Spin-resolved microscopy of $^{87}$Sr SU($N$) Fermi-Hubbard systems

Abstract

Quantum-gas microscopes provide direct access to the phases of the Hubbard model, bringing microscopic insight into the complex competition between interactions, SU(2) magnetism, and doping. Alkaline-earth(-like) fermions extend this spin-1/2 paradigm by realizing higher symmetries and giving access to SU(N) Hubbard models, with rich phase diagrams to be unveiled. Despite its fundamental interest, a microscopic exploration of SU(N) quantum systems has remained elusive. Here we report the realization of a quantum-gas microscope for fermionic $^{87}$Sr. Our imaging scheme, based on cooling and fluorescence on the narrow intercombination line at 689 nm, enables spin-resolved single-atom detection. By implementing a spin-selective optical pumping protocol, we determine the occupation of each of the 10 spin states in a single experimental realization, a crucial capability for probing site-resolved magnetic correlations. We benchmark our method by observing single-particle Larmor precession across the full spin-9/2 ground-state manifold. These results establish $^{87}$Sr quantum-gas microscopy as a powerful approach to study exotic magnetism in the SU(N) Fermi-Hubbard model, and provide a new detection tool for studies in quantum simulation, computation, and metrology.

Figures (8)

• Figure 1: Microscopy of fermionic 87Sr atoms in an optical lattice. (a) Side-view schematic of the setup. Atoms are trapped in a square optical lattice at the center of a glass cell. A coil pair produces a vertical magnetic field $\vec{B}$ aligned with the lattice polarization, $\vec{\epsilon}_{\text{lattice}}$. A 689nm imaging beam enters horizontally with linear polarization, and scattered light is collected by a 0.5-NA objective. A circularly polarized optical-pumping beam is sent vertically. (b) Schematic of an SU(10) Fermi-Hubbard system with tunneling rate $t$ and on-site interaction energy $U$. Only $m_F=-9/2$ (red) atoms are addressed by the imaging light. (c) Energy level diagram of the ground and excited 3P1$F'=11/2$ state, showing the splitting of their Zeeman sublevels. The imaging light has $\sigma^+$ and $\sigma^-$ components but only resonantly addresses the $^1\text{S}_0 \ket{F=9/2, m_F=-9/2}\to \,^3\text{P}_1 \ket{F'=11/2, m_{F'}=-11/2}$ transition, which satisfies the condition for attractive Sisyphus cooling (inset, left panel). Two different transitions within the hyperfine structure of the 3P1 manifold are used for optical pumping and imaging (inset, right panel). (d) Consecutive fluorescence images of a thermal cloud with 300 atoms. Left image: atoms in state $m_F=-9/2$. Right image: atoms in all states taken after a set optical pumping pulses, showing a ten-fold increase in atom number.
• Figure 2: Narrow-line spin-resolved optical pumping. (a) Procedure to characterize the optical pumping fidelity. Left: a reference picture of a spin-polarized cloud is taken via narrow-line fluorescence, represented with a glowing spot. Center: the population of the initial state is depumped through an optical pumping pulse of $\sigma^+$ polarization addressing the $m_{F'} =-7/2$ state. A subsequent image shows practically no atoms, highlighting the spin-resolved nature of the detection. Right: two optical pumping pulses of $\sigma^-$ polarization retrieve the atomic population back to the $m_{F} =-9/2$ state and the cloud is imaged again. (b) Optical pumping with an increasing number of pulses $p$, which bring the atoms to different $m_F$ states and subsequently retrieve them in the stretched $m_F=-9/2$ state (insets). The normalized overlap $\mathcal{O}$ decreases with $p$ (circles) and yields two distinct optical-pumping fidelities $\mathcal{F}_\text{OP,1}$ and $\mathcal{F}_\text{OP,2}$ (boxes). $\mathcal{F}_\text{OP,1}$ is determined by comparing the $p=3$ value to the pinning fidelity $\mathcal{F}_\text{pin}$ (red dashed and gray dotted lines). $\mathcal{F}_\text{OP,2}$ is extracted from a linear fit to the data for $p\geq3$ (dark red solid line).
• Figure 3: Spin-resolved microscopy of the 10 states of 87Sr. (a) The imaging protocol consists of 10 detection blocks, one per spin state, carried out sequentially from the $m_F=-9/2$ to the $m_F=+9/2$ state. Each detection block starts by optically pumping the target spin state to the $m_F=-9/2$ state, followed by spin-resolved fluorescence imaging, and concludes with spin-removal of the population ihe acan $m_F=-9/2$ by means of a blue-detuned pulse on the imaging transition. (b) Merging the reconstructed images yields the spin-resolved occupation for all 10 spin states in a single experimental sequence. Below are the 10 raw snapshots, with the colormap indicating the corresponding spin. The images correspond to $35\times35$ lattice sites containing $253$ atoms evenly distributed among the 10 spin states. (c) Image-to-image detection coincidences between the different images, $N_{\alpha,\beta}$. The 10 bars on the diagonal indicate the spin populations. The presence of off-diagonal coincidences corresponds to the multi-detection events, which are indicated with gray shaded squares in (b). They are primarily caused by off-resonant scattering of the trapping light leading to spin depolarization (see main text).
• Figure 4: Microscopic observation of Larmor precession dynamics in a spin-9/2 system. (a) Scheme of the precession sequence. 1. Preparation: we optically pump all atoms in $m_F = -9/2$ and take a reference image. 2. Precession: a sudden rotation of the magnetic field initiates Larmor precession dynamics. 3. Measurement: after a certain precession time, $t_{\text{p}}$, we quench back the field to its original orientation and perform the spin-resolved imaging protocol. (b) Sample snapshots of the reconstructed spin occupation at different precession times, $t_\text{p}$. The spin evolves from $-9/2$ to $+9/2$ over $T/2$ with $T=$ 40.16(6)ms, from which we determine $B_y=$ 135.0(2)m. (c) Top: Normalized overlap of the image corresponding to each spin, $\mathcal{O}_{m_F}$. Bottom: Total normalized overlap, $\mathcal{O}_\text{tot}$. The solid line shows theoretical predictions accounting for all relevant infidelities, with the period $T$ as the only fitting parameter. The shaded region indicates the effect of the uncertainties in the infidelities. (d) Comparison of the measured $\mathcal{O}_{m_F}$ with theory (solid lines). Data for each $m_F$ are offset by $9/2-|m_F|$ for clarity.
• Figure 5: Atomic polarizability for the 1S0 (black line) and 3P1$F=11/2$ states (red lines) as a function of wavelength for $\pi$-polarized trapping light. The gray dotted-dashed line indicates the trapping wavelength in our experiment, 813.4nm. The inset shows schematically the potential of a certain lattice site for the different states. The polarizabilities are shown in atomic units (a.u.).