# Flat bands and dynamical localization of spin-orbit coupled Bose-Einstein condensates

## TL;DR

This paper explores how spin-orbit coupling and Zeeman field modulation enable flat bands and dynamical localization in shaken optical lattices with Bose-Einstein condensates, revealing conditions for complete tunneling suppression and localized states.

## Contribution

It demonstrates that tuning spin-orbit coupling via Zeeman field modulation is essential for dynamical localization in BECs with shaken optical lattices, extending understanding beyond traditional models.

## Key findings

- Energy bands become flat at specific Zeeman parameters.
- Localization occurs on a dimer with components on different sites.
- Localization persists even with nonlinearity in the system.

## Abstract

Flat bands and dynamical localization of binary mixtures of Bose-Einstein condensates, with spin-orbit coupling subjected to a deep optical lattice which is shaking in time and to a periodic time modulation of the Zeeman field, are investigated. In contrast with usual dynamical localization in the absence of spin-orbit coupling, we find that to fully suppress the tunneling in the system the optical lattice shaking is not enough, and a proper tuning of the spin-orbit term, achievable via the Zeeman field modulation, is also required. This leads to a sequence of Zeeman parameter values where energy bands become flat, the tunneling in the system is suppressed, and the dynamical localization phenomenon occurs. Exact wave functions at the dynamical localization points show that the binary mixture localizes on a dimer with the two components occupying different sites. This type of localization occurs in exact form also for the ground state of the system at the dynamical localization points in the presence of nonlinearity and remains valid, although in approximate form, for a wide range of the Zeeman parameter around these points. The possibility of observing the above phenomena in real experiments is also briefly discussed.