Anisotropy of the Floquet Spectrum of a Dirac Crystal, Induced by a Bichromatic Field

We perform a dynamical symmetry analysis (DSA) of the high-order harmonic generation (HHG) spectrum of an atom interacting with a bichromatic laser field. Within the framework of the conventional Hermitian quantum mechanics (QM), the HHG spectrum calculated using a single Floquet state, or any finite number of Floquet states, is invariant under the inversion of the relative phase of the two-frequency components, $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\phi}$. The asymmetry with respect to the phase inversion seen in the simulated HHG spectra is obtained in the conventional QM only when the Floquet spectrum is continuous and ionization is taken into consideration. However, when the Hamiltonian is complex scaled the description is different. Even a single eigenstate of the complex scaled Floquet operator is enough to describe the breaking of the $\ensuremath{\phi}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\phi}$ symmetry in the HHG spectra. We find that there is a direct correlation between the strength of the asymmetry with respect to the relative phase inversion and the magnitude of the ionization rate. For illustration purposes, the DSA is accompanied by the results obtained for a one-dimensional effective single-electron model Hamiltonian mimicking xenon atom interacting with strong laser field.