Dipole-dipole resonance line shapes in a cold Rydberg gas

We have explored the dipole-dipole mediated, resonant energy transfer reaction, $32p_{3/2}+32p_{3/2}\to 32s+33s$, in an ensemble of cold ${}^{85}\mathrm{Rb}$ Rydberg atoms. Stark tuning is employed to measure the population transfer probability as a function of the total electronic energy difference between the initial and final atom-pair states over a range of Rydberg densities, $2\times {10}^8\le \rho \le 3\times {10}^9 {\mathrm{cm}}^{-3}$. The observed line shapes provide information on the role of beyond nearest-neighbor interactions, the range of Rydberg atom separations, and the electric field inhomogeneity in the sample. The widths of the resonance line shapes increase approximately linearly with the Rydberg density and are only a factor of 2 larger than expected for two-body, nearest-neighbor interactions alone. These results are in agreement with the prediction [B. Sun and F. Robicheaux, Phys. Rev. A 78, 040701(R) (2008)] that beyond nearest-neighbor exchange interactions should not influence the population transfer process to the degree once thought. At low densities, Gaussian rather than Lorentzian line shapes are observed due to electric field inhomogeneities, allowing us to set an upper limit for the field variation across the Rydberg sample. At higher densities, non-Lorentzian, cusplike line shapes characterized by sharp central peaks and broad wings reflect the random distribution of interatomic distances within the magneto-optical trap (MOT). These line shapes are well reproduced by an analytic expression derived from a nearest-neighbor interaction model and may serve as a useful fingerprint for characterizing the position correlation function for atoms within the MOT.

Figure 1

$32p_{3/2}32p_{3/2}\to 32s33s$ DD-resonance line shapes, showing population transfer to $33s$ as a function of applied tuning field at various Rydberg densities: $\rho =2\times {10}^8$ (filled triangles); $\rho =1\times {10}^9$ (filled squares); and $\rho =3\times {10}^9 {\mathrm{cm}}^{-3}$ (crosses). The measured signals are not individually normalized, so the relative heights of the profiles reflects the difference in resonant transition probability. The baseline, corresponding to zero population transfer, is the same for the three data sets. The horizontal axis shows the applied tuning field due to the rods. The resonance line centers are shifted from the expected resonance condition, $F_0=12.7$ V/cm, due to the presence of the MCP offset field described in the text. The three data sets are acquired with the excitation beam focused at (slightly) different locations within the MOT. The relative shifts of the line centers are due to the variation in the offset field within the FWHM of the atom cloud. As described in the text, the inhomogeneity in the offset field is also responsible for the Gaussian line shapes observed at low Rydberg density. The solid (red) line through the lowest density data is the best Gaussian fit of the inhomogeneously broadened line shape. The solid (blue and green) lines through the higher density data are fits to the cusp line shape expected for a random ensemble, as described in the text. The small peaks indicated by arrows on either side of the resonance data appear independent of density. The two features on the high-field side of the resonance can be attributed to the transfer of population to $31d$ (not resolved from $33s$ in the ionization signal) due to $32p_{3/2}+32p_{3/2}\to 31d+29k$ resonances, where the $29k$ states are members of the manifold of $n=29$ Stark states that adiabatically connect to high-$\ell$ states in zero field. The specific resonance(s) responsible for the feature(s) on the low field side of the primary resonance have not been identified.

Figure 2

Resonance width as a function of Rydberg density. Measured widths are shown as points (lower density axis) while the solid line shows the calculated widths (upper density axis) assuming only nearest-neighbor interactions and 15 MHz of inhomogeneous broadening due to the offset field. The filled circles show data taken with a long 3 $\mu \mathrm{s}$ excitation laser pulse, and the filled triangles show data taken with a short 10 ns excitation pulse. No significant difference in the profile widths for the long and short pulse excitations is expected or observed.

Figure 3

Pair-energy detuning from resonance as a function of applied electric field. The filled circles are experimentally determined values of $\delta$ obtained from Stark shift measurements for the $32p_{3/2} |m_j|=3/2, 32s$, and $33s$ levels as a function of the applied field. At these low fields, the energies of all three levels shift quadratically with the field. The solid line is a quadratic fit to the data. Near resonance, the variation in $\delta$ is approximately linear with a slope of 170 MHz/(V/cm) (dashed line).

Figure 4

Comparison of the cusp (solid line) and Lorentzian (dashed line) line shapes expected for ensembles with random [see Eq. (5)] and uniform [see Eq. (3)] atom separation, respectively. The Lorentzian profile assumes the most probable value of $R$ at the Rydberg density used to compute the cusp. Note the cusp's broad, large amplitude wings and relatively narrow central peak.