Dynamical witnesses and universal behavior across chaos and non-ergodicity in the tilted Bose-Hubbard model

Quantum chaos in isolated quantum systems is intimately linked to thermalization and the rapid relaxation of observables. Although the spectral properties of the chaotic phase in the tilted Bose-Hubbard model have been well characterized, the corresponding dynamical signatures across the transition to regularity remain less explored . In this work, we investigate this transition by analyzing the time evolution of the survival probability, the single-site entanglement entropy, and the half-chain imbalance. Our results reveal a clear hierarchy in the sensitivity of these observables: the relaxation value of the entanglement entropy varies smoothly as a function of the Hamiltonian parameters across the chaos-regular transition, while the imbalance exhibits a more pronounced distinction. Most notably, the survival probability emerges as the most robust indicator of the transition between chaos and regularity. When appropriately scaled, all three observables converge onto a common behavior as a function of the Hamiltonian parameters for different numbers of sites and bosons,enabling a universal characterization of the transition between chaotic and regular dynamics.

💡 Research Summary

This paper investigates how dynamical observables can be used to pinpoint the transition between quantum chaotic and regular (non‑ergodic) behavior in the tilted Bose‑Hubbard model (TBH). The authors consider a one‑dimensional lattice with N bosons on M = N sites, open boundary conditions, and three tunable parameters: the hopping amplitude J (set to unity), the on‑site interaction U, and the linear tilt D. By varying U and D they sweep the system through three well‑known integrable limits—Wannier‑Stark localization (U ≈ 0), hard‑core bosons (U ≫ J), and the non‑tilted Bose‑Hubbard model (D ≈ 0)—and the intervening non‑integrable, chaotic regime.

First, static spectral diagnostics are employed. The mean level‑spacing ratio ⟨r⟩, which distinguishes GOE‑type level repulsion (⟨r⟩≈0.535) from Poisson statistics (⟨r⟩≈0.386), is mapped over the (U,D) plane. The authors find that increasing the tilt D rapidly suppresses chaos, driving ⟨r⟩ toward the Poisson value, whereas raising U at fixed D leaves ⟨r⟩ close to the GOE value over a broad range. This asymmetry foreshadows the differing sensitivities of the dynamical probes.

Next, two static quantum‑information measures are examined in the Fock basis: the participation ratio (PR = 1/∑|c_k|⁴) and the single‑site entanglement entropy S = (1/M)∑_i S(i), where S(i)=−Tr