Accurate Mapping of Multilevel Rydberg Atoms on Interacting Spin-$1/2$ Particles for the Quantum Simulation of Ising Models

We study a system of atoms that are laser driven to $n{D}_{3/2}$ Rydberg states and assess how accurately they can be mapped onto spin-$1/2$ particles for the quantum simulation of anisotropic Ising magnets. Using nonperturbative calculations of the pair potentials between two atoms in the presence of electric and magnetic fields, we emphasize the importance of a careful selection of experimental parameters in order to maintain the Rydberg blockade and avoid excitation of unwanted Rydberg states. We benchmark these theoretical observations against experiments using two atoms. Finally, we show that in these conditions, the experimental dynamics observed after a quench is in good agreement with numerical simulations of spin-$1/2$ Ising models in systems with up to 49 spins, for which numerical simulations become intractable.

Figure 1

Mapping a system of multilevel Rydberg atoms onto a spin-$1/2$ model. (a) System: Two atoms separated by a distance $R$; $\theta$ is the angle between the interatomic axis and the quantization axis $z$ defined by a magnetic field $B$. An electric field $E$ can be applied along $z$. (b) A two-photon transition couples coherently the ground state $|g⟩$ to a target Rydberg state $|r⟩$ with an effective two-photon Rabi frequency $\mathrm{\Omega }$. (c) Full energy spectrum of the atom pair. The mapping consists in replacing this complex structure by an effective interaction potential.

Figure 2

Influence of magnetic and electric fields on the interaction potentials around the pair-state $|rr⟩$ where $|r⟩=|61D_{3/2},m_j=3/2⟩$, for $\theta =78°$. The shading encodes the overlap of the eigenstates with the noninteracting state $|rr⟩$. (a) $B=0$ and $E=0$: $|rr⟩$ overlaps with all the degenerate Zeeman pair states. (b) $B=-6.9\mathrm{G}$ and $E=0$: the interaction curves are split due to the Zeeman effect. Some curves still strongly mix with $|rr⟩$ due to the interaction. (c) $B=6.9\mathrm{G}$ and $E=0$: one potential curve dominates. However, (d) the addition of a small electric field $E=20\mathrm{mV}/\mathrm{cm}$ is enough to strongly perturb the pair states. (e),(f) This behavior is absent for $B=3.5\mathrm{G}$.

Figure 3

Approximation of the interaction by an anisotropic van der Waals potential $C_6(\theta )/R^6$. (a) Comparison of the exact interaction energy (solid line) with the asymptotic determination of the van der Waals potential (dashed line) for a fixed angle $\theta =78°$ and $B=3.5\mathrm{G}$. (b) Angular dependence of $C_6(\theta )/R^6$ at $R=9\mu \mathrm{m}$ marked by the cross on (a).

Figure 4

Two-atom blockade experiments. Probability $P_{rr}$ to excite the two atoms as a function of the pulse area $\mathrm{\Omega }\tau$. For $B=6.9\mathrm{G}$ [(a),(b)] increasing $E$ from 0 to $20\mathrm{mV}/\mathrm{cm}$ breaks the Rydberg blockade. At $B=3.5\mathrm{G}$ [(c),(d)] an efficient blockade is maintained, even in the presence of the electric field. The solid lines result from a simulation taking into account the full interaction spectrum (see text). The dashed lines are obtained by modeling the atoms as spin-$1/2$ particles with a single interaction potential for $|rr⟩$, except in case (b) where the pair state is too perturbed. The error bars show the standard error of the mean.

Figure 5

Influence of $\theta$, $B$, $E$ on the mapping onto a spin-$1/2$ system. Calculated probability of double excitations at long times (see text) as a function of the magnetic field $B$ and the angle $\theta$. The interatomic distance is fixed at $R=6.5\mu \mathrm{m}$. The electric field is $E=0$ in (a) and chosen between 0 and $20\mathrm{mV}/\mathrm{cm}$ such that the probability for two Rydberg excitations is maximized in (b).

Figure 6

Dynamics of an ensemble of atoms under Rydberg excitation. (a) 8-atom ring with a nearest neighbor spacing of $6.5\mu \mathrm{m}$. The shaded ellipse illustrates the range of the anisotropic blockaded region $U>\hslash \mathrm{\Omega }$. (b) Evolution of the Rydberg fraction $f_R$ with the pulse area $\mathrm{\Omega }\tau$ for $B=6.9\mathrm{G}$. The inset shows the probability $P_{5+}$ to observe configurations with at least 5 excitations. At large times, the experimental points systematically lie above the results of a simulation of the corresponding spin-$1/2$ model (solid line). (c) Same parameters with $B=3.5\mathrm{G}$. (d) Square lattice of $7\times 7$ traps (lattice spacing $6.1\mu \mathrm{m}$). The blockade extends over nearest and next-nearest neighbors. (e) Evolution of the Rydberg fraction for $B=6.9\mathrm{G}$. Here the data show a slow increase in $f_R$ at long times, while the spin-$1/2$ model predicts a saturation. (f) For $B=3.5\mathrm{G}$, the agreement with the spin-$1/2$ model becomes very good. All figures: error bars depict the standard error of the mean and are often smaller than the symbol size.