Tomographic characterization of non-Hermitian Hamiltonians in reciprocal space

TL;DR

Non-Hermitian Hamiltonians broaden quantum phase diagrams and host exceptional points (EPs) and PT-symmetry phenomena. The authors realize a photonic quantum-walk simulator with reciprocal hopping and a complex on-site term, enabling direct tomographic reconstruction of the non-Hermitian Hamiltonian in reciprocal space: $U(q)=\exp(-i\mathcal{H}_{\rm eff}(q))$ with $\mathcal{H}_{\rm eff}(q)=E(q)\mathbf{n}(q)\cdot\boldsymbol{\sigma}$, and reconstruct $E(q)$ and $\mathbf{n}(q)$ across the Brillouin zone. They identify EPs at critical $\eta_c$ with coalescing eigenvectors $|\psi_1(q)\rangle$ and $|\psi_2(q)\rangle$ (e.g., at $q_c$ and $2\pi-q_c$ for $\delta=1.3$), observe PT-symmetry breaking from real to purely imaginary eigenvalues, and extract a winding number $\nu$ to classify topology. The platform achieves high-fidelity tomographic reconstructions (fidelities ~99%), enabling direct measurements of complex band structures and momentum-space topology—advancing momentum-resolved non-Hermitian physics and sensing applications.

Abstract

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.

Figures (8)

• Figure 1: Non-Hermitian topological quantum walk. (a) At each time step, under the action of the operator $T$ (see Eq. \ref{['eqn:translation']}), the walker maintains the state with probability $\abs{\alpha}^2$, while coupling to neighboring sites with equal probability $\abs{\beta}^2$. The coupling coefficients to the left and right sites are complex, but not related by complex conjugation, which makes the evolution non-unitary. (b) Topological phase diagram as a function of the parameters $\delta$ and $\eta$ (see Eq. \ref{['eqn:mapping']}).
• Figure 2: Experimental process tomography. Three liquid-crystal metasurfaces simulate a single QW step. The application of an external voltage allows for dynamically adjusting the parameters $\delta$ and $\eta$ of dichroic metasurfaces, encoding the model topology. Process tomography is realized by preparing and projecting onto the desired polarization states with a linear polarizer (P), a half-wave plate (H), and a quarter-wave plate (Q). Polarimetric images are then processed to retrieve the model eigenstructure.
• Figure 3: Experimental results and theoretical predictions. Tomographic reconstructions for the cases: (a) ${(\delta,\eta)=(\pi/4,0.9)}$, and (b) ${(\delta,\eta)=(1.3,1.4)}$ across one spatial period $\Lambda$, corresponding to the first BZ. Experimentally reconstructed real and imaginary parts of the energy bands and the $\textbf{n}$-vector components (diamonds) are compared with theoretical predictions (continuous lines). The overlap between the right eigenstates is also reported as infidelity at each quasi-momentum, and the trajectories of the polarization eigenstates are visualized on the Bloch sphere.
• Figure 4: PT-symmetry breaking. The system is driven through a topological phase transition by tuning $\eta$ from 0.3 to 0.6 to 1.3, keeping ${\delta=1.3}$. At the critical value of quasi-momentum $q_c$, PT-symmetry breaking is revealed by the order parameter ${1 - |\langle \psi_{1}^r | VK | \psi_{1}^r \rangle|}$, whose theoretical prediction is drawn as dashed line. The red diamonds are obtained from the experimentally reconstructed Hamiltonian eigenstates (see also Fig. \ref{['fig:pt-appendix']} in Appendix B).
• Figure 5: Experimental results for a weakly non-Hermitian regime. Tomographic reconstructions for the case ${(\delta,\eta)=(\pi,0.25)}$ across one spatial period $\Lambda$, corresponding to the first BZ. Real and imaginary parts of the energy bands and the $\textbf{n}$-vector components are extracted and compared with theoretical predictions. The overlap between the right eigenstates is also reported as infidelity at each quasi-momentum, and the trajectories of the polarization eigenstates are visualized on the Bloch sphere.