Photo-induced phase transitions (PIPT) in quantum systems are the epitome of challenging non-equilibrium many-body phenomena, that also have a wide range of potential applications. Recently interest in light-matter coupled states with an energy gap have yielded evidence for Floquet topological states. In this study we demonstrate nonequilibrium Floquet band formation under ultrafast optical excitation using a one-dimensional topological insulator. As an example, the effects are illustrated using a new Zig-Zag Su-Schrieffer-Heeger model of polyacetylene, which is a paradigmatic Hamiltonian exhibiting nontrivial edge states. Our results indicate short optical pulses feasible in experiments can induce novel topological states, local spectral selection and novel pseudospin textures in polyacetylene. Pump-probe photoemission spectroscopy is able to study these states by measuring Floquet band formation and sizeable energy gaps on femtosecond time scales. We find that optically activated nontrivial variations of sublattice mixing could lead to novel topological phenomenon.The rich variety of states induced by lasers have a wide range of potential applications so that control of these states has become a key objective. We present a computational approach to finding optimal ultrafast laser pulse shapes to induce target states and population inversion in pump-probe PIPT experiments. The Krotov approach for Quantum optimal control theory (QOCT) is combined with a Keldysh Green’s function calculation to describe experimental outcomes such as photoemission, transient single particle density of states and optical responses. Results for a simple model charge density wave (CDW) system are presented, including generation of almost complete population inversion and negative temperature states.
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