Rotational cooling of heteronuclear molecular ions with ${}^1\Sigma$, ${}^2\Sigma$, ${}^3\Sigma$, and ${}^2\Pi$ electronic ground states

The translational motion of molecular ions can be effectively cooled sympathetically to translational temperatures below $100\mathrm{mK}$ in ion traps through Coulomb interactions with laser-cooled atomic ions. The rovibrational degrees of freedom, however, are expected to be largely unaffected during translational cooling. We have previously proposed schemes for cooling of the internal degrees of freedom of such translationally cold but internally hot heteronuclear diatomic ions in the simplest case of ${}^1\Sigma$ electronic ground-state molecules. Here we present a significant simplification of these schemes and make a generalization to the most frequently encountered electronic ground states of heteronuclear molecular ions: ${}^1\Sigma$, ${}^2\Sigma$, ${}^3\Sigma$, and ${}^2\Pi$. The schemes are relying on one or two laser-driven transitions with the possible inclusion of a tailored incoherent far-infrared radiation field.

Figure 1

Rovibrational states of interest in the cooling schemes for ${}^1\Sigma$ states. The cooling concept involves transitions between rovibrational states driven by Raman pulses [solid lines, double arrows in (a) and (b)] via an excited electronic state or direct laser pumping [solid line, single arrow in (c)] and subsequent spontaneous decays (dashed lines).

Figure 2

Top: Born-Oppenheimer electronic potential energy curves of ${\mathrm{MgH}}^{+}\left(X{}^1\Sigma \right)$ calculated by GAUSSIAN in a $6–311++\mathrm{G}$ basis set [31] using Hartree-Fock (HF) theory, Møller-Plesset $n\mathrm{th}$-order perturbation theory $\left(\mathrm{MPn}\right)$, coupled-cluster theories with singles and double excitations (CCD), and single, double, and triple excitations (CCSDT). See Refs. 32, 33, 34, 35, 36 for descriptions of the methods. The curves for the MP4, CCD, and CCSDT calculations are in good agreement close to the equilibrium position, $1.65\mathrm{\AA }$, indicating that these methods give an accurate description of the problem. Bottom: corresponding dipole moment functions ${\mathit{D}}_e^{mol}\left(R\right)$ of Eq. (B3) pointed along the internuclear axis, of ${\mathrm{MgH}}^{+}$, calculated with GAUSSIAN. The MPn and coupled-cluster theories largely agree around the equilibrium distance, although not as well as for the potential curve due to the dependence on electronic wave functions rather than eigenenergies. The classical turning points for the vibrational ground state are marked on the common abscissa at 1.5 and $1.8\mathrm{\AA }$. The result of the MP2 calculation cannot be discriminated from the MP4 result at the internuclear distances of interest.

Figure 3

Population in the rovibrational ground state of ${\mathrm{MgH}}^{+}\left(X{}^1\Sigma \right)$ vs cooling time for the Raman (a) and direct (b) schemes presented in Figs. 1b, 1c with the optimized distribution of incoherent radiation (solid line), quartz-filtered distribution (dashed line), and no incoherent source (dash-dotted line). The same simulation using the scheme of Fig. 1a is depicted for comparison (dot).

Figure 4

Population distribution in ${\mathrm{MgH}}^{+}\left(X{}^1\Sigma \right)$ after $60\mathrm{s}$ of cooling using the Raman (a) and direct (b) schemes with the optimized energy distribution of the incoherent source (black), the quartz-filtered energy distribution (gray), compared to the initial population distribution at $300\mathrm{K}$ (white). The ground-state population after cooling with the optimized incoherent field corresponds to that of a thermal ensemble at $\sim 7\mathrm{K}$.

Figure 5

Cooling scheme for ${}^2\Pi$ states. Each of the two possible values of $\Omega$ results in a series of energy levels and it is necessary to cool the ${}^2\Pi _{1∕2}$ and ${}^2\Pi _{3∕2}$ separately if both are populated. Population is pumped from the first excited rotational state in the $\nu =0$ vibrational ground state to the rovibrational ground state by a laser-induced vibrational transition and subsequent spontaneous decays. All the involved transitions are dipole allowed [cf. Eq. (12) ]. Solid lines indicate laser-pumped transitions while dashed lines indicate spontaneous decay paths. The $\Lambda$ doubling is not shown in the figure.

Figure 6

Cooling efficiency of ${\mathrm{FH}}^{+}\left(X{}^2\Pi \right)$ in the two $\Omega$ substates. In the dipole approximation the two substates are uncoupled if pure Hunds case (a) applies. The cooling scheme will therefore be significantly simplified if one can design the experiment such that only the lowest $\Omega =\frac{3}{2}$ state is populated in the cooling scheme.

Figure 7

Cooling efficiency as function of cooling time for the two $\Omega$ substates of the ${}^2\Pi$ electronic ground state of ${\mathrm{FH}}^{+}$. The cooling is seen to reach steady state after $\lesssim 10\mathrm{s}$ without the inclusion of an incoherent source, largely due to the large permanent dipole moment of ${\mathrm{FH}}^{+}$.

Figure 8

Cooling scheme for ${}^2\Sigma$ states. Due to the spin-rotation coupling, each rotational quantum state $N$ split into two sublevels with $J=\mid N+\frac{1}{2}\mid ,\mid N-\frac{1}{2}\mid$. The dipole-allowed vibrational transitions are indicated on the figure using solid lines for laser-pumped transitions and dashed lines for subsequent spontaneous decay paths.

Figure 9

Cooling scheme for ${}^3\Sigma$ states. Due to spin-spin and spin-rotation coupling, each rotational quantum state $N$ split into sublevels with $J=\mid N+1\mid ,N,\mid N-1\mid$. The dipole-allowed vibrational transitions are included in the figure with solid lines to indicate laser-pumped transitions and dashed lines to indicate spontaneous decay paths.

Figure 10

Population in the lowest rotational state of ${\mathrm{BH}}^{+}\left(X{}^2\Sigma \right)$ as function of cooling time. Simulations are made using the scheme of Fig. 8 with BBR only (dashed line) and with the inclusion of the field from an incoherent source addressing the $N=1\to N=2$ and $N=2\to N=3$ transitions (solid line). We see that a significant improvement is obtainable using the incoherent source. In line with our experience from ${\mathrm{MgH}}^{+}$ there is only a couple of percent loss of cooling efficiency when using a softer low-frequency pass filter, for example, letting the broadband incoherent radiation extend to include transitions up to and including $N=4\to N=5$.

Figure 11

Population in the lowest rotational states of ${\mathrm{BH}}^{+}\left(X{}^2\Sigma \right)$ after cooling in $120\mathrm{s}$ using the incoherent radiation from a lamp addressing the $N=1\to N=2$ and $N=2\to N=3$ transition (black columns) and in BBR only (grey columns). The initial $300\mathrm{K}$ Boltzmann distribution is included for comparison (unfilled columns). We note that slightly better cooling efficiency should can be obtained using longer cooling times (cf. Fig. 10). This is, however, impractical and the obtainable improvements would be rather small. The substructure of the rotational levels is included in the simulation but omitted in the figure.

Figure 12

Top: Born-Oppenheimer potential curves for $X{}^3\Sigma$ ${\mathrm{OH}}^{+}$ calculated by GAUSSIAN with various theoretical models compared to the calculation of Ref. [48]. All our calculations are done in a $6-311++\mathrm{G}$ basis set except the solid black line which is made in the generally more accurate aug-cc-pVTZ basis [31]. The curves agree close to the equilibrium, $1.03\mathrm{\AA }$, for the MP4 (fourth-order Møller-Plesset perturbation theory) and coupled-cluster approaches (CCD, CCSDT) indicating an accurate level of theory. The different methods are described in Refs. 32, 33, 34, 35, 36. Bottom: dipole moment function calculated with GAUSSIAN using similar levels of theory and basis sets. The agreement between the calculations is reasonable and the effect of using the larger basis set for the CCSDT theory is not visible on the given scale, but our results show some discrepancy with the results of Ref. 48. This small discrepancy, however, has very little effect on the cooling scheme. The classical turning points for the vibrational ground state are marked on the common abscissa at 0.95 and $1.15\mathrm{\AA }$. The dipole moment functions are given in center-of-mass coordinates.

Figure 13

Population in the lowest rotational state of ${\mathrm{OH}}^{+}\left(X{}^3\Sigma \right)$ as function of cooling time. Simulations are made using the scheme of Fig. 8 with BBR only (dashed line) and with the inclusion of the field from an incoherent source addressing the $N=1\to N=2$ and $N=2\to N=3$ transitions (solid line). We see that a significant improvement is obtainable using the incoherent source. In line with our experience from ${\mathrm{MgH}}^{+}$and ${\mathrm{BH}}^{+}$ there is only a marginal loss of cooling efficiency when using a softer low-frequency pass filter.

Figure 14

Population in the lowest rotational states of ${\mathrm{OH}}^{+}\left(X{}^3\Sigma \right)$ after cooling in $10\mathrm{s}$ using the incoherent radiation from a lamp addressing the $N=1\to N=2$ and $N=2\to N=3$ transition (black columns) and in BBR only (grey columns). The initial $300\mathrm{K}$ Boltzmann distribution is included for comparison (unfilled columns). The spin substructure of the rotational levels is included in the simulation but omitted on the figure.