Nonlinear optics of Bloch oscillating electrons in semiconductor superlattices

A quantum particle in a periodic potential can exhibit rich dynamics in a driving field. In particular, Bloch oscillation of electrons in a superlattice leads to optical properties that are nonlinear functions of the input fields. In a DC field, we analyze the coupling of Bloch oscillations in a superlattice with plasmons and longitudinal optical (LO) phonons, modelled by a pendulum linearly coupled to an oscillator. In the absence of LO phonons, the pendulum equation predicts a sharp transition from plasma oscillations to Bloch oscillations at a critical density or electric field. Resonant Bloch-phonon coupling enhances the phonon amplitude and generates sidebands, but produces no gap in the Bloch-phonon spectrum. Our predictions qualitatively agree with recent experimental results. Considerably more dramatic is the response of charges to an oscillatory incident field. In an AC field, the resonance between the external frequency and the Bloch oscillation frequency set by the field amplitude makes the optical response a nonlinear function of the field amplitude. At certain discrete values of the AC amplitude the electron is dynamically localized, whereupon the total current density decreases drastically while the power dissipated is maximized. The THz reflection coefficient vanishes at dynamic localization, and thus oscillates with varying AC field amplitude inside the superlattice. At high doping, the nonlinear transformation between the fields inside and outside the superlattice leads furthermore to multistability in the optical properties as a function of the incident field. Similar oscillations and multistability exist for third harmonic power generated by a set of superlattices fabricated into a quasi-optical array. The generated power can be optimized by bringing the harmonics into Fabry-Perot resonance with the substrate. We compare our predictions with recent experimental results for a quasi-optical array. Combining a mixture of DC and AC fields leads to absolute negative conductance, where a charge tunnels against a DC bias by absorbing energy from the photons. We discuss a simple physical picture to describe absolute negative conductance and consider some potential applications.