Hamiltonian Tomography of Photonic Lattices

TL;DR

This paper introduces a spectroscopic method for Hamiltonian tomography of photonic lattices, enabling complete characterization of their properties and topological features through two-port measurements.

Contribution

It presents a novel technique to directly extract Hamiltonian matrix elements from two-port spectra, applicable to various photonic lattice systems.

Findings

Direct measurement of Hamiltonian matrix elements from spectra

Application to topological property measurement like Chern number

Extension to disordered and flat-band systems

Abstract

In this letter we introduce a novel approach to Hamiltonian tomography of non-interacting tight-binding photonic lattices. To begin with, we prove that the matrix element of the low-energy effective Hamiltonian between sites $i$ and $j$ may be obtained directly from $S_{ij}(\omega)$, the (suitably normalized) two-port measurement between sites $i$ and $j$ at frequency $\omega$. This general result enables complete characterization of both on-site energies and tunneling matrix elements in arbitrary lattice networks by spectroscopy, and suggests that coupling between lattice sites is actually a topological property of the two-port spectrum. We further provide extensions of this technique for measurement of band-projectors in finite, disordered systems with good flatness ratios, and apply the tool to direct real-space measurement of the Chern number. Our approach demonstrates the extraordinary potential of microwave quantum circuits for exploration of exotic synthetic materials, providing a clear path to characterization and control of single-particle properties of Jaynes-Cummings-Hubbard lattices. More broadly, we provide a robust, unified method of spectroscopic characterization of linear networks from photonic crystals to microwave lattices and everything in-between.