Anderson localization of one-dimensional hybrid particles

TL;DR

This paper provides an exact analytical solution to the Anderson localization problem in a two-leg ladder with hybrid particles, revealing how localization properties depend on inter-chain coupling and resonance conditions.

Contribution

It extends the DMPK equation to account for eigenvector variables, solving it analytically for a disordered ladder with different intra-chain hopping integrals.

Findings

Localization dominated by slow chain at resonance energy

Localization dominated by fast chain away from resonance

Analytical expressions for Lyapunov exponents as functions of system parameters

Abstract

We solve the Anderson localization problem on a two-leg ladder by the Fokker-Planck equation approach. The solution is exact in the weak disorder limit at a fixed inter-chain coupling. The study is motivated by progress in investigating the hybrid particles such as cavity polaritons. This application corresponds to parametrically different intra-chain hopping integrals (a "fast" chain coupled to a "slow" chain). We show that the canonical Dorokhov-Mello-Pereyra-Kumar (DMPK) equation is insufficient for this problem. Indeed, the angular variables describing the eigenvectors of the transmission matrix enter into an extended DMPK equation in a non-trivial way, being entangled with the two transmission eigenvalues. This extended DMPK equation is solved analytically and the two Lyapunov exponents are obtained as functions of the parameters of the disordered ladder. The main result of the paper is that near the resonance energy, where the dispersion curves of the two decoupled and disorder-free chains intersect, the localization properties of the ladder are dominated by those of the slow chain. Away from the resonance they are dominated by the fast chain: a local excitation on the slow chain may travel a distance of the order of the localization length of the fast chain.