Bloch oscillations, Landau–Zener transition, and topological phase evolution in an array of coupled pendula

Izhar Neder*, Chaviva Sirote-Katz, Meital Geva, Yoav Lahini, Roni Ilan, Yair Shokef

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su–Schrieffer–Heeger model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wave-packet in the bulk of the lattice, we observe Bloch oscillations, Landau–Zener transitions, and coupling between the isospin (i.e., the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the nontrivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 ± 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of p. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure and shed light on the similarities and the differences between quantum and classical topological effects.

Original languageEnglish
Article numbere2310715121
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number9
StatePublished - 27 Feb 2024


FundersFunder number
Ministry of Science and Technology, Israel3-15671
Ministry of Science and Technology, Israel
Israel Science Foundation1790/18
Israel Science Foundation
United States-Israel Binational Science Foundation1899/20, 2018226
United States-Israel Binational Science Foundation


    • Bloch oscillations
    • Landau–Zener transition
    • metamaterials
    • topological phase


    Dive into the research topics of 'Bloch oscillations, Landau–Zener transition, and topological phase evolution in an array of coupled pendula'. Together they form a unique fingerprint.

    Cite this