Quantum-geometric contribution to the Bogoliubov modes in a two-band Bose-Einstein condensate

TL;DR

This paper reveals that in a two-band Bose-Einstein condensate loaded into an optical lattice, the superfluid's effective mass tensor includes a quantum-geometric contribution linked to the quantum metric of Bloch states, beyond the usual band-mass term.

Contribution

It introduces the concept that two-body interactions induce a quantum-geometric contribution to the effective mass tensor in a two-band BEC, connecting it to the quantum metric tensor.

Findings

The inverse effective-mass tensor has both band-mass and quantum-geometric contributions.

Quantum-geometric effects are induced by interband processes via two-body interactions.

Conditions are discussed where the geometric contribution relates to the quantum metric tensor.

Abstract

We consider a weakly-interacting Bose-Einstein condensate (BEC) that is loaded into an optical lattice with a two-point basis, and described by a two-band Bose-Hubbard model with generic one-body and two-body terms. By first projecting the system to the lower Bloch band and then applying the Bogoliubov approximation to the resultant Hamiltonian, we show that the inverse effective-mass tensor of the superfluid carriers in the Bogoliubov spectrum has two physically distinct contributions. In addition to the usual inverse band-mass tensor that is originating from the intraband processes within the lower Bloch band, there is also a quantum-geometric contribution that is induced by the two-body interactions through the interband processes. We also discuss the conditions under which the latter contribution is expressed in terms of the quantum-metric tensor of the Bloch states, i.e., the natural Fubini-Study metric on the Bloch sphere.