Quantum state revivals, time quasicrystals, and entanglement steering in a Fibonacci-driven central-spin model

Abstract

Motivated by the search for non-equilibrium phases of matter beyond the Floquet paradigm, we propose a route to realize perfect revivals and time quasicrystals (TQCs) in a Fibonacci-driven central-spin model with Ising interactions between the central and satellite spins in the absence of disorder. We analytically demonstrate that when the interaction strength, $λ$ is tuned to $2 m π$, no entanglement is generated between the central and satellite spins at stroboscopic times, independent of the strength of the transverse driving field and the drive sequence. This leads to indefinitely long-lived quasiperiodic oscillations in satellite spin magnetization. Furthermore, for initial states with a definite bath magnetization, no entanglement is generated when the transverse driving field, $θ$ is tuned to $π$ for any value of $ λ$, and the satellite spin magnetization oscillations exhibit period-3 revivals at Fibonacci times. Examining the robustness of these oscillations away from this fine-tuned limit, we identify a parameter regime where the quasiperiodic response does not decay over the timescales probed, giving rise to a long-lived $π$-pulse-proximate TQC regime. We further show that at special Clifford points, the system exhibits perfect quasiperiodic oscillations of the satellite spin magnetization, with a parity-dependent Fourier response. This response persists over the timescales probed, even away from the Clifford points for an even number of satellite spins, leading to the formation of a Clifford-proximate TQC. Finally, we demonstrate that at this Clifford points, the system is steered along an entangled trajectory, leading to the dynamical generation of Bell-cat and spin-cat states. These states exhibit magnetization of collective spin fluctuates quadratically with the number of satellite spins, thereby enabling Heisenberg-limited metrology.

The paper studies whether a disorder-free central-spin model under Fibonacci driving can exhibit persistent quasiperiodic responses and generate metrologically useful entanglement. A central spin-$1/2$ couples uniformly through Ising interactions to satellite spins; the drive alternates interaction evolution of strength $\lambda$ and global transverse rotations of angle $\theta$. Two non-entangling regimes are established analytically. At $\lambda=2m\pi$, the interaction unitary factorizes into single-spin operators, so neither drive operation generates entanglement, independently of rotation angle or sequence. At $\theta=\pi$, initial product states with definite bath magnetization remain unentangled, while satellite magnetization exhibits period-3 oscillations when sampled at Fibonacci times. Numerical Fourier analysis finds a finite window near the perfect-$\pi$ pulse with a frequency-locked response, interpreted as a long-lived time-quasicrystal regime. At Clifford points such as $(\lambda,\theta)=(\pi,\pi/2)$, a permutation-symmetric stabilizer description gives exact Fibonacci-index magnetization periods of 18 for even satellite number and 12 for odd satellite number; central-spin entanglement has period 6 for both. Numerical detuning studies find a persistent nearby response for even satellite number. Entropy and collective-spin variances identify Bell-cat and spin-cat states, with pure-state quantum Fisher information scaling as $N_{\rm sat}^2$. The robustness evidence is finite-time: representative entropy and Fourier calculations use 2,000 and 4,000 drive steps, respectively, with around 200 satellites. Away from the exact $\pi$ pulse, the response is expected to be prethermal rather than indefinitely stable. Higher-spin central objects, other aperiodic drives, and interactions among satellites are left for future study.