image: Figure 1 | Conceptual schematic of periodic approximants illustrating light pumping in quasi-periodic lattices. Schematic of Thouless pumping for a probe beam in a longitudinal quasi-periodic optical lattice, along with its first three periodic approximations.
Credit: Peng, R., Yang, K., Fu, Q. et al.
The stable or robust transfer of matter, information, or energy, from one point in space to another is a crucial scientific and technological challenge. In 1983, Nobel laureate D. J. Thouless proposed a seminal solution to this problem, later termed the "Thouless pump". He discovered that in a time-periodically modulated/driving lattice potential, the system can exhibit effects analogous to a magnetic field, enabling the directed transfer of electrons from one lattice unit cell to another. The direction and distance of this transfer are determined by the system's topological invariant—the Chern number. Due to its wave-like nature, the concept of Thouless pumping was rapidly extended to optical, acoustic, mechanical, and cold-atom systems, facilitating stable transport and precise control of various wave phenomena.
As in Thouless's original proposal, research on Thouless pumping has been confined to periodic temporal modulation of the lattice, where the lattice potential must return to its initial state after a time period T. Can this strict temporal periodicity constraint be relaxed? If the lattice modulation is changed from periodic to quasi-periodic, such that the lattice never fully recovers its initial state during evolution, do the system's topological properties persist? If so, how can the topological invariants of such a system be defined or characterized, given that the system never repeats along the evolution axis? Addressing these questions, Professor Fangwei Ye’s team from School of Physics and Astronomy, Shanghai Jiao Tong University, in collaboration with Professor Vladimir Konotop (Faculty of Sciences of University of Lisbon) and Professor Yaroslav Kartashov (Faculty of Institute of Spectroscopy, Russian Academy of Sciences), proposed a quasi-periodically modulated lattice in the time domain. In their modulation scheme, the lattice is simultaneously subjected to two incommensurate temporal modulations with periods T1 and T2 (their ratio is an irrational number, e.g., the golden ratio). Their study revealed that light beams in quasi-periodically driven systems retain topological characteristics, with the transport rate governed by the irrationality of the modulation. This work breaks the constraints of periodic systems in the study of spatial transport of matter or energy, opening new theoretical and practical exploration avenues. While the research has been carried out for the golden ratio and related Fibonacci sequence, the theoretical and experimental approach based on the best rational approximations, are straightforwardly generalizable to other irrational numbers. The findings were published in eLight under the title "Transport of light governed by Fibonacci numbers".
As a concrete example, the researchers selected the golden ratio as the ratio between the two driving frequencies of the lattice. This results in a photonic lattice that exhibits quasi-periodic variation along the beam propagation axis (z-axis), unlike the periodic evolution in conventional Thouless pumping studies. Using the mathematical principle that irrational numbers can be approximated by a series of optimal rational numbers, the team proposed approximating the true quasi-periodic optical lattice with a sequence of periodic lattices Vn (n = 1, 2, 3, ...), as illustrated in Fig. 1. Since Vn is periodic, its band structure and corresponding topological Chern numbers Cn could be computed. Intriguingly, the Chern numbers followed the sequence Cn = 1, 2, 3, 5, 8, ..., adhering to the recurrence relation Cn = Cn-1 + Cn-2—a Fibonacci sequence!
The Fibonacci-derived Chern numbers {Cn} dictate the velocity of the topologically transported beam (measured by the displacement of the center of mass). For the n-th approximation, the velocity is given by vn = (transverse displacement)/(longitudinal evolution) = CnYn/Zn, where Yn and Zn are the transverse and longitudinal periods of Vn (the transverse period Yn was fixed as Y in this study). Remarkably, the longitudinal periods {Zn} also formed a Fibonacci sequence. The researchers demonstrated that as n → ∞ (i.e., the true quasi-periodic lattice), the beam's transport velocity converges to a value proportional to the irrational number governing the quasi-periodic drive—here, the golden ratio. Figures 2(a)-(c) depict the dynamical evolution of the beam’s topological motion for the first six approximations, along with the transverse displacement yn (b) and velocity vn (c), showing rapid convergence to a constant value.
To validate the theory, the team fabricated the first three periodic approximants in a 5×5×20 mm³ strontium barium niobate (SBN) crystal using optical induction. The experimentally realized lattices matched simulations perfectly [Figs. 3(a)-(c)]. The intensity distribution of a probe light beam propagating along z was observed in the (y,z) plane [Figs. 3(d)-(f)], with the Gaussian beam’s centroid motion under different approximants shown in Fig. 3(g). To investigate topological robustness, the lattice amplitude (tunable via external voltage) was varied significantly, yet the transport velocity remained nearly unchanged [Fig. 3(h)], confirming topologically protected transport.
Journal
eLight
Article Title
Topological pumping of light governed by Fibonacci numbers