Engineering Zeeman-manifold quintets using state-dependent light shifts in neutral atoms

TL;DR

The paper introduces a general method to engineer qudits in Zeeman manifolds by leveraging a large linear Zeeman shift together with a state-dependent tensor light shift, lifting degeneracies and placing adjacent level spacings into the RF domain for coherent control. A concrete SU(5) quintet is proposed in the ${}^{88}$Sr ${}^3P_2$ manifold, with a σ^- optical tweezer-induced tensor shift enabling site- and state-selective RF operations, fast multi-photon initialization, and rapid readout. Numerical simulations under realistic parameters show initialization fidelities around $\mathcal{F} \sim 0.99$ in ~1 μs, single-qudit gate fidelities around $\mathcal{F} \sim 0.99$ with $t_{π} \sim 2.5~\mu$s, and fast destructive imaging below $10~\mu$s, enabling complete quintet readout within ~100 μs. The work establishes a broadly applicable framework for high-fidelity qudit control in Zeeman sublevels and points to scalable quantum technologies based on the $^3P_2$ manifold in strontium, with pathways to universal gates and two-qudit interactions via Rydberg coupling.

Abstract

We present a general method for engineering qudits through individually addressable transitions between Zeeman sublevels, achieved by combining a large linear Zeeman shift with a state-dependent light shift. This approach lifts the degeneracy between adjacent states while simultaneously tuning their energy splittings into the radio-frequency (RF) domain, enabling coherent manipulation within the Zeeman manifold using experimentally accessible drive frequencies. As a concrete realization, we investigate the implementation of an $SU(5)$ \emph{quintet} encoded in the Zeeman sublevels of the long-lived $^3\mathrm{P}_2$ state of neutral $\mathrm{^{88}Sr}$ atoms confined in far-detuned, $σ^{-}$-polarized optical tweezers. Using realistic experimental parameters, we numerically demonstrate full control of the \emph{quintet} manifold, including initialization into a specific $SU(5)$ basis state via a multi-photon transfer, coherent state- and site-selective single-qudit rotations driven by RF fields, and fast state-selective optical readout. Our simulations predict state-preparation fidelities of $\mathcal{F} \simeq 0.99$ within $\sim 1~μ\rm{s}$, single-qudit gate fidelities of $\mathcal{F} \simeq 0.99$ with $π$-pulse durations of $\sim 2.5~μ\rm{s}$, and fast destructive imaging with durations below $10~μ\rm{s}$. These results establish a broadly applicable framework for high-fidelity control of Zeeman sublevel-encoded qudits and highlight the $^3\mathrm{P}_2$ manifold in strontium as a promising platform for scalable qudit-based quantum technologies.

Figures (16)

• Figure 1: Linear Zeeman shifts, shown in the left Breit-Rabi level diagram, lift the degeneracy between the five targeted Zeeman sublevels and tune the energy differences between neighboring states, by the applied magnetic field, into a range that is easily accessible with standard RF technology ($\sim 100~\mathrm{MHz}$). The energy differences (or corresponding RF drive frequencies) between neighboring states, indicated by dashed black arrows, remain degenerate under the Zeeman shift alone. By combining the constant Zeeman shift with a state-dependent light shift, induced through the tweezer beams and linearly dependent on the intensity (illustrated in the right panel), the degeneracy is lifted, enabling individual RF drive frequencies for all four transitions (solid arrows).
• Figure 2: (a) Level diagram of $\mathrm{^{88}Sr}$ showing the blue imaging and red cooling transitions Pucher2025_88Sr_Reference, together with the states involved in the multi-photon transfer used for initial-state preparation and readout of the $^3$P$_2$ Zeeman manifold quintet. (b) Proposed experimental setup, consisting of a high-resolution imaging system for preparing $\sigma^-$-polarized optical tweezers with single-atom occupancy at high magnetic field. An RF antenna generates the field required to couple neighboring Zeeman sublevels, which serve as computational basis states. Additional laser beams are indicated for imaging, cooling, and the multi-photon transfer, where $\varphi_{689\mathrm{nm}}$ and $\varphi_{688\mathrm{nm}}$ denote the in-plane angles between the coupling beams chosen to minimize the net momentum transfer, $\hbar |\mathbf{k}_{\mathrm{eff}}| \approx 0$.
• Figure 3: (a) Envisioned timing sequence following initial single-atom preparation and cooling. The full experimental cycle is divided into three stages: State preparation, computation, and final readout. (b) State preparation and readout: atoms are transferred from the ground state $^1$S$_0$ to the $^3$P$_2$, $m_J=1$ state via a multi-photon process, thereby initializing the system in one of the computational basis states. As an example, the bare detunings $\Delta_1$, $\Delta_2$ are shown for this $\ket{g}$$\leftrightarrow$$\ket{1-5}$ transition. $\Delta_3$ is defined in Eq. (\ref{['eq:multi_con']}) but not shown in the figure. After computation, the same multi-photon transfer selectively transfers population from the $^3$P$_2$ manifold back to the ground state, where each state is destructively imaged. For clarity, only $\pi$-polarized light for the $^1$S$_0$$\leftrightarrow$$^3$P$_1$ transition and the $^3$P$_1$, $m_J=0$ intermediate state are depicted (also energy levels are not to scale); however, all polarizations and possible transitions are included in the calculations. Note that the $^3$S$_1$, $m_J=0$ to $^3$P$_1$, $m_J=0$ transition is forbidden by angular-momentum selection rules. A combination of Zeeman and tensor light shift lifts the degeneracy of the $^3$P$_2$ Zeeman sublevels, enabling individual addressing of transitions (black solid arrows) using an RF field.
• Figure 4: State preparation of $^3$P$_2, m_J=1$. Bold left axis and lines: Dynamics of the $^1$S$_0$ ($\ket{g}$) $\leftrightarrow$$^3$P$_2, m_J=1$ ($\ket{4}$) transition, including the Zeeman substructure, under the multi-photon resonance condition for $^3$P$_2, m_J=1$ [Eq.\ref{['eq:multi_con']}]. Faint right axis and colored lines: magnified view, showing rapid oscillations of the $^3\mathrm{P}_1$ populations (not resolvable on this time scale), which are the main limiting factor for the achievable fidelities. Parameters: $\Omega_1/2\pi =$ 14.8 MHz, $\Omega_2/2\pi =$ 3520 MHz, $\Omega_3/2\pi$ = 252 MHz, $\Delta_1/2\pi$ = 60 MHz, $\Delta_2/2\pi$ = 7500 MHz, $\Delta_3/2\pi$$\approx$ 7354.41 MHz, B = 100G and $P_0$ = 30 mW.
• Figure 5: $\pi$-pulse fidelity $\mathcal{F}_\pi$ averaged over 100 simulations for transfer from the computational states to the ground state or vice versa within 1 µs, depending on power fluctuations $f_P$ in both the optical tweezers and the coupling lasers. Error bars represent the standard error. Parameters used can be found in Table \ref{['Tab:params']}.