Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields

Abstract

The quantum evolution of particles under strong fields can be essentially captured by a small number of quantum trajectories that satisfy the stationary phase condition of the Dirac-Feynmann path integral. The quantum trajectories are a key concept in understanding extreme nonlinear optical phenomena, such as high-order harmonic generation (HHG) and highorder terahertz sideband generation (HSG). In contrast to HHG in atoms and molecules, HSG in semiconductors can have interesting effects due to nontrivial 'vacuum' states of band materials. We find that, in a semiconductor with nonvanishing Berry curvature in its energy bands, the cyclic quantum trajectories of an electron-hole pair under a strong elliptically polarized terahertz field can accumulate a Berry phase. Taking monolayer MoS 2 as a model system, we show that the Berry phase appears as a Faraday rotation angle in the pulse emission from the material under short-pulse excitation. This finding reveals an interesting transport effect in the extreme nonlinear optics regime.

Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields

“…A fundamental difference between HHG and HSG is that the electron-hole pair in semiconductors can have nontrivial Bloch states [1]. As a result, when the electron (or hole) is driven by the THz field in the conduction (or valence) band, not only does the quasi-momentum k change [8], but also the Bloch wavefunction evolves with k. Since the THz frequency is much smaller than the band gaps of semiconductors, the evolution is adiabatic (i.e.…”

“…In [1], we studied the Berry phases of quantum trajectories in monolayer MoS 2 with a band gap in the visible wavelength regime. We found that the optical emission delayed by integer multiples of the THz period after the pulse excitation acquired a Faraday rotation, and the rotation angle was exactly the Berry phase of the quantum trajectory (∼0.01 rad for a THz field with frequency 1 THz and strength 8 kV cm −1 ).…”

“…The linear optical response of semiconductors driven by THz fields has been derived in [1]. If the THz field does not mix different pseudo-spin or valley states of the degenerate energy bands, the Berry connections are Abelian and the optical response is simplified to…”

“…The main consequence of equation (7) can be understood using the stationary phase formalism (or the quantum trajectory theory) [1,2,6]. In the path integral, the electron-hole pair moves along all possible trajectories when driven by the THz field, with the phase given by the action…”

“…

High-order terahertz (THz) sideband generation in semiconductors is a phenomenon with physics similar to that of high-order harmonic generation but in a regime of much lower frequency. Our previous paper [1] found that the electron-hole pair excited by a weak optical laser can accumulate a Berry phase along a cyclic trajectory under the driving of a strong elliptically polarized THz field. Furthermore, the Berry phase appears as the Faraday rotation angle of the emission signal under short-pulse excitation in monolayer MoS 2 .

…”