Correlations of Rydberg excitations in an ultracold gas after an echo sequence

We show that Rydberg states in an ultracold gas can be excited with strongly preferred nearest-neighbor distance if densities are well below saturation. The scheme makes use of an echo sequence in which the first half of a laser pulse excites Rydberg states while the second half returns atoms to the ground state, as in the experiment of Raitzsch et al. [Phys. Rev. Lett. 100, 013002 (2008)]. Near the end of the echo sequence, almost any remaining Rydberg atom is separated from its next-neighbor Rydberg atom by a distance slightly larger than the instantaneous blockade radius halfway through the pulse. These correlations lead to large deviations of the atom-counting statistics from a Poissonian distribution. Our results are based on the exact quantum evolution of samples with small numbers of atoms. Finally, we demonstrate the utility of the $\omega$ expansion for the approximate description of correlation dynamics through an echo sequence.

Figure 1

Illustration of an echo excitation sequence and evolution of the Rydberg fraction. For a sequence of duration $T$, we use the fractional time $\tau =t/T$. (a) Sketch of Rabi frequency $\Omega w(t)$ (see Sec. 2a) during the sequence. (b) Fraction of excited state atoms $f$ for a density ${\rho }_0{V}_{b0}=26.2$ (solid line) from a numerical solution of the Schrödinger equation. Here, $V_{b0}=4\pi r_{b0}^3/3$ is a simple estimate of the saturated blockade volume. We also show the noninteracting result for the excited fraction (dashed line). The symbols ($•$) mark times for which we display correlations in Fig. 2.

Figure 2

Spatial atomic correlations during an echo sequence with ${\rho }_0{V}_{b0}=5.4$. (a) Pair correlation function $g^{(2)}(r)$ during the excitation phase at $\tau =0.05$ (solid black line), $\tau =0.37$ (blue dashed line) and $\tau =0.49$ (green dotted line). Below the cutoff length $r_{\mathrm{cut}}$, $g^{(2)}$ is set to zero [22]. (b) The same during the deexcitation phase at $\tau =0.6$ (solid black line), $\tau =0.7$ (blue dashed line) and $\tau =0.75$ (green dotted line). (c) Final shape of the pair correlation function after the pulse ($\tau =1$). (d) Spatial maximum of pair correlations. The location of the time samples in panels (a)–(c) relative to the pulse can be seen clearly in Fig. 1.

Figure 3

Visibility of the pairing effect for a density with $\rho V_{b0}=26.2$. (a) Mandel $Q$ parameter. (b) Fraction $R$ of remnant atoms within the preferred distance peak, using Eq. (16). (c) Total number of paired, excited atoms, extrapolated [24] as if there was a total number of $N_g=1\times {10}^7$ atoms initially.

Figure 4

Effects of inhomogeneities and changes of density on the pairing peak. We show densities ${\rho }_0{V}_{b0}=5.4$, $26.2$, $39$ as black, blue dotted, and green or gray lines, respectively, and an inhomogeneous case $\rho (x)={\rho }_0\exp (-2r^2/{\sigma }^2)$ with ${\rho }_0{V}_{b0}=26.2$ and $\sigma /r_{b0}=1.6$ (red dashed line). (a) Rydberg fraction $f_e$; for increasing homogeneous density the final remnant $f_e(\tau =1)$ gets larger. (b) Paired excited fraction according to Eq. (16). (c) Mandel $Q$ parameter, see Eq. (17). (d) Spatial maximum of pair correlations. (e) $g^{(2)}(r)$ at $\tau =0.75$, for increasing homogeneous density the peak height decreases. (f) Spatial atomic correlations near the end of an echo pulse in a low-density (${\rho }_0{V}_{b0}=5.4$) gas. Shown are snapshots at $\tau =0.96$, $\tau =0.98$, and $\tau =1.0$ in order of increasing peak-height.

Figure 5

Dependence of echo signal and final correlation peak on atomic density. (a) Final Rydberg fraction at $t=T$. These data are very well described by Eq. (11). (b) (▾) Peak height of $g^{(2)}$ at $t=T$. The red dashed curved shows the functional form $g^{(2)}(\rho )=({\rho }_0/\rho ){}^2{g}^{(2)}({\rho }_0)$ with reference density ${\rho }_0=26.2$. This is motivated by a simple model explained in the text.

Figure 6

Comparison of the $\omega$ expansion (green dashed line) with exact solutions of the SE for ${\rho }_0{V}_{b0}=5.4$ and ${\rho }_0{V}_{b0}=26.2$ (solid black line). (a) Excited-state fractions. The higher density case has the higher $f_e(1)$. (b) Pair correlation function $g^{(2)}(r)$ at $\tau =0.49$. The two cases are almost indistinguishable. (c) Spatial maximum of pair correlations. The higher density case has a lower $g_{\max }^{(2)}(1)$. (d) Pair correlation function $g^{(2)}(r)$ at $\tau =0.75$. The higher density case has a lower pairing peak and is cutoff at $r=0.5r_{b0}$.