# Laser Scheme for Doppler Cooling of the Hydroxyl Cation (OH+)

Niccolò Bigagli  Department of Physics, Columbia University, New York, New
York 10027, USA  Daniel W. Savin  Columbia Astrophysics Laboratory, Columbia
University, New York, New York 10027, USA  Sebastian Will  Corresponding
author. Email<EMAIL_ADDRESS>Department of Physics, Columbia
University, New York, New York 10027, USA

###### Abstract

We report on a cycling scheme for Doppler cooling of trapped OH+ ions using
transitions between the electronic ground state $X^{3}\Sigma^{-}$ and the
first excited triplet state $A^{3}\Pi$. We have identified relevant
transitions for photon cycling and repumping, have found that coupling into
other electronic states is strongly suppressed, and have calculated the number
of photon scatterings required to cool OH+ to a temperature where Raman
sideband cooling can take over. In contrast to the standard approach, where
molecular ions are sympathetically cooled, our scheme does not require co-
trapping of another species and opens the door to the creation of pure samples
of cold molecular ions with potential applications in quantum information,
quantum chemistry, and astrochemistry. The laser cooling scheme identified for
OH+ is efficient despite the absence of near-diagonal Franck-Condon factors,
suggesting that broader classes of molecules and molecular ions are amenable
to laser cooling than commonly assumed.

††preprint: APS/123-QED

##  I Introduction

Laser cooling and quantum control of atoms and atomic ions has enabled a
plethora of scientific investigations over the last decades [1, 2, 3, 4]. Over
recent years, the field has expanded towards molecules [5, 6, 7, 8, 9], as
their more complex quantum-state structure opens a broader scope of physics to
be studied, including measurements of fundamental constants [10, 11, 12],
investigations of quantum chemistry [13, 14, 15], applications in quantum
information [16, 7, 6], and access to novel many-body quantum systems [17].
Identifying molecules that are relevant for scientific and technological
applications and at the same time amenable to laser cooling is an active area
of research but, so far, it has almost exclusively focused on neutral
molecules [18, 19, 20, 21, 22, 23].

Molecular ions have broad scientific use cases [6]. Progress on laser cooling
and quantum control schemes for molecular ions promises to open new scientific
avenues, building on the enormous success of atomic ions in quantum science
[24, 1]. For quantum information, the rich internal structure of molecular
ions may allow the realization of efficient gate operations and long qubit
storage times [16, 25] in the same physical system, akin to neutral molecules
[26, 27, 28]. Molecular ions enable Coulomb-mediated two-qubit operations
[29], and have been proposed as an alternative to neutral molecules [30, 31].
They also may allow the realization of qudits, multi-level systems that
constitute a powerful extension of the qubit-based quantum information
paradigm [32, 33]. In addition, cold molecular ions in well-defined quantum
states can be employed in collisional studies in quantum chemistry and
astrochemistry. For example, gas-phase chemistry in cold interstellar clouds
is driven by ion-neutral reactions where the ions are in their lowest
electronic, vibrational, and rotational levels due to the relaxation of
internal excitations close to the 2.7 K cosmic microwave background [34, 35].
Fully quantum mechanical calculations for these reactions are beyond current
computational capabilities for systems with four or more atoms. Therefore,
laboratory measurements with molecules in quantum states similar to those in
interstellar space would be helpful to elucidate the chemical kinetics [36,
37, 38, 39, 40].

Today, the standard approach to the preparation of trapped molecular ions
relies on sympathetic cooling via a co-trapped ionic [41, 28] or neutral [42]
atomic species. Direct laser cooling of molecular ions has not been explored
extensively. Although cooling via co-trapped species has proven effective and
useful [31], it is technically challenging. In addition, for sensitive
applications in quantum information and precision measurements, where high
fidelity and low dephasing are critical [43], the presence of a second species
may eventually be limiting [44, 30]. All-optical cooling schemes should be
highly attractive but, so far, only few theoretical studies on the prospects
of laser cooling of molecular ions exist [44] and most have not accounted for
the full rovibrational structure of the ion under investigation [45, 46, 47].

In this article, we discuss a laser-cooling scheme for the hydroxyl cation,
OH+. We envision the scheme to be applied to a single ion or an ensemble of
ions held in a deep and tightly confining ion trap, as schematically
illustrated in Fig. 1 (a). In this setting, three-dimensional laser cooling
can be achieved with cooling and repumping laser beams incident from a single
direction that has a finite angle with all three Cartesian axes of the trap
[48]. The proposed scheme can provide cooling in the Doppler regime, where the
motional energy of the ion is significantly larger than the trap frequency
[31]. To cool below the Doppler limit, a secondary Raman sideband cooling step
may be employed [44]. However, in this work we solely focus on the description
of the Doppler cooling scheme for OH+.

The OH+ ion is particularly relevant in the context of astrophysics and
astrochemistry. The production of cold, pure, and trapped samples of OH+ would
enable reaction studies that can help shed light on processes such as the
cosmic ray ionization rate of the intersellar medium [49] and the gas-phase
pathway to the formation of water [50]. Additional uses of laser-cooled OH+
molecules may be in quantum information. Due to large rotational spacings
($B\sim 500$ GHz [51]), coherent control of rotational qubits in OH+ would be
challenging but possible via a two-photon Raman transfer. Given that Doppler
cooling schemes have not been widely discussed for molecular ions, this study
also uses OH+ as a proof-of-concept case that may encourage the development of
laser cooling schemes for other molecular ions.

The relevant low-lying potential energy curves of OH+ are shown in Fig. 1 (b)
[52]. OH+ has favorable characteristics for laser cooling. Its energy level
structure is relatively simple, with its first few electronic states having
either a triplet or a singlet nature. Furthermore, OH+ is extremely tightly
bound. As can be seen from the minima of the potential energy curves (see Fig.
1), the bond length is about one $a_{0}$, where $a_{0}$ is the Bohr radius. As
a result, the vibrational and rotational energy spacings are large, of the
order of 3000 cm-1 and 60 cm-1, respectivelyḊue to the large vibrational
spacing, only about a dozen vibrational states exist in the $X^{3}\Sigma^{-}$
potential below the $A^{3}\Pi$ potential, which limits the number of decay
channels for potential transitions of a laser cooling scheme. However, there
is no transition in OH+ with a near-diagonal Franck-Condon factor (FCF), a
characteristic that is often believed necessary for a functional cooling
scheme [44]. The highest branching ratio for a single transition is 0.56
111This branching ratio is observed between the states $A^{3}\Pi$
$|0,1,0\rangle$ and $X^{3}\Sigma^{-}$ $\ket{0,\,0,\,1}$.. Despite this
complication, as we demonstrate below, OH+ supports a laser cooling scheme
that is not more complex than state-of-the-art cooling schemes for neutral
molecules [54]. OH+ also has a non-zero nuclear spin of $I=1/2$ [55], which we
do not explicitly take into account in this study. However, we do not expect
the resulting hyperfine structure to fundamentally prevent the laser cooling
from functioning, as has been seen in earlier demonstrations of laser cooling
[56].

Figure 1: Laser cooling of OH+ ions trapped in an ion trap. (a) Schematic of
an OH+ ion a cylindrical quadrupole ion trap [57]. The left image shows a
three-dimensional sketch of the ion trap; the right image a cross section,
including field lines. The specific trap shown is for illustration purposes
only; the cooling scheme is general and can be implemented in other types of
ions traps. The cooling and repumping beams are incident collinearly from one
direction that has a finite angle with all three Cartesian axes of the trap,
providing cooling in three dimensions. (b) Potential energy curves of OH+ [52]

. Depicted are the low-lying electronic potentials relevant to this study, two
with triplet spin character and two with singlet spin character. The inset
shows a pictorial representation of an OH+ molecular ion.

##  II Laser cycling scheme

The proposed cycling scheme makes use of transitions between the electronic
ground state of the molecule, $X^{3}\Sigma^{-}$, and its first electronically
excited triplet state, $A^{3}\Pi$. The specific transitions of the scheme are
shown in Fig. 2. We explain below how a sufficient degree of closure can be
reached. The required spectroscopic data was extracted from the ExoMol
database [58], which for OH+ relies on the optical transitions published in
Refs. [51, 59]. We utilize the provided energy levels, transition frequencies,
and Einstein $A$ coefficients.

Figure 2: Laser cooling scheme. The green (purple) arrows represent the
cooling and repumping beams from the $X^{3}\Sigma^{-}$ ground electronic state
to the $A^{3}\Pi$ $|0,1,0\rangle$ ($|1,1,0\rangle$) state. The numbers on the
left represent the quantum numbers $|v,N,J\rangle$. Next to each arrow, we
indicate the wavelength of each transition. Transitions from left to right are
ordered from smaller-to-larger transition wavelength and larger-to-smaller
transition strength. The labels above each transition for the first (F) and
second (S) legs of the cooling scheme are introduced for easier comparison to
Fig. 3

The logic of the cooling scheme is as follows: We start from the absolute
molecular ground state, $X^{3}\Sigma^{-}$ $\ket{v=0,\,N=0,\,J=1}$, following
Hund’s case (b) [60], where $v$, $N$ and $J$ are the vibrational quantum
number, the angular momentum excluding electron and nuclear spin, and the
angular momentum excluding nuclear spin, respectively. Due to the $\Delta
J=\pm 1$ selection rule, excitation both to $J=0$ and $J=2$ states is
possible. Excitation to $J=0$ is favorable as the only decay channel is to
$J=1$, reducing the number of needed repumper lasers by a factor of two. In
addition, excitation to a low-lying vibrational state is favorable due to
higher FCFs. Thus, we choose $A^{3}\Pi$ $\ket{0,\,1,\,0}$ as the first excited
state of the scheme. The main decays of this state are to the eight
$X^{3}\Sigma^{-}$ states shown in Fig. 2, with only two other observed decays
with negligible branching ratios ($\sim 10^{-8}$). Overall, the scattering
rate of $A^{3}\Pi$ $\ket{0,\,1,\,0}$ is relatively small at $3.5\times 10^{5}$
s-1 and by itself would lead to long cooling times. By adding a second excited
state to the cycling scheme, the $A^{3}\Pi$ $\ket{1,\,1,\,0}$ state, it is
possible to speed up the cooling time by about a factor of two, as we discuss
below.

Among the decay channels of the excited $A^{3}\Pi$ $\ket{0,\,1,\,0}$ and
$A^{3}\Pi$ $\ket{1,\,1,\,0}$ states are 6 and 8 decay channels, respectively,
with branching ratios above $10^{-3}$, as shown in Fig. 3. This is a practical
threshold for the relevance of a transition in a typical laser cooling scheme
[61]. Branching ratios are calculated using the Einstein coefficients $A_{i}$
for spontaneous decay to a state $i$ [58] through the relation
$\mathrm{BR}_{i}=A_{i}/\sum_{j}A_{j}$, where the sum runs over all possible
radiative decay paths to a lower energy state $j$ for a given $A^{3}\Pi$
state.

We have investigated the question whether loss to other states could
potentially hamper the $X^{3}\Sigma^{-}\leftrightarrow A^{3}\Pi$ cooling
scheme. We find that the effects of state mixing between $A^{3}\Pi$ and the
singlet states $b\,^{1}\Sigma^{+}$ and $a\,^{1}\Delta$ should be minimal. For
the nearby $b\,^{1}\Sigma^{+}$ state, spin-orbit coupling with a strength of
about 75 cm-1 [62] leads to an admixture of 0.5$\%$ $b\,^{1}\Sigma^{+}$
$|v=0\rangle$ to the $A\,^{3}\Pi$ $|v=0,1\rangle$ states, calculated from the
diagonalization of the Hamiltonian of these states with an off-diagonal spin-
orbit contribution. Although this is a non-negligible admixture, it is not
expected to lead to relevant losses out of the cycling scheme (see Appendix).
This is due to the unusual structure of OH+, where decay from
$b\,^{1}\Sigma^{+}$ to $a\,^{1}\Delta$ is suppressed by the
$\Delta\Lambda=0,\pm 1$ selection rules [63]. Furthermore, direct coupling
between $b\,^{1}\Sigma^{+}$ and vibrationally excited $a\,^{1}\Delta$ states
should not be an issue, as the only loss channel via mixing of $a\,^{1}\Delta$
can be due to relaxation into lower vibrational states of $a\,^{1}\Delta$.
However, spontaneous decay between vibrational states is suppressed in the
first order [63] and driving of such transitions by black-body radiation would
be slow compared to the timescale of the cooling scheme. In line with these
arguments, to our knowledge, such decays in OH+ have not been reported.
Finally, we note that loss from predissociation [64] is fully suppressed for
the $X^{3}\Sigma^{-}\leftrightarrow A^{3}\Pi$ cooling scheme.

Figure 3: Branching ratios for decay from the excited states (a) $A\,^{3}\Pi$
$|0,1,0\rangle$ and (b) $A\,^{3}\Pi$ $|1,1,0\rangle$. The decay paths for each
scheme are numbered in order of decreasing branching ratios and the labels and
color coding refer to the transitions shown in Fig. 2. All decays are to the
same states in the $X\,^{1}\Sigma^{-}$ manifold illustrated in Fig. 2.

##  III Results & Discussion

Using the branching ratios from Fig. 3, we quantify the closure that can be
achieved by adding an increasing number of repumping transitions. We calculate
the number of scatterings that will lead to a probability of $10\%$
($n_{10\%}$) and $90\%$ ($n_{90\%}$) for retaining an ion in the cycling
scheme [65]. calculations are made assuming the use of two excited states, as
shown in Fig. 2. We employ a Monte Carlo model in which an ion is initialized
in the ground state and then each scattering event is simulated by updating
the probability of populating each state in the cooling scheme after each
step. Table 1 shows the results of these calculations. The quantity $p$ is the
closure of the scheme, calculated via $n_{x\%}=\ln(x/100)/\ln(p)$ [65]. A
conservative estimate of the time to complete $n_{x\%}$ scattering events,
$t_{x\%}$, is also provided. To calculate this quantity we use the relation
$t_{x\%}=n_{x\%}/R$, where $R=\Gamma/(G+1+2\sum I_{\text{sat},i}/I_{i})$ is
the scattering rate [66, 67]. Here, $\Gamma$ is the excited state linewidth;
$G$ is the number of driven transitions; $I_{\text{sat},i}=\pi
hc\Gamma/3\lambda_{i}^{3}$ is the saturation intensity [67] of the
$i^{\text{th}}$ transition; and $I_{i}$ is the intensity of the laser
addressing said transition, set to $I_{i}=10^{3}\,$mW cm-2, which is an
intensity that is easily achievable in an experiment at the given wavelengths.
In the definition of the saturation intensity, $c$ is the speed of light, and
$\lambda_{i}$ the wavelength of the addressed transition. This results in an
overestimate of $t_{x\%}$ as the expression for $R$ assumes a single excited
state, only $A^{3}\Pi$ $\ket{0,\,1,\,0}$. In an experiment, where several
repumping transitions cycle through a second excited state, here $A^{3}\Pi$
$\ket{1,\,1,\,0}$, the scattering rate is boosted and the cooling time
lowered, as has been experimentally demonstrated in laser cooling schemes for
neutral molecules, e.g. in Ref. [22].

Table 1: Closure, number of scatterings, and scattering time for an increasing number of repumping transitions. Driven transitions | $p$ | $n_{10\%}$ |  $t_{10\%}$ (ms) | $n_{90\%}$ |  $t_{90\%}$ (ms)
---|---|---|---|---|---
F1 | 0.631 | 5 | 0.02 | 1 |
F1-F2 | 0.873 | 17 | 0.1 | 1 |
F1-F3 | 0.950 | 45 | 0.5 | 3 | 0.02
F1-F4 | 0.994 | 361 | 5 | 17 | 0.2
F1-F4, S4 | 0.997 | 918 | 17 | 43 | 0.8
F1-F4, S4, S6 | 0.9986 | 15 941 | 800 | 731 | 36
F1-F4, S4, S6-S7 | 0.99995 | 44 287 | $2.6\times 10^{3}$ | 2 028 | 100
F1-F4, S4, S6-S8 | 0.9999997 | 2 563 381 | $1.9\times 10^{6}$ | 117 295 | $9.0\times 10^{4}$
Table 2: Results of the photon scatterings analysis starting at cryogenic and room temperature. $T=4$ K |  $T=300$ K
---|---
$n_{\mathrm{cool}}$ |  $t_{\text{cool}}$ (ms) | $n_{\mathrm{cool}}$ |  $t_{\text{cool}}$ (ms)
$2.3\times 10^{3}$ | 45 | $2.0\times 10^{4}$ | 400

We calculate the number of scatterings, $n_{\rm cool}$, required to bring the
sample to close-to-zero motional temperature from room (300 K) and cryogenic
(4 K) temperatures. In addition, we calculate the time $t_{\rm cool}$
necessary for this process. For the experimental setup, we assume that the ion
is electrostatically trapped and that cooling and repumping laser beams come
from a single direction that has a finite angle with all three Cartesian trap
axes, as illustrated in Fig. 1 (a). Using a single incoming direction for the
lasers enables cooling in all spatial directions thanks to the confinement
provided by the ion trap, as suggested in Ref. [48], and greatly simplifies
the laser setup. Table 2 summarizes the results. Cooling OH+ ions down from
room temperature, will require $2\times 10^{4}$ scattering events. Comparing
this requirement to the results in Table 1, this can be achieved with 90$\%$
efficiency within $<500$ ms using 8 transitions (F1-F4, S4, and S6-S8).
Cooling OH+ ions from cryogenic temperatures, will require $2.3\times 10^{3}$
scattering events, which can be achieved with 90$\%$ efficiency within $<50$
ms using 6 transitions (F1-F4, S4, and S6). We note that OH+ can be easily
produced at cryogenic temperatures, as demonstrated in earlier work via
electronic impact on water vapor [40, 38]. It is important to note that in
practical experiments less stringent requirements on the probability of
retention should be acceptable. For a $10\%$ probability of retention, cooling
from room temperature can be achieved with 6 to 7 lasers (F1-F4, S4, S6-S7)
within about 1 s, and from cryogenic temperatures with 5 to 6 lasers (F1-F4,
S4, S6) within about 50 ms. Hence, on average it would take 10 s and 0.5 s,
respectively, to successfully initialize the system, a time that is short
compared to the storage time of a molecular ion once it is trapped and cooled.

##  IV Conclusions

In summary, we have presented a direct laser cooling scheme of OH+. The scheme
is expected to work both for a single trapped ion and trapped ensembles. In
the case of an ensemble, the direction of the laser cooling beams could even
be arbitrary, as the Coulomb interactions in the ensemble are expected to
couple all directions of motion [31]. While the scheme requires some technical
effort, laser cooling of OH+ would come with several benefits: Cold samples
can be prepared without the need for sympathetic cooling via neutral or ionic
atoms, and efficiently temperatures can be reached where Raman sideband
cooling can take over to cool ions to the vibrational ground state (of the
trap). Rapid progress in laser technology [68] promises to drastically reduce
the technical complications involved in a cooling scheme with multiple lasers.
Also, due to the confined trapping region of an ion trap, the lasers can be
tightly focused and we expect that only very moderate laser powers are be
necessary. Once cooled, OH+ will enable studies relevant for gas-phase
astrochemistry of interstellar clouds. Furthermore, quantum control of OH+
will be an enabling step for its use as a qubit or qudit in quantum
information experiments. Finally, this work, similar to the case of C2 which
we discussed in earlier work [61], suggests that also molecules with non-
diagonal FCFs can be amenable to laser cooling. We hope this work will inspire
further investigations on the application of laser cooling techniques to more
complex molecules that are scientifically relevant, but have so far been
deemed unsuitable for laser cooling.

##  V Acknowledgements

We thank Peter F. Bernath, Tim de Jongh, Ábel Kálosi, Ian Stevenson, and
Sergey Yurchenko for stimulating discussions, and Octavio Roncero for help
with the potential energy curves. This work was supported by a Columbia
University Research Initiative in Science and Engineering (RISE) award. D.W.
Savin was additionally supported by the NASA Astrophysics Research and
Analysis program under 80NSSC19K0969. S.W. acknowledges additional support
from the Alfred P. Sloan Foundation.

##  VI Appendix

As noted in the main text, a loss channel for the
$X^{3}\Sigma^{-}\leftrightarrow A^{3}\Pi$ cooling scheme could arise from
spin-orbit coupling between $A^{3}\Pi$ and nearby singlet states. Here, we
provide more detail to show that such processes should be sufficiently
suppressed for practical purposes. As mentioned in the main text, the $0.5\%$
admixture of $b^{1}\Sigma^{+}\,|v=0\rangle$ to the $A^{3}\Pi\,|v=0,1\rangle$
states does not open up a loss channel to first order due to prohibitive
selection rules. To second order, decay could happen via the admixture of
${}^{1}\Pi$ character to the $b^{1}\Sigma$ potential, with the closest
${}^{1}\Pi$ state lying about 15,000 cm-1 above the $b^{1}\Sigma$ potential
[52]. With a coupling constant of 75 cm-1 (assuming a coupling similar to
$b^{1}\Sigma^{+}$ and $A^{3}\Pi$; to our knowledge there is no reported
literature value) this leads to a $10^{-5}$ admixture of ${}^{1}\Pi$ character
to the $b^{1}\Sigma$ state. Taken together, this amounts to a loss channel on
the $5\times 10^{-8}$ level. Loss channels allowed through higher order
processes should contribute significantly less. Also direct coupling between
$A^{3}\Pi$ and $a^{1}\Delta$ is not expected to lead to significant loss. The
only decay path out of the cycling scheme would be spontaneous decay into
lower lying vibrational states of $a^{1}\Delta$ which is forbidden [63]. Based
on these arguments it is highly probable that the proposed cooling scheme will
be sufficiently closed, especially taking into account that only closure to
the $10^{-4}$ level will be needed for the scheme to be practically useful
(see main text and Table 2).

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