Analysis of imperfections in the coherent optical excitation of single atoms to Rydberg states

We study experimentally various physical limitations and technical imperfections that lead to damping and finite contrast of optically driven Rabi oscillations between ground and Rydberg states of a single atom. Finite contrast is due to preparation and detection errors, and we show how to model and measure them accurately. Part of these errors originates from the finite lifetime of Rydberg states, and we observe its $n^3$ scaling with the principal quantum number $n$. To explain the damping of Rabi oscillations, we use simple numerical models taking into account independently measured experimental imperfections and show that the observed damping actually results from the accumulation of several small effects, each at the level of a few percent. We discuss prospects for improving the coherence of ground-Rydberg Rabi oscillations in view of applications in quantum simulation and quantum information processing with arrays of single Rydberg atoms.

Figure 1

A typical Rabi oscillation between the ground-state $|g\rangle$ and the Rydberg state $|r\rangle$ (with $n=62$) when all the parameters are optimized on the experiment. The solid line is a fit by a damped sine.

Figure 2

(a) The excitation setup. The tweezers has a $1/e^2$ radius of $1.1\mu \mathrm{m}$, and the blue and red beams have elliptical cross sections with waists $(w_x,w_y)=24\times 50$ and $(w_x,w_z)=50\times 200\mu {\mathrm{m}}^2$, respectively. (b) Relevant levels involved in the Rydberg excitation (see the text).

Figure 3

Effect of small but finite values of $(\varepsilon ,{\varepsilon }^{\prime },\eta )=(0.05,0.08,0.03)$ on the measured probability $P_r$ (solid line) assuming a perfect Rabi oscillation ${\tilde{P}}_r$ (dashed line). The values of SPAM errors are chosen larger than in the experiment to easily visualize their effects.

Figure 4

Detection errors ${\varepsilon }^{\prime }$ caused by the finite Rydberg state lifetime. The black solid line: ${\mathrm{\Gamma }}_R{t}_{\mathrm{recap}}$. The red (light gray) solid line: Eq. (A1). The red dashed line takes into account the $4-\mu \mathrm{s}$ push-out time to remove atoms in $|g\rangle$, see the Appendix.

Figure 5

Influence of the Doppler effect on the Rabi oscillations for a temperature of $T=30\mu \mathrm{K}$ and Rabi frequencies $\mathrm{\Omega }/\left(2\pi \right)$ of (a) 250 kHz, (b) 500 kHz, (c) 1 MHz, and (d) 2 MHz.

Figure 6

Influence of the spontaneous emission from $|p\rangle$. (a) and (b) Calculated Rabi oscillation obtained by solving the OBEs for (a) $\mathrm{\Delta }=2\pi \times 740, {\mathrm{\Omega }}_{\mathrm{b}}=2\pi \times 30$, and ${\mathrm{\Omega }}_{\mathrm{r}}/\left(2\pi \right)=30\mathrm{MHz}$ and (b) 100 MHz. (c) and (d) Comparison between the simulation and the experimental data (with $n=61$) for fixed ${\mathrm{\Omega }}_{\mathrm{b}}=2\pi \times 35$ and ${\mathrm{\Omega }}_{\mathrm{r}}=2\pi \times 210\mathrm{MHz}$ but for decreasing values of the intermediate-state detuning: (c) $\mathrm{\Delta }=2\pi \times 740\mathrm{MHz}$ and (d) $\mathrm{\Delta }=2\pi \times 477\mathrm{MHz}$.

Figure 7

Influence of laser phase noise. (a) Estimated spectral density $S_{\nu }\left(f\right)$ of the frequency noise of the 795-nm (red or light gray) and 950-nm laser (blue or dark gray). The solid lines: usual noise and the dashed line: enhanced noise (see the text). (b) Simulated Rabi oscillation in the presence of usual phase noise. Many Rabi oscillations (thin lines), each for a given realization of the random processes ${\varphi }_i\left(t\right)$, are averaged (thick black line). (c) With intentional extra noise added in the 950-nm laser, the damping is increased, and the simulation compares well with the experimentally observed Rabi oscillation on $n=54$ (red circles).

Figure 8

Including all effects in the simulation. (a) Rabi oscillation (with $n=61$) at $\mathrm{\Omega }/\left(2\pi \right)=4.8\mathrm{MHz}$ with the red solid line showing the result of a parameter-free simulation. The gray-shaded area shows the time window used in panel (b) to quantify the damping (see the text). (b) Influence of the Rabi frequency $\mathrm{\Omega }$ (varied by changing only ${\mathrm{\Omega }}_{\mathrm{r}}$ with ${\mathrm{\Omega }}_b/\left(2\pi \right)=35$ and $\mathrm{\Delta }=740\mathrm{MHz}$) on the Rabi oscillation damping. Simulations are performed with typical experimental parameters (see the text) for each individual source of damping: Doppler effect (dashed-dotted line), spontaneous emission (dashed line), laser phase noise (dotted line), and combining all of them (solid black line) and by adding the SPAM errors (solid red or light gray line). The shading on the latter curve corresponds to standard error on the mean of the Monte Carlo simulation with 600 runs.

Figure 9

Recapture probability $p_{\mathrm{recap}}\left(t\right)$ including antitrapping by the optical tweezers (solid line, red squares) and neglecting the antitrapping (dashed line, red circles). The simulations take parameters $T=20\mu \mathrm{K},U_0=1\mathrm{mK}$, and $\alpha =0.17$, and the experiment is detailed in the text.