Quantum simulation of the Sachdev-Ye-Kitaev model using time-dependent disorder in optical cavities

TL;DR

This work tackles the challenge of realizing the dense all-to-all disorder of the complex $SYK_4$ model in a laboratory setting by proposing a hybrid digital-analog protocol that densifies sparse disorder via Trotterized cycling of time-dependent realizations. A concrete cavity-QED implementation with a single-mode cavity and a speckle-patterned fermionic gas demonstrates how reduced-rank couplings can be summed over $R$ layers to approach the dense $c$SYK$_4$ dynamics, with a KL-divergence based diagnostic quantifying convergence. The paper develops a rigorous framework for convergence: the coupling distribution converges at a rate $\sim 1/R^2$ for a single layer and at least $1/R$ for the trotterized sum, and random-unitary-circuit analysis provides controlled error scaling for chaos probes such as the OTOC and the spectral form factor. Dissipation from photon loss and scattering is treated within a Lindblad/Schwinger–Keldysh formalism, showing that dissipation enters at the same large-$N$ order as unitary dynamics and is tunable via cooperativity, making near-term experiments feasible. Overall, the approach enables scalable quantum simulation of $c$SYK-like chaos and related disorder models on existing cavity-QED platforms, with practical implications for exploring holographic chaos and quantum gravity analogues in the laboratory.

Abstract

The Sachdev--Ye--Kitaev (SYK) model is a paradigm for extreme quantum chaos, non-Fermi-liquid behavior, and holographic matter. Yet, the dense random all-to-all interactions that characterize it are an extreme challenge for realistic laboratory realizations. Here, we propose a general scheme for densifying the coupling distribution of random disorder Hamiltonians, using a Trotterized cycling through sparse time-dependent disorder realizations. To diagnose the convergence of sparse to dense models, we introduce an information-theory inspired diagnostic. We illustrate how the scheme can come to bear in the realization of the complex SYK$_4$ model in cQED platforms with available experimental resources, using a single cavity mode together with a fast cycling through independent speckle patterns. The simulation scheme applies to the SYK class of models as well as spin glasses, spin liquids, and related disorder models, bringing them into reach of quantum simulation using single-mode cavity-QED setups and other platforms.

Figures (2)

• Figure 1: $\textbf{a)}$ Fermionic ${}^6$Li atoms (dark cloud) are trapped at the antinode of a longitudinal mode (red) of a single-mode optical cavity. The system is pumped from the side with a drive (not shown) near-resonant to the cavity field, allowing for atomic scattering from the pump into the cavity and vice versa. Via the use of a light-shift beam (blue), a random phase mask is projected onto the almost two-dimensional trapped cloud of atoms. The resulting speckle pattern with random intensity peaks (white) and valleys (dark blue) is shown above in the zoomed-in circles, together with illustrations of sparse two-body interactions. The speckle pattern is quickly cycled in time through $R$ independent realizations, which enhances the disordered rank-two cavity interactions to SYK-like chaotic interactions. $\textbf{b)}$ Atomic level structure, $|{g;e;\alpha}\rangle$ and respective transition frequencies, $\omega_{\rm a;b}$. Detuning the drive frequency $\omega_{\rm d}$ considerably from $\omega_{\rm a}$ enables adiabatic elimination of the excited states. The speckled intensity $\Omega_{\rm b}$ introduces a position-dependent AC-Stark shift of the excited states and additionally imprints a dependency on time onto the interactions via $\Delta_{\rm da}(t)$. $\textbf{c)}$ Random unitary representation of SYK$_4$. Each layer has a randomly selected number of four-Fermi interactions whose couplings are sampled from a sparse distribution. For visual reasons, we do not show couplings that would overlap with one another in a given layer, though in reality these can occur.
• Figure 2: a): Average error \ref{['eq:DU']} after Trotterizing the random unitary evolution as a function of $J\Delta t$, for $N = 8$ and different total evolution times. b) Average error on the SFF for different values of $N$ and $J \Delta t$, evolving up to $n_{\rm max} = 2000$. c) A sample of the SFF for cSYK for $N = 10$ (solid line) and the SFF from the trotterized time evolution \ref{['eq:trotterized_evolution']} (bullets). The inset shows a brief period of the evolution at late times.