Optical nuclear electric resonance as single qubit gate for trapped neutral atoms

TL;DR

This work introduces optical nuclear electric resonance (ONER) as a fast, high-fidelity single-qubit gate for nuclear spins in neutral atoms, focusing on $^{87}$Sr. By periodically modulating the electronic environment with an amplitude-modulated laser, ONER modulates the electric field gradient and nuclear quadrupole interaction to drive nuclear-spin transitions, enabling $ abla m_I = \pm 1$ and $\pm 2$ channels in a Paschen-Back-like regime. Through detailed simulations of the $^1S_0\rightarrow ^3P_1$ transition and a Floquet analysis, the authors demonstrate spin flips between $m_I=-9/2$ and $m_I=-5/2$ with nuclear Rabi frequencies $\Omega_N/2\pi$ exceeding $10\,\mathrm{kHz}$ and fidelities above $99.9\%$, robust to typical experimental noise. The results indicate ONER can accelerate nuclear-spin operations beyond current Raman/NMR-based approaches, offering scalable, high-resolution control for quantum memories and other neutral-atom quantum technologies, with potential extension to qudit encoding and other atomic species.

Abstract

The precise control of nuclear spin states is crucial for a wide range of quantum technology applications. Here, we propose a fast and robust single-qubit gate in $^{87}$Sr, utilizing the concept of optical nuclear electric resonance (ONER). ONER exploits the interaction between the quadrupole moment of a nucleus and the electric field gradient generated by its electronic environment, enabling spin level transitions via amplitude-modulated laser light. We investigate the hyperfine structure of the 5s$^2$~$^1S_{0}\rightarrow{}$~5s5p~$^3P_1$ optical transition in neutral $^{87}$Sr, and identify the magnetic field strengths and laser parameters necessary to drive spin transitions between the $m_I$ = -9/2 and $m_I$ = -5/2 hyperfine levels in the ground state. Our simulations show that ONER could enable faster spin operations compared to the state-of-the-art oscillations in this 'atomic qubit'. Moreover, we show that spin-flip operations exceeding 99.9\% fidelity can be performed even in the presence of typical noise sources. These results pave the way for significant advances in nuclear spin control, opening new possibilities for quantum memories and other quantum technologies.

Figures (6)

• Figure 1: Schematic illustration of the optical nuclear electric resonance (ONER) protocol applied to the level structure of the (5s$^2$) $^1S_0 \rightarrow$ (5s5p) $^3P_1$ optical transition in $^{87}$Sr, in a Paschen-Back-like regime. The most relevant levels for ONER are plotted in black, less relevant levels are illustrated in gray. A detuned, amplitude-modulated laser field with period $T$ and electronic Rabi frequency $\Omega_\mathrm{E}/2\pi$ drives the system, resulting in an adiabatically modulated occupation of the electronically excited state, illustrated in the gray box. In the $^1S_0$ ground state, the magnetic nuclear spin quantum number $m_I$ remains well-defined. However, in the $^3P_1$ excited states, the non-zero nuclear quadrupole interaction (NQI) tensor $Q$ (represented by an 'orbital-like' tensor plot surface) and the total magnetic dipole moment $\bm{A}$ of the electrons (represented by a vector) couples to the nuclear spin states. A modulated occupation of the excited state, $P_{3P1}(t)$, leads to a modulation of the NQI tensor and the magnetic dipole moment vector of the electrons, which couple to the nuclear spin $I$ via quadrupole interaction and magnetic dipole interaction, respectively. This enables hyperfine nuclear spin transitions in the $^1S_0$ ground state with a nuclear Rabi frequency $\Omega_\mathrm{N}/2\pi$, illustrated by the blue ellipse enclosing the qubit spin levels.
• Figure 2: Magnetic field dependence of the hyperfine states of the $^1S_0$ and $^3P_1$ states. The upper plot displays the energy levels in the ground 5s$^2$$^1S_0$ state (small inset) and the excited 5s5p $^3P_1$ state of $^{87}$Sr for different magnetic fields $B$. The highlighted states with the lowest and third lowest energy both in the ground state and in the excited state are of main interest in our study. The lower plot displays the projections of the three lowest energy states in the excited $^3P_1$ manifold of $^{87}$Sr (one graph for each state, lowest energy state at the bottom) onto the $\ket{^3P_1,m_J,m_I}$ basis vectors. We only find a small mixing of the quantum numbers $(m_J,m_I)$ in the $^3P_1$ state, indicating that they are good quantum numbers. Only the non-zero components are shown. Line style and line color conventions introduced in this plot are used throughout the manuscript.
• Figure 3: Analysis of Rabi oscillations on the $\ket{^1S_0,0,-9/2}$ to $\ket{^1S_0,0,-5/2}$ transition induced by the ONER method. (a) Probability $P_{-5/2}$ (blue lines) for flipping the nuclear spin of $^{87}$Sr in the electronic ground state from $m_I=-9/2$ to $m_I=-5/2$, and observed spin Rabi frequencies $\Omega_\mathrm{N}$ for this transition (red crosses), plotted as a function of the modulation period $T$ of the excitation laser. A total of $6\times{}4=24$ graphs are compiled into a single figure, with the outermost tabular arrangement indicating the chosen values of the electronic Rabi frequency $\Omega_\mathrm{E}/2\pi$, and the magnetic field $B$, respectively. The graphs reveal the amplitude modulation periods $T$ at which spin flips occur, identifiable by equidistant dips and characterized by linearly decreasing nuclear Rabi frequencies. Note the similarities in the graphs along diagonals, indicating similar behavior of the excited state occupation under these conditions. Increasing laser intensity, i.e. larger $\Omega_\mathrm{E}/2\pi$, yields sharper and more tightly packed absorption features, yielding less robust transitions with respect to the modulation period $T$. Incomplete transitions, i.e. dips that do not reach zero, are mainly caused by limited resolution in $T$, limited overall simulation time $t\leq50\;\mathrm{\mu s}$, or, in rare cases, by spin mixture. (b) Time evolution of the spin level occupation $P$ for selected transitions between the $m_I=-9/2$ and $m_I=-5/2$ levels (absorption features marked by black triangles in \ref{['fig:trans']}). If two absorption features are selected, the lower modulation period is shown below. The sum over the remaining occupations $m_I\notin\{-9/2,-5/2\}$ (other states) reveals the spin mixing, which occurs if the nuclear Rabi frequency is too large with respect to the effective spin level splitting. We observe high-fidelity Rabi oscillations across various magnetic fields and laser parameters, particularly also for parameters readily achievable in standard laboratories.
• Figure 4: Close-up of the Rabi oscillations for parameter combinations: (200 G, 60 MHz, first peak), (300 G, 40 MHz, first peak), and (1000 G, 40 MHz, first peak) from left to right. The population $P$ is shown over a time interval from 0 to 30 $\mathrm{\mu s}$. At high laser intensity, the excited state is strongly populated. Besides the global Rabi oscillation caused by the amplitude modulation that 'drives' the system further towards the transition, small local oscillations appear. These are caused by the amplitude modulation and the occupation of the excited state, as visible in the left plot. Additionally, hyperfine interaction can lead to oscillations within one external modulation period, due to small mixing of spin states, as can be seen in the middle plot. For a low population of the excited state and a small amplitude modulation period, smooth Rabi oscillations are achieved, as exemplified in the right plot.
• Figure 5: Comparison of the results from the Floquet analysis with the full time evolution (solid blue line, left axis). The state mixture of $\ket{^1S_0,0,-9/2}$ and $\ket{^1S_0,0,-5/2}$ in the Floquet mode basis is given by Equation \ref{['eq:state mixture']} (dashed orange line, right axis). Two cases are considered: $B=300\;\mathrm{G}$ and $\Omega_\mathrm{E}/2\pi = 40\;\mathrm{MHz}$ (top) and $B=500\;\mathrm{G}$ and $\Omega_\mathrm{E}/2\pi = 20\;\mathrm{MHz}$ (bottom). The peak shape and position are well described by the state mixture measure $\mathcal{M}$ (Equation \ref{['eq:state mixture']}) for the Floquet mode basis.