Observation and analysis of resonant coupling between nearly degenerate levels of the $2{}^1{\Sigma }_g^{+}$ and $1{}^1{\Pi }_g$ states of ultracold ${}^{85}$Rb${}_2$

We report on the anomalously high line strength of a single rotational level in the ultracold photoassociation of two ${}^{85}$Rb atoms to form ${}^{85}$Rb${}_2$. The $v^{\prime }=111$, $J^{\prime }=5$ level belongs to the Hund's case (c) $2\left(0_g^{+}\right)$ state, which correlates to the Hund's case (a) $2{}^1{\Sigma }_g^{+}$ state. Its strength is caused by coupling with a very near-resonant long-range state. The long-range component is the energetically degenerate $v^{\prime }=155$, $J^{\prime }=5$ level of the case (c) $2\left(1_g\right)$ state, correlating to the case (a) $1{}^1{\Pi }_g$ state. The line strength is enhanced by an order of magnitude through this coupling, relative to nearby vibrational levels and even to nearby rotational levels of the same vibrational level. This enhancement is in addition to the enhancement seen in all $J^{\prime }=3$ and 5 levels of the $2\left(0_g^{+}\right)$ state due to an $l=4$ shape resonance in the $a{}^3{\Sigma }_u^{+}$ state continuum, which alters the distribution of levels formed by photoassociation.

Figure 1

Potential-energy curves for the $2\left(0_g^{+}\right)$ and $2\left(1_g\right)$ states as calculated by [16]. The levels of the $2\left(0_g^{+}\right)$ state observed in [17] are shown, as is the $v^{\prime }=155$ level of the $2\left(1_g\right)$ state that is energetically nearly degenerate with $v^{\prime }=111$.

Figure 2

Three REMPI spectra of the same region, using different PA levels to form the molecules. At top (black): a spectrum taken using PA to the resonantly coupled $v^{\prime }=111,J^{\prime }=5$ level of the $2\left(0_g^{+}\right)$ state. At center (red): a spectrum using a typical vibrational level, $v^{\prime }=107$, $J^{\prime }=3$, which is not resonantly coupled. At bottom (blue): the spectrum taken using a small satellite feature of the $v^{\prime }=111,J^{\prime }=5$ line, which is marked in Fig. 4b. This spectrum is the average of three scans and is enlarged by a factor of 10 for visibility. We believe that this satellite feature represents a level of coupled $2\left(0_g^{+}\right)$ and $2\left(1_g\right)$ character. Vibrational quantum numbers of the initial $a{}^3{\Sigma }_u^{+}$ and intermediate $2{}^3{\Sigma }_g^{+}$ states are assigned as indicated.

Figure 3

(a) Rotational spectrum of $v^{\prime }=110$ of the $2\left(0_g^{+}\right)$ state as seen in ion detection from the $a{}^3{\Sigma }_u^{+}$, $v^{\prime }=20$ level, a fairly typical example of a strong PA transition to the $2\left(0_g^{+}\right)$ state. (b) The spectrum of $v^{\prime }=111$ detected from the $a{}^3{\Sigma }_u^{+}$, $v^{\prime }=22$ level, showing the strongly enhanced $J^{\prime }=5$ line. This spectrum has been scaled based on Lorentzian fitting of other spectra, which exhibited clipping. The resulting $J^{\prime }=3$ line strength is comparable to that of $v^{\prime }=110$ in panel (a). At the top of panel (b) is the MOT fluorescence trap loss spectrum of the long-range $2\left(1_g\right)$ state. High-resolution REMPI and trap loss scans of these coupled states are shown in Figs. 4b and 4a, respectively. In both panels, the lines marked with an asterisk (*) are hyperfine “ghosts” from the lower $F=2$ ground-state hyperfine level of ${}^{85}$Rb${}_2$.

Figure 4

(a) Hyperfine structure of the $v^{\prime }=155$ level of the $2\left(1_g\right)$ state, averaged from six trap loss scans. The energy zero point is the position of the $v^{\prime }=111$, $J^{\prime }=5$ level of the $2\left(0_g^{+}\right)$ state. Below the data is a model fit using a hyperfine Hamiltonian from Ref. [23] and described in the text. The model is vertically offset from the experimental data, for clarity. (b) A series of scans at progressively lower PA intensities through the $v^{\prime }=111,J^{\prime }=5$ level of the $2\left(0_g^{+}\right)$ state. The scans are vertically offset and smoothed for clarity. To the left of the main peak is the satellite feature whose REMPI spectrum is shown in Fig. 2. Both panel (a) and panel (b) are on the same scale, and referenced to the position of the $2\left(0_g^{+}\right)$, $v^{\prime }=111$, $J^{\prime }=5$ level at $12798.17\left(3\right)$ cm${}^{-1}$.