In a groundbreaking advance at the intersection of optics and topological physics, researchers have unveiled a novel approach to manipulate light transport using quasi-periodic lattices governed by Fibonacci numbers. Traditionally, the concept of topological pumping, a robust means of moving matter or energy across a system, has relied heavily on temporal periodicity. This fundamental assumption, dating back to the pioneering work of Nobel laureate D. J. Thouless in 1983, stipulated that the system’s driving lattice potential must repeat identically after a fixed time period. Challenging this longstanding constraint, scientists from Shanghai Jiao Tong University, in collaboration with leading experts from Portugal and Russia, have demonstrated that quasi-periodic temporal modulation—where the lattice never reverts exactly to its initial state—can also sustain topologically protected transport of light.
For decades, Thouless pumping served as a cornerstone in condensed matter physics by enabling quantized displacement of electrons in periodic lattices subjected to cyclic driving. The topological invariants in these systems, specifically Chern numbers, are integer values that govern the direction and magnitude of particle transport, signaling a profound link between abstract mathematical concepts and physical phenomena. Extension of this principle into optical platforms brought about stable, controllable transfer of photonic wave packets, underpinned by robust topological order. However, because of the strict periodicity fundamentally embedded in the original theory, explorations of aperiodic or quasi-periodic drives remained largely uncharted territory—until now.
The research team proposed an ingenious scheme involving two incommensurate temporal modulations acting simultaneously on the optical lattice potential, characterized by periods T₁ and T₂ whose ratio is an irrational number. The golden ratio, widely regarded as the archetype of irrationality, was chosen as their primary example. This setup diverges radically from prior models; instead of the lattice returning to its initial configuration after a finite time, it follows a never-repeating quasi-periodic trajectory along the propagation axis (z-axis) of the light beam. This required the researchers to rethink how topological invariants could be defined in a system devoid of exact cyclicity.
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To tackle this challenge, the scientists leveraged a profound mathematical strategy rooted in number theory, approximating the quasi-periodic lattice by a sequence of “periodic approximants.” Each approximant corresponds to a rational approximation of the golden ratio derived from the Fibonacci sequence—a series renowned for its recursive elegance and appearance in natural patterns. These approximants constitute periodic systems that admit well-defined band structures and consequently computable Chern numbers. Strikingly, the resulting topological invariants traced out a Fibonacci-like progression themselves, following the rule Cₙ = Cₙ₋₁ + Cₙ₋₂, where Cₙ denotes the Chern number for the nth approximant.
This remarkable discovery links the topological properties of the system directly with the arithmetic nature of the irrational number governing the drive, establishing a bridge between abstract mathematical sequences and tangible physical effects. The velocity of the light beam’s topological transport, measured as transverse displacement per unit longitudinal evolution, was found to adhere to the same Fibonacci scaling laws. By analyzing the transverse and longitudinal periods of each rational approximation, the team revealed that both periods form Fibonacci sequences, providing a beautifully self-consistent framework that culminates in a limit governed by the golden ratio itself.
The investigation extended into the dynamical realm, with simulations portraying the evolution of beam propagation across successive approximants. By the sixth approximation, the system’s transport velocity converged almost perfectly to a constant proportional to the golden ratio. The hallmark of topological protection emerged here: robustness in the face of perturbations, underscoring the fact that the observed transport does not hinge on delicate parameter tuning but instead arises from the system’s fundamental topological nature.
To validate their theoretical predictions, the team engineered the first three periodic approximants experimentally using a strontium barium niobate (SBN) crystal fabricated via optical induction. The fabricated photonic lattices, measuring 5×5×20 mm³, faithfully reproduced the expected spatial patterns. A probe Gaussian light beam was launched along the lattice, and its intensity distribution was meticulously recorded in the (y, z) plane. The observed centroid trajectories matched numerical predictions with impressive fidelity, constituting compelling evidence for the realization of quasi-periodic topological pumping in a tangible platform.
A critical test of topological phenomena resides in their resilience against external disturbances. In this case, the researchers manipulated the lattice amplitude by varying an external voltage across a wide range. Despite these considerable changes, the transport velocity of the beam remained steadfastly stable, an unmistakable fingerprint of topological robustness. Such immunity to deformations and parameter fluctuations holds profound implications for future photonic devices where reliable wave transport is paramount.
Beyond confirming the physical realization of quasi-periodic topological pumping, this study opens several avenues for theoretical and practical advances. By disentangling the strict periodicity requirement, it broadens the landscape in which topological effects can be harnessed, potentially impacting fields ranging from energy transport to information processing. The lessons drawn here may inspire novel schemes in acoustics, cold atoms, and mechanical systems, where the interplay of non-periodic driving and topology remains largely unexplored.
Importantly, although the research focuses on the golden ratio and its related Fibonacci sequence, the conceptual and experimental methods extend naturally to other irrational numbers and associated rational approximants. This universality hints at a vast class of novel systems where quasi-periodicity injects rich topological phenomena, which could be tuned and engineered by selecting different irrational ratios. The interdisciplinary nature of these findings—weaving together advanced mathematics, materials science, and photonics—highlights the vibrant, interconnected fabric of modern physics.
The implications of this breakthrough are especially exciting given the burgeoning interest in topological photonics as a platform for robust information transmission, low-loss waveguiding, and photonic circuitry immune to disorder. Real-world technological components often face imperfections and nonidealities that compromise performance. Harnessing quasi-periodic topological pumping predicates new device architectures that defy these limitations, potentially revolutionizing optical communication and computation technologies.
This seminal work, titled “Topological pumping of light governed by Fibonacci numbers,” was published in the journal eLight. It presents both a theoretical framework and its experimental verification, underscoring a crucial paradigm shift in how topological invariants may be conceived and applied in aperiodically driven physical systems. By marrying number theory with experimental ingenuity, the study pioneers a new frontier where light not only traverses space but also encodes deep mathematical order within its journey.
As researchers worldwide delve into the fertile domain of non-periodic topological physics, this discovery stands as a beacon demonstrating that the union of quasi-periodicity and topology yields unexpected grace and control. The elegance of Fibonacci sequences, etched into the very patterns of light propagation, may soon inspire a generation of technologies tapping into the subtle symmetries and invariants hidden within complex temporal evolutions.
Subject of Research: Topological photonic pumping in quasi-periodic optical lattices
Article Title: Topological pumping of light governed by Fibonacci numbers
Web References: http://dx.doi.org/10.1186/s43593-025-00095-9
Image Credits: Peng, R., Yang, K., Fu, Q. et al.
Keywords
Topological pumping, quasi-periodic lattices, Fibonacci sequence, golden ratio, Chern numbers, photonic lattices, optical induction, topological robustness, irrational modulation, light transport, non-periodic driving, topological photonics
Tags: condensed matter physics advancementsFibonacci numbers in opticsinterdisciplinary physics researchmanipulating light transportnovel approaches in light manipulationoptical platforms for photonic transportquasi-periodic lattices in physicsrobust transport of light wavesShanghai Jiao Tong University studytemporal modulation in light systemstopological invariants and Chern numberstopological light pumping breakthroughs