Optical Feshbach resonances and ground-state-molecule production in the RbHg system

We present the prospects for photoassociation, optical control of interspecies scattering lengths, and, finally, the production of ultracold absolute ground-state molecules in the Rb+Hg system. We use the state-of-the-art ab initio methods for the calculations of ground- [CCSD(T)] and excited-state (EOM-CCSD) potential curves. The RbHg system, thanks to the wide range of stable Hg bosonic isotopes, offers possibilities for mass tuning of ground-state interactions. The optical lengths describing the strengths of optical Feshbach resonances near the Rb transitions are favorable even at large laser detunings. Ground-state RbHg molecules can be produced with efficiencies ranging from about 20% for deeply bound to at least 50% for weakly bound states close to the dissociation limit. Finally, electronic transitions with favorable Franck-Condon factors can be found for the purposes of a STIRAP transfer of the weakly bound RbHg molecules to the absolute ground state using commercially available lasers.

Figure 1

Potential-energy curves near the ${}^{2}P+{}^{1}S$ and ${}^{2}S+{}^{1}S$ asymptotes of RbHg in Hund's case (a) representation calculated using state-of-the-art ab initio methods; see Sec. 2 for details.

Figure 2

Potential-energy curves near the ${}^{2}P+{}^{1}S$ and ${}^{2}S+{}^{1}S$ asymptotes of RbHg in the $L-S$ coupled Hund's case (c) representations calculated using Eq. (4) from the nonrelativistic ab initio potentials shown in Fig. 1 and the spin-orbit matrix elements shown in Fig. 3.

Figure 3

(a) Relevant spin-orbit matrix elements for Hund's case (a) potential curves correlating to the ${}^{2}P-{}^{1}S$ asymptote of RbHg; (b) and (c) transition dipole moments from excited states to the ground state in Hund's case (a) and (c) representations, respectively.

Figure 4

Example $s$-wave scattering lengths $a$ for collisions of Rb and Hg atoms in their respective atomic ground states, ${}^{2}S_{1/2}$ and ${}^{1}S_0$. Two scenarios are shown: one calculated with the unmodified $X{}^2\mathrm{\Sigma }$ ab initio potential (blue solid lines) and one where the potential was modified to increase its total WKB phase $\varphi$ by $\pi /2$, which corresponds to half a vibrational state (yellow dashed lines). In the former, the available RbHg isotopic combinations would span a wide range of positive and negative scattering lengths with magnitudes both large and small. In the latter case, however, most of the scattering lengths would be positive and of magnitude similar to the mean scattering length $\overline{a}\approx 57a_0$ (horizontal dotted gray line). The unusually slow variation of scattering lengths with the reduced mass $\mu$ is due to the shallowness of the RbHg $X{}^2\mathrm{\Sigma }$ potential curve (see Fig. 1) which only supports 44 bound states (for ${}^{87}\mathrm{Rb}$ and ${}^{202}\mathrm{Hg}$).

Figure 5

Example optical lengths $l_{\mathrm{opt}}$ of Feshbach resonances near the Rb atomic transition ${}^{2}S_{1/2}\to {}^{2}P_{1/2}$ at 795 nm as a function of excited bound-state energy $E_b$ shown for characteristic $s$-wave scattering lengths $a$, in order: a large positive scattering length of $204a_0$, one close to the mean scattering length ($\overline{a}\approx 57a_0$), one close to zero, and a large negative scattering length. We have marked some of the excited vibrational states $v_{\mathrm{PA}}^{\prime }$ supported by the (2) $j=1/2\left|\mathrm{\Omega }\right|=1/2$ potential (Fig. 2) with arrows in order to discuss the products of their optical decay; see Fig. 8 and Sec. 6 for details.

Figure 6

Thermally averaged photoassociation-induced Hg atom loss rates ${\tilde{K}}_{\mathrm{in}}{n}_{\mathrm{Rb}}$ near the 795 nm transition to the RbHg (2) $j=1/2 \left|\mathrm{\Omega }\right|=1/2$ state calculated using Eq. (16) for sample temperature of $T=100\mu \text{K}$, laser intensity $I=500\mathrm{W}/{\mathrm{cm}}^2$, and the number density of the Rb cloud of $n_{\mathrm{Rb}}={10}^9{\mathrm{cm}}^{-3}$. For many photoassociation lines the two-body loss rates reach $1{\mathrm{s}}^{-1}$, which is more than the one-body losses of the Hg MOT reported in Ref. [33]. This will enable direct detection of heteronuclear photoassociation by monitoring the steady-state populations in a two-species MOT much like in the previous RbYb experiment [24].

Figure 7

Effective Franck-Condon factors calculated for transitions between vibrational states $v^{\prime }$ of appropriate excited electronic potentials near the ${}^{2}P+{}^{1}S$ asymptote and vibrational states $v$ in the $X{}^{2}S+{}^{1}S$ ground state of the RbHg molecule. The internuclear distance variability of the transition dipole moment was taken into account; see Eq. (19). It is interesting to note that the Franck-Condon factors for the $\left(3\right)j=3/2,\left|\mathrm{\Omega }\right|=1/2\to \left(1\right)j=1/2,\left|\mathrm{\Omega }\right|=1/2$ transition are unusually diagonal for the lowest 20 vibrational states—which may be the result of both curves having similar equilibrium distances $R_e$.

Figure 8

Effective Franck-Condon factors for transitions between selected (2) $j=1/2, \left|\mathrm{\Omega }\right|=1/2$ vibrational states $v^{\prime }$ and ground vibrational states $v$. For many of the excited-state vibrational levels $v^{\prime }$ the product states are dominated by one ground energy level $v$. For example, photoassociation of Rb and Hg atoms to the $v_{\mathrm{PA}}^{\prime }=125$ state could, through spontaneous emission, produce ground-state RbHg molecules in the $v=37,l=0$ rovibrational state at an efficiency of over 25%.

Figure 9

Products of Franck-Condon factors describing the transition probabilities in STIRAP processes between a starting ground vibrational level $v$ of energy $E_b$, an intermediate vibrational level $v^{\prime }$ in the (2) $j=1/2, \left|\mathrm{\Omega }\right|=1/2$ excited electronic state and the rovibrational ground state.