Photodissociation of long-range Rydberg molecules

We present photodissociation of ultracold long-range Rydberg molecules as a tool to characterize their electronic properties. We photoassociate ${}^{39}\mathrm{K}_2 37{}^{2}P$ molecules with highly entangled electronic and nuclear spins of the two bound atoms and quantify the entanglement by projection of the molecular state onto noninteracting atoms using radio-frequency photodissociation. By comparison of experimental photodissociation rates with theoretical predictions, we further characterize the electronic and nuclear wave function of the photoassociated molecules. Based on the complete characterization of the formed long-range Rydberg molecules, we demonstrate a full hyperfine-spin flip of a free ground-state atom through the interaction with a Rydberg atom.

Figure 1

Energy-level scheme (not to scale) of the relevant molecular asymptotes involved in the photoassociation and photodissociation of $|37,1,j\rangle |F\rangle$ LRMs. Arrows I and II indicate photoassociation from ground-state atoms prepared in $F=1$ and $F=2$, respectively. The dash-dotted arrows III and IV indicate rf photodissociation of the LRMs to $|35,2,j^{\prime }\rangle |F^{\prime }\rangle$. The potential-energy curves in the regions shaded in light blue and light red are depicted in Figs. 2 and 3, respectively. See the text for a definition of the state labels $|n,l,j\rangle |F\rangle$.

Figure 2

(a) PECs of states associated with the near-degenerate $|37,1,3/2\rangle |1\rangle$ and $|37,1,1/2\rangle |2\rangle$ asymptotes. Shown are all states for $\mathrm{\Omega }=1/2$ with ${}^3\mathrm{\Sigma }^{+}$ (dark and light green), ${}^{1,3}\mathrm{\Sigma }^{+}$ (red), and ${}^{1,3}\mathrm{\Pi }$ (blue) symmetry. The red-filled curve depicts the vibrational wave function of the vibrational ground state in the outermost well of the ${}^{1,3}\mathrm{\Sigma }^{+}$ state correlated to the $|37,1,3/2\rangle |1\rangle$ asymptote, ${\mathrm{\Psi }}_e$. Its $R$-dependent absolute square of the coefficient ${\alpha }_{1/2;2}^e$ is depicted in the inset. (b) Photoassociation spectra recorded close to the $|37,1,3/2\rangle |1\rangle$ asymptote after preparing the ground-state atoms in (I) the $F=1$ and (II) the $F=2$ hyperfine level. For measurement II, the intensity of the photoassociation laser was increased by a factor of 4.7(1) to improve the signal-to-noise ratio. The laser frequency is given relative to the frequency of the atomic $|37,1,3/2\rangle |1\rangle \leftarrow |1\rangle |1\rangle$ transition. Solid bars indicate the calculated positions and strengths of photoassociation resonances [color coding as in (a)], as described in Appendix pp1. The red star marks the resonance probed by photodissociation (Fig. 3).

Figure 3

(a) PECs ($\mathrm{\Omega }=1/2$) correlated to the $|35,2,j^{\prime }\rangle |2\rangle$ asymptotes. States with ${}^3\mathrm{\Sigma }^{+}, {}^{1,3}\mathrm{\Sigma }^{+}$, and $\mathrm{\Pi }$ symmetry are depicted by light green, red, and blue lines, respectively. The spectral region in which bound states can exist is highlighted in gray [see, also, (b)]. The gray dashed line marks the equilibrium distance of ${\mathrm{\Psi }}_e$. (b) The rf spectra of ${}^{1,3}\mathrm{\Sigma }^{+}{K}_2$ molecules [UV-laser detuning marked by a red star in Fig. 2] in the vicinity of the transition to the spin-orbit-split (III) $|35,2,j^{\prime }\rangle |1\rangle$ and (IV) $|35,2,j^{\prime }\rangle |2\rangle$ asymptotes after a $30\mu \mathrm{s}$ photoassociation pulse followed by a $10\mu \mathrm{s}$ rf pulse. Zero rf detuning is set to $86.667\mathrm{GHz}, 11.85\mathrm{MHz}$ below the atomic transition $35{}^{2}D_{3/2}\leftarrow 37{}^{2}P_{3/2}$, corresponding to the transition from the bound molecular state to the dissociation continuum associated with the $|35,2,3/2\rangle |1\rangle$ asymptote. The gray-shaded areas mark spectral regions where transitions to bound states are possible [see (a)]. The vertical thin blue lines mark the transition frequencies from the initial molecular state to the indicated dissociation asymptotes. The gray and blue lines show the simulated spectrum, where the intensities of the bound (gray) and continuum (blue) parts have been scaled separately to the experimental data because of the different normalization conditions.

Figure 4

Fraction of ions $\mathcal{Q}$ detected in the time-of-flight window set for $37{}^{2}P_j$ (open symbols, left axis) and $35{}^{2}D_j$ (filled symbols, right axis) as a function of the applied rf power at the frequencies 86.6941 GHz (black), 86.6669 GHz (blue), and 86.2052 GHz (red), corresponding to the transitions from the molecular state to the respective dissociation thresholds specified in the inset. The rf powers are given as a fraction $I_{\mathrm{rel}}$ of the maximal applied power. Also shown are the global fit (solid and dashed lines for $37{}^{2}P_j$ and $35{}^{2}D_j$, respectively), fit parameters, and statistical uncertainties of the theoretical model to the combined experimental data (see text for details).

Figure 5

Energy dependence of the photodissociation rates from the ${\mathrm{\Psi }}_e$ state to the $|35,2,3/2\rangle |1\rangle$ (green), $|35,2,5/2\rangle |1\rangle$ (blue), and $|35,2,3/2\rangle |2\rangle$ (brown) asymptotes obtained from fits to power-dependent measurements at different rf detunings above the dissociation threshold. The error bars indicate the standard deviation of the fitted rates. The simple model explained in the text predicts the energy dependence $\gamma \left(E_k\right)$ given in the figure. Here, $A_{j;F}$ is an asymptote-dependent amplitude, $\mu$ is the reduced mass of the molecule in atomic mass units, $\nu =E_k/h$ is the detuning from the threshold in megahertz, and $\sigma$ is the Gaussian width of the initial molecular state in Bohr radii. The red curve is a global fit of this model to the data points, weighted by their standard deviations, with free parameters $A_{3/2;1}, A_{3/2;2}, A_{5/2;1}$, and $\sigma$. The shown photodissociation rates are normalized by the fitted amplitudes $A_{j;F}$. The rates at $0\mathrm{MHz}$ rf detuning (gray data points) were not included into the fit because of the divergence of the theoretical model at this point.

Figure 6

The rf spectra of the ${}^3\mathrm{\Sigma }^{+}$ ($v=0$) ${\mathrm{K}}_2$ molecules [dark green bar in the upper panel of Fig. 2] in the vicinity of the transition to the spin-orbit-split (III) $|35,2,j^{\prime }\rangle |1\rangle$ and (IV) $|35,2,j^{\prime }\rangle |2\rangle$ asymptotes after a $30\mu \mathrm{s}$ photoassociation pulse followed by a $10\mu \mathrm{s}$ rf pulse. Zero rf detuning is set to $86.662\mathrm{GHz}, 16.31\mathrm{MHz}$ below the atomic transition $35{}^{2}D_{3/2}\leftarrow 37{}^{2}P_{3/2}$, corresponding to the transition from the bound molecular state to the dissociation continuum associated with the $|35,2,3/2\rangle |1\rangle$ asymptote. The gray-shaded areas mark regions where transitions to bound states are possible. The vertical thin blue lines mark the transition frequencies from the initial molecular state to the indicated dissociation asymptotes.