# Molecular-ion quantum technologies

Mudit Sinhal  Stefan Willitsch<EMAIL_ADDRESS>Department of
Chemistry, University of Basel,
Klingelbergstrasse 80, 4056 Basel, Switzerland.

###### Abstract

Quantum-logic techniques for state preparation, manipulation and non-
destructive interrogation are increasingly being adopted for experiments on
single molecular ions confined in traps. The ability to control molecular ions
on the quantum level via a co-trapped atomic ion offers intriguing
possibilities for new experiments in the realms of precision spectroscopy,
quantum information processing, cold chemistry and quantum technologies with
molecules. The present article gives an overview of the basic experimental
methods, recent developments and prospects in this field.

Molecular ions, quantum-logic spectroscopy, precision measurements, quantum
technologies

††preprint: APS/123-QED

###### Contents

  1. I Introduction
  2. II Experimental Techniques
    1. II.1 Ion trapping
    2. II.2 Generation of molecular ions and their state initialization
    3. II.3 Cooling of trapped ions
  3. III Destructive state-readout techniques
  4. IV Quantum-logic experiments on single trapped molecular ions
    1. IV.1 Quantum-nondemolition molecular state readout by state-dependent coherent motional excitation
    2. IV.2 Molecular state detection and spectroscopy via a motional qubit
    3. IV.3 Molecular quantum logic spectroscopy using resolved-sideband Raman transitions
  5. V Outlook on future developments and conclusions

##  I Introduction

Over the past years, the development of experimental methods which enable the
control of single isolated quantum systems has made impressive progress. While
sophisticated techniques for the cooling and coherent manipulation of atomic
systems are now well established and form the basis for their application in
modern quantum science including quantum computation, quantum simulation and
precision measurements [1, 2, 3, 4, 5], their adaption to molecular systems
has remained a persistent challenge. The complexity of the molecular energy-
level structure reflecting electronic, vibrational and rotational motions as
well as electronic- and nuclear-spin degrees of freedom poses, in general,
severe difficulties for their translational cooling as well as the
preparation, manipulation and readout of individual quantum states. While
direct laser cooling of diatomic molecules like SrF [6] and CaF [7, 8] as well
as polyatomic molecules like H2CO [9], CaOH [10], CaOCH3 [11] has been
demonstrated, the vast majority of molecules cannot be laser cooled because
they lack closed optical cycling transitions. Thus, alternative cooling
schemes have to be implemented [12]. By the same token, sensitive optical-
cycling methods which are used for state readout in atomic systems [13, 14,
15] cannot readily be applied.

While the additional degrees of freedom offered by molecules, in particular
rotations and vibrations, impose challenges for their quantum control, they at
the same time provide a rich playground for new applications. Spectroscopic
transitions in molecular systems span a wide range of frequency bands from the
kHz to the PHz regime many of which are not easily accessible in atomic
systems. Within the high density of interacting molecular levels, transitions
with excellent coherence properties, which are scarce in atomic systems, can
readily be identified. Such transitions are attractive candidates for qubits
and for precision measurements [16]. Moreover, molecules offer prospects as
novel platforms for tests of fundamental physical concepts such as the
electric dipole moment of electron (eEDM) [17, 18] and possible parity
violation effects at the molecular level [19, 20, 21], for precise
determinations of fundamental constants [22, 23, 24, 25, 26], for precision
spectroscopic measurements [27, 28], for tests of ab-initio calculations and
molecular quantum theory [29, 30, 26], and for investigations of state- and
energy-controlled atom-molecule [31, 32] and ultracold molecule-molecule [33]
collisions. Molecules are also being increasingly considered for new frequency
standards and clocks [34, 16, 35, 36]. Last but not least, molecules are the
building blocks of chemistry giving access to a vast variety of chemical and
biochemical phenomena the study of which immensely benefits from precise
experimental methods [37, 38, 39].

In this chapter, we discuss upcoming quantum technologies for the coherent
control and manipulation of molecular ions in radiofrequency (RF) traps and
their application in various domains of molecular science. Quantum-logic
schemes in which single molecular ions are trapped together with single atomic
ions in ion traps have recently emerged as general and flexible methods for
non-destructive quantum-state manipulation of single molecules. In such
experiments, operations that cannot directly be performed on the molecular ion
due to its complex energy-level structure can be implemented via the co-
trapped atomic ion. This new methodology constitutes a paradigm change in how
molecules are interrogated: single molecules are used instead of large
ensembles, investigations can be performed in a completely non-destructive and
even quantum-non-demolition manner, and dramatic improvements in measurement
sensitivity and precision can therefore be realised. As will be reviewed in
this chapter, such protocols have so far successfully been employed for the
cooling of molecular ions to the motional ground state of the trap, for the
preparation and non-destructive detection of molecular quantum states, for
extremely sensitive spectroscopic measurements and for the implementation of
coherent operations [40, 41, 42, 43, 44, 45, 46], while a variety of further
applications has been proposed.

We begin by discussing the basic concepts and methods used in ion trapping and
cooling. To contrast with recent developments, we briefly summarise
destructive techniques that have been used in the past to probe trapped
molecular ions. We then give an overview of the most recent developments in
molecular-ion quantum technologies with a focus on advances in the non-
destructive manipulation and coherent control of single molecular ions.
Finally, we conclude with an outlook on future developments in the field.

##  II Experimental Techniques

###  II.1 Ion trapping

The experiments discussed here all rely on the confinement of ions in RF
traps, also referred to as Paul traps. The theory of RF traps has been
extensively treated in text books and review articles, see, e.g., Refs. [47,
48, 13, 49, 50, 51, 52]. Here, we briefly discuss the principles of linear-
quadrupole RF traps (LQTs) which are of particular relevance in the present
context.

Figure 1: Schematic representation of a linear-quadrupole radiofrequency ion
trap consisting of four cylindrical electrodes arranged in a quadrupolar
configuration. Ions are confined in the centre of the trap by applying
radiofrequency voltages $V_{\text{RF}}$ and static voltages $V_{\text{DC}}$ to
the electrodes as indicated.

Figure 1 shows a schematic of a generic linear Paul trap consisting of four
cylindrical electrodes arranged in a quadrupolar configuration. Note that LQTs
can be configured in a variety of three- and two-dimensional electrode
geometries including cylinders, wires, blades, wafers and surface structures
[50, 51]. Because Earnshaw’s theorem forbids the generation of an
electrostatic potential minimum, ion traps employ RF fields in order to
confine ions dynamically. A dynamic trapping field in the radial ($x$, $y$)
directions is generated by applying a time-varying voltage
$V_{\text{RF}}=V_{0}\cos(\Omega_{\text{RF}}t)$ to a pair of diagonally opposed
electrodes (shown in red in Fig. 1). Spatial confinement of the ions along the
axial ($z$) direction is provided by applying static potentials,
$V_{\text{DC}}$, on the endcaps of the trap (shown in blue in Fig. 1).

An LQT is designed to generate an approximately quadrupolar electric potential
close to the trap centre. The equations of motion for a single ion in such a
trap take the form of Mathieu equations [53, 47, 49],

$\frac{d^{2}u}{d\tau^{2}}+\left[a_{u}-2q_{u}\cos(2\tau)\right]u=0,$ (1)

where $u\;\in\;\\{x,y,z\\}$. The Mathieu parameters, $a_{u}$ and $q_{u}$, are
given by,

$\displaystyle a_{x}$
$\displaystyle=a_{y}=-\frac{1}{2}a_{z}=-\kappa\,\frac{4eV_{\text{DC}}}{mz_{0}^{2}\Omega_{\text{RF}}^{2}},$
(2a) $\displaystyle q_{x}$
$\displaystyle=-q_{y}=\frac{2eV_{\text{RF}}}{mr_{0}^{2}\Omega_{\text{RF}}^{2}},\;\;q_{z}=0,$
(2b) and, $\displaystyle\qquad\quad\tau=\frac{\Omega_{\text{RF}}t}{2},$ (2c)

where, $m$ and $e$ are the mass and the charge of the trapped ion,
respectively, 2$z_{0}$ is the distance between the endcaps, 2$r_{0}$ is the
distance between two diagonally opposed electrodes and $\kappa$ is a geometry
parameter describing the stiffness of the static axial trapping potential
which is considered to be harmonic [49]. Stable trapping of ions is possible
for certain ranges of the $a_{u}$ and $q_{u}$ parameters which is usually
represented in terms of stability diagrams [49].

In the limit of $|a|$, $|q|$ $\ll$ 1, the motion of a trapped ion according to
Eq. 1 can approximately be described in terms of two superimposed components,
a slow, ‘secular’ harmonic motion with frequency $\omega_{u}$ and a fast
‘micromotion’ at the frequency $\Omega_{\text{RF}}$ of the RF drive. The
trajectory of the ion can thus be expressed as [54],

$u(t)\approx
A_{u}\cos(\omega_{u}t+\phi)\left[1-\frac{q_{u}}{2}\cos(\Omega_{\text{RF}}t)\right],$
(3)

where $\phi$ is a phase describing the ion’s initial position and velocity and
$A_{u}\equiv A_{x}$, $A_{y}$, $A_{z}$ are the amplitudes of the motion. In Eq.
3, the first term corresponds to the secular and the second term to the
micromotion. The secular motional frequencies are given by,

$\omega_{u}=\frac{\Omega_{RF}}{2}\sqrt{a_{u}+\frac{q_{u}^{2}}{2}}.$ (4)

In the quantum limit, the secular motion of the ion can be represented by a
quantum harmonic oscillator with discrete motional states
$\left|n_{u}\right\rangle$ characterised by motional quantum numbers
$n_{u}=0$, $1$, $2$, $3$, … and energies
$E_{n_{u}}=(n_{u}+1/2)\hbar\omega_{u}$. For an ion in the motional ground
state $\left|n_{u}=0\right\rangle$, the extent of its wavefunction is given
by,

$u_{0}=\sqrt{\frac{\hbar}{2m\omega_{u}}}.$ (5)

In the interaction of the ion with light, the quantity $\eta=ku_{0}$ (the
“Lamb-Dicke parameter”) represents the ratio between the wavelength of the
radiation with wave vector $k=2\pi/\lambda$ and $u_{0}$ [13]. In the
Lamb–Dicke regime, defined as $\eta^{2}(2\,\bar{n}_{u}+1)<<1$, where
$\bar{n}_{u}$ represents the average motional quantum number of the ion in the
trap, the extent of the ion’s wavefunction is much smaller than the wavelength
of the radiation coupling to the ion. In this regime, first-order Doppler
shifts due to the motion of the ion vanish and transitions between different
motional states appear as discrete sidebands modulated onto an ionic spectrum
[13].

Figure 2: A Illustration of the combined internal-motional energy levels of a
two-level system trapped in a harmonic potential. In the Lamb-Dicke regime,
transitions between the quantised motional states of the particle appear as
red and blue sidebands (red and blue arrows respectively) modulated onto
carrier transitions (grey arrows). B Probability
($\text{P}_{\left|g\right\rangle\rightarrow\left|e\right\rangle}$) for
exciting the
$\left|g\right\rangle=\text{(4s)}^{2}\text{S}_{1/2}(m=-1/2)\rightarrow\left|e\right\rangle=\text{(3d)}^{2}\text{D}_{5/2}(m=-5/2)$
transition in Ca+ (also see Fig. 4) for a Ca+–N${}_{2}^{+}$ two-ion Coulomb
crystal as a function of the laser frequency, $f$, around the carrier
frequency, $f_{\text{Carrier}}$. Red (-) and blue (+) sidebands of the in-
phase (IP) and out-of-ohase (OP) axial motional modes of the ion crystal are
observed. Reproduced from Ref. [42] with permission from the Royal Society of
Chemistry.

Figure 2A depicts the combined internal-motional energy levels of a
hypothetical two-level ion, with internal ground ($\left|g\right\rangle$) and
an excited ($\left|e\right\rangle$) state, trapped in a harmonic potential. As
the ion’s motional frequencies (of order MHz) are usually much smaller than
its internal transition frequencies (hundreds of THz for electronic
transition), transitions involving motional energy levels appear as
modulations on electronic excitations. Transitions of the form
$\left|g,n_{u}\right\rangle\rightarrow\left|e,n_{u}\right\rangle$ or
$\left|e,n_{u}\right\rangle\rightarrow\left|g,n_{u}\right\rangle$ in which the
ion is excited or de-excited without a change in the motional quantum number
are termed as carrier transitions (grey arrows in Fig. 2A). Transitions in
which the motional quantum number of the trapped ion changes are known as
sideband transitions. When a sideband appears at a higher frequency than the
carrier (blue arrows), i.e.,
$\left|g,n_{u}\right\rangle\rightarrow\left|e,n_{u}+1\right\rangle$ or
$\left|e,n_{u}\right\rangle\rightarrow\left|g,n_{u}-1\right\rangle$, the
transition is termed a blue sideband (BSB). By contrast, a transition of the
form $\left|g,n_{u}\right\rangle\rightarrow\left|e,n_{u}-1\right\rangle$ or
$\left|e,n_{u}\right\rangle\rightarrow\left|g,n_{u}+1\right\rangle$ is termed
a red sideband (RSB) featuring a lower frequency than the carrier (red
arrows). In Fig. 2A, only first-order sidebands with $\Delta n_{u}=\pm 1$ are
indicated, but it should be noted that also higher-order sidebands with
$|\Delta|n_{u}>1$ exist [13].

In the context of experiments with multiple trapped ions, the Coulomb
interaction between the charged particles couples the motion of the individual
ions. The collective motions of a string of ions is then described in terms of
normal modes [55, 56]. For quantum-logic experiments with two ions of mass
$m_{1}$ and $m_{2}$, the axial secular motion of the two-ion string can be
described by an in-phase (IP) mode with a frequency $\omega_{-}$ and an out-
of-phase mode (OP) with a frequency $\omega_{+}$ given by [56],

$\omega_{\mp}^{2}=\frac{\omega_{z}}{m_{1}}\left(1+\frac{1}{M_{12}}\mp\sqrt{1+\frac{1}{M_{12}^{2}}-\frac{1}{M_{12}}}\>\right),$
(6)

where $\omega_{z}$ is the axial secular frequency of the ion with the higher
mass, $m_{1}$, and $M_{12}=m_{1}/m_{2}$ is the mass ratio of the two ions.
Figure 2B shows a typical spectrum of a Ca+– N${}_{2}^{+}$ two-ion chain in an
ion trap depicting the carrier and the red- and blue-sideband transitions
(represented by “-” and “+” signs, respectively) corresponding to the IP and
OP axial modes.

###  II.2 Generation of molecular ions and their state initialization

Figure 3: Rovibrational quantum-state preparation techniques for molecular
ions: A Trapped (i) MgH+ [57] and (ii) HD+ [58] molecular ions are prepared in
the rovibrational ground state ($N=0,v=0$) by continuously pumping population
from selected rotational levels in the ground vibrational state ($v=0$) to
excited levels using infrared lasers (red arrows). The excited molecules can
decay to the ground state ($N=0,v=0$) by spontaneous emission (blue wavy
arrows). Coupling to the ambient blackbody-radiation field (BB; black arrows)
leads to a redistribution of the level populations which ultimately accumulate
in the rotational ground state. Reproduced from Ref. [59]. B Two variants of
state-selective photoionisation schemes for the preparation of N${}_{2}^{+}$
molecular ions in the rovibronic ground state [60, 61]. Both schemes start
from rotational levels of the vibrationless ($v=0$) $X^{1}\Sigma^{+}_{g}$
ground electronic state, but differ in the choice of intermediate electronic
and vibrational states for the resonance-enhanced photoionisation sequence. In
(i), the $a^{\prime\prime 1}\Sigma^{+}_{g},v=0$ state is employed [60] while
in (ii) the $a^{1}\Pi_{g},v=6$ state is chosen [61]. In both cases, the
wavelength of the ionisation laser is selected so as to only populate the
rotational ground state of the cation.

For quantum experiments with trapped molecular ions, the clean chemical
preparation of the ions in the trap and their initialisation in the required
internal state is imperative. Molecular hydrides like CaH+ and MgH+, which are
frequently used species in the present context, are usually produced by
chemical [62, 63, 44, 45, 46, 43] or photo-chemical reactions [64] of the
relevant laser-cooled alkaline-earth ions with hydrogen gas lekaed into the
vacuum chamber. Homonuclear ions like O${}_{2}^{+}$, N${}_{2}^{+}$,
H${}_{2}^{+}$ and their isotopomers can be generated by photoionisation [65,
66, 60, 67, 61] or electron-impact ionization [68] of their neutral precursor
molecules.

The initialisation of the molecular ions in specific internal, i.e.,
electronic, vibrational, rotational and even hyperfine-Zeeman, states (usually
the absolute ground state) can be achieved by a variety of methods. Polar
molecular ions whose rotational-vibrational (rovibrational) degrees of freedom
couple to dipole radiation can be prepared in specific rovibrational levels by
blackbody-radiation-assisted optical pumping techniques [57, 58, 43, 44], see
Fig. 3A for examples. Extensions of these schemes have also been used for
hyperfine-state preparation [69]. As a further development, broadband optical-
pumping schemes using femtosecond lasers which address multiple rotational
transitions simultaneously have been shown to achieve state preparation on
considerably reduced timescales [70].

By contrast, homonuclear diatomic ions like N${}_{2}^{+}$ and O${}_{2}^{+}$ do
not feature a permanent electric dipole moment. Consequently, their
rovibrational degrees of freedom do not couple to dipole radiation which
renders direct optical-pumping schemes inefficient. As an alternative, state-
selective photoionisation schemes can be implemented for producing the ions in
well-defined rotational states. The technique relies on optical selection and
propensity rules which lead to the population of a limited number of
rotational states in the ion upon photoionisation of its neutral precursor
molecule. By exciting the neutral molecule to a well-defined intermediate
electronic state in a first step and subsequently setting the photoionization
energy slightly above the lowest ionic state accessible due to selection rules
(threshold photoionisation), molecular ions could be generated in single
rotational levels, see Fig. 3B [60, 67].

Finally, a probabilistic state preparation of the molecular ions can also be
achieved by the state-readout schemes discussed in Sec. IV below.

###  II.3 Cooling of trapped ions

Figure 4: Energy-level scheme for the laser cooling of $~{}^{40}$Ca+ ions.
Cooling to temperatures close to the Doppler limit is achieved by the repeated
scattering of photons on the
$\text{(4s)}^{2}\text{S}_{1/2}\rightarrow\text{(4p)}^{2}\text{P}_{1/2}$
transition at 397 nm. A second laser beam at 866 nm is required to repump
population from the meta-stable $\text{(3d)}^{2}\text{D}_{3/2}$ state in order
to close the laser cooling cycle. Red sidebands can be addressed repeatedly
on, e.g., the narrow
$\text{(4s)}^{2}\text{S}_{1/2}(m=-1/2)\leftrightarrow\text{(3d)}^{2}\text{D}_{5/2}(m=-5/2)$
electric-quadrupole transition in order to cool the ion to the motional ground
state. In this case, an additional re-pumping laser at 854 nm is employed in
order to drive the
$\text{(3d)}^{2}\text{D}_{5/2}\rightarrow\text{(4p)}^{2}\text{P}_{3/2}$
transition (omitted for clarity in the above representation) from which the
ion can relax to the ground state by spontaneous emission. Adapted with
permission of American Association for the Advancement of Science, from Sinhal
et al., [40], 367(6483), 2020; permission conveyed through Copyright Clearance
Center, Inc.

Ions loaded into a trap are typically hot with secular translational
temperatures vastly exceeding room temperature. Thus, multi-stage cooling
schemes are usually required to prepare the ions in their motional ground
state which is the starting point for the advanced experimental protocols
discussed in this article.

In a first Doppler-cooling stage, photons are repeatedly scattered from a
closed optical transition with a laser frequency slightly detuned to the red
from resonance [71]. Alkaline-earth ions like Be+, Mg+, Ca+, Ba+ and Sr+
exhibit simple energy-level structures which enable the implementation of the
required closed optical cycles. As an example, Fig. 4 depicts the Doppler-
cooling scheme employed for Ca+ ions. As most molecular ions do not possess
optical cycling transitions suitable for laser cooling, their motion has to be
cooled sympathetically by the interaction with simultaneously trapped, laser-
cooled atomic ions [64]. The limiting temperature achievable by Doppler laser
cooling is given by [71],

$T_{\text{D}}\approx\frac{\hbar\gamma}{2k_{\text{B}}},$ (7)

where $\gamma$ is the natural linewidth of the cooling transition, $\hbar$ is
the reduced Planck constant and $k_{\text{B}}$ is the Boltzmann constant.
$T_{\text{D}}$ is typically on the order of 1 mK, which under typical trapping
conditions is usually sufficient for the establishment of the Lamb-Dicke
regime. Under these conditions, the ions localise in the trap forming “Coulomb
crystals” in which individual particles can be observed, addressed and
manipulated, see Fig. 5 [64, 72, 73].

Following Doppler cooling, a second cooling stage needs to be implemented that
repeatedly addresses red-sideband transitions to optically pump the ions into
their motional ground state. This can either be direct sideband cooling on a
narrow optical transition, e.g., the
$\text{(4s)}^{2}\text{S}_{1/2}(m=-1/2)\leftrightarrow\text{(3d)}^{2}\text{D}_{5/2}(m=-5/2)$
transition in Ca+ (Fig. 2) [74], Raman sideband cooling as is frequently used
in Be+ [75], or electromagnetically-induced-transparency (EIT)-based sideband
cooling [76, 77]. In a mixed-species ion chain, sideband cooling in the Lamb-
Dicke regime needs to address specific collective modes of the ion motion
[78]. For many protocols, it is sufficient to cool only one mode of motion to
its ground state, e.g., the IP or OP axial mode.

##  III Destructive state-readout techniques

Figure 5: Destructive state-detection and spectroscopic techniques for trapped
molecular ions: A Fluorescence images of a Coulomb crystal of Ca+ and
N${}_{2}^{+}$ ions before and after laser-induced charge transfer (CT) with Ar
atoms following their infrared excitation to the first excited vibrational
state [79]. The images were obtained by recording the laser-induced
fluorescence produced during Doppler cooling of the Ca+ ions. Molecular ions
appear as a dark, non-fluorescing region in the centre of the crystal. The
number of molecular ions disappearing from the crystal due to CT was found by
comparison of the experimental fluorescence images (red panel) with molecular-
dynamics simulations (green panel). Reproduced from Ref. [79]. B Resonance-
enhanced multiphoton dissociation (REMPD) state-detection scheme implemented
for rovibrational spectroscopy of HD+ at 1.4 $\mu$m [80]. A single photon at
266 nm was employed to dissociate HD+ ions from the $v=4$ vibrational state
following excitation with the spectroscopy laser. Reprinted figure with
permission from [80]. Copyright 2007 by the American Physical Society. C REMPD
of sympathetically cooled MgH+ ions [57]. The images on the left-hand side
show (I) the initial pure Mg+ Coulomb crystal, (II) the crystal after loading
MgH+ ions (visible as the non-fluorescing region at the extremities of the
crystal), and the crystal after (III) one and (IV) four REMPD cycles. The
figure on the right-hand side shows the number of ions deduced from the images
as a function of the experiment time. Reprinted by permission from the
Springer Nature Customer Service GmbH: Springer Nature for Staanum et al.,
[57], Copyright 2010. D REMPD of Coulomb-crystallised CaH+ ions produced by
the chemical reaction of laser-cooled Ca+ ions with H2 gas [63]. The CaH+ ions
are photodissociated thus regenerating the original Ca+ crystal. The figure on
the right-hand side shows the total fluorescence yield of the Ca+ ions as a
function of the exposure time to the dissociation laser for various laser
wavenumbers. Reprinted with permission from [63]. Copyright 2018 American
Chemical Society.

Prior to the advent of quantum-logic-spectroscopic readout schemes, various
action-spectroscopy techniques have been used in order to probe Coulomb-
crystallized molecular ions. As chemical changes of the trapped molecular ions
can, in principle, be detected with unit probability using mass spectrometry
[81], these methods are highly sensitive and have been widely used in various
spectroscopic experiments and for the measurement of state populations.

As an example, laser-induced charge transfer (LICT) was implemented for
performing electronic and vibrational spectroscopy of N${}_{2}^{+}$ molecular
ions co-trapped with atomic Ca+ ions (Fig. 5A) [60, 67, 79]. These experiments
capitalised on a state-dependent charge-transfer reaction of N${}_{2}^{+}$
with Ar atoms according to $\text{N}_{2}^{+}{}(v\geq
1)+\text{Ar}\rightarrow\text{N}_{2}+\text{Ar}^{+}$ which only occurs in
excited vibrational states of N${}_{2}^{+}$ with vibrational quantum numbers
$v\geq 1$ on energetic grounds [82]. Starting from the vibrational ground
state, vibrationally excited states were populated either in direct infrared
excitation [79] or indirectly via excitation to the first excited electronic
state followed by spontaneous emission [60, 67]. The latter scheme was used to
map out rotational state populations following the generation of the ions by
state-selective threshold photoionisation.

A complementary technique is resonance-enhanced multiphoton dissociation
(REMPD) which relies on the dissociation of the molecular ions following their
excitation to a suitable intermediate molecular state. Such a scheme was
employed, for instance, in the high-resolution infrared spectroscopy of
trapped HD+ molecular ions sympathetically cooled by laser-cooled Be+ ions
[80, 83, 26, 25], as shown schematically in Fig. 5B, and in the rotational
spectroscopy of HD+ [84]. The technique was also employed for the read-out of
rotational state populations following rotational laser cooling and buffer-gas
cooling of sympathetically cooled MgH+ ions [57, 85] (Fig. 5C). REMPD
techniques were also used for the vibronic [62] and rovibronic [63]
spectroscopy of sympathetically cooled CaH+ ions (Fig. 5D).

A major drawback of action-spectroscopic methods is their inherently
destructive nature. They rely on a chemical change of the molecular ion
following photoexcitation and thus necessitate a complete re-initialisation of
the experiment every experimental cycle. The duty cycle of such experiments is
therefore low which impairs measurement statistics and, therefore, sensitivity
and precision. As a consequence, these experiments typically employ large
Coulomb-crystallised ensembles of ions to improve statistics [85, 79, 26, 25].
This invariably entails higher translational temperatures and thus Doppler and
other line broadening effects due to ensemble averaging (except in special
cases in which spectroscopic experiments can be performed in the Lamb-Dicke
regime under these conditions, see, e.g., Ref. [84]). However, in the context
of advanced applications such as coherent and ultrahigh-resolution experiments
on molecular ions, it is advantageous, and in many cases necessary, to work
with single particles. It thus became expedient to develop novel techniques
capable of interrogating single molecular ions which do not destroy the
molecule and, ideally, also preserve its quantum state.

##  IV Quantum-logic experiments on single trapped molecular ions

Figure 6: Quantum-nondemolition state detection of molecular ions: A
Simplified energy-level scheme of the N${}_{2}^{+}$ ion relevant for the
quantum-logic-based state determination technique implemented in Refs. [40,
41]. B Schematic of the protocol implemented for the state readout of
N${}_{2}^{+}$. Two counter-propagating laser beams form a one-dimensional
optical lattice interacting with a Ca+-N${}_{2}^{+}$ two-ion string cooled to
the motional ground state of the in-phase (IP) mode in a harmonic trap. One of
the lattice beams is detuned by the frequency of the IP mode, $f_{\text{IP}}$,
to resonantly excite its motion by an optical dipole force (ODF) depending on
the internal state of the N${}_{2}^{+}$ ion. C Calculated magnitude of the ac-
Stark shift, $|\Delta E|$, which gives rise to the ODF, experienced by the
N${}_{2}^{+}$ ion as a function of the laser-frequency detuning, $\Delta$,
from a strong spin-rovibronic transition in the molecule. At the lattice-laser
wavelength indicated by the vertical black dotted line, the N${}_{2}^{+}$ ion
experienced a larger ac-Stark shift in its ground rovibronic state
($\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$) compared to the all other
states ({$\left|\uparrow\right\rangle$${}_{\text{N}_{2}}$}) leading to the
state selectivity of the detection scheme. In the experiments, the Ca+ ion was
shelved in an excited state $\left|\uparrow\right\rangle$${}_{\text{Ca}}$ in
which it experienced a negligible ac-Stark shift (green dashed line) in
comparison to the N${}_{2}^{+}$ ion. Conversely, a Ca+ ion in its ground state
would experience a considerably increased ac-Stark shift (purple dashed line)
leading to a spurious background signal in the experiment. D Rabi spectroscopy
on the Ca+ $\left|\uparrow\right\rangle$${}_{\text{Ca}}$ $\rightarrow$
$\left|\downarrow\right\rangle$${}_{\text{Ca}}$ blue-sideband (BSB) transition
for N${}_{2}^{+}$ ion in the
$\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ state (blue) and in one of
the {$\left|\uparrow\right\rangle$${}_{\text{N}_{2}}$} states (red) as a
function of the BSB pulse length $t_{729}$, following coherent motional
excitation by the state-dependent ODF.
$P(\left|\downarrow\right\rangle_{\text{Ca}})$ denotes the population in the
$\left|\downarrow\right\rangle_{\text{Ca}}$ state. The green trace represents
a background measurement of the Rabi oscillation signal without ODF. E Time
trace of N${}_{2}^{+}$ state-detection attempts (blue and orange dots) for
measuring the molecular state. A threshold of 0.25 (grey shaded area) was used
to distinguish between N${}_{2}^{+}$ in the
$\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ or
{$\left|\uparrow\right\rangle$${}_{\text{N}_{2}}$} states. A histogram of the
state detection attempts is shown in the inset. The figure has been
republished with permission of American Association for the Advancement of
Science, from Sinhal et al., [40], 367(6483), 2020; permission conveyed
through Copyright Clearance Center, Inc.

Over the past few years, a range of coherent protocols have been developed for
manipulating and probing trapped molecular ions. These schemes can be regarded
as variants and extensions of the “quantum-logic spectroscopy” which was
originally conceived in the framework of the Al+ ion optical clock [86] and
adapted to molecular species. These schemes are inherently non-destructive,
thus mitigating the problems associated with the destructive techniques
discussed in the previous section, and pave the way for a range of new
applications of trapped molecular ions within the realms of quantum science
and precision measurements.

In the context of experiments with single trapped molecular ions, we discuss
the following three different implementations of quantum-logic protocols. As a
first example, we consider the quantum-nondemolition state detection of a
single N${}_{2}^{+}$ ion performed via a co-trapped Ca+ ion [42, 40]. The
scheme relies on the coherent motional excitation of the ions using molecular-
state-dependent optical-dipole forces (ODFs). This approach enabled the
tracking of molecular quantum state with a high time resolution as well as
rovibronic molecular spectroscopy. In an extension of this scheme [41],
specific molecular states embedded within a dense energy-level structure could
be identified. As a second example, we discuss the non-destructive state
detection of MgH+ ions [43]. In these experiments, an ODF was implemented in
order to map the molecular state onto a motional qubit which was then read out
on a co-trapped Mg+ ion. As a final example, we present molecular state
manipulation schemes implemented for CaH+ ions via co-trapped Ca+ ions [44,
45]. These experiments relied on exciting motional sidebands with Raman
transitions on the molecule and allowed for coherent spectroscopy of selected
molecular rotational states. Moreover, entanglement between the molecule and
the atom could be demonstrated [46].

###  IV.1 Quantum-nondemolition molecular state readout by state-dependent
coherent motional excitation

Fig. 6 illustrates a quantum-non-demolition experiment for the measurement of
the spin-rovibronic state of a single N${}_{2}^{+}$ via a co-trapped Ca+ ion
[42, 40]. In the first step of the experimental protocol, the IP mode of the
Ca+–N${}_{2}^{+}$ two-ion string was cooled to the ground state of the trap.
An optical-dipole force (ODF) the strength of which depended on the molecular
state was then used to map the state information onto the motion of the two-
ion string. In case the N${}_{2}^{+}$ was in the state targeted by the ODF
($\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ in Fig. 6 A), the
Ca+–N${}_{2}^{+}$ string experienced a large ODF which was used to coherently
excite the motion of the previously cooled IP mode. By contrast, for all other
molecular states (summarily labeled
{$\left|\uparrow\right\rangle$${}_{\text{N}_{2}}$} in Fig. 6), the magnitude
of the ODF was significantly reduced so that only negligible motional
excitation occurred. Subsequently, the motional excitation of the two-ion
string was detected by Rabi sideband thermometry on the Ca+ ion thus revealing
the information about the internal state of the N${}_{2}^{+}$ ion.

The ODF for the state detection was implemented by two counter-propagating
laser beams which formed a one-dimensional optical lattice. In order to
coherently excite the IP mode of motion, the amplitude of the ODF was
modulated by detuning one of the lattice beams by the mode frequency,
$f_{\text{IP}}$ (Fig. 6B). The magnitude of the ODF experienced by the
Ca+–N${}_{2}^{+}$ string was dependent on the ac-Stark shift experienced by
the N${}_{2}^{+}$ ion due to the lattice laser. Fig. 6C depicts the calculated
ac-Stark shift, $\Delta E$, for
$\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ (here designated as the
rovibronic ground state of the molecule) and the
{$\left|\uparrow\right\rangle$${}_{\text{N}_{2}}$} states as a function of the
lattice-laser detuning from a rovibronic transition (here the
$\text{X}^{2}\Sigma_{g}^{+}(v^{\prime\prime}=0)-\text{A}^{2}\Pi_{u}(v^{\prime}=2)~{}R_{11}(1/2)$
line [87], where $v^{\prime}(v^{\prime\prime})$ represent the vibrational
quantum number of the upper(lower) electronic state originating from the
$\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ state of N${}_{2}^{+}$. The
frequency of the lattice laser, $f_{\text{lattice}}$ (black-dotted line), was
set such that the a molecule in the
$\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ state experienced a large
ac-Stark shift and hence a large ODF. Conversely, molecules in excited
rotational or vibrational states experienced a much weaker ODF due to the
lattice laser being farther detuned from any possible spectroscopic transition
originating from these states thus resulting in a smaller motional excitation.
For the lattice intensity and polarization chosen in these experiments, a Ca+
ion in the ground
$\left|\downarrow\right\rangle$${}_{\text{Ca}}$$=(4\text{s})^{2}\text{S}_{1/2}(m=-1/2)$
state (Fig. 4B) experienced a large, nearly detuning-independent ac-Stark
shift (purple-dashed line in Fig. 6C) leading to a background ODF originating
from Ca+ which impaired the signal-to-noise (SNR) ratio for the detection of
the state of the N${}_{2}^{+}$ ion. Thus, in order to minimize the coupling of
the atomic ion to the lattice laser, the Ca+ ion was shelved in the
$\left|\uparrow\right\rangle$${}_{\text{Ca}}$$=(3\text{d})^{2}\text{D}_{5/2}(m=-5/2)$
state which experienced a negligible ac-Stark shift (green dashed line)
resulting in no appreciable motional excitation due to the Ca+ ion.

Fig. 6D shows the results of Rabi spectroscopy experiments performed on Ca+
ion after the implementation of the ODF. For N${}_{2}^{+}$ in the
$\left|\downarrow\right\rangle$${}_{\text{N}_{2}}$ state, significant motion
was excited resulting in Rabi oscillations (blue trace) on the blue sideband
of the IP mode of the Ca+
$\left|\uparrow\right\rangle_{\text{Ca}}{}\rightarrow\left|\downarrow\right\rangle_{\text{Ca}}{}$
transition. By contrast, for molecules in the
{$\left|\uparrow\right\rangle$${}_{\text{N}_{2}}$} states, no Rabi
oscillations were observed (orange trace) as blue sideband transitions could
not be excited when the ion string remained cooled to the ground motional
state of the IP mode. Indeed, the signal obtained was comparable to a
background measurement of the blue-sideband oscillations on a ground-state
cooled Ca+– N${}_{2}^{+}$ chain without the application of any ODF (green
trace). Each of the traces in Fig. 6D resulted from thousands of measurements
performed on the same molecular ion which was possible due to the non-
destructive nature of the state-detection scheme.

Fig. 6E depicts the results of the experiments as a function of molecular-
state detection attempts. Each data point represents the average of 22 Rabi
experiments for which the pulse time was set within the region of maximum
contrast indicated by the light-blue area in Fig. 6D. In this case, a single
state-detection attempt took only few 100 ms which allowed for tracking of the
molecular state with a high time resolution. For the measurements recorded at
the beginning of the experiment (blue dots), the molecule was determined to be
in the rovibronic ground state 105 times with zero false events when setting
the detection threshold above the grey-shaded area in Fig. 6E. Subsequently,
the molecule changed its state and was detected not to be in the rovibronic
ground state 163 times with zero false detection attempts (orange dots). In
this experiment, the fidelity for the faithful detection of the rovibronic
ground state amounted to $>$99%.

Figure 7: Non-invasive molecular spectroscopy of single molecular ions: A Line
shape of the
$\text{X}^{2}\Sigma_{g}^{+}(v^{\prime\prime}=0)-\text{A}^{2}\Pi_{u}(v^{\prime}=2)~{}R_{11}(1/2)$
rovibronic transition in N${}_{2}^{+}$ measured by determining the ac-Stark
shift experienced by the ion at various lattice-laser detunings (blue data
points) [40]. The line center was found by a fit (orange trace) to the
measurements in very good agreement with the literature values (green dashed
lines). B Vibronic Einstein $A$ coefficients, $A_{\text{vibronic}}$, extracted
from the measured ac-Stark shifts in panel A at various detunings (blue data
points). The mean of the measurements (orange line) was found to be in very
good agreement with previous literature values (green dashed lines).
Republished with permission of American Association for the Advancement of
Science, from Sinhal et al., [40], 367(6483), 2020; permission conveyed
through Copyright Clearance Center, Inc.

An application of this state-detection protocol is shown in Fig. 7. Since the
magnitude of the excited motion is dependent on the ac-Stark shift giving rise
to the ODF which in turn depends on the detuning of the lattice laser from a
spectroscopic transition in the molecule, the scheme can be employed for non-
invasive spectroscopy on the molecular ion. By extracting the magnitude of the
ac-Stark shift from the observed Rabi oscillations as a function of the
lattice-laser detuning, the spectral lineshape of the resonance could be
mapped out (Fig. 7A). From the determined ac-Stark shifts, quantitative
spectroscopic parameters like vibronic Einstein-A coefficients,
$A_{\text{vibronic}}$, were obtained (Fig. 7B). Both line positions and
Einstein coefficients were in very good agreement with literature values
(green dashed lines in Fig. 7) thus validating the accuracy of the scheme.

Figure 8:  Phase-sensitive forces for molecular state identification. A Same-
phase (SP) and opposite-phase (OP) configurations of the positions of a
N${}_{2}^{+}$ (orange dots) and Ca+ (blue dot) ion in an optical lattice in
order to measure the amplitude and phase of the ODF experienced by the
N${}_{2}^{+}$ ion. The Ca+ ion was always red-detuned from its closest
resonance and thus always experienced an approximately constant negative ac-
Stark shift, $-|\Delta_{\text{a}}|$ and therefore a positive ODF. Depending on
the molecular quantum state, the N${}_{2}^{+}$ ion was either red
($-|\Delta_{\text{m}}|$) or blue ($+|\Delta_{\text{m}}|$) detuned from its
nearest resonance resulting in attractive and repulsive optical forces,
respectively. B If the ODF on N${}_{2}^{+}$ and Ca+ have the same sign (sub-
panel a), a stronger combined force and therefore stronger Rabi signal is
measured on the Ca+ ion after motional excitation in the SP configuration
(blue trace) compared to the OP configuration (magenta trace). Conversely, if
the ODF on N${}_{2}^{+}$ and Ca+ have opposite signs, the OP configuration
gives a stronger Rabi signal (sub-panel b). The relative sign of the ODF on
both ions thus yields additional information on the molecular state which is
particularly valuable for state-identification within dense energy-levels
structures. Reproduced from Ref. [41].

In a variation of this scheme, not only the magnitude, but also the sign of
the ac-Stark shift were retrieved [41]. As the ac-Stark shift changes sign
across a resonance, its sign encodes the direction of the detuning from
resonance (red or blue) and thus valuable additional information for
molecular-state identification which is particularly useful in molecules with
a dense energy-level structure. Fig. 8 depicts an experiment on a Ca+–
N${}_{2}^{+}$ chain in which in addition to the N${}_{2}^{+}$ ion also the Ca+
ion experienced a finite and constant detuning-independent ODF. An
interferometric measurement as shown in Fig. 8A was then performed in order to
detect the molecular state. As before, the measurement started with preparing
the Ca+– N${}_{2}^{+}$ string in the ground state of the IP mode of motion.
The Ca+ ion was then prepared in the
$\left|\uparrow\right\rangle$${}_{\text{Ca}}$ state such that it was always
red-detuned from its closest resonance and experienced an ODF in the direction
of higher lattice intensity. The N${}_{2}^{+}$ ion was generated in an
arbitrary excited rotational state with the aim to identify this state within
the dense Zeeman-spin-rotational energy-level structure in this region. Thus,
for N${}_{2}^{+}$ the amplitude and sign of the ac-Stark shift (and hence the
ODF) depended on its specific rovibronic, hyperfine and Zeeman state, while it
remained constant and negative for Ca+. In order to identify the molecular
state, two combined measurements were then performed. In the first experiment,
the two ions were placed in the trap such that their separation corresponded
to an integer multiple of the lattice nodes. This lattice configuration was
denoted as “same phase” (SP). In the SP configuration, the ODFs on the
molecule and the atom acted in the same (opposite) direction when the
molecular ion was red (blue) detuned from its closest resonance
(SP${}_{\text{R}}$ and SP${}_{\text{B}}$ in Fig. 8A). In the second
experiment, the distance between the ions was changed by half a lattice
spacing, such that the two ions always experienced an opposite gradient of the
lattice field, i.e., the “opposite phase” (OP) of the lattice. In the OP
configuration, the forces acted in the same (opposite) direction when the
molecular ion was blue (red) detuned (OP${}_{\text{R}}$ and OP${}_{\text{B}}$
in Fig. 8A). Fig. 8B depicts Rabi oscillation signals on the
$\left|\uparrow\right\rangle$${}_{\text{Ca}}$$\rightarrow$$\left|\downarrow\right\rangle$${}_{\text{Ca}}$
BSB transition for the SP (blue) and OP (purple) lattice configurations after
an ODF pulse for two different molecular states (panels a and b). In panel a,
the stronger SP signal (larger frequency and amplitude of the Rabi
oscillation) suggested that the lattice laser was red detuned from the closest
molecular transition. By comparing the measured amplitude and sign of the ac-
Stark shift with theory, the spin-rotational state of the molecule could be
identified as $N=6$, $J=11/2$, where in this case $N$ and $J$ are the
rotational and spin-rotational quantum numbers, respectively. In panel b, the
stronger OP signal suggested that the lattice laser was blue detuned from the
closest molecular transition. In this case, all rotational levels $N\leq 4$
could be excluded.

###  IV.2 Molecular state detection and spectroscopy via a motional qubit

Figure 9: A Schematic representation of the quantum-logic spectroscopy
implemented for the state detection of MgH+ in Ref. [43]. See text for
details. Reproduced with permission from F. Wolf for [88]. B Circuit diagram
of the state-mapping process from the molecular to the motional state (panel
a). A Bloch-sphere representation depicting the final motional qubit state for
the $J\neq 1$ (panel b) and $J=1$ (panel c) rotational states is also shown. C
Theoretical estimation (green shaded region) and experimental observations
(data points) for the population measurement in the ground state
$\left|\downarrow\right\rangle_{\text{Mg}}$ of the Mg+ ion provided that the
molecule is in the $J=1$ rotational state, after the implementation of the
state-dependent force. D Mapping of a rovibronic transition in the MgH+ ion by
application of the non-destructive state detection technique. Panels B, C, and
D have been adapted by permission from the Springer Nature Customer Service
GmbH: Springer Nature for Wolf et al., [43], Copyright 2016.

Fig. 9 shows a complementary approach to molecular-state identification in the
presence of a spurious detuning-independent ODF experienced by the atomic ion.
In these experiments [43], which represented the first quantum-logic
spectroscopy performed on a molecular ion, a specific motional state was
engineered on a Mg+– MgH+ two-ion chain. An ODF was then implemented in order
to map the state of the molecular ion onto the motion which was subsequently
detected by sideband measurements on the atomic ion. The experimental sequence
is shown in Fig. 9A. Here, both the IP and the OP motional modes of the two-
ion chain were cooled to the ground state by resolved-sideband cooling [89].
In a first step, the motion was initialized in a state
$\left|\downarrow\right\rangle$${}_{\text{motion}}=\left|1\right\rangle_{\text{IP}}\left|0\right\rangle_{\text{OP}}$
corresponding to one quantum of excitation in the IP mode. The atomic ion was
prepared in an excited state $\left|\uparrow\right\rangle$${}_{\text{atom}}$.
An ODF was then applied using a 1D optical lattice with parameters chosen such
that the motional state of the two-ion crystal was changed to
$\left|\uparrow\right\rangle$${}_{\text{motion}}=\left|0\right\rangle_{\text{IP}}\left|1\right\rangle_{\text{OP}}$
by the ODF acting solely on the atom in case the molecular ion was not in the
$J=1$ rotational state (Fig. 9Bb). By contrast, if the MgH+ ion was in the
targeted rotational state, the Mg+– MgH+ two-ion chain experienced an
additional state-dependent ODF due to the molecular ion resulting in a
different dynamics on the Bloch sphere of the motional qubit (Fig. 9Bc). For
suitably chosen experimental parameters, the protocol corresponds to a CNOT
logic gate performed on the motional qubit with the molecular state as the
control allowing to detect the $J=1$ rotational state in the ground vibronic
manifold. In a subsequent step, a sideband measurement on the atomic ion was
used to detect the motional state of the two-ion chain resulting in the
determination of the molecular state. Fig. 9C shows the probability of
exciting the sideband on the Mg+ ion after the ODF pulse as a function of the
detuning, $\Delta_{\text{MgH}}$, of the lattice laser frequency on both the
red and blue side of a molecular transition starting from the $J=1$ rotational
state. The theoretical estimates, including the measurement uncertainties, are
given by the green areas and the experimental realisations are represented by
the data points. By determining the strength of the ODF acting on the
molecular ion, a spectroscopic measurement on the relevant molecular
transition was performed as shown in Fig. 9D.

Figure 10: Non-destructive state determination and coherent manipulation of
CaH+ ions. A Schematic of the quantum-logic technique implemented for
molecular state detection and projective preparation in Ref. [44]
(MI=molecular ion, AI=atomic ion). See text for details111The definition of a
red sideband as indicated in the third panel of Fig. 1A adopted by the authors
in Ref. [44] differs from the one used throughout this article (for example
see Fig. 2A where such a transition would be the labelled as a blue
sideband).. B Coherent spectroscopy and manipulation on the $J=1$ (panels 1, 2
and 3) and $J=2$ (panels 4, 5 and 6) rotational states of the CaH+ ion on
transitions of the form
$\left|J,-J-1/2,-\right\rangle\leftrightarrow\left|J,-J+1/2,-\right\rangle$.
Frequency scans are shown in panels 1 and 4, Rabi flops are depicted in panels
2 and 5, Ramsey fringes as a function of the relative phase between two
$\pi/2$ pulses with a wait time of 15 ms are shown in panels 3 and 6.
Reprinted by permission from the Springer Nature Customer Service GmbH:
Springer Nature for Chou et al., [44], Copyright 2017. C High-resolution
frequency spectra (panels 1 and 3) and Rabi flops (panels 2 and 4) driven by
an optical frequency comb for $\Delta J=2$ rotational transitions in a single
CaH+ ion. Republished with permission of American Association for the
Advancement of Science, from Chou et al., [45], 367(6485), 2020; permission
conveyed through Copyright Clearance Center, Inc. D Parity measurements of
entangled states between an atom and a low-frequency
($\left|\psi_{\text{l}}\right\rangle$) as well as a high-frequency
($\left|\psi_{\text{h}}\right\rangle$) molecular qubit. Reprinted by
permission from the Springer Nature Customer Service GmbH: Springer Nature for
Lin et al., [46], Copyright 2020.

###  IV.3 Molecular quantum logic spectroscopy using resolved-sideband Raman
transitions

A different approach to molecular-state preparation and detection was
implemented by Chou et al. [44] in CaH+ (see Fig. 1). As in the previously
discussed protocols, the experiment was started by cooling the shared motion
of a Ca+– CaH+ two-ion chain and preparing the Ca+ ion in an excited state. In
the room-temperature environment of the experiment in Ref. [44], the molecules
were prepared in their electronic and vibrational ground state and preparation
of a specific rotational state relied on blackbody-assisted population
redistribution. Raman laser beams were then applied in order to drive specific
BSB transitions in the molecule such that a change in the molecular state was
accompanied by an excitation of the shared motion of the two-ion chain (Fig.
1A). The excited motion was subsequently detected by driving a red sideband on
the atomic ion followed by fluorescence detection leading to projection of the
molecule into a specific quantum state by the measurement. By allowing
sufficient time for the molecule to re-equilibrate with the blackbody
environment, Raman spectra of several molecular transitions could be recorded
by sideband measurements on the atomic ion.

Since the times for population redistribution of the rotational states by BBR
were long (about 100 ms to more than 2 s at room temperatures for $J<8$)
compared to the time required to drive transitions ($<5$ ms), the population
of the rotational states could be accumulated in specific magnetic sublevels
by optical pumping and the targeted quantum states could then be coherently
manipulated. The eigenstates of CaH+ are labelled by the quantum numbers
$\left|J,m,\xi\right\rangle$, where $J=0,1,2,3,..$ is the rotational angular
momentum quantum number in this case, $m\in\\{-J-1/2,-J+1/2,J-1/2,J+1/2\\}$
denotes the sum of the components of the rotational angular momentum and the
proton spin along the magnetic field direction. $\xi\in\\{+,-\\}$ indicates
the relative sign of a superposition of product states with the same $m$ but
opposite proton spin. Fig. 1B depicts coherent spectroscopy and operations on
the molecular ion on the
$\left|J,-J-1/2,-\right\rangle\leftrightarrow\left|J,-J+1/2,-\right\rangle$
transitions for $J=1$ (panels 1, 2 and 3) and $J=2$ (panels 4, 5 and 6).

Based on these state-detection and preparation techniques, a high-resolution
rotational spectrum of a single CaH+ molecule was recorded with a linewidth
$<1$ kHz and an accuracy of $\sim 1$ part per billion [45]. The spectra were
obtained by coherently driving stimulated-Raman rotational transitions using
an optical frequency comb (Fig. 1C). Here, the molecule was prepared in the
$\left|\mathcal{J}^{\prime\prime}\right\rangle=\left|2,-5/2,-\right\rangle$
state followed by probing the
$\left|\mathcal{J^{\prime}}\right\rangle\leftarrow\left|\mathcal{J}^{\prime\prime}\right\rangle$
transition where
$\left|\mathcal{J^{\prime}}\right\rangle=\left|4,-7/2,-\right\rangle$. The
probabilities of the molecule being in either states as a function of the
Raman difference frequency from the resonance is shown in panels 1 and 3. When
prepared in the $\left|\mathcal{J^{\prime}}\right\rangle$ state, the coherent
Rabi flops to and from the $\left|\mathcal{J^{\prime\prime}}\right\rangle$
state are shown in panels 2 and 4.

Moreover, by applying tailored pulse sequences on the the Ca+ and and CaH+
ion, a quantum entanglement was generated between the atom and the molecule
[46]. Fig. 1D shows parity measurements of entangled states of the atom and a
low-frequency ($\left|\psi_{\text{l}}\right\rangle$) as well as a high-
frequency ($\left|\psi_{\text{h}}\right\rangle$) molecular qubit. The state
parities, $\mathit{\Pi}$, were measured as a function of the phase of an
‘analysis’ pulse, $\phi_{\text{a}}$, and oscillate as
$C\cos(2\phi_{\text{a}}+\phi_{0})$ where $\phi_{0}$ is an offset in phase and
$C\geq 0$ is the observed contrast. The fidelities for the generation of the
low-frequency and high-frequency entangled states in this experiment were
estimated to be $F_{\text{l}}=0.87(3)$ and $F_{\text{h}}=0.76(3)$ indicating
bipartite entanglement for both cases [90].

##  V Outlook on future developments and conclusions

In addition to the already realised experiments discussed above, alternative
quantum-logic protocols have been proposed for molecular ions. Mur-Petit et
al. [91] have suggested temperature-independent geometric-phase gates based on
state-dependent forces for non-destructive molecular-state detection. These
operations are insensitive to the motional state of a trapped atom-molecule
two-ion crystal and can thus be applied to Doppler cooled ions possibly
leading to simpler and faster experimental implementations.

Loh et al. [92] have proposed a state-detection technique relying on state-
dependent magnetic $g$ factors which lead to distinct Zeeman splittings for
electronic, rotational and hyperfine quantum states of a molecule. Such a
Zeeman-splitting-assisted quantum logic spectroscopy (ZSQLS) scheme would
employ lasers far detuned from one-photon transitions in the molecule to drive
Raman transitions between Zeeman sublevels without scattering photons and thus
preserve the quantum state of the molecular ion. Apolar molecules like
O${}_{2}^{+}$ and polar molecules like SO+, CO+ and SiO+ could be probed by
the ZSQLS technique under suitable experimental conditions [92].

As an extension of the CaH+ state-detection scheme implemented by Chou et. al
[44], Wolf et al. [93] proposed a quantum-logic technique for non-destructive
state detection of O${}_{2}^{+}$ ions that would allow for amplification of
the state detection signal which is particularly important if single-shot
readout of the atomic ion is not possible for technical reasons. The scheme
employs bichromatic far-detuned Raman lasers to drive red and blue sideband
transitions between selected Zeeman states simultaneously. For specific
rotational states, the phase of the bichromatic drive can be chosen such that
a Schrödinger cat state of the form
$\left|\psi\right\rangle_{\text{SC}}=\left|+\right\rangle\left|\alpha\right\rangle\pm\left|-\right\rangle\left|-\alpha\right\rangle$
is produced, where
$\left|\pm\right\rangle=(\left|1/2\right\rangle\pm\left|3/2\right\rangle)/\sqrt{2}$
and $\left|\pm\alpha\right\rangle$ denotes a coherent state with complex
amplitude $\pm\alpha$. By detecting the depletion of the motional ground state
due to the emergence of the Schrödinger cat state after application of the
Raman pulses, the successful drive of the transition can be detected on the
atomic ion making it possible to infer the molecular state.

Another series of proposals explored the potential of using the dipole moments
of trapped polar molecular ions as a resource for quantum logic, either by
capitalising on direct dipole-dipole couplings between the molecules [94] or
the coupling of the dipoles to the normal modes of a Coulomb crystal [95, 96].
The latter approach offers possibilities for implementing protocols which do
not require a direct optical addressing of the molecules and their cooling to
the motional ground state. Proposals for further developments of the dipole-
phonon toolbox include the application of microwave fields to address both the
internal states and the motions of the trapped polar ions [97].

A variant of quantum-logic spectroscopy makes use of the recoil imparted by
the absorption or emission of photons to the motion of ground-state cooled
ions. The resulting change of motional state can then be detected by the usual
sideband-spectroscopy techniques. This approach has proven useful in the
precise measurement of strong, dipole-allowed transitions in atomic ions [98].
Extensions of this method to infrared spectroscopy [99] and to probing the
vibrational dynamics dynamics of trapped molecular ions [100] have recently
been proposed.

The new coherent methods for molecular ions discussed are not only of interest
for applications like quantum science and precision spectroscopy, but also
have a considerable potential in completely different realms such as collision
studies and chemistry. Proof-of-principle experiments were presented in Refs.
[40, 16] in which changes of the internal state [40] and of the chemical
identity [16] of single N${}_{2}^{+}$ ions were tracked as a consequence of
collisions demonstrating the utility of the present methods as non-invasive
probes for investigating inelastic and reactive collisions of molecular ions.

In parallel to the methodological advances outlined above, new technological
developments are addressing an ever growing variety of molecular systems.
Recently laser cooling and optical-pumping schemes for molecular cations like
BH+ [101], SiO+ [102, 103], TeH+ [104], AlCl+ and AlF+ [105], and anions like
C${}_{2}^{-}$ [106] have been explored theoretically. Apart from the systems
already used in current experiments, i.e., MgH+, CaH+ and N${}_{2}^{+}$,
quantum-logic techniques for molecular ions like H${}_{2}^{+}$ and its
isotopologues [23, 35, 36], O${}_{2}^{+}$ [107, 93] and polyatomics such as
cyanoacetylene [100] and propanediol [108] haven been explored in various
proposals. Clearly, with the vast number of molecular systems available, the
possibilities are virtually endless.

The coherent experimental tools available for trapped molecular ions have now
reached a similar level of sophistication as the ones employed for atomic
ions. Although the field of molecular-ion quantum technologies is still in its
infancy, the presently realised experiments as well as the current proposals
for future advances discussed in this review illustrate the immense potential
for the quantum control and manipulation of molecular ions and the ensuing
applications in precision spectroscopy, quantum science and even chemistry.

## Acknowledgements

Our work is funded by the Swiss National Science Foundation (SNSF) under
project grants CRSII5_183579 and 200021_204123. It is also supported as a part
of NCCR QSIT, a National Centre of Competence in Research, funded by the Swiss
National Science Foundation (grant number 51NF40-185902).

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