Engineering frustrated Rydberg spin models by graphical Floquet modulation

Abstract

Arrays of Rydberg atoms interacting via dipole-dipole interactions offer a powerful platform for probing quantum many-body physics. However, these intrinsic interactions also determine and constrain the models -- and parameter regimes thereof -- for quantum simulation. Here, we propose a systematic framework to engineer arbitrary desired long-range interactions in Rydberg-atom lattices, enabling the realization of fully tunable $J_1$-$J_2$-$J_3$ Heisenberg models. Using site-resolved periodic modulation of Rydberg states, we develop an experimentally feasible protocol to precisely control the interaction ratios $J_2/J_1$ and $J_3/J_1$ in a kagome lattice. This control can increase the effective range of interactions and drive transitions between competing spin-ordered and spin liquid phases. To generalize this approach beyond the kagome lattice, we reformulate the design of modulation patterns through a graph-theoretic approach, demonstrating the universality of our method across all 11 planar Archimedean lattices. Our strategy overcomes the inherent constraints of power-law-decaying dipolar interactions, providing a versatile toolbox for exploring frustrated magnetism, emergent topological phases, and quantum correlations in systems with long-range interactions.

Figures (4)

• Figure 1: Schematic illustration of $J_1$-$J_2$-$J_3$ dipolar interactions between Rydberg atoms arrayed in programmable geometries, and the corresponding atomic levels relevant for controlling the interaction strengths. The two spin-$1/2$ states are encoded in Rydberg levels, with $\ket{\downarrow} = \ket{nS}$ and $\ket{\uparrow} = \cos\theta\ket{n'S}- \sin\theta\ket{nP}$. The microwave-dressed superposition state $\ket{\uparrow}$ is used to realize the XXZ model, achieved by coupling $\ket{nP}$ to $\ket{n'S}$ using a microwave field with Rabi frequency $\Omega_{\text{mw}}$ and detuning $D$ (with $2\theta = \textrm{arctan}(2\Omega_{\text{mw}}/D)$). Addressing beams couple the Rydberg state $\ket{\downarrow}$ to an intermediate state $\ket{p}$ off-resonantly, inducing a local energy shift (dashed line). In the limit of large detuning $\Delta_{\text{addr}} \gg \Omega_{\text{addr}}$, this shift goes as $\delta \approx \Omega_{\text{addr}}^2 / \Delta_{\text{addr}}$. Precise control of the addressing beams allows for the design of desired $J_1$-$J_2$-$J_3$ interactions.
• Figure 2: Renormalization of $J_1$-$J_2$-$J_3$ spin interactions on the kagome lattice via dual-frequency modulation. Site-resolved periodic modulations (amplitude $\delta_q$, frequency $\omega_q$, $q=1,2$) imprint phase (color) patterns $\phi_{mq}$ to engineer $J_1$-$J_2$-$J_3$ couplings independently. (a) Tuning $J_2/J_1$: modulation at $\omega_1$ assigns phases $\phi_{m1} = 0$ (orange), $2\pi/3$ (gray), $4\pi/3$ (blue). Nearest- ($J_1$) and third-neighbor ($J_3$) interactions between distinct colors are renormalized by $\mathcal{N}_1$, while the $J_2$ interactions of same-color pairs remain unmodified. (b) Tuning $J_3/J_1$: modulation at $\omega_2$ applies phases $\phi_{n2} = 0$ (red), $2\pi/3$ (green), $4\pi/3$ (yellow). First- ($J_1$) and second-neighbor ($J_2$) couplings between differing colors acquire a factor of $\mathcal{N}_2$, while preserving $J_3$ (shared-color third neighbors). Combined, these protocols independently adjust the interaction ratios $J_2^{\text{eff}}/J_1^{\text{eff}}$ and $J_3^{\text{eff}}/J_1^{\text{eff}}$, enabling controlled exploration of frustrated regimes.
• Figure 3: The required frequencies for engineering long-range interactions in 11 Archimedean lattices. All lattices are vertex-transitive, with their mathematical descriptions (in brackets) provided by sequences of numbers $n_i$, separated by commas (e.g., $n_1,n_2,\cdots,n_r$), representing the number of vertices in the polygons surrounding each vertex of the lattice. Below each lattice graph, the number of modulated frequencies needed for controlling $J_1$-$J_2$ and $J_1$-$J_2$-$J_3$ interactions is listed.
• Figure 4: Strategies for controlling $J_1$-$J_2$-$J_3$ spin interactions on the honeycomb lattice via multifrequency modulation. Each colored circle represents a distinct local modulation applied to the corresponding lattice site, expressed as $\sum_{m,q} \delta_q \sin(\omega_q t + \phi_{mq})$, where $m$ indexes the lattice sites, $q$ indexes the modulation frequencies, and $\phi_{mq}$ denotes the modulation phase associated with site $m$ and frequency $q$. (a) The two processes for modulating $J_1$-$J_2$-$J_3$ interactions. In process I, a two-color pattern is designed to tune $J_2$/$J_1$: modulation at frequency $\omega_1$ assigns phases $\phi_1'$ for blue, and $\phi_2'$ for orange. $J_1$ and $J_3$ are renormalized by a factor $\mathcal{N}_1$ (Eq. \ref{['eqj']}) while $J_2$ is unchanged. In process II, a four-color pattern is designed to tune $J_3/J_1$, modulated by three different frequencies. (b) The designed modulation frequencies for the four-color lattice in process II. To ensure uniform modulation across each pair, we use three distinct frequencies $\omega_{2,A}$, $\omega_{2,B}$, and $\omega_{2,C}$. The corresponding modulation phases are listed in the table. The color pairings between modulated lattice sites are indicated, with a “$\checkmark$” denoting modulated interactions and a blank space indicating unmodulated ones. Each frequency modulates four connections, with the tiling ensuring uniform modulation across all six connections. (c) The corresponding color pattern for three different frequencies in process II, wherein we decompose process II into three color patterns and use black and white to represent the phase difference for each frequency modulation.