Geometry of Bloch states probed by St\"uckelberg interferometry

TL;DR

This paper explores how St"uckelberg interferometry applied to Bloch oscillations in a honeycomb lattice reveals detailed geometric properties of Bloch states, including phase shifts and transition probabilities influenced by lattice effects.

Contribution

It introduces a lattice-specific St"uckelberg interferometer model that uncovers unique effects like Landau-Zener tunneling twisting and non-periodic behaviors, advancing the understanding of Bloch state geometry.

Findings

Lattice effects cause twisting of Landau-Zener tunneling.

Inter-band transition probability saturates at high forces.

Interferometry reveals the quantum metric tensor of Bloch states.

Abstract

Inspired by recent experiments with cold atoms in optical lattices, we consider a St\"uckelberg interferometer for a particle performing Bloch oscillations in a tight-binding model on the honeycomb lattice. The interferometer is made of two avoided crossings at the saddle points of the band structure (i.e. at M points of the reciprocal space). This problem is reminiscent of the double Dirac cone St\"uckelberg interferometer that was recently studied in the continuum limit [Phys. Rev. Lett. 112, 155302 (2014)]. Although the two problems share similarities -- such as the appearance of a geometric phase shift -- lattice effects, not captured by the continuum limit, make them truly different. The particle dynamics in the presence of a force is described by the Bloch Hamiltonian $H(\boldsymbol{k})$ defined from the tight-binding Hamiltonian and the position operator. This leads to many interesting effects for the lattice St\"uckelberg interferometer: a twisting of the two Landau-Zener tunnelings, saturation of the inter-band transition probability in the sudden (infinite force) limit and extended periodicity or even non-periodicity beyond the first Brillouin zone. In particular, St\"uckelberg interferometry gives access to the overlap matrix of cell-periodic Bloch states thereby allowing to fully characterize the geometry of Bloch states, as e.g. to obtain the quantum metric tensor.