Vibronic transitions in the alkali-metal (Li, Na, K, Rb) – alkaline-earth-metal (Ca, Sr) series: A systematic analysis of de-excitation mechanisms based on the graphical mapping of Frank-Condon integrals

Research on ultracold molecules has seen a growing interest recently in the context of high-resolution spectroscopy and quantum computation. After forming weakly bound molecules from atoms in cold collisions, the preparation of molecules in low vibrational levels of the ground state is experimentally challenging, and typically achieved by population transfer using excited electronic states. Accurate potential energy surfaces are needed for a correct description of processes such as the coherent de-excitation from the highest and therefore weakly bound vibrational levels in the electronic ground state via couplings to electronically excited states. This paper is dedicated to the vibrational analysis of potentially relevant electronically excited states in the alkali-metal (Li, Na, K, Rb)– alkaline-earth metal (Ca,Sr) diatomic series. Graphical maps of Frank-Condon overlap integrals are presented for all molecules of the group. By comparison to overlap graphics produced for idealized potential surfaces, we judge the usability of the selected states for future experiments on laser-enhanced molecular formation from mixtures of quantum degenerate gases.

Figure 1

Potential energy curves for NaSr obtained by different methods.

Figure 2

Spectroscopic regions with vibronic states for the different molecules. The zero axis corresponds to the dissociation limit of the ground state. The marked region corresponds to energies that can be addressed by a Ti:sapphire laser.

Figure 3

PECs for the alkali-metal–(Li, Na, K, Rb) Ca series as determined by MRCI up to 20 000 ${\mathrm{cm}}^{-1}$.

Figure 4

PECs for the alkali-metal (Li, Na, K, Rb)–Sr series as determined by MRCI up to 20000 ${\mathrm{cm}}^{-1}$.

Figure 5

Potential energy curves and transition dipole moments for KSr and RbSr including spin-orbit interaction.

Figure 6

Franck-Condon factors calculated from Morse potentials. Varying of the parameters for the excited state shows possible configurations for efficient transfer schemes.

Figure 7

Depiction of the nuclear wave functions involved in the transfer schemes for the cases of a tight bound excited state (case b) and a loosely bound excited state (case c).

Figure 8

Franck-Condon factors as a function of the ground-state vibrational number ${\nu }^{\prime \prime }$ and the excited state vibrational number ${\nu }^{\prime }$ for all four molecules (rows) and the three most relevant excited states $3{}^2\mathrm{\Sigma }^{+}, 1{}^2\mathrm{\Pi }$, and $2{}^2\mathrm{\Pi }$ in the doublet manifold (columns).

Figure 9

Franck-Condon factors as a function of the ground-state vibrational number ${\nu }^{\prime \prime }$ and the excited state vibrational number ${\nu }^{\prime }$ for all four molecules (rows) and two excited states ($1{}^4\mathrm{\Sigma }^{+}$ and $1{}^4\mathrm{\Pi }$) in the quartet manifold (columns).

Figure 10

Comparison of Einstein $A$ coefficient graphics obtained with different strategies for the inclusion spin-orbit coupling. (a) In Hund's case (a) the spin-orbit coupling is completely neglected. (b) Spin-orbit coupling is introduced via effective matrix elements as described in Sec. 2b. (c) For intermediate coupling we used the duo program and included spin-orbit coupling effects via the distance-dependent spin-orbit matrix elements obtained from ab initio calculations.

Figure 11

Einstein $A$ coefficients for the three most relevant excited states $3{}^2\mathrm{\Sigma }^{+}, 1{}^2\mathrm{\Pi }$, and $2{}^2\mathrm{\Pi }$ in the doublet manifold. Darker colors are used for the Einstein $A$ coefficients of transitions from the highest vibrational level of the ground state to an arbitrary vibrational level of the denoted excited state. The corresponding transition energies can be read from the $x$ axis. Brighter colors are used for the Einstein $A$ coefficients of transitions between an arbitrary vibrational level of the excited state into the lowest vibrational level of the ground states. To simplify the interpretation, we shifted their positions by the energy difference between the highest and lowest vibrational state of the ground state (see text).