Measuring spectral functions of doped magnets with Rydberg tweezer arrays

Abstract

Spectroscopic measurements of single-particle spectral functions provide crucial insight into strongly correlated quantum matter by resolving the energy and spatial structure of elementary excitations. Here we introduce a spectroscopic protocol for single-charge injection with simultaneous spatial and energy resolution in a Rydberg tweezer array, effectively emulating scanning tunneling microscopy. By combining this protocol with single-atom-resolved imaging, we go beyond conventional spectroscopy by not only measuring the single-particle spectral function but also directly imaging the microscopic structure of the excitations underlying spectral resonances in frustrated $tJ$ Hamiltonians. We reveal resonances associated with the formation of bound magnetic polarons -- composite quasiparticles consisting of a mobile hole bound to a magnon -- and directly extract their binding energy, spatial extent, and spin character. Finally, by exploiting the spatial tunability of our platform, we measure the local density of states across different lattice geometries. Our work establishes Rydberg tweezer arrays as a powerful platform for spectroscopic studies of strongly correlated models, offering microscopic control and direct real-space access to emergent quasiparticles in engineered quantum matter.

Figures (6)

• Figure 1: Measurement of spectral functions by local hole injection.A, Holes and spins are encoded in Rydberg states of each atoms, coupled by the dipolar interaction. The resulting many-body spectrum is probed with the combination of a global microwave and local modulations of the energy level encoding the hole. B, The combined action of a global microwave and locally modulated light shifts induces the local injection of a hole at frequencies $\omega_{\rm MW}\pm \omega_{\rm LS}$. This local injection couples to all available states with arbitrary momentum $q$ at the corresponding energy (left panel). As a result, the protocol provides access to the local density of states (right panel). C, Effective Rabi frequency of the oscillation (inset) driven by the first sideband on a single atom, as a function of the modulation depth $\kappa =\Delta_\mathrm{LS}/\omega_\mathrm{LS}$, compared to the expected Bessel function $J_1(\kappa)$ (solid line). D, Spectroscopy of a two-atom system with only a global microwave. The hole can only be injected at zero momentum and thus only the $\ket{+}$ state is probed. E, Spectroscopy of the same system with one of the two site exposed to a modulated light shift. The resulting local sideband couples to both eigenstates. Vertical grey lines: expected positions of the $\ket{\pm}$ resonances. Solid lines: numerical simulations of an ideal system with an offset to account for detection and preparation errors. Error bars correspond to one standard deviation.
• Figure 2: Sideband spectroscopy of a triangular plaquette.A, Mechanism of kinetically-induced binding on a triangle. Left: A single hole with positive tunneling ($t>0$) experiences kinetic frustration. When a nearby magnon forms a singlet state with an adjacent spin, it effectively reverses the tunneling sign ($t^{\text{eff}} \approx -t < 0$). This relieves the frustration, lowering the kinetic energy. Right: Eigenstates and band structures of one hole hopping on 3 sites in a background forming a spin triplet (blue) or spin singlet (red). B, Sideband spectroscopy of the triangular plaquette starting from $\ket{\downarrow \downarrow \downarrow}$ (top) or$\ket{\downarrow \downarrow h}$. Without magnon two peaks at $-t$ and $2t$ correspond to accessible triplet states. With one magnon additional peaks at $t$ and $-2t$ appears associated to singlet states; the latter is the origin of the hole-magnon bound state in triangular lattices. B, Sideband spectroscopy with measurement along $x$ to reconstruct spin-spin correlations (see SM). Singlet (triplet) states showcase negative (positive) correlations. B, C, Solid lines: simulations of a perfect system with an offset to account for detection and preparation errors.
• Figure 3: Binding energy of a frustrated triangular ladder.A, Fluorescence image of a 10-site triangular ladder. B, Eigenstates of the interaction Hamiltonian with (left) and without (right) a magnon. The presence of the magnon next to the hole relieves the frustration allowing the formation of a hole-magnon bound state with binding energy $E_b$. C, Left: Spectroscopy of a 10-atom triangular ladder for the two initial states. Initial configurations are illustrated above and show the two modulated sites. We subtract an offset corresponding to the baseline due to experimental errors, measured far-detuned from the spectrum. Solid lines correspond to a simple phenomenological model, see main text. Right: Difference bewteen the two spectra on the left corresponding to the contribution of the bound states (purple) and the spectrum without magnon vertically shifted to remove the noise floor (blue). Both are fitted with a half-lorentzian (lines). D, Measured hole-magnon correlations, obtained from postselecting on two missing majority spins mu_kinetic_bound_state, for two values of the detuning $\delta_\mathrm{MW}$ corresponding to $E=-1.8\hbar t$ (bound part of the spectrum) and $E=-0.6\hbar t$ (continuum).
• Figure 4: Atomic scanning tunneling spectroscopy.A, 1D ring, B, triangular lattice, and C, kagome lattice. Left panels: Single-hole band structures for the different lattices calculated for the dipolar $1/r^3$ (solid lines) and nearest-neighbor tunnelings (dashed lines). For the 2D geometries, a cut along the $\Gamma$-K-M points of the Brillouin zone is shown (see inset B, left). Right panels: Red circles: Measured average number of holes $\langle \hat{N}^h \rangle$ as a function of the energy, which is proportional to the LDOS (error bars represent one standard error of the mean). We include a shift in the vertical axis to account for imperfect calibration. Insets show spatial maps of the atomic arrays where the color scale indicates the measured hole density $\langle \hat{n}^h_j \rangle$ at the energy shown by the gray line. The dashed circles identify the sites subjected to the local light-shift modulation, effectively acting as the "STM tip" for local hole injection. Solid lines represent theoretical LDOS from the band structure with a phenomenological Gaussian broadening to account for the probe's energy resolution. (A) shows an asymmetric density of states due to dipolar interactions; the triangular lattice (B) displays a singularity associated with saddle points at the M-points; and the kagome lattice (C) exhibits a strong peak at high energy, associated the the two peaks in the band structure due to finite energy resolution, and a peak at lower energy.
• Figure 5: Experimental setup for generating the local light-shift modulations.A, Two far detuned beams (offset and modulated) are focused onto the atoms to generate an offset negative light shift and a modulated positive light shift. B, The two beams are combined at a beam splitter before reflecting off a SLM. The SLM imprints a holographic phase pattern with a defined pupil size to address specific atoms in the array with nearly uniform intensity. The AOM, driven by two frequency tones, generates the time-periodic modulation of the light-shift potential.