to sensitively probe $\mu$
variation in a platform offering ultracold temperatures, long trap lifetimes,
and full quantum control. Kozyryev et al. (2021) identified a promising near-
degeneracy in the energy levels of SrOH that could be used for such an
experiment.

The third, and final, feature of polyatomic molecules that we note for
precision measurements is chirality. Because chirality requires three distinct
molecular axes, it can only be found in molecules with at least four atoms.
Parity-violating effects in the Standard Model are predicted to result in
energy splittings between chiral molecules at the level of $\sim$mHz to Hz in
various species of experimental interest (Cournol et al. (2019)). In
principle, even in the absence of parity violation, any chiral molecule could
convert to its enantiomer via the tunneling of some set of nuclei through a
vibrational energy barrier, resulting in a double-well-type energy structure
and associated splitting $\Delta E_{\pm}$ between molecular eigenstates. In
many molecules such as hydrogen peroxide (HOOH) or the chiral isotopologue of
ammonia (NHDT), the tunneling splitting dominates parity-violating effects,
$\Delta E_{\pm}\gg\Delta E_{\rm{pv}}$, and the symmetry breaking between left-
and right-handed molecules occurs only “ _de facto_ ,” i.e. due to initial
conditions described by Quack et al. (2022). Nevertheless, in this case it is
possible to measure the effect of parity-violating interactions, for example
by observing an initial parity eigenstate acquire a non-zero amplitude of an
opposite-parity eigenstate upon free evolution (Quack (1986)). The other
limiting case is where the parity-violating energy shifts are large compared
to the tunneling splitting, $\Delta E_{\rm{pv}}\gg\Delta E_{\pm}$, so that the
symmetry breaking between enantiomers is “ _de lege_ ,” i.e. due to intrinsic
dynamics of the molecular energies. In this case, the energies of two
enantiomers can be directly spectroscopically measured (e.g., via high-
sensitivity infrared absorption experiments) and compared. Most molecules with
heavier constituents, including species of interest such as CHFClBr and S2Cl2,
are expected to exhibit “ _de lege_ ” parity violation due to the large
reduced tunneling mass (Quack et al. (2008)). Chiral molecules are also
sensitive to parity-violating cosmic fields associated with certain dark
matter candidates, as pointed out by Gaul et al. (2020). Thus precise
measurements of enantiomeric energy splittings can be a definitive probe of
parity-violating weak interactions and beyond-Standard Model interactions in
suitably chosen molecules. Laser-cooled polyatomic chiral molecules (for
example CaOCHDT, a chiral analogue of CaOCH3) could potentially enable
measurements of these parity-violating effects for the first time due to the
possibility of longer interaction times and full quantum control (Augenbraun
et al. (2020a)).

###  1.3 Collisions and chemistry

Collisions of polyatomic molecules—both with atoms and other molecules—are of
great scientific interest to a number of disciplines across physics and
chemistry. A distinction is typically made between two different temperature
regimes, namely “cold” collisions at temperatures of $\lesssim$$1~{}\text{K}$,
and the “ultracold” collision regime, defined by temperatures sufficiently low
that only a single partial wave participates in the collision. The ultracold
collision regime is sometimes defined informally as $\lesssim$$1~{}\text{mK}$,
though the actual temperature for collisions to only include a single partial
wave can be much lower for heavier molecules. In addition, there are
subtleties involved in defining the single-partial-wave regime for dipolar
collisions where all partial waves may contribute even at low collisional
energies, see Chomaz et al. (2023). Numerous techniques have been developed
for studying collisions at cold temperatures, and cold collision dynamics for
a number of molecules, including polyatomic species, have already been
investigated. These experiments have typically relied on beam-based
approaches, though a variety of different experimental techniques have been
employed, including slowing techniques to reduce collisional temperatures,
e.g., Stark (van de Meerakker and Meijer (2009)), Zeeman (Plomp et al.
(2020)), and “cryofuge” (Wu et al. (2017)) deceleration. Trapping of the
molecular species—from slowed or buffer-gas-cooled (Hummon et al. (2011))
molecular beams—has allowed for increased molecular interaction times, and has
lead to novel studies of cold dipolar collisions. Stark deceleration followed
by magnetic trapping has, for example, been used to study dipolar collisions
between OH and $\text{ND}_{3}$ molecules in a magnetic trap (Sawyer et al.
(2011)), and cryofuge deceleration has recently been combined with electric
trapping to study dipolar collisions between $\text{CH}_{3}\text{F}$ molecules
(Koller et al. (2022)). Collision studies at cold temperatures play a
particularly important role in astrophysics, specifically for studying the
astrochemical reactions that result in the molecular compositions observed in
interstellar environments, see Herbst and van Dishoeck (2009). For more
extensive reviews of cold collision studies, we refer interested readers to
the articles by Toscano et al. (2020); Heazlewood and Softley (2021). The
ultracold regime remains largely unexplored due to the difficulty in producing
ultracold molecules, especially for polyatomic molecules. The extension of
direct laser cooling techniques to polyatomic molecules could potentially
provide a path towards realizing collision studies at ultracold temperatures.
Below we provide a few key highlights of some of the novel collision dynamics
and chemistry that are expected to arise at ultracold temperatures and which
could potentially be explored with ultracold polyatomic molecules.

In the ultracold regime, the de Broglie wavelength of the colliding molecules
exceeds their classical size, requiring a quantum-mechanical description of
the collision process. Molecular interactions in this regime are therefore
characterized by quantum statistics and quantum threshold behavior, and a host
of different quantum collision and chemistry phenomena are expected to arise
as a result (Carr et al. (2009)). For example, due to the delocalized wavelike
nature of molecules in this regime, long-range dipolar interactions are
expected to be essential for determining collision rates between polar
molecules at ultracold temperatures. This bears important implications for
chemical reaction rates. While reaction rates are “classically” expected to
increase with temperature, many chemical reactions instead proceed at much
accelerated rates in the limit of absolute zero (Richter et al. (2015)). Some
of these quantum effects are already starting to be explored with ultracold
diatomic molecules. Seminal experiments with ultracold KRb molecules created
by photoassociation of laser-cooled atoms have, for example, probed long-range
interactions in ultracold $\text{KRb}{-}\text{KRb}$ (Ospelkaus et al. (2010);
Ni et al. (2010); Hu et al. (2021)) and $\text{KRb}{-}\text{Rb}$ (Nichols et
al. (2022)) collisions. Ultracold collision experiments have since also been
performed with a range of other bialkali species (Ye et al. (2018); Gregory et
al. (2021); Bause et al. (2021)). Similar experiments could be imagined with
polyatomic molecules, which would grant still broader access to new quantum
collision and chemistry phenomena. As an example, it has been proposed that
collective many-body effects in a Bose degenerate gas of triatomic molecules
may lead to a “Bose-stimulated” photodissociative process in which branching
to either of two decay channels, $ABC\rightarrow AB+C$ or $ABC\rightarrow
A+BC$, can be significantly amplified (Moore and Vardi (2002)). Ultracold
collision experiments with polyatomic molecules would also, along with
experiments with diatomic molecules, provide important experimental benchmarks
for new scattering theories and quantum chemistry at ultracold temperatures.

An essential implication of the quantum nature of the ultracold regime is that
both elastic and inelastic collision rates are expected to have a strong
dependence on the exact molecular quantum states (Carr et al. (2009)).
External fields, too, can have strong and potentially very different effects
on elastic and inelastic rates, for example by changing the relative
orientation of two colliding molecules (Tscherbul and Krems (2006); Tscherbul
et al. (2009); Brouard et al. (2014); Tscherbul and Krems (2015)). This
provides the capability for separately “tuning” elastic as well as inelastic
collision rates, which, as we discuss below, may be important for the
realization of degenerate gases of polyatomic molecules. In the context of
chemical reactions, this provides a path to controlled chemistry at ultracold
temperatures in which chemical reactions can be very precisely studied and,
possibly, engineered (Krems (2008); Balakrishnan (2016); Bohn et al. (2017)).
These points are especially pertinent to polyatomic molecules, whose many
internal degrees of freedom may provide additional tools for external field
control, and, in particular, allow for easy orientation of the molecules.
Recent experiments with CaF molecules have demonstrated that direct laser
cooling in combination with optical trapping is a feasible approach for
realizing the level of control required to characterize the quantum state and
field dependencies of collision rates (Cheuk et al. (2020); Anderegg et al.
(2021)). Optical trapping of ultracold polyatomic molecules additionally opens
the door to probing collisional dynamics in confined geometries in which
collision dynamics take on qualitatively distinct behavior compared to that of
an unconfined gas (Carr et al. (2009)).

Finally, understanding the collisional processes of polyatomic molecules at
ultracold temperatures is likely to have a fundamental impact on the potential
realization of Bose or Fermi degenerate polyatomic gases. In particular, the
experiments with bialkali molecules mentioned earlier have demonstrated that,
in the ultracold regime, even molecules that are not chemically reactive can
exhibit large inelastic collision losses due to forming long-lived complexes
in so-called “sticky collisions” (Bause et al. (2022)). This prevents
efficient evaporative cooling, which is the typical approach used for creating
degenerate atomic gases (Pethick and Smith (2002)). The tunability of the
reaction rates mentioned above is in this regard important and can enhance the
elastic-to-inelastic collision ratio by several orders of magnitude using
external fields. Several “shielding” techniques have already been demonstrated
for diatomic molecules (Matsuda et al. (2020); Valtolina et al. (2020);
Anderegg et al. (2021); Li et al. (2021)). Recently, Schindewolf et al. (2022)
showed that quantum degeneracy could be reached for diatomic species using
this shielding technique. Similar methods are likely required for the
realization of degenerate gases of polyatomic molecules. A scheme for field-
induced shielding in collisions between CaOH molecules has been proposed by
Augustovičová and Bohn (2019). Alternative prospects for collisional cooling
using sympathetic molecule-atom collisions have also recently been explored by
Wójcik et al. (2019) for complex polyatomic molecules such as benzene and
azulene.

###  1.4 Experimental approaches besides direct laser cooling

Achieving any of the diverse goals summarized above requires exquisite control
over the molecules to be probed. In order to reap the benefits of polyatomic
molecules, we must cool them to cold ($\lesssim 4$ K) or ultracold ($\lesssim
1$ mK) temperatures. This cooling is required to “compress” the molecular
population into a small number of quantum states and to slow their thermal
velocities to tens of meters per second; the former increases an experiment’s
signal-to-noise ratio and the latter enables long interrogation times to
improve achievable precision and control. Because polyatomic molecules contain
many internal degrees of freedom, cooling them can be a very difficult task.
Many research groups have tackled this problem, and we review some relevant
methods here. A variety of approaches have been explored to realize this
control.

A very successful method of producing ultracold diatomic molecules involves
laser cooling atoms and then binding these pre-cooled atoms together (using
photoassociation, magnetoassociation, etc.); see Ni et al. (2008); Molony et
al. (2014); Park et al. (2015); Guo et al. (2016); Rvachov et al. (2017);
Cairncross et al. (2021). While this has produced a number of ultracold
diatomic molecules in single quantum states, including degenerate gases, it is
not clear it can be generalized to produce polyatomic molecules. However, very
recent work by Yang et al. (2022) has shown some evidence for production of
triatomic molecules in a mixture of 23Na40K and 40K. It is also unclear
whether this exciting result can be extended to other species (or larger
ones), especially those containing difficult-to-cool atomic species such as O,
C, and/or H.

Optoelectrical Sisyphus cooling, described by Zeppenfeld et al. (2012); Prehn
et al. (2016), uses state-dependent energy shifts and repeated microwave
transitions/optical pumping steps to cool molecules as they move through an
electric trap. Energy is removed by ensuring molecules move away from the
trap’s center along a “steep” potential and return to the center along a
“shallow” potential. This technique relies on the linear Stark shifts that can
be achieved in symmetric or asymmetric top molecules. It has been used to
produce trapped samples of CH3F and H2CO at $\sim$mK temperatures and could
potentially be used to observe molecule-molecule collisions in the trap. To
date, the molecules that have been cooled using this method do not offer
convenient optical transitions, so vibrational transitions with relatively
long lifetimes have been used instead. The method could likely be adapted to
make use of the optical transitions offered by some of the molecules discussed
in this review, speeding up the cooling rates considerably.

###  1.5 Direct laser cooling

Direct laser cooling is a promising approach because it may be widely
applicable to diverse structures of molecules and brings with it the
possibility of high-efficiency quantum state preparation and readout using
optical photons. The direct laser cooling approach is the focus of this review
paper. Figure 2 presents a schematic overview of an idealized molecular laser
cooling experiment. Because many of these techniques were honed in the context
of diatomic molecules, the interested reader should refer to the reviews by
Hutzler et al. (2012); Tarbutt (2018); McCarron (2018); Fitch and Tarbutt
(2021).

Figure 2:  Overview of an idealized molecular laser cooling sequence.
Molecules are produced in a cryogenic buffer-gas beam (CBGB) source,
decelerated, trapped in a magneto-optical trap, and then transferred into a
conservative trap for a particular science goal. Reproduced from Augenbraun
(2020).

####  1.5.1 Forming closed cycling transitions

The photon scattering process is one in which molecules go through a series of
photon absorption and spontaneous emission cycles that can be described as a
Bernoulli sequence. Suppose we have applied laser repumpers such that a
molecule has a probability $p$ to decay to a state that is not addressed by
laser light. The probability $P_{n}$ for a molecule to experience $n$
absorption-emission cycles is given by $P_{n}=(1-p)^{n}$. The average number
of photons scattered by molecules is $\bar{n}=\frac{1}{1-p}$. We often refer
to this as the “photon budget,” and it sets the (exponential) scale for how
many photons can be scattered before significant fractions of the population
are lost. For example, if laser slowing requires 10,000 photon scattering
events and we would like 90% of the initial population to remain after
slowing, we require $p\approx 10^{-5}$. Clearly, understanding branching
ratios as small as $1$ part in $10^{5}$ can be crucial to achieve efficient
laser cooling.

A typical laser cooling experiment requires scattering many thousands of
optical photons. To repeatedly scatter this number of photons without the
molecules accumulating in states that do not couple to applied laser light
(“dark states”), it is necessary to form a “closed cycling transition.” In a
closed cycle, a molecule that is driven to an electronically excited state is
guaranteed to decay back to the same state that it started in. In reality, no
optical transition is fully closed and the molecule has a finite probability
to decay to a state that is different from the initial state. Such a molecule
needs to be “repumped” into the cycling transition. The higher the probability
of decaying to other states, the more repumping transitions must be addressed.
This quickly becomes an experimental limitation, as each decay typically
requires a separate laser to be added to the experiment. This points to the
importance of selecting atoms and molecules whose branching ratios are
favorable. The general guidelines for selecting molecules and transitions for
laser cooling were first pointed out by Di Rosa (2004), who highlighted the
need for (1) strong transitions, (2) highly diagonal vibrational branching,
and (3) no intermediate (i.e., metastable electronic) states between the two
states used for optical cycling.

The electronic transitions driven in laser cooling are typically electric
dipole (E1) transitions. Since the dipole operator is odd under parity
transformations, the parity of the excited and ground states must be opposite.
As a result, selection rules for changes to the total angular momentum $J$ are
$\Delta J=0,\pm 1$ (with $J^{\prime}=0$ to $J=0$ forbidden). We can use these
selection rules to our advantage by cycling from $J=1$ to $J^{\prime}=0$, as
originally recognized by Stuhl et al. (2008), which must decay back to $J=1$,
leading to rotational closure.

One consequence of driving from $J=1$ to $J^{\prime}=0$ to attain rotational
closure is the presence of dark states. This is a generic problem in the laser
cooling of molecules, where the excited state often has the same number of
sublevels (or fewer) than the ground state. This means that for any fixed
laser polarization, dark states exist. Molecules that collect in these dark
states can be returned to bright states in a number of different ways, most
commonly using DC magnetic fields, polarization modulation, or microwave
pulses (Berkeland and Boshier (2002)).

####  1.5.2 Effects of multilevel systems on laser cooling

For a system that can decay to multiple states, the prototypical two-level
scattering rate equation is modified. For resonant light where all the
transitions are driven with equal intensity, Williams et al. (2017) show that
the maximum achievable scattering rate is modified to

$R_{\text{scat}}^{\text{max}}=\frac{n_{e}}{n_{g}+n_{e}}\Gamma$ (5)

where $n_{e}$ is the number of excited states and $n_{g}$ is the number of
ground states. As an example, in CaF there are 12 Zeeman sublevels in the
ground state, 4 in the excited state, and 12 in the $v$=1 ground state. Hence
$n_{e}=4$ and $n_{g}=24$. If lasers are tuned to drive both $\tilde{X}(v=0)$
and $\tilde{X}(v=1)$ to the $\tilde{A}$ state, the achievable scattering rate
will reduce by a factor of about 4. This decrease in the scattering rate would
significantly hinder the laser cooling of a molecule. A commonly employed
technique to circumvent this limitation is to repump the molecules through a
different excited state than the one used for the “main” transition. This
approach is also crucial for polyatomic molecules as more states must be
repumped in a laser cooling scheme.

With closed cycling transitions established, diatomic molecules have been
laser slowed and cooled (Shuman et al. (2009, 2010); Barry et al. (2012);
Zhelyazkova et al. (2014); Hemmerling et al. (2016); Truppe et al. (2017c);
Lim et al. (2018)) and trapped in red-detuned magneto-optical traps (Hummon et
al. (2013); Barry et al. (2014); McCarron et al. (2015); Norrgard et al.
(2016); Steinecker et al. (2016); Chae et al. (2017); Truppe et al. (2017a);
Anderegg et al. (2017); Williams et al. (2017); Collopy et al. (2018)) and
blue-detuned magneto-optical traps (Burau et al. (2022)), cooled to sub-
Doppler temperatures (Truppe et al. (2017a); Cheuk et al. (2018); Caldwell et
al. (2019); Ding et al. (2020)), and loaded into magnetic (Williams et al.
(2018); McCarron et al. (2018)) and optical traps (Anderegg et al. (2018);
Langin et al. (2021); Wu et al. (2021); Lu et al. (2022)) and tweezers
(Anderegg et al. (2019); Lu et al. (2022); Holland et al. (2022); Bao et al.
(2022)).

##  2 Molecular structure for laser cooling experiments

Compared to diatomic molecules, polyatomic molecules can have significantly
more complex internal structures. Here, we review the structure of polyatomic
molecules as is relevant to laser cooling and trapping experiments. We
generally work within the Born-Oppenheimer approximation, separating the
electronic, vibrational and rotational motions, but molecular physics beyond
the Born-Oppenheimer approximation is also introduced as necessary to
understand its impact on laser cooling and trapping experiments.

###  2.1 Electronic structure

For the molecules of interest to laser cooling experiments, the largest
relevant energy scale corresponds to excitation of a valence electron that is
used for optical cycling (changing the principle quantum number). Typical
electronic excitation energies for several classes of molecules are in the
visible region of the electromagnetic spectrum (400–800 nm). Understanding the
origin and nature of these electronic states is needed to predict which
molecules are favorable for laser cooling experiments.

We will focus, for the moment, on molecules in which an alkaline-earth atom,
$M$, bonds to some electronegative ligand, $L$ (e.g., F, OH, OCH3, SH, etc.),
because these molecules are expected to have the structure desired for laser
cooling (Kozyryev et al. (2016a); Augenbraun et al. (2020a); Isaev et al.
(2017); Ivanov et al. (2019); Isaev and Berger (2016)). As explained by Ellis
(2001), the alkaline-earth atoms tend to form ionically-bonded molecules due
to their low ionization energies. Whether $M$ transfers one or both of its
valence electrons to the bonding partner depends on whether $L$ can form
singly or doubly charged anions. For the examples of $L$ listed above, singly-
charged anions form, so $M$ retains one electron even after forming a bond.

Figure 3: Illustration of orbital mixing reducing the interaction between
valence electron and negatively charged ligand. (a) Mixing of $s\sigma$ and
$p\sigma$ orbitals to generate the $\tilde{X}\,^{2}\Sigma^{+}$ state. (b)
Mixing of $p\pi$ and $d\pi$ orbitals to generate the $\tilde{A}\,^{2}\Pi$
state. In both cases, the rightmost image shows quantum chemical calculations
of the electronic distribution confirming this simple orbital mixing picture,
using CaOH as an example. Reproduced from Augenbraun (2020).

The relevant structural details can largely be extracted from a simple picture
involving just three ingredients: a positively charged metal ion, $M^{2+}$, an
optically active “valence” electron near the metal, and a negatively charged
ligand, $L^{-}$. The presence of an unpaired electron leads to low-lying,
metal-centered electronic excitations that can be used for optical cycling and
laser cooling. One would naively expect the unpaired electron to have dominant
$s$ orbital character in the electronic ground state. More detailed ligand-
field theory calculations (e.g., those by Rice et al. (1985); Allouche et al.
(1993)) show that this is largely true, but also that the interaction with the
negatively charged ligand deforms the valence electron to minimize electron-
electron repulsion (Ellis (2001)). For example, in the ground state this
deformation is realized via mixing of $s\sigma$ and $p\sigma$ orbitals on $M$,
a process shown schematically in Fig. 3. The orbital notation will be
described more in the next paragraph. It has been found that in CaF, the
ground electronic state ($\tilde{X}\,^{2}\Sigma^{+}$) arises from a mixture of
approximately $80\%$ of the $4s\sigma$ orbital and about $20\%$ of the
$4p\sigma$ orbital, while the lowest electronic state $\tilde{A}\,^{2}\Pi$ is
made up of about $70\%$ $4p\pi$ and $25\%$ $3d\pi$ character (Rice et al.
(1985)). Similar values are found for Ba-containing monohalides (Allouche et
al. (1993)) and larger monomethoxide species (Augenbraun et al. (2021b)).

An energy level diagram of the low-lying electronic states can be constructed
from these ideas (Dick (2007); Ellis (2001)). The basic idea is to consider
the Hamiltonian
$\mathbf{H}=\mathbf{H}_{M^{+}}+\mathbf{H}_{L^{-}}+\mathbf{H}^{\prime}$, where
$\mathbf{H}_{M^{+}}$ and $\mathbf{H}_{L^{-}}$ describe the energy levels of
the free ions and $\mathbf{H}^{\prime}$ describes the interaction between the
optically active valence electron and the ligand (Rice et al. (1985); Allouche
et al. (1993)). When the ligand is treated as a point charge perturbation, it
has three qualitative effects on the spectrum (Dick (2007)):

  1. 1.

It can shift the atomic ion’s energy levels.

  2. 2.

It can split the $m_{l}$ components of each atomic $nl$ state. We can think of
this as arising from strong electrostatic forces along the bond, producing a
Stark effect that resolves $m_{l}$ components along the bond axis. As is
typical for the Stark effect, we do not resolve $+m_{l}$ and $-m_{l}$
components, and therefore label states as $\lambda=\left\lvert
m_{l}\right\rvert=0,1,2,\ldots$. States with $\lambda=0,1,2,\ldots$ are
denoted by $\sigma,\pi,\delta,\ldots$, respectively.111We are using lower-case
letters here because we are describing the single valence electron. Below, we
will use capital letters to describe the total electronic state.

  3. 3.

It can mix orbitals obeying the selection rule $\Delta m_{l}=0$. This may seem
innocuous at first, but is actually quite important because this effect means
each molecular orbital is a linear combination of atomic ion orbitals. Not
only does this greatly affect the ordering of molecular states, it also means
the molecular parameters of a given state will appear as an “average” of the
atomic states that it comprises.

Figure 4: Correlation of low-lying electronic states from atomic ion ($M^{+}$)
to linear ($C_{\infty v}$) and nonlinear ($C_{s}$) molecules. The labeled
regions correspond to the following qualitative processes: (i) Shifting of
atomic ion levels, (ii) Splitting of atomic $m_{l}$ components, (iii) Mixing
of energy levels with the same $m_{l}$, (iv) Cylindrical symmetry breaking due
to a nonaxial ligand. Modeled after diagrams in Ellis (2001); Dick (2007).

Figure 4(i-iii) shows the development of the energy levels as these effects
are sequentially added. This diagram also shows how we name electronic states.
For a linear molecule ($C_{\infty v}$ point group symmetry), we label
electronic states by letters: $\tilde{X}$ for the ground state,
$\tilde{A},\tilde{B},\ldots$ for the first, second, …, electronically excited
states, respectively. Note that (for historical reasons) the alphabetical
ordering usually, but does not always, match the energetic ordering of energy
levels.222Electronic states with multiplicity different from than the ground
state are labeled by lower case letters $\tilde{a},\tilde{b},\ldots$, usually
in order of increasing energy. In addition, because in most cases the
electronic states can be identified by their electronic orbital angular
momentum ($\Lambda$) and spin multiplicity ($2S+1$), we use the labeling
scheme ${}^{2S+1}\Lambda$. For states with definite projection of spin onto
the molecular axis, we can also add a subscript ($\Omega$) to specify the
total angular momentum projection on the internuclear axis (see also
discussion of angular momentum coupling below).

Figure 5: Molecular orbitals of the lowest several electronic states for Ca-
containing molecules as symmetry is systematically lowered from $C_{\infty v}$
(CaOH) to $C_{3v}$ (CaCH3) to $C_{2v}$ (CaNH2) to $C_{s}$ (CaSH). The
distortion of the valence electron as a function of ligand asymmetry is
clearly visible. Panels (a-d) show, respectively, the HOMO, LUMO, LUMO+1, and
LUMO+2 for CaOH; (e-h) show these for CaCH3; (i-l) for CaNH2; and (m-p) for
CaSH. Figure reproduced from Augenbraun et al. (2020a).

This description has so far assumed the ligand can be treated as a point
charge, meaning the $M^{+}L^{-}$ system has cylindrical symmetry about the
$M-L$ axis. Such symmetry is exact only for linear molecules (diatomic or
polyatomic). If $ML$ is a nonlinear molecule but still retains some axial
symmetry (e.g., MOCH3) then the picture is largely unchanged except that there
will be no formal distinction between the symmetries of degenerate electronic
states (e.g., for $C_{3v}$ symmetry, $\Pi,\Delta,\ldots$ states are all
classified as having $E$ symmetry). If the ligand breaks the axial symmetry
(e.g., MSH), then the orbital degeneracy described in steps (2) and (3) just
above is no longer guaranteed, and degenerate electronic states can split. For
example, a $\Pi$ electronic state of CaOH will correlate to states of
$A^{\prime}$ and $A^{\prime\prime}$ symmetry for CaSH, as shown in Fig. 4(iv).
Figure 5 shows the clear role that asymmetry of the ligand has on slightly
deforming the valence electron that remains localized on the metal optical
cycling center. Despite the asymmetry, in all cases plotted the optical
cycling properties are preserved.

###  2.2 Vibrational structure

During laser cooling of a polyatomic molecule, a number of vibrational states
may be excited due to the lack of a perfectly closed cycling transition. These
states are generally close to the bottom of the potential energy surface,
which can be Taylor expanded as

$\displaystyle V=V_{0}$
$\displaystyle+\frac{1}{2}\sum_{i=1}^{3N}\sum_{j=1}^{3N}\left(\frac{\partial^{2}V}{\partial
q_{i}\partial q_{j}}\right)_{q_{i}=q_{j}=0}q_{i}q_{j}$
$\displaystyle+\frac{1}{3!}\sum_{i}\sum_{j}\sum_{k}\left(\frac{\partial^{3}V}{\partial
q_{i}\partial q_{j}\partial q_{k}}\right)_{0}q_{i}q_{j}q_{k}+\ldots$ (6)

Here we express the potential energy as a function of the $3N$ nuclear
coordinates $q_{i}$, which describe the 3-dimensional motion of all $N$
nuclei. To a good approximation, the low-lying vibrational states in this
potential may be described within the harmonic approximation, in which case
the vibrational Hamiltonian is (Demtröder (2003); Bernath (2017))

$H_{v}=\sum_{i=1}^{3N-6(5)}\left(-\frac{\hbar^{2}}{2}\frac{\partial^{2}}{\partial
Q_{i}^{2}}+\frac{1}{2}\lambda_{i}Q_{i}^{2}\right)$ (7)

where $Q_{i}$ is a mass-weighted normal coordinate describing the $i$th normal
mode of vibration and $\sqrt{\lambda_{i}}$ is the vibrational frequency of the
mode. Notice that for a polyatomic molecule there are $3N-6$ (or $3N-5$ for a
linear molecule) normal vibrational coordinates; the other 6(5) normal
coordinates are taken up by the 3 translational and 3(2) rotational degrees of
freedom of the molecule.

Vibrations along each of the normal coordinates are fully separable, so the
vibrational wavefunction can be expressed as

$\psi(Q_{1},Q_{2},\ldots,Q_{3N-6})=\prod_{i=1}^{3N-6}\psi_{v_{i}}(Q_{i})$ (8)

where $\psi_{v_{i}}(Q_{i})$ is the simple harmonic oscillator wavefunction
(Bernath (2017)) with vibrational quantum number $v_{i}$. The energy of the
state in the harmonic approximation is

$E_{v}=\sum_{i=1}^{3N-6}\hbar\omega_{i}\left(v_{i}+\frac{1}{2}\right)$ (9)

Reintroducing the anharmonic terms in the potential energy surface has two
effects. The first is to couple the harmonic oscillator eigenstates from Eq.
8, as discussed below. The second effect is to alter the state energies, which
can be expressed as (Demtröder (2003); Bernath (2017); Herzberg (1966))

$\displaystyle
G(v_{1},v_{2},\ldots,v_{p})=\sum_{i}\omega_{i}\left(v_{i}+\frac{d_{i}}{2}\right)$
$\displaystyle+\sum_{j\leq
i}x_{ij}\left(v_{i}+\frac{d_{i}}{2}\right)\left(v_{j}+\frac{d_{j}}{2}\right)$
$\displaystyle+\sum_{j\leq i}g_{ij}\ell_{i}\ell_{j}+\ldots$ (10)

where $\omega_{i}$ is the frequency of the $i$th mode, $d_{i}$ is its
degeneracy, $x_{ij}$ and $g_{ij}$ are anharmonicity constants, and $\ell$ is a
quantum number describing the vibrational angular momentum along the molecular
axis.

####  2.2.1 Vibrational state notation

Linear triatomic molecules. Linear triatomic molecules, including the alkaline
earth monohydroxides CaOH, SrOH, and YbOH, have 4 normal vibrational modes,
one of which (the bending mode) is doubly degenerate. The vibrational state is
labeled by $(v_{1}v_{2}^{\ell}v_{3})$, where $v_{1}$ is the quantum number for
the symmetric stretching mode, $v_{2}$ describes the bending mode, $v_{3}$
describes the antisymmetric stretching mode, and $\ell$ gives the vibrational
angular momentum in the bending mode. Note that this angular momentum arises
because linear combinations of the degenerate bending vibrations along two
perpendicular (linear) axes can be formed which have elliptical trajectories.
This can be thought of as producing a nuclear orbital angular momentum about
the molecular axis. Doubly degenerate bending modes such as this one are
formally described by the 2D harmonic oscillator, and $\ell$ can take the
values $v_{2},v_{2}-2,\ldots,-v_{2}+2,-v_{2}$ (see Bernath (2017)).

Larger polyatomic molecules. For larger molecules, labeling vibrational states
as $(v_{1},v_{2},\ldots,v_{N})$ becomes unwieldy. Instead, we use the notation
$i_{v_{i}}$, where $i$ labels the vibrational mode and $v_{i}$ is the
vibrational quantum number for that mode. Only modes with $v_{i}\neq 0$ are
labeled, and $i$ is enumerated from 1 to $n$, where $n$ is the total number of
modes. Typically, the modes are ordered first by the symmetry of the
vibrational mode and then in order of decreasing energy, as described in
Herzberg (1966). For example, in the symmetric top molecule CaOCH3, the
vibrational state $4_{1}$ corresponds to one excitation of the 4th vibrational
mode (Ca–O stretch), while $3_{1}4_{2}$ corresponds to 2 excitations of the
Ca–O stretch mode and one excitation of the O–C stretch mode (3rd vibrational
mode). In this notation, unspecified vibrational modes are assumed to have
$v_{i}=0$.

####  2.2.2 Anharmonic coupling

As mentioned above, one effect of the anharmonic terms in the potential energy
surface (Eq. 6) is to mix the harmonic oscillator wavefunctions, such that the
eigenstates of the vibrational Hamiltonian are admixtures of different
harmonic oscillator basis states. This effect is most prominent for
vibrational states of the same symmetry that are nearby in energy, and is
often referred to as _Fermi resonance_. The Fermi resonance interaction is
described in detail in Hougen (1962).

One example of Fermi resonance occurs in CaOH, whose bending mode frequency is
approximately half the stretching mode frequency. This means that, for
example, the $(100)$ and $(02^{0}0)$ states are nearly degenerate and strongly
mixed by a cubic term in the potential energy surface,
$V_{122}=k_{122}q_{1}q_{2}^{2}$, which mixes states with $\left\lvert\Delta
v_{1}\right\rvert=1$ and $\left\lvert\Delta v_{2}\right\rvert=2$. Here
$k_{122}$ is an anharmonic force constant. One practical implication of this
interaction is that excited electronic states that would decay to
$\widetilde{X}(100)$ can also decay to $\widetilde{X}(02^{0}0)$ at an enhanced
rate; this is important for understanding vibrational branching ratios in
polyatomic molecules (see Sec. 2.4.2). In this example, $(100)$ mixes with
$(02^{0}0)$ but not $(02^{2}0)$ because states with different $\ell$ have
different symmetry. For example, in a $\Sigma$ electronic state, $\ell=0$
vibrational levels have $\Sigma$ symmetry, $\ell=1$ vibrational levels have
$\Pi$ symmetry, and so on. For a more detailed discussion of symmetry in
polyatomic molecules, see Bunker and Jensen (2006).

###  2.3 Rotational structure

The rotational structure of molecules can be described, to leading order, by a
rigid rotor model. The Hamiltonian is $H_{{\rm
rot}}=\frac{R_{a}^{2}}{2I_{a}}+\frac{R_{b}^{2}}{2I_{b}}+\frac{R_{c}^{2}}{2I_{c}}$,
where $a,b,c$ denote the three principal axes in the molecular body-fixed
frame, $I_{a}\leq I_{b}\leq I_{c}$ are their corresponding moments of inertia,
and $R_{a},R_{b},R_{c}$ are the corresponding projections of the molecule’s
rotational angular momentum. It is useful to introduce the molecular constants
$A=(2I_{a})^{-1}$, $B=(2I_{b})^{-1}$, and $C=(2I_{c})^{-1}$. The energy level
structure resulting from this Hamiltonian depends on the relative values of
$I_{a},\,I_{b},$ and $I_{c}$. We describe each case in turn, initially
neglecting the effects of electronic, nuclear, and vibrational angular
momentum. In this scenario, the quantum number $N=J-S$ is equivalent to the
rigid-body rotational angular momentum $R$. We will denote the rotational
angular momentum by $N$ to make the notation compatible with the more general
case treated later.

####  2.3.1 Linear molecules

A linear molecule has $I_{a}=0$ and $I_{b}=I_{c}$ so that $B=C$. All diatomic
molecules, and a small but important class of polyatomic molecules including
alkaline-earth hydroxides like CaOH and SrOH, are linear. In the limit that
$I_{a}\rightarrow 0$, any angular momentum about the $a$-axis requires
infinite energy, so we take $N_{a}=0$. Then $H_{{\rm
rot}}=B(N_{b}^{2}+N_{c}^{2})=BN^{2}$. The energy levels of this Hamiltonian
form a quadratic ladder with eigenvalues $E_{N}=BN(N+1)$. Each $N$ manifold
contains $2N+1$ degenerate states, distinguished by the quantum number
$M\equiv N_{Z}$, the projection of $N$ on the lab-fixed $Z$-axis.

####  2.3.2 Spherical top molecules

In the special case that $I_{a}=I_{b}=I_{c}$, so that $A=B=C$, the Hamiltonian
also reduces to the form $H_{{\rm rot}}=BN^{2}$, again with eigenvalues
$E_{N}=BN(N+1)$. In this case, however, the additional structure leads to a
set of $(2N+1)^{2}$ degenerate states, distinguished by independent quantum
numbers $M$ and $K\equiv N_{a}$. Spherical top molecules include species such
as CH4 (methane), SF6 (sulfur hexafluoride), and C60 (buckminsterfullerene),
but to our knowledge no laser-coolable spherical top molecules have been
proposed to date and we do not consider them further here.

####  2.3.3 Symmetric top molecules

A molecule with exactly two equal moments of inertia is classified as a
symmetric top. This case is further subdivided into the prolate (“cigar-
shaped”) symmetric top, where $I_{a}<I_{b}=I_{c}$, and the oblate (“pancake-
shaped”) symmetric top, where $I_{a}=I_{b}<I_{c}$. We explicitly consider the
prolate case first; in the oblate case, one must only substitute $a\rightarrow
c$ and $A\rightarrow C$ in all formulas. Then the Hamiltonian is $H_{{\rm
rot}}=BN^{2}+(A-B)K^{2}$, where as before $K\equiv N_{a}$. The energy levels
are $E_{N,K}=BN(N+1)+(A-B)K^{2}$, with the restriction that $K\leq N$. There
are $2N+1$ degenerate states with energy $E_{N,0}$, distinguished by $M$. When
$K\neq 0$, there are $2(2N+1)$ degenerate states distinguished by both $M$ and
the sign of $K=\pm|K|$. Examples of prolate symmetric top molecules include
CH3F (methyl fluoride) and the laser-coolable species CaOCH3 (calcium
monomethoxide). Examples of oblate molecules include C6H6 (benzene) and NH3
(ammonia); to date, no oblate symmetric top molecules have been proposed for
laser cooling. For molecules where $A\gg B$, typical of known laser-coolable
molecules, there is a quadratic ladder of widely separated “$K$-stacks,” each
of which contains a quadratic ladder of more finely separated $N$ states.

####  2.3.4 Asymmetric top molecules

In an asymmetric top molecule, all moments of inertia are unequal. The
resulting energy level structure is quite complex, and no projection of $N$ on
the molecule-frame axes is a good quantum number. For each value of $N$, there
are $2N+1$ non-degenerate states. The degree of asymmetry can be characterized
by the Ray’s asymmetry parameter, $\kappa=\frac{2B-A-C}{A-C}$. In the prolate
(oblate) symmetric top limit, $B=C(A)$, we obtain $\kappa=1(-1)$. In the
maximally asymmetric case, $B=(A+C)/2$, we obtain $\kappa=0$. Eigenstates can
be designated by labels $N_{K_{a}K_{c}}$, where $K_{a}$ and $K_{c}$ are the
projections of $N$ on the $a$-axis and $c$-axis in the limit that a molecule
is deformed to the prolate and oblate symmetric top limits, respectively. In
the general case, $-1<\kappa<1$, neither $K_{a}$ nor $K_{c}$ are rigorously
good quantum numbers.

Laser-coolable asymmetric top molecules proposed in the literature to date
(for example, CaSH, CaOCHDT, and SrOC6H5) are prolate ($\kappa<0$), though
there is no apparent reason that oblate molecules ($\kappa>0$) are necessarily
inconsistent with laser cooling. An example of an oblate asymmetric top
molecule is C4H4N2 (pyrimidine). In the case of an asymmetric top molecule
near the prolate symmetric top limit, $\kappa\approx-1$, the energy level
structure closely resembles that of a prolate symmetric top. A similar
situation holds for asymmetric top molecules near the oblate limit.

####  2.3.5 Dependence on spin and electronic angular momenta

The ground states of proposed laser-coolable polyatomic molecules generally
have vanishing electronic angular momentum, and can be described in a Hund’s
case (b) basis where $N$ is a good quantum number. Corrections to the
rotational structure described above occur due to electron spin, nuclear spin,
and vibrational angular momentum (for example in the bending vibrational mode
of a linear triatomic molecule). The last of these shifts the overall energy
of the ladder of eigenstates but does not otherwise change the rotational
energy progression.

Coupling of the electron spin and molecular rotation is generally more
complicated. The spin-rotation Hamiltonian takes the general form

$H_{{\rm
SR}}=\frac{1}{2}\sum_{\alpha,\beta}\epsilon_{\alpha\beta}(N_{\alpha}S_{\beta}+S_{\beta}N_{\alpha}),$
(11)

where $\alpha,\beta$ takes values $a,b,c$ and $\epsilon$ is a symmetric tensor
described in, e.g., Hirota (1985). In a linear molecule, only the components
$\epsilon_{bb}=\epsilon_{cc}\equiv\gamma$ take nonzero values and the simpler
form

$H_{{\rm SR}}=\gamma N\cdot S$ (12)

is obtained. For linear molecules with $S=1/2$, this term mixes adjacent $N$
levels, and both $N$ and $S$ cease to be good quantum numbers, though
$\vec{J}=\vec{N}+\vec{S}$ is preserved. When $\gamma\ll B$, as is typical of
the alkaline-earth pseudohalides, the effect is to split $N$ into a pair of
states with $J=N\pm 1/2$.

The situation is more complicated for nonlinear molecules. In a symmetric top
molecule, the spin-rotation Hamiltonian (neglecting contributions that can mix
different values of $\Lambda$ in a ${}^{2}E$ state) reduces to

$H_{\text{SR}}=\epsilon_{aa}N_{a}S_{a}+\frac{1}{4}(\epsilon_{aa}+\epsilon_{bb})(N_{+}S_{-}+N_{-}S_{+}),$
(13)

assuming the prolate case (see Hougen (1980)). The asymmetric top case is
generically much more complicated, and spin-rotation interactions can mix
states that differ in $N$ by up to 1, and differ in $K$ by up to 2 (Hirota
(1985)). In alkaline-earth pseudohalides, the ground-state spin-rotation
splittings are typically no larger than $\sim$100 MHz at low $N$, and increase
with larger $N$. In most laser-coolable molecules used so far, the size of the
spin-rotation constant is dominated by a second-order perturbation via spin-
orbit coupling rather than the direct coupling of electron spin and molecular
rotation. Excited states with spin-rotation structure, for example the
$\tilde{B}\,^{2}\Sigma^{+}$ state in MOH molecules, can have spin-rotation
constants comparable to the rotational constant due to the closer proximity to
electronic states with $\Delta\Lambda=\pm 1$. Because excited-state spin-
rotation structure is large compared to the natural linewidth, and because the
ground-state spin-rotation splittings are $\sim$100 MHz or less, the spin-
rotation structure is easily addressed with frequency sidebands added to the
laser beam, e.g. using acousto-optic or electro-optic modulation.

In states with orbital angular momentum, for example a ${}^{2}\Pi_{1/2}$
electronic state of a linear molecule, electrostatic interactions couple the
orbital angular momentum $L$ to the internuclear axis and spin-orbit
interactions couple $L$ with $S$. In this case, $N$ is not a good quantum
number. The rotational energies follow a quadratic ladder in $J$ as $H_{{\rm
rot}}=BJ(J+1)$, with each $J$ value split into a pair of opposite-parity
states by a $\Lambda$-doubling Hamiltonian, which arises from spin-orbit
mixing with states of different $\Lambda$. In symmetric top and asymmetric top
molecules, similar interactions split pairs of opposite-parity states in
electronic manifolds with non-zero orbital angular momentum.

####  2.3.6 Hyperfine effects and nuclear spin statistics

In many of the polyatomic laser cooling experiments pursued to date, the
valence electron responsible for optical cycling is localized on the spin-0
nucleus of an alkaline-earth (or alkaline-earth-like) metal atom, and a spin-0
oxygen nucleus serves as a “linker” to a ligand such as H and CH3. As a
result, the nuclei with non-zero spin are far from the optically active
valence electron, and hyperfine interactions are on the order of only
$\lesssim 1$ MHz in both the ground and excited electronic states. Hyperfine
interactions therefore play a negligible role in laser cooling, though they
might be important in single-quantum-state preparation and readout as needed
for many applications (see Sec. 3.9).

Some applications, for example studying nuclear-spin-dependent parity
violation (Norrgard et al. (2019a)) or nuclear magnetic quadrupole moments
(Hutzler (2020)), require the optically active electron to be localized near a
nucleus with spin $I>0$. This complicates the structure by splitting each $J$
level into $2I+1$ states of distinct $F$ levels. Generally, this should have
no substantial effect on laser cooling as long as each $F$ level is optically
addressed by either spectral broadening or the addition of frequency sidebands
to optical cycling lasers.

Another consequence of nuclear spins in polyatomic molecules stems from the
connection between nuclear spin symmetry and molecular rotation that arises
due to symmetry requirements on the total wavefunction, as described in Bunker
and Jensen (2006). In the simplest case of a molecule composed of two
identical atoms with vanishing nuclear spin, some rotational levels do not
exist because the wave function must be even under exchange of the identical
nuclei–for example, the ground ${}^{3}\Sigma_{g}^{-}$ manifold of 16O2 only
has odd $J$ levels. Generically, rovibronic levels occur with a “statistical
weight” that corresponds to the number of nuclear spin states that give the
required totally symmetric state. For example, in CaOCH3 the “para” nuclear
spin configuration (in which two hydrogen nuclear spins are aligned) occurs in
$K=1,2,4,5,\ldots$ rotational states, but the “ortho” spin configuration (in
which all nuclear spins are aligned) occurs in $K=0,3,\ldots$ rotational
states. Because of the weak hyperfine coupling in CaOCH3, conversion between
para and ortho configurations is highly suppressed; because of the lack of
interconversion, the two nuclear spin configurations behave as though they
were essentially independent species. Experimentally, laser cooling schemes
for both nuclear spin isomers were demonstrated by Mitra et al. (2020) where
the isomer cooled was selected spectroscopically, by driving transitions out
of either the $K=0$ or $K=1$ rotational state.

###  2.4 Transitions

In the following sections, we describe the properties and intensities of
transitions between the various energy levels present in polyatomic molecules.

####  2.4.1 Electronic transitions

Transition intensities can be calculated from the square of the transition
moment integral (Bernath (2017))

$\mathbf{M}=\int\psi^{\prime}(\mathbf{r},\mathbf{R})^{\ast}\,\bm{\mu}\,\psi(\mathbf{r},\mathbf{R})\,d\tau,$
(14)

where integration with respect to $\tau$ implies integrating over all
electronic and nuclear coordinates, and single primes denote excited states.
Invoking the BO approximation, we can write
$\psi(\mathbf{r},\mathbf{R})=\psi_{n}(\mathbf{R})\psi_{e}(\mathbf{r};\mathbf{R})$.
Then the transition moment integral becomes

$\mathbf{M}=\int\psi_{n^{\prime}}(\mathbf{R})^{\ast}\left(\int\psi_{e^{\prime}}^{\ast}(\mathbf{r};\mathbf{R})\,\bm{\mu}\,\psi_{e}(\mathbf{r};\mathbf{R})\,d\tau_{e}\right)\psi_{n}(\mathbf{R})\,d\tau_{n}.$
(15)

We have separated integration over all nuclear coordinates ($d\tau_{n}$) and
electronic coordinates ($d\tau_{e}$). Let us now define an electronic
transition dipole moment as
$\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}(\mathbf{R})=\langle\psi_{e^{\prime}}|\bm{\mu}|\psi_{e}\rangle$,
and we explicitly denote its dependence on the nuclear coordinates. We can
imagine expanding
$\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}(\mathbf{R})$ in a Taylor
series about some value of $\mathbf{R}$ and retaining only the first
(constant) term, which we will denote by
$\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}$.333More details about the
choice of point on which to center the expansion can be found in the textbook
by Bernath (2017). Often, the value of
$\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}$ is inferred from
measurements or $\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}(\mathbf{R})$
can be found via ab initio calculations and used in conjunction with numerical
vibrational wavefunctions to compute the necessary integral. This simplifies
Eq. 15 and allows us to write

$\displaystyle\mathbf{M}$ $\displaystyle=$
$\displaystyle\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}\int\psi_{n^{\prime}}(\mathbf{R})^{\ast}\psi_{n}(\mathbf{R})\,d\tau_{n}.$
(16)

The transition moment has factored into an electronic and nuclear portion, as
would be expected from the BO approximation treatment. If we explicitly
include both vibrational and rotational nuclear motions, we can write the
intensities of rovibronic transitions as (Bernath (2017))

$I_{(e^{\prime}v^{\prime}J^{\prime})\rightarrow(evJ)}=\left\lvert\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}\right\rvert^{2}q_{v^{\prime}\rightarrow
v}S^{J^{\prime}}_{J^{\prime\prime}},$ (17)

where $\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}$ is the electronic
transition dipole moment between electronic states $e^{\prime}$ and $e$,
$q_{v^{\prime}\rightarrow v^{\prime\prime}}$ is a Franck-Condon factor (FCF)
between vibrational states $v^{\prime}$ and $v$, and
$S^{J^{\prime}}_{J^{\prime\prime}}$ is a Hönl-London factor between states
$J^{\prime}$ and $J$. We can interpret this factorization as stating that the
intrinsic strength of some transition is set by
$\left\lvert\bm{\mathcal{R}}^{e^{\prime}}_{e^{\prime\prime}}\right\rvert^{2}$
while the distribution of that intensity among vibrational and rotational
lines are set by $q_{v^{\prime}\rightarrow v^{\prime\prime}}$ and
$S^{J^{\prime}}_{J^{\prime\prime}}$.

####  2.4.2 Vibrational transitions

Figure 6: The Franck-Condon principle, as illustrated for diatomic molecules
using CaF. (a) The $X\,^{2}\Sigma^{+}$ and $B\,^{2}\Sigma^{+}$ electronic
orbitals of the prototypical laser cooling molecule CaF. Localization of the
electron density near the Ca metal in both the ground and excited states leads
to favorable Franck-Condon overlap. (b) Vibrational wavefunctions of CaF. (c)
The corresponding vibrational decay strengths for the
$X\,^{2}\Sigma^{+}-B\,^{2}\Sigma^{+}$ and $X\,^{2}\Sigma^{+}-C\,^{2}\Pi$
transition of CaF.

Unlike rotational transitions, where angular momentum conservation rigorously
constrains the allowed decay channels, in vibrational transitions no selection
rules are absolute. Instead, vibrational branching is governed in the Born-
Oppenheimer approximation by the overlap of the nuclear wave functions of the
initial and final vibronic states, as characterized by the Franck-Condon
Factor,

$q_{v^{\prime}\rightarrow
v^{\prime\prime}}=\left\lvert\int\psi_{e,v^{\prime}}^{\ast}\psi_{g,v}d\tau_{n}\right\rvert^{2}.$
(18)

where $v^{\prime}$ denotes a vibrational state in the excited electronic
manifold $e$, and $v$ denotes a vibrational state in the ground electronic
manifold $g$. If the Franck-Condon factors of an electronic transition are
“diagonal” then $q_{v^{\prime}\rightarrow
v^{\prime\prime}}\approx\delta_{v^{\prime},v}$; in other words, a vibrational
level $e(v^{\prime})$ decays (almost) only to $g(v)$. Geometrically, this will
occur when all bond lengths, bond angles, and vibrational constants are
approximately identical between the ground and excited electronic states. In
practice, off-diagonal FCFs are more sensitive to small fractional changes in
bond lengths compared to similarly small fractional changes in harmonic
constants. This can be understood from a simple model of wave function overlap
between displaced 1D harmonic oscillators with distinct harmonic constants. An
example of this, for the diatomic molecule CaF, is shown in Fig. 6.

The vibrational branching ratios from a given vibronic excited state are
proportional to the FCFs, but contain an additional factor of
$\omega_{v^{\prime},v}^{3}$:

$b_{v^{\prime}\rightarrow v}=\frac{q_{v^{\prime}\rightarrow
v}\omega^{3}_{v^{\prime},v}}{\sum_{v}q_{v^{\prime}\rightarrow
v}\omega^{3}_{v^{\prime},v}}.$ (19)

Thus, transitions to lower-lying vibrational states are slightly favored,
relative to what one might expect from considering only the FCF. In practice,
the VBRs and FCFs are quantiatively similar, but only the VBR is important for
determining and achieving a nearly-closed optical cycle.

In the Born-Oppenheimer (BO) approximation, the symmetry of a vibrational
state is conserved in a vibronic transition. For example, in linear molecules
like CaOH and SrOH, $|\ell|$ would not be expected to change. In practice,
non-BO perturbations induce weak $\left\lvert\ell\right\rvert$-changing
transitions at the level of $\sim$$10^{-3}$ vibrational branching probability;
see Sec. 2.5 for details.

####  2.4.3 Measuring vibrational branching ratios

Figure 7: Demonstration of optical cycling with the polyatomic molecule SrOH.
Molecular beam fluorescence with (blue) and without (red) addressing both
spin-rotation components of the optical cycling rovibronic transition. When
both spin-rotation components are addressed, population cycles between the
ground $N=1$ and excited $J^{\prime}=1/2$ manifolds, and repeated photon
scattering increases the detected fluorescence by over an order of magnitude.
Reproduced from Kozyryev et al. (2016b).

Vibrational branching ratios can be measured in any of several ways. The
simplest method conceptually is to add repumping lasers (lasers that address
vibrational loss channels and return them to the set of bright states) one at
a time and observe the total fluorescence collected. To determine the diagonal
vibrational branching ratio, one can measure the fluorescence induced when
addressing all spin-rotation and hyperfine components of a rotationally-closed
vibronic transition, and then compare this to the fluorescence induced when
only a single quantum state is addressed (for which typically $\sim$1 photon
is scattered before populating an unaddressed quantum state, depending only on
theoretically well-known rotational branching ratios). See Fig. 7 for an
example in SrOH. In a similar manner, observing the fluorescence collected
when repumping $v$ in an optical cycle, as opposed to exhausting a lossy
optical cycle without repumping $v$, reveals the probability that $v$ is
populated among all possible loss channels. By sequentially adding repumpers,
in principle all excited vibrational branching ratios can be determined in
this way. In practice, this method may be suitable to measure vibrational
branching ratios in a long-lived trap (e.g., a MOT), but it is impractical in
a molecular beam for VBRs smaller than $\sim$1% because molecules exit the
fluorescence region before exhausting a highly closed optical cycle.

An improved method to measure vibrational branching ratios below $\sim$1% is
to optically cycle $\sim$100 photons in an “interaction region,” e.g. by
addressing the diagonal vibronic transition and dominant one or two repumping
pathways, and to measure the fraction of molecular population that is
recovered to low-lying states when repumping a state $v$ in a “cleanup region”
(between the interaction region and fluorescence detection region). If
recovery of $\sim$1% of the molecular population can be resolved after
$\sim$$10^{2}$ photons are scattered in the interaction region, then
vibrational branching ratios of $\sim$$10^{-4}$ can be measured. This method
was used by Baum et al. (2021) to determine an optical cycling scheme for CaOH
with approximately $5\times 10^{3}$ photons scattered per molecule.

A major limitation of the methods described above is that a vibrational
branching ratio can only be measured for a state with known high-resolution
repumping transitions. Most polyatomic molecules that might be of interest for
future laser-cooling experiments have little preexisting spectroscopic data,
especially in excited vibrational states. Therefore, an alternative method
should be used to screen a potential laser-coolable molecule for favorable
VBRs without requiring high-resolution spectroscopy of up to a dozen
vibrational states.

Figure 8: Dispersed laser-induced fluorescence apparatus, reproduced from
Zhang et al. (2021). CaOH and YbOH molecules are produced in a CBGB via
ablation of a metal precursor in the presence of water vapor. 40 cm
downstream, molecules are excited via an optical cycle with 50$-$100 photons
scattered per molecule on average, increasing the fluorescence yield.
Fluorescence is collimated by an in-vacuum lens and directed toward a Czerny-
Turner monochromator, which disperses light emitted in different vibronic
transitions (generally separated by many nm in wavelength) onto different
regions of an EMCCD camera. Figure 9: Dispersed fluorescence measurements of
YbOH $\tilde{A}(000)$, showing 89% probability of decay on the diagonal
vibronic transition, with progressively weaker decays to higher-lying
vibrational states. Decays with strength as small as $2(1)\times 10^{-5}$
relative probability can be measured due to the increased fluorescence yield
arising from optical cycling excitation. Reproduced from Zhang et al. (2021).

The standard approach to measure VBRs without directly repumping vibrationally
excited states is dispersed laser-induced fluorescence (DLIF). See Fig. 8 for
a typical experimental configuration. Molecules in a molecular beam are driven
to an excited state of interest, $e$, and subsequently fluoresce to the
vibronic ground states $\\{v_{i}\\}$ with probabilities $\\{P_{i}\\}$. The
fluorescence is collected and focused into a spectrometer, in which a
diffraction grating is used to spatially disperse the fluorescence
wavelengths. The light is then imaged onto an EMCCD, producing an image that
shows the relative intensity of each decay feature. Example data are shown for
emission from the $\tilde{A}(000)$ state of YbOH in Fig. 9.

With $\sim$$10^{4}$ ablation pulses, vibronic decays with relative
probabilities of $10^{-2}-10^{-3}$ may be detected. By optically cycling
molecules in the detection region, up to 100 photons may be scattered per
molecule, directly increasing the number of fluorescence photons collected. In
cases where the detection sensitivity is limited by camera read-noise or
clock-induced-charge, the sensitivity of the measurement is increased
proportionally to the number of photons scattered per molecule. In this way,
branching ratios on the order of $10^{-5}$ have been measured for CaOH, YbOH,
and SrOH by Zhang et al. (2021); Lasner et al. (2022). The unambiguous
assignment of weak decays is greatly aided by high-quality theoretical
predictions, which must account for perturbations like those described in Sec.
2.5.

DLIF measurements of VBRs are a powerful method to directly observe all decay
channels (inside an observed wavelength range) without requiring high-
resolution spectroscopy for optical pumping and repumping of many vibrational
states. It is especially important for measuring vibrational branching ratios
of polyatomic molecules, where numerous vibrational states may have VBRs
around $10^{-5}-10^{-4}$, a level that is weak in an absolute sense but
sufficiently strong to require repumping during radiative slowing and magneto-
optical trapping. For a polyatomic molecule with many vibrational modes, it is
not necessarily clear in the absence of measurements what the dominant loss
channel will be for an optical cycle scattering many more than $\sim$100
photons per molecule. This makes a brute-force search for vibrational
repumpers inefficient in all but the exceptional cases where the molecule is
extensively well-understood from the outset.

####  2.4.4 Repumping transition spectroscopy

Figure 10: Beamline configuration for pump-repump spectroscopy of
vibrationally excited states. An upstream interaction populates vibrationally
excited states either through optical cycling or direct off-diagonal
excitation. A clean-up region recovers population into a detected state (or
set of states), which is excited by a known laser transition to produce
fluorescence that is imaged onto an EMCCD camera. When a laser in the
interaction region is scanned over an excited state with the clean-up lasers
off or out of resonance, a dip in detected fluorescence is observed. This
enables the discovery of excited-state repumping pathways. When a laser in the
clean-up region is scanned over a repumping transition, detected fluorescence
is partially recovered.

Even in the best-studied laser-coolable molecules such as CaOH and SrOH, the
vibrational states that become populated after a molecule has scattered around
$10^{4}$ photons have not all been fully analyzed in the literature. It is
thus necessary to conduct spectroscopic searches for repumping pathways. These
measurements are often conducted in one of two ways: fluorescence in a
molecular beam, or absorption in a cryogenic buffer-gas cell.

When repumper spectroscopy is conducted using laser-induced fluorescence from
a molecular beam (see Fig. 10), a vibrationally excited state can be prepared
either by optically cycling until vibrational dark states are populated, or by
direct off-diagonal excitation that decays preferentially to the vibrational
state of interest in the ground electronic manifold. Due to the rotationally-
closed nature of the transition used for optical cycling, often only the $N=1$
level of the target vibrational state is populated. (Additional rotational
levels may be populated in nonlinear polyatomic molecules where the rotational
selection rules are slightly relaxed; see Tab. 1.) The frequency of a laser in
a downstream “clean-up” region can then be scanned to repump population into
(often lower-lying) detectable states, which are excited in a region even
farther downstream. Fluorescence in the detection region is observed on a PMT
or EMCCD camera. When the frequency of the repumping laser passes through a
resonance, an increase in detected population will be observed. Since CBGBs
sometimes do not efficiently thermalize the molecular vibrational
distribution, in some cases a molecular beam may have sufficient natural
population of a vibrationally excited state to detect, provided the correct
repumping frequency is addressed in the clean-up region. The same geometry
shown in Fig. 10 can also be used to search for excited vibrational states in
the electronically excited manifold by scanning the frequency of a laser in
the interaction region and observing a dip in the detected population of low-
lying vibrational states.

This pump-repump method is best-suited to discovering the frequency of
_specific_ rovibronic transitions, since in practice at most a few states can
be simultaneously detected in the downstream region. This means that, for a
given experimental configuration, only a few transitions (including the “laser
cooling transition”) in the clean-up region can produce a spectroscopic
signal, dramatically simplifying the data analysis. A corresponding
disadvantage, however, is that the signal is spectroscopically sparse, and a
good initial estimate of the repumping frequency is required. This estimate
can be made either from the observed decay wavelengths obtained via DLIF, or
by high-quality theoretical predictions (e.g., made by extrapolating from the
known positions of lower-lying vibrational states). Repumper frequency
uncertainties obtained by high-resolution DLIF measurements are typically on
the order of 5 cm-1, but can be improved by using a diffraction grating with
higher line density or by narrowing the width of the entrance slit to the
spectrometer.

A related technique can be used to locate repumping transitions after a MOT
has been achieved. Namely, one can apply laser light near a suspected
repumping transition in the MOT region, with all other trapping laser beams
on. While scanning the frequency of the (unknown) repumping laser, one
monitors the MOT lifetime. When a resonance is reached, the lifetime of the
MOT should increase. This is analogous to the pump-repump method but with an
interaction time of tens or even hundreds of ms, which would be impractical in
a molecular beam.

An alternative approach to identifying vibrational repumpers is high-
sensitivity absorption measurements inside a buffer gas cell (Pilgram (2023)).
Frequency-modulated (FM) absorption spectroscopy has been used to observe the
weak $\Delta v=-2$ repumping transition for states as high as $(300)$ in YbOH.
Unlike the pump-repump spectroscopy described above, all thermally populated
rotational states will be probed in this way. This may be desirable in order
to make a full spectroscopic assignment of the molecular transitions and
constants in a vibronic transition, or to more quickly locate a
spectroscopically active frequency region (since a denser “forest” of lines
appears, compared with the signal in pump-repump measurements). On the other
hand, the spectrum must be fully assigned in order to identify which
transitions are connected to the laser cooling state. Aside from these broader
considerations, the experimental signal-to-noise ratio may favor either the
pump-repump or high-sensitivity absorption measurements depending on factors
like the vibrational quenching efficiency in the buffer gas cell, the degree
of scattered light suppression in the downstream fluorescence region, the
strengths of the probed transitions, and other technical factors.

####  2.4.5 Designing an optical cycle

With known vibrational branching ratios (e.g., from high-resolution DLIF
measurements), one must carefully design the optical cycle. Several factors
are important:

  * 1.

The number of lasers should be minimized, to reduce cost and experimental
complexity

  * 2.

The excited state that is coupled to the ground vibrational state–e.g., (000)
in an MOH molecule–should not be coupled to any other vibrational states if
possible, so as to maximize the photon scattering rate and optical forces (see
Sec. 1.5.2)

  * 3.

Lasers should be at convenient optical wavelengths and available at high power

  * 4.

Optical transitions should be strong (e.g., $\Delta v=-1$ repumpers) to
increase the optical pumping or repumping rates for fixed laser powers

  * 5.

Decays to states without existing spectroscopy should be eliminated where
possible

Not all of these factors are mutually compatible, and some balancing is
required (e.g., the number of lasers is generally minimized when only a single
excited state is used). Typically, only one or two “diagonal enough” excited
states are available to choose from for the dominant excitation, and one of
these may result in significantly fewer states populated after the $\sim
10^{4}$ photon scattering events required for magneto-optical trapping. For
example, in SrOH, both $\tilde{A}(000)$ and $\tilde{B}(000)$ appear to have
reasonably diagonal VBRs, but high-resolution DLIF measurements show that
$\tilde{B}(000)$ populates $(03^{1}0),$ $(12^{0}0)$, $(13^{1}0)$, and
$(05^{1}0)$ at the $\sim 10^{-4}$ level; none of these are significantly
populated by $\tilde{A}(000)$, as demonstrated in Lasner et al. (2022).
Therefore, many additional repumpers would need to be added to scatter
$10^{4}$ photons primarily through $\tilde{B}(000)$, as compared to scattering
the same number of photons through $\tilde{A}(000)$. For this reason, an
optical cycle coupling $\tilde{A}(000)\leftarrow\tilde{X}(000)$ is favored.
Because vibrationally excited states are populated only after approximately 20
photons are scattered through $\tilde{A}(000)$, the $\tilde{B}(000)$ state may
be used as a repumping pathway without limiting the optical cycle to less than
$1.5\times 10^{4}$ photons scattered. Wherever possible, the optical cycle
proposed in Lasner et al. (2022) favors repumping through the $\tilde{B}$
manifold due to the much more inexpensive and convenient high-power sum-
frequency generation (SFG) sources around 630$-$650 nm compared with 690$-$710
nm. The only exception is that the $(02^{2}0)$ state cannot be repumped
through $\tilde{B}$ due to a near-vanishing transition strength, and is
instead coupled to $\tilde{A}(100)$. Wherever possible, the strongest
transition that decreases vibrational quantum numbers from the ground to
excited state is chosen, under the constraint that $\tilde{A}(000)$ may not
coupled strongly to any state except $\tilde{X}(000)$. The resulting optical
cycle is shown in Fig. 11, along with the optical cycle used to magneto-
optically trap CaOH.

To model the number of scattered photons before loss to unaddressed
vibrational states occurs, it is useful to construct an absorbing Markov chain
model similar to that described in Baum et al. (2021): the states of the
Markov chain represent vibrational levels in the electronic ground manifold.
For a given optical cycling scheme, the transition probabilities from a state
$v$ are given by the VBRs of the excited state to which $v$ is coupled. Each
step of the Markov chain represents a single photon scatter. Unaddressed
vibrational states “transition” only to themselves, and are absorbing states
in the Markov chain. In this model, it is straightforward to calculate many
properties of interest to laser cooling, including the average number of
scattered photons before an absorbing (i.e., dark vibrational) state is
reached, how many times each transient state is visited on average, and the
distribution of population among dark states.

Figure 11: (a) Optical cycling scheme proposed for laser cooling and trapping
SrOH, reproduced from Lasner et al. (2022). (b) Optical cycling scheme used to
produce a MOT of CaOH.

####  2.4.6 Rotational transitions

Rotational transitions follow a generally well-behaved set of selection rules
governed by angular momentum algebra. Unless otherwise stated, in this section
we consider only electric dipole (E1) transitions. Higher-order transitions
(e.g. M1, E2, etc.) are significantly weaker and are typically irrelevant for
molecular laser cooling.

The general formula for generating a closed cycling transition in molecules
was first proposed by Stuhl et al. (2008), and relies on rotational and parity
selection rules ($\Delta J=0,\pm 1$; $\Delta p=\pm 1$; $J=0\rightarrow
J^{\prime}=0$ forbidden). The key observation was that, while a traditional
“type-I” $J\rightarrow J+1$ transition cannot be rotationally closed (since a
rotational level with $J+2$ necessarily exists in the ground state), driving
the “type-II” $J=1\rightarrow J^{\prime}=0$ transition (or its closest
analogue within the molecule of choice) forms a closed transition. One
consequence of this choice of transition is that there are more ground states
than excited states, $m_{J}>m_{J}^{\prime}$, meaning that there will be dark
states regardless of the laser polarization chosen. These can be remixed with
magnetic fields or polarization modulation (Berkeland and Boshier (2002)).

Geometry | Transition axis | Basis∗ | Selection rules∗
---|---|---|---
Linear | $\parallel$ | b $\rightarrow$ b |  $\Delta\Lambda=0$; $\Delta\ell=0$; $\Delta N=0,\pm 1$
Linear | $\perp$ | b $\rightarrow$ a |  $\Delta\Lambda=\pm 1$; $\Delta\ell=0$
STM | $\parallel$ | b $\rightarrow$ b |  $\Delta K=0$; $\Delta N=0,\pm 1$
STM | $\perp$ | b $\rightarrow$ a |  $\Delta K_{R}=0$; $\Delta K=\pm 1$
ATM | a type | b $\rightarrow$ b |  $\Delta K_{a}=0$; $\Delta K_{c}=\pm 1$; $\Delta N=0,\pm 1^{**}$
ATM | b type | b $\rightarrow$ b |  $\Delta K_{a}=\pm 1$; $\Delta K_{c}=\pm 1$; $\Delta N=0,\pm 1^{**}$
ATM | c type | b $\rightarrow$ b |  $\Delta K_{a}=\pm 1$; $\Delta K_{c}=0$; $\Delta N=0,\pm 1^{**}$
Table 1: Summary of the main selection rules discussed in this review. All
transitions follow the selection rules $\Delta J=0,\pm 1$ and $\Delta p=\pm 1$
in addition to those listed. Also note that at least one angular momentum
quantum number must change, so transitions with $\Delta K=\Delta N=\Delta J=0$
(or analogous) are forbidden.
∗Note that the selection rules described here apply only when the states are
exactly described by the basis listed; in many real scenarios the selection
rules are only approximate.
${}^{**}\Delta N=2$ transitions may be allowed in ATMs where the electronic
angular momentum is not fully quenched.

All polyatomic molecules that have been directly laser cooled (and most
polyatomics proposed for laser cooling) have a single valence electron and
therefore have spin-doublet ground states ($S=1/2$). In this case, the
rotational angular momentum $N$ couples to the electron spin to form a total
angular momentum quantum number $J$, which takes half-integer values.444Here
we ignore hyperfine degrees of freedom, which are often unresolved in
polyatomic molecules, though the selection rules discussed below can be easily
generalized to include hyperfine structure. In such a system, the electric
dipole selection rules $\Delta J=0,\pm 1$ and $J=0\nrightarrow J^{\prime}=0$
always apply, as well as the requirement that the parity of the state changes,
$\Delta p=\pm 1$. In this case, the excited state used to achieve rotational
closure is no longer $J^{\prime}=0$ but $J^{\prime}=1/2$.

In addition to the universal selection rules on $J$ and parity, other
selection rules exist for specific molecular geometries and angular momentum
coupling cases. These are summarized in Table 1. The selection rules are
sorted by molecule geometry, as well as the axis along which the transition
dipole moment is induced. For molecules with cylindrical symmetry, transitions
can be either parallel ($\parallel$) or perpendicular ($\perp$), meaning the
transition dipole moment is induced either along the principal molecular axis
or perpendicular to it, respectively. For asymmetric top molecules, a
transition can be induced along any of the three molecular axes, as described
further below. Finally, different selection rules apply depending on the
coupling of angular momenta within each electronic state involved. Typically,
in nondegenerate states with zero electronic angular momentum Hund’s case (b)
applies, where both $N$ and $J$ are good quantum numbers. However, degenerate
states with electronic angular momentum $\Lambda\neq 0$ are typically
described by Hund’s case (a), where the electron spin is strongly coupled to
the molecular axis and $N$ is no longer a good quantum number.

Rotational transition strengths are described by the Hönl-London factors
$S_{J}^{J^{\prime}}$ in eqn. 17. They can be calculated using angular momentum
algebra by taking matrix elements of the dipole operator (see, e.g., Hirota
(1985)), or looked up in tables.

Figure 12: Closed cycling transitions in alkaline-earth monohydroxides (e.g.
CaOH) for laser cooling on the
$\widetilde{X}^{2}\Sigma^{+}\rightarrow\widetilde{A}^{2}\Pi_{1/2}$ and
$\widetilde{X}^{2}\Sigma^{+}\rightarrow\widetilde{B}^{2}\Sigma^{+}$ electronic
transitions. (a) For nondegenerate ($\ell=0$) vibrational levels of the ground
state, two laser frequencies are required to address the $J=1/2^{-}$ and
$J=3/2^{-}$ levels of the $N=1$ rotational state. (b) For repumping the
$\widetilde{X}(010)$ bending mode, which has vibrational angular momentum
$|\ell|=1$, an additional color addressing the $N=2,J=3/2^{-}$ state is
required, regardless of which excited state is chosen. The dashed line refers
to a transition which is allowed only when the excited state does not adhere
to the case (b) limit, i.e. when $N$ is not a good quantum number.

Linear molecules. In nondegenerate vibrational states ($\ell=0$) of
${}^{2}\Sigma^{+}$ linear polyatomic molecules like CaOH, SrOH, and YbOH, a
closed optical cycling transition is formed by driving the
${}^{P}Q_{21}(J=1/2)$ and $P_{1}(J=3/2)$ transitions, which address both spin-
rotation components of the $\widetilde{X}\,^{2}\Sigma^{+}(N=1^{-})$ ground
state and excite them to either the
$\widetilde{A}\,^{2}\Pi_{1/2}(J^{\prime}=1/2^{+})$ or
$\widetilde{B}\,^{2}\Sigma^{+}(N^{\prime}=0,J^{\prime}=1/2^{+})$ state (Fig.
12a). The key ingredient for achieving rotational closure in this scheme is
not the specific electronic state but the fact that it have a well-resolved
$J^{\prime}=1/2^{+}$ level. In this case the transition is guaranteed to be
closed by the E1 selection rules $\Delta J=0,\pm 1$ and $\Delta p=\pm 1$.

The inclusion of hyperfine structure threatens to void these selection rules
because the $(J^{\prime}=1/2,F=1^{+})$ level of the excited state may, in
principle, decay to the $(N=3,J=5/2,F=2^{-})$ ground state level. However,
this requires that the hyperfine interaction mix the $N=1$ and $N=3$ states.
The mixing fraction is on the order of $\sim c^{2}/(10B)^{2}\sim 10^{-10}$ for
CaOH, where $c$ is the dipolar hyperfine constant and $B$ is the rotational
constant. While this decay mechanism is negligible for CaOH and alkaline-earth
monohydroxides with similar structure, it may be of some significance for
species with large hyperfine structure, e.g., with a nuclear spin on the
optical cycling center (see, e.g., Pilgram et al. (2021)).

In degenerate vibrational states ($\ell\neq 0$) rotational closure is
complicated by the appearance of parity doublets in the ground state (Fig.
12b). In this case, the excited $J^{\prime}=1/2^{+}$ state can not only decay
to $N=1^{-}$ as before, but also to the $N=2,J=3/2^{-}$ sublevel. For the
linear polyatomic molecules which have been laser cooled to date, the most
important cases are $\ell=1$ and $\ell=2$. For example, in CaOH, YbOH, and
SrOH, the $\widetilde{X}(01^{1}0)$ and $\widetilde{X}(02^{2}0)$ states
typically require repumping at or above the $10^{-4}$ level (Zhang et al.
(2021); Lasner et al. (2022)). As shown in Fig. 12, $\ell=1$ states require
both an $N=1^{-}$ and an $N=2^{-}$ repumping laser. Unlike the spin-rotation
splitting between the $J=1/2$ and $J=3/2$ states in $N=1$, which are typically
spaced by $\sim 10-100$ MHz and can be addressed with rf modulation (e.g. AOMs
or EOMs), the $N=1$ and $N=2$ repumpers in $\ell=1$ bending modes are split by
10s of GHz in the alkaline-earth monohydroxides, meaning that either two
separate lasers or high frequency EOMs need to be used to bridge the gap. The
$\ell=2$ states, meanwhile, require just a single repumping laser, which
addresses the $N=2,J=3/2^{-}$ state.

For $\ell\neq 0$ bending modes, the excited state used for repumping also
merits careful consideration. The primary concern is that many candidate
states are best described by a Hund’s case (b) basis, meaning that both
$N^{\prime}$ and $J^{\prime}$ are good quantum numbers. Accordingly, while the
$N=1^{-}$ ground state levels can always be repumped through the excited
$J^{\prime}=1/2^{+}$ state, in some cases repumping of $N=2^{-}$ is forbidden
by an approximate $\Delta N=0,\pm 1$ selection rule. For example, in the laser
cooling scheme used for CaOH by Vilas et al. (2022),
$\widetilde{X}(01^{1}0)(N=1^{-})$ is repumped through the
$\widetilde{B}\,^{2}\Sigma^{+}(000)(N^{\prime}=0,J^{\prime}=1/2^{+})$ state,
but $N=2^{-}$ is not because of $\Delta N$ selection rules. Instead, the
$N=2,J=3/2^{+}$ state is repumped through the $\widetilde{A}(010)$ electronic
manifold, which has two components $\mu^{2}\Sigma^{(+)}$ and
$\kappa^{2}\Sigma^{(-)}$ (Fig. 12b; this state is described in detail in Li
and Coxon (1995)). These states are intermediate between case (a) and case
(b), so $\Delta N$ selection rules are somewhat weakly enforced; however, the
$\kappa^{2}\Sigma^{(-)}$ is the best candidate for $N=2^{-}$ repumping because
its $J^{\prime}=1/2^{+}$ state has dominantly $N^{\prime}=1$ character, while
the $\mu^{2}\Sigma^{(+)}(J^{\prime}=1/2^{+})$ state has predominantly
$N^{\prime}=0$ character. In CaOH, the
$\mu^{2}\Sigma^{(+)}(J^{\prime}=1/2^{+})$ state has only a $\sim 7\%$
transition strength to $N=2^{-}$, while the
$\kappa^{2}\Sigma^{(-)}(J^{\prime}=1/2^{+})$ state connects in approximately
equal proportion to $N=1^{-}$ and $N=2^{-}$ in the $\ell=1$ bending modes.

In practice, it is often necessary to repump degenerate bending modes using
transitions that do not satisfy the $\Delta\ell=0$ selection rule (see section
2.4.5). In this case, rotational transition strengths are challenging to
calculate because they may be altered by vibronic perturbations. These must be
considered on a molecule-by-molecule basis. For repumping the $\ell=1$ bending
modes in CaOH, it is empirically known that repumping on the
$\widetilde{X}^{2}\Sigma^{+}(010)(N=1^{-})\rightarrow\widetilde{B}^{2}\Sigma^{+}(000)(N^{\prime}=0,J^{\prime}=1/2^{+})$
transition is sufficiently strong (Baum et al. (2020, 2021); Vilas et al.
(2022)). However, in CaOH the $N=2^{-}$ states are repumped using
$\Delta\ell=0$ transitions (Vilas et al. (2022)).

Figure 13: Rotationally closed cycling transitions for symmetric top molecules
with structure similar to CaOCH3 on the (a)
$\widetilde{X}^{2}A_{1}\rightarrow\widetilde{A}^{2}E$ and (b)
$\widetilde{X}^{2}A_{1}\rightarrow\widetilde{B}^{2}A_{1}$ electronic
transitions. Cycling transitions exist within both the $K_{R}=0$ stack and the
$K_{R}=1$ stack, though in the latter case an $N=2$ rotational repumping laser
is required.

Symmetric top molecules. The rotational cycling transitions for
${}^{2}\\!A_{1}$ symmetric top molecules (STMs) (e.g. CaOCH3 or YbOCH3) are
shown in Fig. 13. In addition to the $J$ and parity selection rules discussed
above, there are also $K$ selection rules for STMs.

For parallel transitions between two nondegenerate electronic states (e.g.
$\widetilde{X}\,^{2}A_{1}\rightarrow\widetilde{B}\,^{2}A_{1}$), the selection
rule $\Delta K=0$ holds. It is useful to divide the molecule into “$K$
stacks”, each of which is well isolated during optical cycling. A canonical
$N=1^{-}\rightarrow N^{\prime}=0^{+}$ cycling transition can therefore be
formed in the $K=0$ stack, while cycling in the $K=1$ stack requires an
additional $N=2$ repumping laser, as shown in Fig. 13b.

For perpendicular transitions between a nondegenerate and a degenerate
electronic state (e.g.
$\widetilde{X}\,^{2}A_{1}\rightarrow\widetilde{A}\,^{2}E$), the photon adds
one unit of electronic angular momentum ($\Lambda$) about the molecular axis,
while the rigid body rotation of the molecule (i.e. $K_{R}$, the projection of
$R$ onto the molecular axis) is unchanged. Therefore $\Delta K=\Delta
K_{R}+\Delta\Lambda=\pm 1$, but $\Delta K_{R}=0$.555Note that the electronic
angular momentum is quenched in symmetric top molecules, so $\Lambda$ is not
necessarily an integer. Instead, the orbital angular momentum about the
symmetry axis is labeled by $\zeta_{e}$, and both it and $K_{R}$ are only
approximate quantum numbers. Nonetheless, the selection rules discussed above
based on $K_{R}$ stacks are expected to hold in general, though the $K_{R}$
notation is simply a convenience for molecules where $\zeta_{e}\approx 1$ (see
Brazier and Bernath (1989)). In this case we can instead divide the structure
into isolated “$K_{R}$ stacks”, noting that $K=K_{R}$ in the nondegenerate
ground state. A closed $N=1^{-}\rightarrow J^{\prime}=1/2^{+}$ cycling
transition can be found in the $K_{R}=0$ stack, while cycling in the $K_{R}=1$
stack requires an $N=2$ repumping laser because each rotational level contains
parity doublets (Fig. 13a). Note that the degenerate excited state is
typically well described by a Hund’s case (a) basis, so $N^{\prime}$ is not a
good quantum number. See Herzberg (1966); Brown (1971); Cerny et al. (1993);
Brazier and Bernath (1989) for additional details on rotational selection
rules and $K_{R}$ stacks in STMs.

The selection rules described above have been tested in experiments with
CaOCH3 (Mitra et al. (2020)), where $\sim$120 photons were scattered in the
$K_{R}=0$ stack and $\sim$30 photons were scattered in the $K_{R}=1$ stack. It
is empirically unknown whether the $K_{R}$ selection rules hold beyond this
number of photons.

Figure 14: Rotationally closed cycling transitions for $a$, $b$, and $c$-type
bands in asymmetric top molecules, as described in the text. Dashed lines
correspond to transitions that are allowed in ATMs with unquenched electronic
angular momentum, which is expected to be true for molecular geometries near
the symmetric-top limit. Reproduced from Augenbraun et al. (2020a).

Asymmetric top molecules. In asymmetric top molecules (ATMs), there are three
types of electronic transitions to consider, corresponding to induced dipole
moments along either the $a$, $b$, or $c$ principal axis of the molecule.
These are analogous to parallel and perpendicular transitions in linear
molecules or STMs, where parallel transitions become $a$ ($c$)-axis
transitions for prolate (oblate) molecules, and the other two axes play the
role of perpendicular transitions. Each transition axis has its own selection
rules, shown in Tab. 1, and closed cycling transitions for each transition
type are shown in Fig 14. The transition dipole moment $\mu$ will in general
have a projection onto each of the principal axes and inherit some of each of
the selection rules (Augenbraun et al. (2020a)). It is therefore advisable to
choose molecules whose transition dipole moments are well aligned with the
molecular axis to limit the number of rotational states that require
repumping.

###  2.5 Perturbations

Figure 15: Schematic illustration of how perturbations among electronically
excited states can lead to an “intensity borrowing” effect that induces
nominally forbidden transitions (or increases the intensities of transitions
that were expected to be very weak).

We have thus far described the energy eigenstates of polyatomic molecules
using a basis of well-defined quantum numbers. These, in turn, led to strict
selection rules governing vibrational and rotational branching. In real
molecules, however, there are mechanisms that perturbatively couple these
basis states, e.g. via mixing of electronic and vibrational angular momentum.
The result is an effect known as “intensity borrowing”: nominally forbidden
transitions become allowed because the energy eigenstates of the molecule
contain a small admixture of basis states with different quantum numbers
and/or symmetry. The effect is illustrated schematically in Fig. 15. While
these effects are typically small ($\sim 10^{-3}$ level or below), they can
become important when forming cycling transitions capable of scattering many
thousands of photons, or when certain vibronic levels of electronically
excited states have energy gaps that are “accidentally” small.

While perturbations can take many forms and must in general be considered on a
molecule-to-molecule basis, below we will discuss a few known effects for
molecules presently being laser cooled or proposed for laser cooling.

The Renner-Teller (RT) effect describes vibronic mixing between the electronic
angular momentum $\Lambda$ and the vibrational angular momentum $\ell$ in
linear polyatomic molecules with degenerate vibrational modes. In particular,
it allows $\Lambda$ and $\ell$ to change while conserving the total spinless
angular momentum projection $K=\Lambda+\ell$. The physical origin of this
effect is that bending vibrations reduce the cylindrical symmetry of the
molecule and can therefore break the degeneracy of the in-plane and out-of-
plane electronic orbitals in states with $\Lambda>0$. See Hirota (1985) for a
detailed description of this interaction.

The Renner-Teller effect is responsible for vibrational branching that
violates the $\Delta\ell=0$ selection rule in linear polyatomic molecules, as
previously studied in detail for CaOH, SrOH, BaOH, and YbOH (Baum et al.
(2021); Zhang et al. (2021); Lasner et al. (2022); Kinsey-Nielsen et al.
(1986)). In these molecules, there are two important types of RT-induced
branching. The first is $\Delta\ell=\pm 1$ branching enabled by first-order RT
coupling, which mixes states according to the selection rule
$\Delta\Lambda=-\Delta\ell=\pm 1$. This enables direct vibronic coupling
between the $\widetilde{A}^{2}\Pi(000)$ and
$\widetilde{B}^{2}\Sigma^{+}(01^{1}0)$ states in alkaline earth
monohydroxides, thereby allowing $\widetilde{A}(000)$ to decay directly to
$\ell=1$ ground states (e.g. $\widetilde{X}(01^{1}0)$) via intensity borrowing
from the $\widetilde{B}(01^{1}0)$ state. Likewise, $\widetilde{B}(000)$ can
decay to $\widetilde{X}(01^{1}0)$ via RT coupling with
$\widetilde{A}(01^{1}0)$. This coupling typically contributes at the $\sim
10^{-3}$ to $10^{-4}$ range in the alkaline earth monohydroxides (Baum et al.
(2021); Zhang et al. (2021); Lasner et al. (2022)). A smaller effect mixes
$\widetilde{A}(000)$ and $\widetilde{A}(01^{1}0)$ directly via contributions
from both first-order RT and spin-orbit coupling, but it is not discussed
further here. See Baum et al. (2021); Zhang et al. (2021) for more details.

The second effect of RT mixing is $\Delta\ell=\pm 2$ branching induced by
second-order RT coupling, which mixes states according to the selection rule
$\Delta\Lambda=-\Delta\ell=\pm 2$. This term directly couples
$\widetilde{A}^{2}\Pi(000)$ to $\widetilde{A}^{2}\Pi(02^{2}0)$, as it can mix
the $|\Lambda=1,\ell=0\rangle$ and $|\Lambda=-1,\ell=2\rangle$ components of
the $\widetilde{A}$ state. Observed decays to $\widetilde{X}(02^{2}0)$ and
$\widetilde{X}(12^{2}0)$ in alkaline earth monohydroxides are attributed to
this mechanism (Baum et al. (2021); Zhang et al. (2021); Lasner et al. (2022);
Vilas et al. (2022)).

An analogous interaction, called the (pseudo-)Jahn-Teller (JT) effect, is
possible in nonlinear symmetric top molecules. A detailed review of Jahn-
Teller physics is provided by Barckholtz and Miller (1998). For the purposes
of laser-coolable polyatomic molecules such as CaOCH3 or YbOCH3, we can regard
the (pseudo-)JT effect in the ${}^{2}E$ state of a nonlinear molecule as
analogous to the RT effect in a ${}^{2}\Pi$ linear molecule. This effect can
lead to mixing between a ${}^{2}E$ and ${}^{2}A_{1}$ electronic state which
alters the vibrational structure of the ${}^{2}E$ state and changes the
vibronic emission intensities associated with spontaneous emission from this
state. In a molecule such as CaOCH3 or YbOCH3, the first electronically
excited state ($\tilde{A}\,^{2}\\!E$) is affected by mixing with the
$\tilde{B}\,^{2}\\!A_{1}$ level. Especially due to second-order spin-
orbit–vibronic coupling, this interaction can dramatically increase the
intensity of decays to vibrational bending modes that would have been
symmetry-forbidden within the BO approximation. Experimentally, these decays
have been observed in CaOCH3 (Augenbraun (2020)) and in YbOCH3 (Augenbraun et
al. (2021b)). In both cases, it was possible to model the intensity of decay
to vibrational bending modes on the basis of quantum chemical predictions of
the JT parameters provided by Paul et al. (2019). In both cases, the nominally
symmetry-forbidden decays were significantly stronger than decays to symmetry-
allowed levels of the same vibrational mode, directly indicating the role that
vibronic coupling plays in this process.

These observations point to two important considerations in the selection of
laser-coolable nonlinear molecules. First, one must consider beyond-BO-
approximation effects when deciding whether simple estimates of FCFs are
justified in selecting a molecule for future experimentation. Second,
experimental measurements of vibrational branching ratios are crucial to
identify weak decays that violate expectations based on molecular symmetry.
See Sec. 2.4.3 for details on such measurements.

##  3 Experimental techniques

###  3.1 Cryogenic buffer-gas beams

Almost all applications of laser-cooled molecules benefit from long
interaction times, achieved either through a trap (enabling, in principle,
arbitrarily long hold times) or a molecular beam propagating over a large
distance (tens of centimeters to a few meters, enabling probe times of
$\sim$1$-$100 ms). In both cases, a large flux of initially slow molecules is
essential. In order to trap molecules, any motion at the time of production
must be removed in a manner whose difficulty typically scales with the initial
momentum or kinetic energy, whereas for a fixed beam line the interaction time
is inversely proportional to the beam velocity.

An additional requirement for a practical molecular source is a low rotational
temperature. Rotationally excited states of small molecules like CaF and SrOH
obtain significant thermal population at temperatures of $\sim$1 K, and the
population in any single quantum state at room temperature is suppressed by
orders of magnitude compared to the low-temperature limit. This problem grows
exponentially for large asymmetric top molecules, with multiple rotational
modes that have small rotational constants (owing to the large moments of
inertia).

Figure 16: A representative cryogenic buffer gas cell, modified from Mitra et
al. (2020). Hot reagent CH3OH (methanol) gas is introduced via a capillary,
and cold 4He buffer gas is introduced via a second capillary. A pulsed Nd:YAG
laser ablates a “target” of Ca metal, which reacts with methanol to form
CaOCH3 in the first-stage cell. Additional windows at the downstream end of
the first-stage cell may be used to monitor molecular production via optical
absorption measurements. The helium thermalizes the molecules and entrains
them through a hole in the front of the cell. A second-stage cell, stood off
from the first stage to lower the buffer gas density, reduces the forward
velocity of the molecular beam. Buffer gas and molecules are extracted through
the front of the second-stage cell.

Buffer-gas cooling in closed cells was developed by the atomic hydrogen
community in the late 1970s/early 1980s. Early examples with molecules include
the De Lucia group, which studied CO molecules (and, later, polyatomic
molecules) in the presence of a $\sim\\!4$ K He buffer gas, as in Messer and
De Lucia (1984); Willey et al. (1988); Mengel and De Lucia (2000). The Doyle
group pioneered the use of buffer-gas cooling to load atoms and molecules into
(superconducting) magnetic traps in Doyle et al. (1995); Weinstein et al.
(1998); Campbell et al. (2007); Doret et al. (2009). The first molecular beam
source generated from buffer-gas cells was developed in the Doyle group in
collaboration with Prof. David DeMille, as described in Maxwell et al.
(2005b). Hydrodynamic cryogenic buffer-gas beams (CBGBs) were created by
Patterson and Doyle (2007) and developed by both the Doyle and DeMille groups
and the Hinds/Tarbutt groups. Reviews by Hutzler et al. (2012); Barry et al.
(2011) discuss the key features of such sources, which include low forward
velocities and hydrodynamic enhancement effects that led to beams several
orders of magnitude brighter than previous realizations.

A primary benefit of a CBGB (beyond the low velocity and high flux) is that it
allows the experiment to separate molecule production (occurring in a
cryogenic region with poor optical access) from molecule manipulation (usually
occurring in a room-temperature, ultra-high vacuum region). The CBGB was a key
technological development that enabled direct laser cooling of molecules, as
supersonic beams produce velocities so high as to make the deceleration of
molecules to trappable velocities extremely difficult.

A CBGB source typically consists of four nested regions. The innermost region
is a “buffer gas cell,” usually made of high-purity copper and cooled to $\sim
1-4$ K. A representative example is shown in Fig. 16. There, stable reagent
molecules (methanol) are introduced into the cell via a hot capillary, while
Ca metal is ablated via a pulsed Nd:YAG laser (typically with $\sim$10$-$40
mJ/pulse). The ablated atoms and methanol gas react to form CaOCH3.
Simultaneously, He buffer gas is flowed through a second capillary that pre-
cools the He gas before it enters the cell. Inside the cell, the He
thermalizes with the cold walls of the cell and collisions between molecules
and He atoms cool the molecules. The flow of helium entrains molecules through
a hole in the front of the cell (typically $\sim$$3$-$7$ mm diameter). The
helium and molecules then flow into a lower-density “second-stage cell” that
is stood off from the first stage of the buffer gas cell with a small gap for
buffer gas to escape. This lowers the buffer gas density in the second-stage
cell and reduces the forward velocity of molecules, at the expense of some
reduction in the molecular extraction (typically a factor of order unity).
Molecules then emerge from the second cell with forward velocities in the
range of $\sim$40$-$200 m/s, depending on the molecular mass, cell
temperature, and buffer gas flow rate.

A number of variations on the buffer gas cell are used, depending on
experimental requirements and the molecular species of interest. To produce
cold beams of stable molecules such as ammonia or formaldehyde, no ablation is
necessary (Buuren et al. (2009)). On the other hand, even complex radical
molecules can be produced directly from ablation of pressed-powder pellets (or
“targets”) of stable constituents, without reagent gases. For example, a
mixture of SrH2 and naphthol powders produces SrO-naphthyl radicals (Mitra et
al. (2022)). Because of the versatility in molecular production methods, a
wide variety of polyatomic molecules including both radical (Augenbraun et al.
(2020b); Baum et al. (2020); Mitra et al. (2020); Zhu et al. (2022); Mitra et
al. (2022) and non-radical Maxwell et al. (2005a); Buuren et al. (2009);
Herschbach (2009); Sawyer et al. (2011); Eibenberger et al. (2017); Spaun et
al. (2016); Satterthwaite et al. (2019); Patterson and Doyle (2013); Piskorski
(2014)) species can be created in a CBGB. Molecules as large as nile red
(C20H18N2O2) have been produced and spectroscopically studied in a cryogenic
buffer gas cell without forming helium-molecule clusters (Piskorski et al.
(2014)).

Both helium and neon buffer gas are commonly used in CBGBs, with neon
requiring cell temperatures above approximately 16 K to maintain a suitable
vapor pressure. Depending on the requirements for molecular flux and the
initial beam velocity, the second-stage cell may be omitted. The lowest
forward velocities can be achieved by cooling the second stage with a He-3 pot
at temperatures around or below 1 K (Augenbraun et al. (2021a)). In that work,
the heat loads arising from ablation and gas flows, in the range of tens to
hundreds of mW, made it impractical to cool the first-stage cell using a He-3
refrigerator. Nevertheless, by cooling the second-stage alone, beams of Yb
(Ca) with peak forward velocities as low as about 20 m/s (40 m/s) have been
observed by Augenbraun (2020); Augenbraun et al. (2021a); Sawaoka et al.
(2022).

Surrounding the buffer gas cell is a cold box, usually constructed of high-
purity copper and held at $\sim$4 K by a pulse tube cryocooler to serve as a
cryopump. When helium buffer gas is used, charcoal sorbs are thermally
anchored to the 4 K box in order to achieve adequate helium (cryo)pumping
speeds; other buffer gasses, such as neon, are efficiently cryopumped directly
using copper surfaces held at 4 K.

The cryopumping box is contained within another box, typically made of
aluminum or copper, held at $\sim$50 K by the first stage of a pulse tube
cryocooler. This box shields the cryopumping box and buffer gas cell from
room-temperature black body radiation. The radiation shields are housed within
a vacuum chamber at $\sim$$10^{-7}$ Torr or better.

We summarize several important molecular beam parameters under typical
conditions in Tab. 2. Experiments seeking to trap molecules usually operate
toward the lower range of buffer gas flow rates, ablation energy, and ablation
repetition rate, to enable lower temperatures and forward molecular beam
velocities at the expense of beam brightness and experimental duty cycle.

Parameter | Typical range
---|---
Forward velocity | 40$-$200 m/s
Solid angle FWHM | 0.2$-$1 sr
Rotational temperature | 1$-$4 K
Brightness |  $10^{8}$-$10^{11}$ sr-1 pulse-1
Ablation repetition rate | 1$-$50 Hz
Ablation energy | 10$-$40 mJ
Buffer gas flow rates | 2$-$40 sccm
Table 2: Representative operating and performance parameters of molecular
CBGBs.

###  3.2 Optical cycling

Figure 17: Optical cycle for an ideal two-level system, reproduced from
Augenbraun et al. (2020a). Three phenomena repeat up to tens of thousands of
times: (1) A photon is absorbed, driving a molecule from the ground to excited
state and imparting a momentum recoil to the molecule, (2) subsequently, a
photon is spontaneously emitted in a random direction, and (3) the molecule
returns to its ground state.

The key ingredient to laser cooling is the near-closure of an optical cycle,
so as to approximate an ideal two-level system, as depicted in Fig. 17. A
molecule initially in its ground state absorbs a photon with energy