Electric Dipole Echoes in Rydberg Atoms

We report the first observation of echoes in the electric dipole moment of an ensemble of Rydberg atoms precessing in an external electric field $F$. Rapid reversal of the field direction is shown to play a role similar to that of a $\pi$ pulse in NMR in rephasing a dephased ensemble of electric dipoles resulting in the buildup of an echo. The mechanisms responsible for this are discussed with the aid of classical trajectory Monte Carlo simulations.

Figure 1

Behavior of the pseudospins ${\overrightarrow{J}}_{\pm }$. To first-order these precess with angular velocities $\pm {\omega }_s/2$ in the electric field. The electric dipole is proportional to $n⟨\overrightarrow{A}⟩=({\overrightarrow{J}}_{+}-{\overrightarrow{J}}_{-})$.

Figure 2

Time evolution of the classical trajectory of a single Rydberg electron ($n=350$) in an external electric field $F=20\mathrm{mV}/\mathrm{cm}$ applied along the $z$ direction: (a) projection of the trajectory onto the ($z$, $x$) plane during a Stark precession period ($T_s\approx T_k/2\approx 40\mathrm{ns}$) equal to about seven Kepler periods ($T_n=6.5\mathrm{ns}$). (b) Slow rotational motion in the ($A_x$, $A_y$) plane for $0\le t\le 1000\mathrm{ns}$ ($T_m\simeq 2500\mathrm{ns}$). All coordinates are depicted in scaled units ($r_0=r/n^2$, $p_0=pn$).

Figure 3

(a) Time-dependent field used to study the dipole echo (see text). The probe HCP, with scaled amplitude $n\Delta p_p=0.53$, is applied after a variable delay time. (b) Echo in the time-resolved survival probability (experiment: red, CTMC simulation: blue) for $n=350$ atoms initially polarized along the $x$ axis and a field $|F|=20\mathrm{mV}/\mathrm{cm}$ applied in the $z$ direction. The field is reversed at $\tau =500\mathrm{ns}$. (c) Time-resolved survival probability without field reversal.

Figure 4

Time evolution of the probability density in the ($J_{+}^{x^{\prime }}$, $J_{+}^{y^{\prime }}$) plane. The snapshots are taken at $t=0$, $\tau$, and $2\tau$, where $\tau =500\mathrm{ns}$. Upper panels: evolution for initial state with maximal transverse polarization (see text). Lower panels: evolution for the state realized in the experiment (see text). The arrows denote the precessional motion of the wave packet. $J_{+}^{x^{\prime }}$ and $J_{+}^{y^{\prime }}$ are expressed in scaled units.

Figure 5

CTMC simulations of the width $\Delta {\varphi }_{+}(t)$ of the wave packet in the azimuthal angle, ${\varphi }_{+}$. The initial linear diffusion $\Delta {\varphi }_{+}\propto t$ is reversed at $t=\tau =500\mathrm{ns}$ due to field reversal. Results are included for three different cases: the maximally polarized initial state (dashed line), the experimentally realized initial state (solid line), and without field reversal (dotted line). The magnitude of the field is $20\mathrm{mV}/\mathrm{cm}$. To compare only the diffusive nature of $\Delta {\varphi }_{+}$, the initial spread is subtracted from the results.