Cold hybrid ion-atom systems

Hybrid systems of laser-cooled trapped ions and ultracold atoms combined in a single experimental setup have recently emerged as a new platform for fundamental research in quantum physics. This paper reviews the theoretical and experimental progress in research on cold hybrid ion-atom systems which aim to combine the best features of the two well-established fields. A broad overview is provided of the theoretical description of ion-atom mixtures and their applications, and a report is given on advances in experiments with ions trapped in Paul or dipole traps overlapped with a cloud of cold atoms, and with ions directly produced in a Bose-Einstein condensate. This review begins with microscopic models describing the electronic structure, interactions, and collisional physics of ion-atom systems at low and ultralow temperatures, including radiative and nonradiative charge-transfer processes and their control with magnetically tunable Feshbach resonances. Then the relevant experimental techniques and the intrinsic properties of hybrid systems are described. In particular, the impact is discussed of the micromotion of ions in Paul traps on ion-atom hybrid systems. Next, a review of recent proposals is given for using ions immersed in ultracold gases for studying cold collisions, chemistry, many-body physics, quantum simulation, and quantum computation and their experimental realizations. The last part focuses on the formation of molecular ions via spontaneous radiative association, photoassociation, magnetoassociation, and sympathetic cooling. Applications and prospects are discussed of cold molecular ions for cold controlled chemistry and precision spectroscopy.

Figure 1

Schematic representation of an example hybrid ion-atom experiment: a linear radio-frequency Paul trap is used to store ions, whereas the ultracold atoms are confined in a crossed optical dipole trap. By precisely overlapping the positions of these two traps, a single ion can be immersed into the center of the ultracold neutral atom cloud. From [312].

Figure 2

Nonrelativistic molecular potential energy curves of the $(\mathrm{Li}+\mathrm{Yb}{)}^{+}$ ion-atom system. From [367].

Figure 3

Transition electric dipole moments between singlet states of the $(\mathrm{Li}+\mathrm{Yb}{)}^{+}$ ion-atom system. Adapted from [367].

Figure 4

The ground-state potential energy surface for the ${\mathrm{C}}_2{\mathrm{H}}^{-}$ molecular anion interacting with the Rb atom. The inset shows the corresponding Legendre components. Adapted from [366].

Figure 5

Long-range ion-atom potential for the lowest partial waves in units of the characteristic energy $E^{\star }$ and the characteristic length $R^{\star }$. Dots show the positions of the maxima of the centrifugal barriers. The inset shows the two lowest potentials.

Figure 6

Elastic collision rate constant in the singlet channel $A^1{\mathrm{\Sigma }}^{+}$ (upper panel) and rate constant of radiative $A^1{\mathrm{\Sigma }}^{+}\to X^1{\mathrm{\Sigma }}^{+}$ charge transfer (lower panel) vs collision energy in the ${\mathrm{Ca}}^{+}+\mathrm{Na}$ system. Both the elastic and the total reactive rate constants are decomposed into selected partial waves. From [164].

Figure 7

Sensitivity of the rate constants for (a) elastic $K_{\mathrm{el}}$ and reactive $K_{\mathrm{R}}$ scattering and of the (b) ratio of elastic to reactive rate constants at collision energies of $10\mu \mathrm{K}$ (black solid lines) and 1 mK (red dashed lines) in the ${\mathrm{Yb}}^{+}+\mathrm{Li}$ system to a scaling factor $\lambda$ applied to the interaction potential $V(R)\to \lambda ·V(R)$. From [367].

Figure 8

Schematic representation of possible charge-transfer processes in cold ion-atom collisions: RA—radiative association, RCT—radiative charge transfer, and nRCT—nonradiative charge transfer. Exemplary electronic states and vibrational wave functions are presented.

Figure 9

Selected possible arrangements of the charge and electronic states in ion-atom systems. (a) Typical for alkali-metal ions interacting with alkali-metal atoms and (b) and (c) for alkaline-earth-metal ions interacting with alkali-metal atoms. Magnetically tunable Feshbach resonances are expected for $A^{+}+B$ and $A+B^{+}$ collisions in (b) and (c), respectively.

Figure 10

Bound-state energies of the polarization potential as a function of the short-range phase $\varphi$ for the first few lowest partial waves. From [167].

Figure 11

Radiative charge-transfer rate constants for the ${\mathrm{Ca}}^{+}+\mathrm{Na}$ system assuming a scattering length $a=R^{\star }$. The predictions of the quantum defect model (solid black lines) with fitted reaction probability are compared with the full numerical calculation (blue triangles). The red squares show the contribution of the free-free transitions (charge transfer), while the green circles stand for the free-bound transitions (radiative association). The dashed line gives a thermal average of the rate constants, and the thin gray line is the Langevin rate constant. From [167].

Figure 12

Diagrams of an atom capture by an ion. (a) The spontaneous capture in level $v$ is followed by phonon emission (with corresponding rates). (b) The stimulated rate from the condensate $c$ to the bound level $v$ may also produce heating, i.e., population of noncondensate atoms $nc$ (c). From [64].

Figure 13

Differential Lamb shift (see text for definition) for ${\mathrm{CN}}^{-}$ in either a ${}^{87}\mathrm{Rb}$ (solid lines) or ${}^{84}\mathrm{Sr}$ (dashed lines) BEC as a function of the atomic density $n$. The upper panel corresponds to a total impurity-BEC angular momentum $L=2$, whereas the lower panel corresponds to $L=1$. From [241].

Figure 14

The single-particle densities of a Tonk-Girardeau gas of 20 particles in the presence of a central ion for ${{}^{135}\mathrm{Ba}}^{+}+{}^{87}\mathrm{Rb}$ and ${{}^{174}\mathrm{Yb}}^{+}+{}^{87}\mathrm{Rb}$ systems with a typical trapping frequency $\omega =70\mathrm{Hz}$ (thick black line). The short-range phases in each case are chosen to be ${\varphi }_e=\pi /4$ and ${\varphi }_o=-\pi /4$. The thin red lines in the plots represent the result of a pseudopotential approximation for the ion, using a large value for the scattering length. From [119].

Figure 15

Upper panels: Normalized atomic density profiles for different values of the atom-atom interaction strength $g_{1\mathrm{D}}/E^{\star }{R}^{\star }=10$, 40, and 160 (solid, dashed, and dotted lines) with the static ion located in the trap center. In addition, the density profile of the Tonk-Girardeau (TG) gas as a gray shaded area is shown. Note that we show only the densities along the positive semiaxis, because of the symmetry of the ground state. Middle panels: Free expansion in the waveguide for the TG gas in a harmonic trap (i.e., without ion). Bottom panels: Free expansion in the waveguide for the TG gas after the sudden removal of the ion at time $t=0$. In all panels the units of the length and energy scales are calculated with respect to the atom mass, namely, $R^{\star }=\sqrt{\alpha e^2{m}_{\mathrm{a}}/{\hslash }^2}$ and $E^{\star }={\hslash }^2/2m_{\mathrm{a}}(R^{\star }{)}^2$, and the trap length is $R^{\star }/\sqrt{8}$. Adapted from [327].

Figure 16

Time evolution of the density for $N=10$ atoms after the sudden creation of a tightly trapped ion in the center of the atom trap ($\hslash /E^{\star }=0.39\mathrm{ms}$ for ${}^{87}\mathrm{Rb}$ atoms). Adapted from [325].

Figure 17

The $s$-wave scattering length as a function of the magnetic field for collisions of a Cr atom with a (a) ${\mathrm{Sr}}^{+}$, (b) ${\mathrm{Ba}}^{+}$, and (c) ${\mathrm{Yb}}^{+}$ ion, obtained from coupled-channel calculations. Typical scattering lengths of $0.1R^{\star }$, $R^{\star }$, and $2R^{\star }$ for the $X^6{\mathrm{\Sigma }}^{+}$ electronic state and $-0.1R^{\star }$, $-R^{\star }$, and $-2R^{\star }$ for the $a^8{\mathrm{\Sigma }}^{+}$ electronic state are assumed, respectively. Adapted from [365].

Figure 18

Magnetic field dependence of the elastic and reactive (radiative charge exchange and association) collision rate constants for the ${\mathrm{Ca}}^{+}+\mathrm{Na}$ system calculated with MQDT and close-coupled model with the same scattering lengths at collision energy $E/k_B=1\mu \mathrm{K}$. Scattering lengths of $R^{\star }$ and $-R^{\star }$ are assumed for the $X^1{\mathrm{\Sigma }}^{+}$ and $a^1{\mathrm{\Sigma }}^{+}$ electronic states, respectively. From [167].

Figure 19

Energy spectrum of the relative motion for an atom and an ion confined in identical spherically symmetric harmonic traps with frequency $\omega$ vs the distance $d$ between traps, calculated for $\phi =-\pi /4$, and $R^{\star }=3.48a_{\mathrm{HO}}$, where $a_{\mathrm{HO}}=\sqrt{\hslash /\mu \omega }$. The blue dashed line shows the approximate dependence of the molecular state energy on $d$, while symbols at $d=0$ depict the angular momentum of the molecular states. From [165].

Figure 20

Energy spectrum for an atom and an ion confined in harmonic traps with ${\omega }_{\mathrm{i}}=5.5{\omega }_{\mathrm{a}}$ as a function of the distance $d$. Calculations are performed for ${\phi }_{\mathrm{e}}=-\pi /4$, ${\phi }_{\mathrm{o}}=\pi /4$, and $l_{\mathrm{a}}=0.9R^{\star }$. From [165].

Figure 21

Even and odd short-range phases ${\phi }_{\mathrm{e}}$ and ${\phi }_{\mathrm{o}}$ calculated for $a_{\perp }=R^{\star }$. Numerical results (solid red lines) are compared with predictions of the model replacing the ion-atom interaction with the energy-dependent pseudopotential. Adapted from [165].

Figure 22

Ratio $a_{\perp }/a_{3\mathrm{D}}(E_{\parallel })$ calculated at the location of the ion-atom CIR position as a function of the longitudinal collision energy $E_{\parallel }$ for three different values of the ratio $R^{\star }/a_{\perp }$ for the ion-atom pair ${{}^{171}\mathrm{Yb}}^{+}+{}^6\mathrm{Li}$. The circles represent the calculated values of $a_{\perp }/a_{3\mathrm{D}}(k)$ via the integration of Eq. (63) with the boundary condition (64). The solid curves correspond to $a_{\perp }/a_{3\mathrm{D}}(k)=1.4603-0.6531(m_{\mathrm{a}}/\mu )(E_{\parallel }/\hslash {\omega }_{\perp })$ at $R^{\star }/a_{\perp }=0.025$, where the effective-range approximation for the energy-dependent scattering length $a_{3\mathrm{D}}(k)$ has been applied, and Eq. (65) at higher values of $R^{\star }/a_{\perp }$. From [238].

Figure 23

Ratio $a_{\perp }/a_{3\mathrm{D}}$ in which the CIR occurs as a function of $R^{\star }/a_{\perp }$ calculated for different ion-atom pairs in the zero-energy limit $E_{\parallel }/E^{\star }={10}^{-6}$. Shaded areas indicate the frequency range ${\omega }_{\perp }=2\pi \times (10–100)\mathrm{kHz}$ of the atomic transverse trap for each of the ion-atom pairs. Since $R^{\star }$ relies on the reduced ion-atom mass as well as the trap width $a_{\perp }$, the frequency ranges (i.e., the shaded areas) depend on the specific pair, too. Adapted from [238].

Figure 24

Schematic drawing of the experimental setup used by [302]. The upper part shows the atom chamber, whereas the lower part presents the ion chamber. ${}^6\mathrm{Li}$ atoms are supplied from the right side of the atom chamber and are decelerated by a Zeeman slower. First, ${}^6\mathrm{Li}$ atoms are trapped by a conventional MOT, and then, they are trapped by an optical dipole trap at the center of the atom chamber. The atoms are transported to the position of the ion trap by moving one lens placed on a translation stage. ${\mathrm{Ca}}^{+}$ ions are trapped at the center of the ion trap electrode placed inside the ion chamber. The 397-nm cooling laser for the ions is incident along both the radial and the axial directions of the ion trap, and the 866-nm repump laser is incident along the axial direction. Fluorescence of the ions is collected by an objective lens and detected by a photomultiplier tube and an electron multiplying charge coupled device (EMCCD) camera. The energy-level diagram of the ${{}^{40}\mathrm{Ca}}^{+}$ ion is shown in the inset. From [302].

Figure 25

Normalized equilibrium energy distributions for different mass ratios $\xi =m_{\mathrm{a}}/m_{\mathrm{i}}$ in a Paul trap. The buffer-gas cloud distribution is given by a Gaussian of size ${\sigma }_{\mathrm{a}}=R_0/100$ and temperature of $T_{\mathrm{a}}=200\mu \mathrm{K}$. Also shown is the energy distribution in the Boltzmann regime (red curve) and the energy distribution for $\xi =34$ (purple curve), according to the expressions in Table 5. The inset compares the exponents $\kappa$ of the power law in the energy distribution (as defined in Table 5), for different models. The condition $\kappa =-2$ separates the regimes of stable from unstable ion motion. From [157].

Figure 26

Carrier Rabi spectroscopy between the $S_{1/2}$ ground state and the metastable $D_{5/2}$ state in a ${\mathrm{Sr}}^{+}$ ion. (a)–(f) Each graph corresponds to a different interaction time between the ion and ultracold Rb atoms (0, 0.5, 2, 3.5, 5, 6.5 ms, respectively). The solid lines indicate a fit using a Tsallis distribution, while the dashed lines show the fit of the data to a thermal distribution. (g) The ions temperature (filled colored circles) as a function of the interaction time (top $x$ axis) and the average number of Langevin collisions (bottom $x$ axis). Open circles come from a simulation that takes into account the polarization potential. Black dots are simulation results taking into account only hard-sphere collisions. (h) Ion power-law parameter $n$, which measures to what thermal extent a distribution can be used. The ion energy distribution starts with $n\gg 1$, consistent with a Maxwell-Boltzman distribution, and converges to $n=4.0(2)$ after $\sim 10$ collisions. For $n>10$, thermal and Tsallis distributions are almost indistinguishable. From [236].

Figure 27

Trajectories of an ion $r_{\mathrm{i}}(t)$ and an atom $r_{\mathrm{a}}(t)$ during a classical one-dimensional low-energy collision. The atom of mass $m_{\mathrm{a}}$ approaches the ion of mass $m_{\mathrm{i}}=2m_{\mathrm{a}}$ held in the center of a rf trap with secular frequency $\omega =2\pi /T$ and Mathieu parameter $q=0.1$, leading to a hard-sphere collision at $r_{\mathrm{i}}=r_{\mathrm{a}}=r_{\mathrm{c}}$, $t=0$ and rf phase $\varphi$. For $\varphi =\pi /2$ (dotted lines), the trap field adds energy to the system, causing heating. For $\varphi =3\pi /2$ (solid lines), the rf field removes energy, binding the atom to the ion and causing further collisions at various rf phases until enough energy is accumulated to eject the atom. From [48].

Figure 28

(a) The effect of micromotion on the quantum dynamics of an atomic double-well system interacting with a single ion shown. (b) The quasienergy spectrum is shown as a function of interwell separation $d$. The spectrum shows the symmetric and antisymmetric ground states of the double-well system, and its energy difference is proportional to the atomic tunneling rate. It can be seen that for large $d$, there is almost no tunneling, whereas as $d$ gets smaller tunneling (i.e., energy splitting) occurs. Since the atom needs to pass the ion during tunneling, the rate depends on the internal state of the ion and can thus be used to entangle the ion and atom ([103]). Taking micromotion into account leads to many additional quasienergies that run almost vertically up as a function of $d$. As $d$ gets smaller, the coupling between the two ground states and the high energy Floquet states get larger, until for very small $d$, the two ground states disintegrate which signals the breakdown of the double-well system. For the calculation, ${}^7\mathrm{Li}$ and ${{}^{171}\mathrm{Yb}}^{+}$ were assumed with atomic trap frequency ${\omega }_{\mathrm{a}}=2\pi \times 98\mathrm{kHz}$ and ${\mathrm{\Omega }}_{\mathrm{rf}}=2\pi \times 967\mathrm{kHz}$ with $q=0.4$ and $a=0$ for the ion such that ${\omega }_{\mathrm{i}}=2\pi \times 137\mathrm{kHz}$. For the calculation the Floquet classes $j=-2,\dots ,2$ were taken into account. Adapted from [178].

Figure 29

The cooling rate in one spatial direction for an ${{}^{138}\mathrm{Ba}}^{+}$ immersed in ${}^{87}\mathrm{Rb}$ gas with density ${10}^{12}{\mathrm{cm}}^{-3}$ and temperature of $200\mu \mathrm{K}$ calculated using a master equation formalism as a function of Paul trap parameters $a$ and $q$ with $\mathrm{\Omega }=2\pi \times 1\mathrm{MHz}$. Adapted from [201].

Figure 30

Schematic of a linear rf trap with $n=4$. The atoms are indicated in red; the ion is shown in blue. The eight radio-frequency electrodes are indicated by the gray cylinders. The two insets on the right illustrate two elastic collisions, one close to the trap center and the other close to the ion’s turning point. The trap radius $r_0$ is indicated by the arrow. From [157].

Figure 31

Single Rydberg electron immersed in a BEC: (a) comparison of the size of the Rydberg states and BEC cloud, and (b) corresponding interaction potentials. From [8].

Figure 32

Adiabatic potentials for a ground-state atom and an ion (solid black), for a dressed atom with $\mathrm{\Omega }=2\pi \times 10\mathrm{MHz}$ and $\mathrm{\Delta }=2\pi \times 1\mathrm{GHz}$ (red dash-dotted) and $2\pi \times 0.4\mathrm{GHz}$ (blue dashed) assuming coupling to the $|30S_{1/2}⟩$ state of lithium. From [330].

Figure 33

Charge-exchange rate coefficient measured for the ${\mathrm{Yb}}^{+}+\mathrm{Yb}$ system as a function of average collision energy. Circles and green and blue diamonds represent ${{}^{172}\mathrm{Yb}}^{+}+{}^{174}\mathrm{Yb}$, ${{}^{172}\mathrm{Yb}}^{+}+{}^{171}\mathrm{Yb}$, and ${{}^{174}\mathrm{Yb}}^{+}+{}^{172}\mathrm{Yb}$, respectively. The solid line gives the Langevin rate, while the dashed line accounts for the contribution of the electronically excited Yb present due to MOT lasers. The black arrow indicates the ionization isotope shift between ${}^{174}\mathrm{Yb}$ and ${}^{172}\mathrm{Yb}$. From [123].

Figure 34

(a) Atom loss from the magnetic trap as a function of the ion energy in the ${{}^{172}\mathrm{Yb}}^{+}+\mathrm{Rb}$ experiment. The solid line results from the numerical simulation based on the full collision cross section, while the dashed line gives the predictions of the Langevin model. Points are experimental results. (b) Temperature increase of the atomic cloud vs mean ion energy. From [405].

Figure 35

(a) False-color fluorescence images of a ${\mathrm{Ba}}^{+}$ Coulomb crystal (upper panels) and their molecular dynamics simulation (lower panels) before (left) and after (right) reactions with Rb atoms. Product ions, dark in the experiment, are depicted (blue: ${{}^{138}\mathrm{Ba}}^{+}$, green: lighter isotopes of ${\mathrm{Ba}}^{+}$, yellow: ${{}^{87}\mathrm{Rb}}^{+}$, red: ${\mathrm{BaRb}}^{+}$). (b) Resonance excitation mass spectra before and after the reactions. From [132].

Figure 36

Hyperfine spin relaxation in ${{}^{171}\mathrm{Yb}}^{+}$ interacting with ultracold Rb atoms. (a) The probability $p_{\mathrm{Bright}}$ after preparation of the ion in the $|F=1,m_F=0{⟩}_i$ (full symbols) or the $|F=0,m_F=0{⟩}_i$ (open symbols) state vs the interaction time for atoms in the $|F=1,m_F=-1{⟩}_{\mathrm{a}}$ state. Error bars denote 1 standard deviation uncertainty intervals resulting from a total of 5000 measurements. The fit values at $t=0$ are limited by detection errors. (b) Similar data for collisions with ${}^{87}\mathrm{Rb}$ atoms in the $|F=2,m_F=2{⟩}_{\mathrm{a}}$ state and a total of 19 000 measurements. (c)–(e) Zeeman-resolved detection within the $F=1$ manifold after preparation in $|F=1,m_F=0{⟩}_i$ (atoms in $|F=2,m_F=2{⟩}_{\mathrm{a}}$) with 1200 measurements per Zeeman state. The measurements are performed by applying resonant $\pi$ pulses, exchanging the population of the dark state $|F=0,m_F=0{⟩}_i$ with $|F=1,m_F=-1{⟩}_i$, $|F=1,m_F=0{⟩}_i$ or $|F=1,m_F=1{⟩}_i$ immediately before the detection of the probability of the ion being in the dark state $p_{\mathrm{Dark}}$. From [288].

Figure 37

Dynamics of the decay of the atomic cloud under the influence of the ion observed in the ${\mathrm{Rb}}^{+}+\mathrm{Rb}$ experiment. (a) Remaining atom number as a function of time for ion energy $E\approx 35\mathrm{mK}$ and initial atomic density $n_{\mathrm{at}}={10}^{11}{\mathrm{cm}}^{-3}$. (b) The same but for $E\approx 0.5\mathrm{mK}$ and $n_{\mathrm{at}}\approx 1.1\times {10}^{12}{\mathrm{cm}}^{-3}$. (c) Histograms containing the data from (b). From [144].

Figure 38

Experimentally tuning the ion trapping allows for the observation of a Peierls-like phase transition when atoms are confined in an effective 1D dipole trap at a transverse separation ${\mathrm{\Delta }}_r$ from the ion string. The discrete translational symmetry of the linear chain (a) is spontaneously broken toward a zigzag pattern of the ions (c) and a band gap opens at $k=\pi /2d$ for the atoms. (b) At different atomic fillings $n$ (per ion) and below a critical temperature $T_c$, a phase transition occurs. From [23].

Figure 39

Sketch of the atomic bosonic Josephson junction controlled by the internal spin state of a tightly trapped ion (large blue sphere).

Figure 40

Concept for quantum computation with atoms and ions: Atoms are prepared in an optical lattice in a Mott insulator phase. A movable ion is trapped in a radio-frequency trap. From [75].

Figure 41

Energy spectrum for the ${{}^{135}\mathrm{Ba}}^{+}$ ion and the ${}^{87}\mathrm{Rb}$ atom with the singlet and triplet scattering lengths: $a_s=0.90R^{\star }$ and $a_t=0.95R^{\star }$, respectively. The left panel shows the complete diagram for the $|11⟩$ channel, while the right panel presents the close-up around $d=0.6R^{\star }$ for all the channels of the computational basis. Small differences between channels that can be observed are the basis for realizing the ion-atom phase gate. From [75].

Figure 42

The number of remaining ${}^{87}\mathrm{Rb}$ atoms as a function of the ${{}^{87}\mathrm{Rb}}^{+}$ ion position relatively to the trap center: (a) thermal cloud, (b) partially condensed cloud, and (c) an almost pure condensate. The solid lines correspond to fits, where the ion energy and atom temperature have been used as free parameters. From [312].

Figure 43

Fourier transform of the temporal single-particle Green’s function $⟨{\widehat{c}}_{\sigma ,j}^{†}(t){\widehat{c}}_{{\sigma }_j}(0)⟩$ for an $s$-wave and a $d$-wave superconductor. From [196].

Figure 44

Left panel: The Raman level scheme used for coupling an atom and an ion with initial energy $E$ such that a stable ion-atom molecule can be prepared with the aim of inferring the single-particle energy distribution of a Fermi gas. For further details see the text. Right panel: Photoassociation rate for an ion in an atomic Fermi gas as a function of the detuning $\tilde{\delta }={\delta }_2-\eta$ for different regimes of the gas: Free fermions at $T=0$ (dash-dotted line), BCS regime at $T=0$ for $k_F^0{a}_s^{\mathrm{aa}}=-0.5$ (energy gap ${\mathrm{\Delta }}_0=0.047{\epsilon }_F$) with $k_F^0$ being the Fermi wave number and $a_s^{\mathrm{aa}}$ the scattering length (dashed line), the unitarity limit at $T=0$ (solid line) and at $T=T_c$ (dotted line). The major peak is due to the breaking of fermionic pairs of atoms, while the minor one is attributed to thermally excited atoms. The inset shows the photoassociation rate density $G(\epsilon ,E)$ (at a fixed initial ion energy) for the transition $|0⟩\to |2⟩$ for ${\delta }_2=2.5{\delta }_1$. The low peak is determined by the width ${\gamma }_2$ and can be controlled by the laser parameters (see text), whereas the maximal peak occurs when $E=\tilde{\delta }$. From [333].

Figure 45

Radiative association rate constant vs binding energy of the final rovibrational level in the $X^1{\mathrm{\Sigma }}^{+}$ electronic ground state for an ${\mathrm{Yb}}^{+}$ ion colliding with a Li atom in the $A^1{\mathrm{\Sigma }}^{+}$ electronic state plotted at a temperature corresponding to (left panel) 100 nK and (right panel) $10\mu \mathrm{K}$. Rates to states with different rotational angular momentum $l$ are presented using different symbols and colors. From [367].

Figure 46

Transition electric dipole moments between vibrational levels of the $X^2{\mathrm{\Sigma }}^{+}$ ground and $A^2{\mathrm{\Sigma }}^{+}$ excited electronic states of the ${\mathrm{LiRb}}^{+}$ molecular ion. Adapted from [368].

Figure 47

Photoassociation rate constants into rovibrational levels of the $X^1{\mathrm{\Sigma }}^{+}$ electronic ground state for collisions of ${{}^{174}\mathrm{Yb}}^{+}$ ions with ${}^6\mathrm{Li}$ atoms in the $A^1{\mathrm{\Sigma }}^{+}$ state (upper panel) and into rovibrational levels of the $c^3{\mathrm{\Sigma }}^{+}$ and $b^3\mathrm{\Pi }$ excited electronic states for collisions in the $a^3{\mathrm{\Sigma }}^{+}$ state (lower panel). From [367].

Figure 48

Schematic representation of the optical pathways to produce ${\mathrm{NaCa}}^{+}$ molecular ions in the electronic ground state. From [92].

Figure 49

Total electronic energies of ${\mathrm{N}}_2^{+}+\mathrm{Rb}$ reactant (entrance) and ${\mathrm{Rb}}^{+}+{\mathrm{N}}_2$ product channels. The channels are characterized by the electronic states of the species as indicated. Charge transfer is most efficient in the case of a near resonance between the entrance and product channels. The product channels closest in energy to the ${\mathrm{N}}_2^{+}+\mathrm{Rb}$ ($5s$) and ($5p$) entrance channels are indicated in green and red, respectively. From [135].