Kinetically-induced bound states in a frustrated Rydberg tweezer array

Abstract

Understanding how particles bind into composite objects is a ubiquitous theme in physics, from the formation of molecules to hadrons in quantum chromodynamics and the pairing of charge carriers in superconductors. The formation of bound states usually originates from attractive interactions between particles. However, the binding can also arise purely from the motion of dopants due to kinetic frustration, which is potentially related to unconventional pairing in moiré materials. Here, we report the first direct observation of kinetically-induced bound states between holes and magnons using a Rydberg atom array quantum simulator of the bosonic $t$-$J$ model in frustrated ladders and 2D lattices. First, we demonstrate the formation of mobile one-hole-one-magnon bound states. We then construct three-particle one-hole-two-magnon bound states and reveal the underlying binding mechanism by observing kinetically-induced singlet correlations. Finally, we investigate how mobile dopants structure their magnetic environment in a spin-balanced 2D triangular lattice, showing that a hole induces $120^\circ$ antiferromagnetic order, while a doublon dopant generates in-plane ferromagnetic correlations. Our results demonstrates compelling evidence of kinetically-induced binding, opening a new avenue to understand novel pairing mechanisms in correlated quantum materials like superconductors in moiré superlattices.

Figures (6)

• Figure 1: Experimental setup and the mechanism of kinetically-induced binding.a, Experimental setup and sequence. The spin-down $\ket{\downarrow}$, hole $\ket{h}$, and spin-up $\ket{\uparrow}$ states are mapped onto the $\ket{60S_{1/2}, m_{J}=1/2}$, $\ket{60P_{3/2}, m_{J}=1/2}$, and $\ket{61S_{1/2}, m_{J}=1/2}$ Rydberg levels of $^{87}$Rb atoms. The atoms are arranged in a triangular ladder geometry, characterized by intra-leg spacing $a$ and inter-leg spacing $h$. The sequence consists of (i) initial state preparation using local light shifts, (ii) time evolution during which we ramp down the light shift, and (iii) measurement of final state occupations. The populations in the spin-down ($\hat{n}^{\downarrow}$), hole ($\hat{n}^{h}$), and spin-up ($\hat{n}^{\uparrow}$) bases are measured in separate experimental runs, where the state at each site is mapped to the presence or absence of an atom. The image below shows a single shot from a spin-down measurement capturing a bound pair. b, Fluorescence image of a 37-site 2D triangular lattice. c, Mechanism of kinetically-induced binding. (i) A magnon forms a singlet state with adjacent spins surrounding a hole, creating a bound state. (ii) The hole moves to magnon's position, an internal dynamic of the bound state that remains kinetically unfrustrated. (iii) The hole hops to a neighboring spin-down site, effectively moving the bound state by one lattice site. d, Theoretical calculation of the binding energy ($E_b/t$) versus the ladder's aspect ratio ($h/a$) for a single hole bound to one magnon (1H1M, blue) and two magnons (1H2M, red). The dot marks a set of parameters used in the experiments.
• Figure 2: Kinetically-induced one-hole-one-magnon bound state in ladders.a, Initial state on a 19-site triangular ladder, with a hole (white) and magnon (red) on adjacent sites. b, Time evolution of the hole-magnon connected correlations, $\langle n_i^h n_j^\uparrow \rangle^{s}_{c}$. c, Cut of the COM probability distribution, $\mathcal{P}_{\text{COM}}$, along the center of the ladder (see inset). Experimental data (blue circles) is compared to the corresponding $\sin^2$ distribution (solid grey line). The shaded area represents numerically simulated distribution including errors. d, Initial state on a 37-site 2D triangular lattice. e, Hole-magnon connected and symmetrized correlations in the hole's reference frame, $C^{c, s}(\mathbf{d})$, at time $T=5\,\mu\text{s}$. f, The delocalized COM probability distribution, $\mathcal{P}_{\text{COM}}$, at $T=5\,\mu\text{s}$ of the 1H1M bound pair. g, Top: Hole-magnon non-connected and symmetrized correlations $C^s(d)$ versus distance for ladders (blue circles) and 2D arrays (tan circles). Solid lines represent exponential fit of the correlation. Shaded areas are the numerical simulation including errors. Bottom: Normalized connected correlations $C^{c,s}(d)/C^{c,s}(1)$. Dashed lines correspond to the theoretical ground state. The inset shows the dominant kinetic contributions of the binding, and the red plaquettes are kinetically frustrated while the green one's frustration is alleviated. Error bars denote one standard error estimated via bootstrap, and are smaller than marker size.
• Figure 3: Observation of a 1H2M bound state and kinetically-induced antiferromagnetism.a, Time evolution of the magnon-magnon correlation, $\braket{\hat{n}^\uparrow_i \hat{n}^\uparrow_j}$. The system is initialized at $T=0.0\,\mu\text{s}$ with two localized magnons as shown in the inset. As the system evolves, the correlations delocalize across the ladder but remain concentrated near the main diagonal indicating that the two magnons move together as a bound pair. The gray dashed lines indicate the position of strongest correlations of ground state. b, Hole-magnon connected correlations $\braket{\hat{n}_i^h \hat{n}_j^\uparrow}^{s}_{c}$ at different times. c, Probabilities of particle configurations at $T=6\,\mu\text{s}$. The left panel displays the probability of finding the two magnons in various relative arrangements, as measured in the $\uparrow$ basis, confirming their tendency to remain close. The right panel plots the three-body correlator $\braket{\not{\hat{n}}_{\downarrow}\not{\hat{n}}_{\downarrow}\not{\hat{n}}_{\downarrow}}$, measuring the likelihood of different configurations of the 1H2M state. The hole's position (dashed circle) is inferred by comparing these configurations with the two-magnon arrangements shown in the left panel. d, Spatial map of hole-induced magnetic order. The plots show the connected three-body hole-spin-spin correlators for the transverse, $\braket{\hat{n}^h \hat{S}^x \hat{S}^x}_c^s$ (left), and longitudinal, $\braket{\hat{n}^h \hat{S}^z \hat{S}^z}_c^s$ (right), components. At $T=2.0\,\mu\text{s}$, strong antiferromagnetic correlations (blue links) develop on the bonds around the hole's initial location (red dot), demonstrating the kinetically-induced binding requires formation of singlets around the hole. e, Time evolution of the nearest-neighbor hole-spin-spin correlators in the hole frame, $C_{hxx}^{c,s}(\mathbf{d}_i,\mathbf{d_j})$ (transverse) and $C_{hzz}^{c,s}(\mathbf{d}_i,\mathbf{d_j})$ (longitudinal), summed over the sites adjacent to the hole. Dashed (shaded) lines indicate ideal simulation (simulation including experimental imperfections). Error bars denote one standard deviation extracted via bootstrap.
• Figure 4: Spin bag induced by mobile dopants in a 2D triangular lattice.a, The top panel shows the initial state of the 2D triangular lattice, which can be doped with a single hole or particle (white circle) in a background of spin-up (red) and spin-down (blue) atoms. The bottom panel illustrates the quasi-adiabatic protocol, showing the time-dependent light shift used to create a hole (positive values, red) or a particle (negative values, blue). The correlations are measured at $T=2\,\mu\text{s}$. b, c, Spatial maps of connected three-body correlators for hole doping in the hole frame, where $k,l$ are the index of sites. Both the longitudinal correlator $\braket{\hat{n}^{h} \hat{S}^{z} \hat{S}^{z}}_c^s$ (b) and the transverse correlator $\braket{\hat{n}^{h} \hat{S}^{x} \hat{S}^{x}}_c^s$ (c) reveal strong antiferromagnetic correlations (blue) on the bonds surrounding the hole. d, Radially averaged correlators for hole doping, plotted as a function of distance $r$ from the hole. Here $r$ is defined as the distance between the hole and the center of bonds, as depicted in the inset. e, f, Spatial maps of correlators for a doublon dopant in the particle frame. The particle induces strong ferromagnetic transverse correlations $\braket{\hat{n}^{h} \hat{S}^{x} \hat{S}^{x}}_c^s$ (f), while the longitudinal correlations $\braket{\hat{n}^{h} \hat{S}^{z} \hat{S}^{z}}_c^s$ (e) are significantly suppressed. g, Radially averaged correlators for doublon doping, highlighting the transverse ferromagnetic correlations and a weak longitudinal correlations. Error bars denote one standard error estimated via bootstrap, and are smaller than marker size.
• Figure 5: Kinetic magnetism. Experimentally measured structure factor $|S(\mathbf{k})|$ of two-body correlation $\braket{\hat{S}^x_i\hat{S}^x_j}_c$ for hole doping (a, b) and doublon doping (c, d), where $\mathbf{k}$ is the momentum. The white hexagon outlines the first Brillouin zone. a, For hole doping at $T=0\,\mu\text{s}$, the initial state shows weak magnetic correlations. b, After evolving for $T=2\,\mu\text{s}$, correlations emerge at the corners of the Brillouin zone (K points), signaling the formation of $120^{\circ}$ antiferromagnetic order induced by the mobile hole. c, For doublon doping at $T=0\,\mu\text{s}$, the system has negligible magnetic correlations. d, By $T=2\,\mu\text{s}$, a strong peak develops at the center of the Brillouin zone ($\Gamma$ point), indicating the emergence of ferromagnetic order driven by the particle's motion.