Topological nature of nonlinear optical effects in solids

Abstract

There are a variety of nonlinear optical effects including higher harmonic generations, photovoltaic effects, and nonlinear Kerr rotations. They are realized by strong light irradiation to materials that results in nonlinear polarizations in the electric field. These are of great importance in studying the physics of excited states of the system as well as for applications to optical devices and solar cells. Nonlinear properties of materials are usually described by nonlinear susceptibilities, which have complex expressions including many matrix elements and energy denominators. On the other hand, a nonequilibrium steady state under an electric field periodic in time has a concise description in terms of the Floquet bands of electrons dressed by photons. We show theoretically, using the Floquet formalism, that various nonlinear optical effects, such as the shift current in noncentrosymmetric materials, photovoltaic Hall response, and photo-induced change of order parameters under the continuous irradiation of monochromatic light, can be described in a unified fashion by topological quantities involving the Berry connection and Berry curvature. We found that vector fields defined with the Berry connections in the space of momentum and/or parameters govern the nonlinear responses. This topological view offers a route to designing nonlinear optical materials.

Keywords: Berry connection; Berry curvature; Floquet bands; Keldysh formalism; nonequilibrium steady states; nonlinear Kerr rotation; nonlinear optical effects; shift current.

Fig. 1. Schematic picture of the Floquet two band model.

Under the drive of monochromatic light, energy bands evolve into Floquet bands, which describe Bloch states dressed with photons. When two Floquet bands cross, they show an anticrossing. The nonequilibrium steady state (and hence, NLORs) can be captured by studying this anticrossing of two Floquet bands.

Fig. 2. Vector field A~for the 1D model, which preserves TRS and breaks inversion symmetry.

We plot (${\tilde{\mathcal{A}}}_{\mathit{k}},{\tilde{\mathcal{A}}}_{{\mathit{Q}}_2}$) in the parameter space (k, Q₂) with Q₁ = 1, ${\mathit{Q}}_1^{\prime }$ = 0. Inset is a plot of distribution of the “flux” $\tilde{\mathcal{F}}$ defined in the text, which is related to the third-order nonlinear responses.

Fig. 3. Vector fields A~ for the 1D model, which break both TRS and inversion symmetry.

(A and B) We plot (${\tilde{\mathcal{A}}}_{\mathit{k}},{\tilde{\mathcal{A}}}_{{\mathit{Q}}_2}$) with Q₁ = 1, ${\mathit{Q}}_1^{\prime }$ = 0.1 (A) and (${\tilde{\mathcal{A}}}_{\mathit{k}},{\tilde{\mathcal{A}}}_{{\mathit{Q}}_1^{\prime }}$) with Q₁ = 1, Q₂ = 0.4 (B).

Fig. 4. Transformation laws of Berry connection and Berry curvature.

We consider the geometry in the space spanned by the momentum k and the parameter Q quantifying the inversion breaking and the geometry in the momentum space with k_x and k_y (or ${\mathit{Q}}_1^{\prime }$). Here, Q is even under T, whereas k and ${\mathit{Q}}_1^{\prime }$ are odd under T. All Q, ${\mathit{Q}}_1^{\prime }$ and k are odd under P.