Edge states and quantum optical high-harmonic generation from topological insulators

Abstract

The strong-field process of high-harmonic generation (HHG) has, in recent years, been treated from a quantum optical perspective in the emerging research area of strong-field quantum optics. These investigations show that HHG radiation is, in general, in a nonclassical state of light. However, the quantum optical treatment of HHG from topological nontrivial materials is missing. Here, we aim to address this gap in current knowledge and consider the quantum optical HHG response from the Su-Schrieffer-Heeger model, a finite chain of atoms with both a topologically trivial and nontrivial insulating phase, the latter supporting edge states. We find that HHG from both topological phases is squeezed at the band-gap frequency. Interestingly, while the harmonic spectrum discriminates the two topological phases of the system, the degree of squeezing only discriminates the phases for smaller chain lengths. We attribute this difference to a relative increase in overlap between bulk and edge states in the topological nontrivial phase for smaller systems. Our findings reveal how the strength of dipole couplings governs the nonclassical HHG response and define new research questions on topologically protected generation of quantum light in strong-field physics.

Figures (2)

• Figure 1: (a) Illustration of the real-space configuration of the SSH model in its different phases with physical interpretation of relevant parameters introduced in he main text. (b), (c) Energy spectra for the trivial and topological phases for the long (b) and short (c) SSH chains. The topological phase supports states at zero energy. (d), (e) Eigenstates of the SSH model in the topological phase for both the ground state (top row), the highest-energy state below the band gap (middle row), and the zero-energy state (bottom row), localized at the edges of the chain.
• Figure 2: (a) Spectra obtained from Eq. (\ref{['eq:spectrum']}) from both the long (blue and orange) and short (yellow and red) chains for both the trivial ($\delta = 0.15$) and topological ($\delta = -0.15$) phase. We note how the spectra discriminate the topological phases. The degree of squeezing for the long (b) and short chain (c), respectively. We note that excluding edge-state transition currents changes the degree of squeezing for the short chain at harmonics below the gap (c, green dashed) while it leaves the degree of squeezing from the long chain (b, purple dashed) indifferent. The coherent spectra in (a) are shown without $N$ scaling and have been shifted for visual clarity, and the degree of squeezing is shown for $N=1$ SSH chain. The parameters for the system are stated in the main text.