Highly Resolved Measurements of Stark-Tuned Förster Resonances between Rydberg Atoms

We report on experiments exploring Stark-tuned Förster resonances between Rydberg atoms with high resolution in the Förster defect. The individual resonances are expected to exhibit different angular dependencies, opening the possibility to tune not only the interaction strength but also the angular dependence of the pair state potentials by an external electric field. We achieve a high resolution by optical Ramsey interferometry for Rydberg atoms combined with electric field pulses. The resonances are detected by a loss of visibility in the Ramsey fringes due to resonances in the interaction. We present measurements of the density dependence as well as of the coherence time at and close to Förster resonances.

Figure 1

(a) Single shot Ramsey spectrum for a pulsed electric field of $E_z=0.3\mathrm{V}/\mathrm{cm}$ (blue dots) and least square fit to the data (red line). $\Delta$ is the detuning of the exciting laser to the atomic resonance. (b) Pulse sequence used throughout this Letter. (c) Color-coded Ramsey spectra for varying pulsed electric fields. With increasing electric field a phase shift of these fringes occurs that depends quadratically on the electric field. A loss in visibility at $0.08\mathrm{V}/\mathrm{cm}$, $0.13\mathrm{V}/\mathrm{cm}$, and $0.21\mathrm{V}/\mathrm{cm}$ is visible, marked by the arrows in (c). (a),(c) Parameters for Ramsey measurements: ${\tau }_p=0.15\mu \mathrm{s}$, ${\tau }_d=1.0\mu \mathrm{s}$, $\rho =1.2\times {10}^{12}{\mathrm{cm}}^{-3}$.

Figure 2

The uppermost graph shows the phase $\varphi$ of the Ramsey fringes. The lower panels show the visibility normalized to the mean value $\overline{V}$ of each data set against electric field for different densities $\rho$ of ground state atoms. The magnetic field is 13.55 G and the Rabi frequency is constant for all measurements. The lower electric field axis, valid for all graphs, denotes the calibrated component of the electric field, $E_z$. The dashed vertical lines indicate the resonant electric fields obtained from measurements. On the upper electric field axis (total electric field $|\overrightarrow{E}|$, including an experimentally not accessible radial component) the calculated resonant electric fields are indicated by diamonds.

Figure 3

Magnetic field dependence of the electric fields $|\overrightarrow{E}|$ required to tune the pair states into Förster resonance for parallel electric and magnetic field (dots and lines). Additional splittings occur in finite magnetic field due to the splitting into magnetic sublevels. The angular maximum of the interaction strength $U_{\max }$ of the resonances is indicated by the color and the diameter of the dots, that linearly increases with the strength of the resonance. The strongest resonance is at $|\overrightarrow{B}|=0\mathrm{G}$ and $|\overrightarrow{E}|=0.234\mathrm{V}/\mathrm{cm}$ with a strength of $1556\mathrm{MHz}{\mu \mathrm{m}}^3$. The crosses indicate the measured resonance positions in $E_z$ at 10.50 G and 13.55 G, i.e., uncorrected for the radial electric field.

Figure 4

Ramsey experiment with variable delay times between ${\tau }_d=0\mu \mathrm{s}$ and ${\tau }_d=2\mu \mathrm{s}$ at an atom density of $1.2\times {10}^{12}{\mathrm{cm}}^{-3}$. (a) Color-coded experimental data and (b) data obtained by a numerical fit, both at $E_z=0.21\mathrm{V}/\mathrm{cm}$. (c) Dephasing time $T_2$ obtained from the numerical fits for different electric fields $E_z$. The error bars denote the standard deviation of the fit parameter.