Bloch oscillation in a Floquet engineering quadratic potential system

We investigate the quantum dynamics of a one-dimensional tight-binding lattice driven by a spatially quadratic and time-periodic potential. Both Hermitian ($J_1 = J_2$) and non-Hermitian ($J_1 \neq J_2$) hopping regimes are analyzed. Within the framework of Floquet theory, the time-dependent Hamiltonian is mapped onto an effective static Floquet Hamiltonian, enabling a detailed study of the quasi-energy spectrum and eigenstate localization as function of the driving frequency $ω$. We identify critical frequencies $ω_c$ at which nearly equidistant quasi-energy ladders emerge, characterized by a pronounced minimum in the normalized variance of level spacings. This spectral regularity, which coincides with a peak in the mean inverse participation ratio (\textrm{MIPR}), leads to robust periodic revivals and Bloch-like oscillations in the time evolution. Numerical simulations confirm that such coherent oscillations persist even in the non-Hermitian regime, where the periodic driving stabilizes an almost real and uniformly spaced quasi-energy ladder.

💡 Research Summary

This paper presents a comprehensive study of quantum dynamics in a one-dimensional tight-binding lattice subjected to a spatially quadratic and time-periodic potential. The research investigates both Hermitian (J1 = J2) and non-Hermitian (J1 ≠ J2) hopping regimes, leveraging Floquet theory to map the time-dependent problem onto an effective static Hamiltonian in an extended Sambe space. This framework allows for a detailed analysis of the quasi-energy spectrum and the localization properties of Floquet eigenstates as functions of the driving frequency ω.

The model Hamiltonian consists of a static hopping term with amplitudes J1 and J2, and a driving term given by an on-site potential F(l,t) = F0 l² cos(ωt). This setup is relevant to experimental platforms like ultracold atoms in modulated traps or photonic waveguide arrays with engineered curvature.

The analysis begins by establishing two contrasting limits. In the static limit (ω=0), the quadratic confinement dominates. For the Hermitian case, this creates harmonic-oscillator-like states near the potential minimum and Wannier-Stark-like localized states at higher energies near the boundaries. In the non-Hermitian case, the asymmetric hopping induces a skin effect, leading to boundary accumulation, which competes with the central confinement of the quadratic potential. In the high-frequency limit (ω→∞), the time-averaged Hamiltonian reduces to the uniform hopping term H0. Here, the Hermitian system supports extended Bloch states, while the non-Hermitian one exhibits a complex energy spectrum and skin localization under open boundaries.

For finite driving frequencies, the core of the analysis employs the Floquet formalism. The time-periodic Schrödinger equation is transformed into an eigenvalue problem in Sambe space, where the Floquet Hamiltonian is represented as an infinite block matrix. Diagonal blocks contain the static part shifted by integer multiples of the photon energy (nω), while off-diagonal blocks couple different photon sectors. In practice, a truncated matrix with a finite number of photon sectors (|m| ≤ M) is diagonalized numerically.

The central finding is the identification of critical driving frequencies ω_c at which a remarkable spectral reorganization occurs. Portions of the quasi-energy spectrum form nearly equidistant ladders. This spectral regularity is quantitatively captured by a pronounced minimum in the normalized variance of nearest-neighbor quasi-energy spacings. Concurrently, the mean inverse participation ratio (MIPR) of the corresponding Floquet eigenstates peaks, indicating their enhanced spatial localization. This conjunction of a regular energy ladder and localized states has a direct dynamical consequence: an initial wave packet prepared within this ladder subspace exhibits robust periodic revivals and Bloch-like oscillations in its time evolution. The revival period is given by t_c = 2π/ΔE, where ΔE is the uniform spacing of the quasi-energy ladder.

A significant result is that these coherent oscillations persist even in the non-Hermitian regime. Typically, non-Hermitian Hamiltonians possess complex eigenvalues, leading to decay or amplification that disrupts coherent periodic motion. However, this work demonstrates that the periodic drive can dynamically stabilize a quasi-energy ladder that is almost entirely real and uniformly spaced. This effective “Hermitization” of a spectral subspace within a non-Hermitian system enables long-lived, Hermitian-like revival dynamics despite the underlying nonreciprocal hopping.

The study combines analytical Floquet arguments (high-frequency expansion, Sambe space structure) with systematic numerical diagnostics, including spectral variance, MIPR calculations, and direct wave packet propagation simulations. These methods collectively identify the frequency windows where ladder formation is robust and confirm the resulting revival dynamics. In summary, this research unveils a mechanism for generating and protecting coherent quantum oscillations via Floquet engineering, which operates in both Hermitian and non-Hermitian settings. It highlights the potential of temporal modulation to sculpt complex spectra and control dynamics, opening avenues for quantum state control and the exploration of non-equilibrium phases in driven-dissipative platforms.