Tomographic characterization of non-Hermitian Hamiltonians in reciprocal space

Non-Hermitian Hamiltonians enrich quantum physics by extending conventional phase diagrams, enabling novel topological phenomena, and realizing exceptional points with potential applications in quantum sensing. Here, we present an experimental photonic platform capable of simulating a non-unitary quantum walk generated by a peculiar type of non-Hermitian Hamiltonian, largely unexplored in the literature. The novelty of this platform lies in its direct access to the reciprocal space, which enables us to scan the quasi-momentum across the entire Brillouin zone and thus achieve a precise tomographic reconstruction of the underlying non-Hermitian Hamiltonian, indicated by the comparison between theoretical predictions and experimental measurements. From the inferred Hamiltonian, it is possible to retrieve complex-valued band structures, resolve exceptional points in momentum space, and detect the associated parity-time symmetry breaking through eigenvector coalescence. Our results, presented entirely in quasi-momentum space, represent a substantial shift in perspective in the study of non-Hermitian phenomena.

💡 Research Summary

In this work the authors present a photonic quantum‑walk platform that enables full tomographic reconstruction of a non‑Hermitian Hamiltonian directly in reciprocal (quasi‑momentum) space. Unlike most previously studied non‑Hermitian lattice models, which rely on non‑reciprocal hopping amplitudes (e.g., the Hatano‑Nelson or non‑Hermitian SSH models), the implemented Hamiltonian features reciprocal hopping of equal magnitude while the non‑Hermitian character originates from inter‑cell hopping terms that are not complex conjugates of each other, together with a complex on‑site potential. The discrete‑time evolution operator is U = T W, where the coin rotation W = (σ₀ + iσₓ)/√2 and the translation T = α I + iβ( |m + 1, B⟩⟨m, A| + h.c.) contain two complex parameters α and β. By parametrising α = cos δ + iη/2 and β = sin δ + iη/2, the authors can independently tune the Hermitian phase δ and the degree of non‑Hermiticity η.

Experimentally the walk is realized with three liquid‑crystal metasurfaces (LCMS). One uniform LCMS implements the coin rotation, while two dichroic g‑plates provide the translation. The g‑plates are patterned with a periodic optic‑axis modulation that maps the transverse spatial coordinate onto the quasi‑momentum q, so that each pixel of the camera corresponds to a discrete q‑value across the first Brillouin zone. By preparing six different input polarisation states (using a linear polariser, a half‑wave plate and a quarter‑wave plate) and projecting onto six output states, the intensity I_{ij}=I₀|⟨j|U|i⟩|² is recorded for every q. From these polarimetric data the authors perform a numerical minimisation to extract the quasi‑energy E(q) and the components of the complex Bloch vector n(q) that define the effective Hamiltonian H_eff(q)=E(q) n(q)·σ.

The reconstructed evolution operators match the theoretical predictions with average fidelities of 98.8 % ± 0.5 % and 99 % ± 2 % for two representative parameter sets. Diagonalising the reconstructed H_eff(q) yields the right eigenstates |ψ₁(q)⟩ and |ψ₂(q)⟩; their overlap |⟨ψ₁|ψ₂⟩|² exhibits sharp minima at specific q‑values, signalling the presence of exceptional points where eigenvectors coalesce. By integrating the winding number ν = (1/2π)∮(n×∂ₙn)·s dq, the authors confirm a topological transition: for (δ, η) = (π/4, 0.9) the winding number is ≈0 (trivial phase), whereas for (δ, η) = (1.3, 1.4) it approaches 1 (non‑trivial phase). The transition line in the (δ, η) plane is associated with the emergence of exceptional points and a simultaneous PT‑symmetry breaking. Increasing η drives the system from a weakly non‑Hermitian regime with small imaginary parts of the bands to a strongly non‑unitary regime where the imaginary components dominate, cusps appear in the band structure, and the PT‑order parameter ⟨ψ|V K|ψ⟩ abruptly changes at the critical quasi‑momentum q_c.

Overall, the paper demonstrates (i) a method to access the full Brillouin zone in a photonic quantum‑walk experiment, (ii) precise tomographic reconstruction of complex band structures and exceptional points, and (iii) direct observation of PT‑symmetry breaking and topological winding in a non‑Hermitian system. The approach circumvents the need for elaborate real‑space engineering or post‑selection schemes required in superconducting‑qubit or trapped‑ion platforms, offering a versatile and high‑fidelity tool for exploring non‑Hermitian physics, topological phases, and enhanced quantum sensing.