State-selective all-optical detection of Rydberg atoms

We present an all-optical protocol for detecting population in a selected Rydberg state of alkali-metal atoms. The detection scheme is based on the interaction of an ensemble of ultracold atoms with two laser pulses: one weak probe pulse which is resonant with the transition between the ground state and the first excited state, and a pulse with high intensity which couples the first excited state to the selected Rydberg state. We show that by monitoring the absorption signal of the probe laser over time, one can deduce the initial population of the Rydberg state. Furthermore, it is shown that—for suitable experimental conditions—the dynamical absorption curve contains information on the initial coherence between the ground state and the selected Rydberg state. We present the results of a proof-of-principle measurement performed on a cold gas of ${}^{87}\mathrm{Rb}$ atoms. The method is expected to find application in quantum computing protocols based on Rydberg atoms.

Figure 1

Illustration of the atom-laser interaction as used for the model of time-resolved EIT. States $|1\rangle , |2\rangle$, and $|3\rangle$ denote the ground state, the first excited state, and a Rydberg state of the atom, respectively. In our experimental setup, these states correspond to the $5S_{1/2}(F=2), 5P_{3/2}(F=3)$, and $35S_{1/2}(F=2)$ states of ${}^{87}\mathrm{Rb}$. The atomic transition $|1\rangle \leftrightarrow |2\rangle$ is driven by a weak probe laser with the Rabi frequency ${\mathrm{\Omega }}_{\mathrm{p}}$ (red), while the transition $|2\rangle \leftrightarrow |3\rangle$ is driven by a stronger coupling laser with the Rabi frequency ${\mathrm{\Omega }}_{\mathrm{c}}$ (blue). ${\gamma }_{\mathrm{p}}$ and ${\gamma }_{\mathrm{c}}$ denote coherence decay terms. ${\delta }_{\mathrm{p}}$ and ${\delta }_{\mathrm{c}}$ are the detunings of each laser to the corresponding atomic resonance. The radiative decay from the selected Rydberg state $|3\rangle$ to the neighboring states is accounted for by including a reservoir state $|4\rangle$. ${\mathrm{\Gamma }}_{ij}$ denote the respective incoherent decays, consisting of spontaneous emission as well as transitions induced by blackbody radiation.

Figure 2

Pulse sequence for the coupling (blue) and probe (red) lasers. At $t=0$ a fraction of the atomic population is prepared in the Rydberg state. In the simulations this fraction is an input value, while in the experiment it is determined by the length of the excitation pulse (Exc.). Before this, an optical pumping pulse (Pump) is used to pump the atoms to the correct polarization. The time evolution of the optical density is monitored after the excitation pulse with the probe laser (A, B, C). The coupling laser is added in time interval B (EIT pulse). The Rabi frequencies are taken from the experimental values in Sec. 4.

Figure 3

Dynamics of the populations ${\rho }_{kk}\left(t\right)$ of the atomic states (dotted lines) and the relative absorption ${\alpha }_{\mathrm{rel}}\left(t\right)$ (solid lines) of the probe laser induced by the pulse sequence A, B, C (see Fig. 2) for atoms initially prepared in the ground state (a), the selected Rydberg state (b), and the reservoir state (c). Since the population ${\rho }_{22}$ is almost zero at all times, except for a transient population in the first 100 ns of B, it is not shown here.

Figure 4

Numerically calculated relative absorption ${\alpha }_{\mathrm{rel}}$ of the probe laser in the beginning of part B for atoms prepared in states ${\rho }_{\mathrm{m}}(t=0)=\frac{1}{2}(|1\rangle \langle 1|+|3\rangle \langle 3|)$ and ${\rho }_{\pm }(t=0)=\frac{1}{4}(|1\rangle \pm |3\rangle )(\langle 1|\pm \langle 3|)$. The parameters used for the calculation are the same as for Fig. 3 with ${\mathrm{\Omega }}_{\mathrm{c}}<{\mathrm{\Gamma }}_{21}$ in (a) and ${\mathrm{\Omega }}_{\mathrm{c}}\approx {\mathrm{\Gamma }}_{21}$ in (b). This distinct signature of coherent states is expected to be experimentally observable and even more pronounced if ${\mathrm{\Omega }}_{\mathrm{c}}$ is larger.

Figure 5

Experimental setup for time-resolved EIT measurements. The counterpropagating coupling (blue) and probe (red) beams are superimposed on the atom cloud and separated with dichroic mirrors. The transmission of the probe laser is detected by an avalanche photo diode (APD). The intensities of the lasers are controlled with acousto-optic modulators (AOM). Both laser frequencies are stabilized using a frequency comb.

Figure 6

Measured optical density for the dynamics during the probe sequence. The shaded areas are 95% confidence intervals for the relative optical densities ${\mathrm{OD}}_{\mathrm{rel}}\left(t\right)$ obtained from the measurements by applying Eqs. (10) and (11) for three durations of the excitation pulse ($0\mu \mathrm{s}$ for I, $0.5\mu \mathrm{s}$ for II, and $2\mu \mathrm{s}$ for III). Solid lines represent fit results for the simulated relative absorption $\alpha \left(t\right)/{\alpha }_0$. The reversed order of the absorption signals in part B can be explained by dipole-dipole interactions.

Figure 7

Dynamics of the population of the four states (${\rho }_{11}\widehat{=}5S_{1/2}, {\rho }_{22}\widehat{=}5P_{3/2}, {\rho }_{33}\widehat{=}35S_{1/2}, {\rho }_{44}\widehat{=}\mathrm{Reservoir}$) retrieved from the fit to the dataset with high excitation (see Fig. 6). The colors match those of the states in Fig. 3. The population of the Rydberg state ($35S_{1/2}$) starts to decay immediately after excitation at $t=0$.