Hyperfine-resolved optical spectroscopy of ultracold $^{87}$Rb$^{133}$Cs molecules: the $\mathrm{b}^3Π_0$ metastable state

TL;DR

This work achieves hyperfine-resolved optical spectroscopy of the spin-forbidden X$^1\Sigma^+$ to $\mathrm{b}^3\Pi_0$ transitions in ultracold $^{87}$Rb$^{133}$Cs, across vibrational and rotational manifolds under magnetic fields. By combining precise spectroscopy with an effective Hamiltonian that includes $0^+$/$0^-$ mixing and Zeeman/hyperfine interactions, the authors extract the excited-state rotational constant $B_0(0^+)$, hyperfine couplings $A_{\mathrm{Rb}}$, $A_{\mathrm{Cs}}$, the quadrupole coupling $(eqQ)_{\mathrm{Rb}}$, and the $0^+/0^-$ splitting $\Delta$, achieving sub-MHz agreement. They further quantify transition dipole moments via Rabi oscillations, measure excited-state lifetimes, and show that the $v'=0$ level decays predominantly back to $v''=0$ (probability >79%), supporting near-closed optical cycles. These results refine the spectroscopic model for RbCs and inform strategies for magic-wavelength trapping and direct imaging or cooling of ultracold molecules. Collectively, the findings enhance the control of molecular internal states and advance prospects for quantum simulation, precision measurement, and quantum-controlled chemistry with bialkali molecules.

Abstract

Using an ultracold gas of $^{87}$Rb$^{133}$Cs molecules, we perform hyperfine-resolved spectroscopy of transitions from the vibronic ground state to the lowest rovibrational states of the electronic state $\mathrm{b}^3Π_0$, as a function of magnetic field. These transitions are spin forbidden, resulting in narrow linewidths, and feature near-diagonal Franck-Condon factors. We develop a model of the hyperfine and Zeeman structure that includes coupling between the $0^+$ and $0^-$ components of $\mathrm{b}^3Π_0$. We fit the spectra to obtain rotational and hyperfine coupling constants. We measure transition dipole moments associated with specific transitions by directly observing Rabi oscillations as a function of a resonant laser pulse duration. Using resonant $π$ pulses, we prepare molecules in the electronically excited state and directly measure the spontaneous emission rate.

Figures (7)

• Figure 1: The relevant electronic potentials for RbCs. In the excited state, the potentials for the mixed $\mathrm{A}^1\Sigma$ and $\mathrm{b}^3\Pi_0$ states are shown by black lines, while those for $\mathrm{b}^3\Pi_1$ and $\mathrm{b}^3\Pi_2$ are shown in grey. The shaded areas indicate regions of the mixed states that have been investigated by (i) photoassociation in magneto-optical traps Kerman2004, (ii) Laser-induced fluorescence and Fourier transform spectroscopy in heat pipes Docenko2010Kruzins2014, (iii) absorption spectroscopy with ultracold molecules produced via magnetoassociation Debatin2011. The energy region studied in this work is shown in green, with the vertical green arrow labelled ‘probe’ indicating the $\mathrm{X}^1\Sigma^+$$\rightarrow$$\mathrm{b}^3\Pi_0$ transitions. The inset illustrates the rotational structure of the $\mathrm{X}^1\Sigma^+$, $v = 0$ ground state.
• Figure 2: Characterisation of the vibrational structure. The $\mathrm{b}^3\Pi_0$ electronic potential with the energies of $v' = 0, \,1, \, \textrm{and} \,2$ shown on the left. The green arrow indicates the probe laser driving transitions from $(1,6)_\mathrm{G}$ to the $(0,5)_\mathrm{E}$ state in each vibrational level. Spectroscopy on each of these transitions performed at 181.6 G is shown on the right, with the laser detuning ($f-f_0$) plotted relative to the centre frequency $f_0$ of each of the observed transitions. We attribute the sloped background in the spectra for $v' = 2$ to nearby hyperfine structure combined with imperfect polarisation of the probe laser. In the analysis of this measurement only, we fit the results with a linearly varying background.
• Figure 3: An overview of rotational and hyperfine structures of the transitions at 181.6 G for $v' = 0$. (a) The coarse rotational structure. The inset shows the level diagram of the transitions we drive. The colours of the vertical arrows match the corresponding data points. (b), (c) and (d) show the observed hyperfine structure. (b) The $\sigma^{-}$ transition from $(1,6)_\mathrm{G}$ to $(0,5)_\mathrm{E}$ (also shown in Fig. \ref{['fig:vib_spectra']}(d)). (c) $\pi$ and $\sigma^{\pm}$ transitions from the $(0,5)_\mathrm{G}$ state to $(1,M'_F)_\mathrm{E}$ states. (d) $\pi$ transitions from the $(1,6)_\mathrm{G}$ state to $(2,6)_\mathrm{E}$ states. (e) -- (k) show the zoomed-view of all transitions we observed in (c). All transitions in (e) -- (k) are marked with the character of the hyperfine state $(J',M'_F)_\mathrm{E}$. The x axis scale of (e) -- (k) are the same. For a clearer view of the transition profiles, we have shown the error bars only on (e), while all other data have similar error bars.
• Figure 4: Shift in frequency with magnetic field for the transition $(1,6)_\mathrm{G} \rightarrow (0,5)_\mathrm{E}$ transition between $44$ and $369$ G. To characterise the shift we fit the data (black markers in upper panels) with both linear (blue dashed line) and quadratic functions (red solid line). The residuals of both fit functions are shown in the bottom panel, where the blue and red markers correspond to the linear and quadratic fits, respectively. The red solid curve overlaid on the linear-fit residuals represents the difference between the quadratic and linear fitted values, indicating the quadratic nature of the shift.
• Figure 5: Comparison between experiment and theory for the transitions from $J"=0$ to $J'=1$. The upper panel shows the relative transition frequencies and the relative transition strengths extracted from the model (shown in Table \ref{['tab:rot_hf_transitions']}), while the lower panel shows the experimentally observed relative frequencies. The x-axis for the observed ones are calculated with relative to the lowest energy state with $M'_F = 5$. The y-axis for the experimental frequencies is arbitrary and shows the values of $M'_F$. Solid lines (in the upper panel) corresponding to upper states with $M'_F = 4,\, 5$ and $6$ are shown in green, red and blue, respectively, while the observed one are shown by the markers accordingly. The transitions that were initially unobserved in the experiment are shown as open green markers.