Long-lived coherence between ground and Rydberg levels in a magic-wavelength lattice

By confining atoms in a state-insensitive optical lattice, the lifetime of the ground-state–Rydberg coherence is increased to $\ge 20\mu \mathrm{s}$, an order of magnitude improvement over previous experiments using freely diffusing atoms. Using these enhanced lifetimes, we measure the so-called magic lattice wavelengths for Rb and use them to extract the $6p_{3/2}\text{−}n{s}_{1/2}$ reduced electric dipole matrix elements. Good agreement is found with values obtained using an effective one-electron potential for principal quantum numbers $n$ between $n=30$ and $n=70$. We develop a theoretical model based on quantized motion to map out the ground-state–Rydberg coherence as a function of time that is in good agreement with the experimental results. The availability of long coherence times may present new opportunities for high-resolution spectroscopy and quantum information science.

Figure 1

(a) A cold sample of ${}^{87}\mathrm{Rb}$ gas is trapped in a 0.5-$\mu$m-period one-dimensional optical lattice formed by a retroreflected beam $E_L$. Two nearly counterpropagating beams $E_1$ and $E_2$ excite a spin wave between the $|5s_{1/2},F=2\rangle$ and $|n{s}_{1/2}\rangle$ levels. After a storage time $T_s$, a retrieval pulse $E_R$ is applied, creating an array of atomic dipoles which give rise to a phase-matched emission from the sample. The actual geometry used in the experiment differs somewhat from that shown schematically in the figure. (b) Relevant ${}^{87}\mathrm{Rb}$ energy levels and corresponding fields, with $\mathrm{\Delta }={\omega }_L-{\omega }_{ns,6p_{3/2}}$ and ${\mathrm{\Delta }}_1={\omega }_{E_2}-{\omega }_{ns,6p_{3/2}}$. (c) Schematic diagram indicating transitions between the ground- and excited-state motional levels. (d) Timing diagram showing the excitation and retrieval pulse sequence.

Figure 2

(a) Graph of the analytic approximation and exact expressions of $g_{nl}$ (dashed and solid curves, respectively) as a function of storage time $T_s$ for $U_0/k_B=40\mu \mathrm{K}$ and different sample lengths: blue, $L=1\mu \mathrm{m}$; dark green, $L=50\mu \mathrm{m}$; light green, $L=100\mu \mathrm{m}$; orange, $L=150\mu \mathrm{m}$; and red, $L=500\mu \mathrm{m}$. (b) Graph of $g_{nl}$ for sample length $L=100\mu \mathrm{m}$ and trap depths $U_0/k_B=5$, 10, 20, and $40\mu \mathrm{K}$, represented by increasing line thickness.

Figure 3

Graph of $g_l\left(T_s\right)$ as a function of ${\omega }_0{T}_s$: red solid curve, ${cos}^2\left(k_{L}X\right)$ potential; black dashed curve, harmonic potential.

Figure 4

(a)–(d) Normalized signal $\eta \left(T_s\right)$ at storage time $T_s$ around the first revival ($10–12\mu \mathrm{s}$) as a function of lattice detuning $\mathrm{\Delta }$ for principal quantum numbers 30, 51, 60, and 65. The solid curves, based on the model described in the text, are used to extract the values of ${\mathrm{\Delta }}_{m,n}$. The dashed red and solid green vertical lines represent the theoretically expected and the extracted values of the magic detuning, respectively. Blue and red bands represent fits using temperatures 20% lower and higher than the best-fit value, respectively. (e) ${\mathrm{\Delta }}_{m,n}$ as a function of the principal quantum number $n$, with the solid curve based on our theoretical model. (f) Extracted values of the scaled reduced matrix elements as a function of $n$.

Figure 5

Normalized signal $\eta$ as a function of storage time for several principal quantum numbers: (a) $n=30$, (b) $n=40$, (c) $n=51$, and (d) $n=65$. The solid black curve is based on our theoretical model. Blue and red bands represent temperatures 20% lower and higher than the best-fit value, respectively. The gray curve shows loss attributable to blackbody and spontaneous decay from the Rydberg state. The dashed red curve adds in the contribution of spontaneous decay from the $6P$ level. The dashed blue curve additionally includes the dephasing attributable to the nonlattice potential. Most experimental error bars are smaller than the shown markers.

Figure 6

Normalized signal $\eta$ as a function of storage time for $n=40$ for (420–1018)-nm (green circles) and (795–475)-nm (orange diamonds) excitation, with the corresponding atomic transitions shown in the inset. The solid curves are the result of a numerical simulation of atomic motion using the model described in the text. The black curve is the same as in Fig. 5. Most experimental error bars are smaller than the shown markers.