Quantum Dynamics in Quasi-Periodic Systems

Quantum dynamics in quasi-periodic systems explores the evolution of quantum states in media where underlying potentials or driving forces exhibit quasi-periodic order, lying between strictly periodic crystals and random environments. Such systems are exemplified by electrons in quasicrystals, cold atoms in optical lattices with incommensurate modulations and quantum walks under external fields. The interplay between inherent aperiodicity and quantum coherence gives rise to rich spectral and dynamical phenomena, including metal–insulator transitions, anomalous transport, fractal spectra and topologically protected states. Key theoretical frameworks draw on the spectral theory of Schrödinger operators, cocycle dynamics and renormalisation schemes, while experimental platforms invoke photonic lattices, ultracold atomic gases and superconducting circuits. Understanding the balance of localisation and transport in quasi-periodic media informs condensed-matter physics, quantum information processing and materials science, offering routes to engineer robust quantum devices and explore fundamental aspects of wave propagation beyond conventional periodic order. Recent advancements have deepened insights into high-dimensional dynamics, established rigorous dynamical bounds and characterised transitions in spectral measures, underlining the global significance of quasi-periodic quantum systems in both foundational research and prospective applications.

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Technical terms

Quasi-periodic potential: A potential function composed of two or more incommensurate periodic components, lacking translational periodicity but exhibiting deterministic aperiodic order.

Lyapunov exponent: A measure of the exponential rate of divergence or convergence of nearby trajectories in a dynamical system, here quantifying localisation and stability of wavefunctions.

Anderson localisation: The absence of diffusive transport in a disordered or quasi-periodic medium due to destructive interference, leading to spatially localised eigenstates.

Quantum transport exponent: A parameter describing the asymptotic scaling of wavepacket spreading with time, distinguishing ballistic, diffusive or sub-diffusive dynamics.

Diophantine frequency: An irrational number satisfying specific arithmetic conditions that ensure bounded small-denominator estimates in perturbation and KAM theory.

Schrödinger operator: A linear operator governing the quantum evolution of a particle, typically of the form −Δ + V(x), where V(x) is the potential energy function.

References

1. Upper bounds on quantum dynamics in arbitrary dimension. Journal of Functional Analysis (2023).
2. Anderson localization for the unitary almost Mathieu operator. Nonlinearity (2024).
3. Schrödinger operators with potentials generated by hyperbolic transformations: I—positivity of the Lyapunov exponent. Inventiones Mathematicae (2022).