Optical Conductivity From Pair Density Waves

We present a theory of optical conductivity in systems with finite-momentum Cooper pairs. In contrast to the BCS pairing where AC conductivity is purely imaginary in the clean limit, there is nonzero AC absorption across the superconducting gap for finite-momentum pairing if we break the Galilean symmetry explicitly in the electronic Hamiltonian. Vertex correction is crucial for maintaining the gauge invariance in the mean-field formalism and dramatically changes the optical conductivity in the direction of the pairing momentum. We carried out a self-consistent calculation and gave an explicit formula for optical conductivity in a simple case. This result applies to the Fulde-Ferrell-Larkin-Ovchinnikov state and candidates with pair density waves proposed for High- Tc cuprates. It may help detect PDW and determine the pairing gap as well as the direction of the pairing momentum in experiments.