Imperfect blockade in Rydberg superatoms

Abstract

Ensembles of atoms interacting via their Rydberg levels, known as "superatoms" for their ability to encode qubits and to emit single photons, attract increasing attention as building blocks for quantum network nodes. Assessing their performance requires an accurate, physically informative and numerically scalable description of interactions in a large and disordered ensemble. We derive such a description from first principles and successfully test it against brute-force numerics and experimental data. This model proves essential to make quantitative predictions about gate fidelities or photon emission efficiencies, and to guide experiments towards large-scale superatom-based systems.

Figures (3)

• Figure 1: Effective blockade model. (a) In the "microscopic" model of Eq.\ref{['eq:HDriveDiscrete']}, the collective state $\left| G \right\rangle$ is coupled to the exchange-symmetric singly-excited state $\left| R \right\rangle$. $\left| R \right\rangle$ is off-resonantly coupled to asymmetric states via the doubly-excited space, where the eigenstates correspond to atomic pairs at well-defined distances. (b) The effective model treats the cloud as a continuous medium and uses the eigenstates of a 3D harmonic oscillator as a basis, including a few of them in the Hamiltonian, and treating the others as a memory-less continuum $\left| C \right\rangle$. The laser coupling between singly- and doubly-excited states preserves the total number of spatial excitations.
• Figure 2: Populations of the qubit's excited state $\left| R \right\rangle$ (red), of the doubly-excited symmetric state $\left| \Psi_{0,0,0} \right\rangle$ (blue), and of all excited states (black). A brute-force integration of Eq. \ref{['eq:HDriveDiscrete']} (full lines) is compared with our model where $z_e=z_\Omega$ (dashed) or $z_e=z_0$ (dotted). The shaded areas around the full lines correspond to the standard deviations obtained from $8$ brute-force simulations. To speed up the latter, doubly-excited states with interaction energies exceeding $32\times\Omega/2$ are considered as far off-resonant and discarded from the Hilbert space.
• Figure 3: Experimental validation. (a) Rabi oscillations between the collective states $\left| G \right\rangle$ and $\left| R \right\rangle$ of a $^{87}$Rb superatomic qubit are driven via a two-photon transition coupling the atomic states $g=5S_{1/2}$ and $r=nS_{1/2}$ with n=(109,95,80). The collective Rabi frequency $\Omega=\sqrt{N_0}\Omega_1\Omega_2/2\Delta$ involves the effective number of atoms $N_0\approx 620$, the Rabi frequencies $\Omega_1\in\unit{[0,2\pi\times 22]}{\mega\hertz}$ and $\Omega_2=2\pi\times\unit{7.8}{\mega\hertz}$ on the two branches of the transition, and the detuning $\Delta=-2\pi\times\unit{500}{\mega\hertz}$ from the intermediate state $e=5P_{1/2}$ (see text and Appendix \ref{['app:ExptProtocol']}). (b) Temporal shapes of the Rabi driving pulses, and those of the readout pulse and of the emitted photon used in the R/not(R) measurement shown in panel (d). (c) G/not(G) measurement. If, after the driving sequence, the superatom is in $\left| G \right\rangle$, a control beam coupling $e$ to the Rydberg state $r'=78S_{1/2}$ makes the cloud transparent to a probe resonant on the $g-e$ transition. Otherwise, Rydberg blockade shifts the $e-r'$ resonance and destroys the transparency. The contrast in the probe's transmission is enhanced by a resonant medium-finesse optical cavity. Circles show the not(G) probability, measured as a function of $\Omega$ for the states $109S_{1/2}$ (blue), $95S_{1/2}$ (green), and $80S_{1/2}$ (red). Error bars indicate the standard error. Solid lines represent the corresponding theoretical predictions, while dashed lines show the expected populations in the doubly excited manifold. (d) R/not(R) measurement. Quasi-adiabatically increasing the Rabi frequency $\Omega_m$ of a mapping beam resonant on the $r-e$ transition coherently converts $\left| R \right\rangle$ into a photon, emitted from the cavity in free space. The graph shows the measured number of emitted photons, corrected for detection losses outside the vacuum chamber, together with the theoretical predictions, with the same conventions as the plot above.