Programmable non-Hermitian photonic quantum walks via dichroic metasurfaces

TL;DR

This work addresses the challenge of simulating non-Hermitian quantum dynamics in photonics by implementing a programmable non-unitary quantum walk (QW) in the synthetic space of light's transverse momentum. The approach combines dichroic liquid-crystal metasurfaces (g-plates) to realize a coin-dependent translation and a unitary coin rotation, while polarization-dependent absorption (characterized by $\eta=\eta_e-\eta_o$) induces non-unitarity, yielding a non-Hermitian evolution operator $T_{\delta,\eta}$. The authors demonstrate up to five time steps, with tunable dissipation via an external voltage that adjusts $\delta$ and $\eta$, and show that the NH QWs map to a two-band non-Hermitian tight-binding model with reciprocal but non-conjugate nearest-neighbor couplings. This platform provides a flexible optical testbed for exploring open-system NH quantum dynamics and NH topological phenomena, expanding the toolkit of photonic simulators for dissipation-driven quantum processes, including potential studies of NH topology and quantum geometry.

Abstract

The evolution of a closed quantum system is described by a unitary operator generated by a Hermitian Hamiltonian. However, when certain degrees of freedom are coupled to an environment, the relevant dynamics can be captured by non-unitary evolution operators, arising from non-Hermitian Hamiltonians. Here we introduce a photonic platform that implements non-unitary quantum walks, commonly used to emulate open-system dynamics, in the synthetic space of light transverse momentum. These walks are realized by propagating light through a series of dichroic liquid-crystal metasurfaces, that impart polarization-dependent momentum shifts. The non-unitary behavior stems from dichroic dye molecules with polarization-dependent absorption, whose orientation is coupled to that of the liquid crystals. We demonstrate multiple walks up to five time steps, with adjustable levels of dichroism set by the metasurface voltage, which is controlled remotely. This discrete-time process maps onto two-band tight-binding models with reciprocal yet non-Hermitian nearest-neighbor couplings, corresponding to a less-studied class of non-Hermitian systems. Our platform broadens the range of optical simulators for controlled investigations of non-Hermitian quantum dynamics.

Figures (5)

• Figure 1: Transverse momentum modes and dichroic $g$-plates. (a) Optical modes for the QW simulation are copropagating Gaussian beams with a slightly tilted propagation direction. The spatial period of their transverse phase profile is $\Lambda/m$, where $m$ is the mode index. A $g$-plate with spatial period ${\Lambda}$ is used to create and manipulate these modes, which can be sorted in the focal plane of a lens. (b) In dichroic metasurfaces, the dye molecules align with the LC substrate. Accordingly, the application of an electric field allows for simultaneously tuning the birefringence and the dichroic power of the device. (c) Measured values of the parameter ${\eta}$ at six voltages corresponding to ${\delta = \pi \pmod{2\pi}}$ for five dichroic ${g\text{-plates}}$.
• Figure 2: Experimental implementation. (a) Sketch of the experimental setup for the simulation of NH QWs. A Gaussian beam is expanded and spatially filtered with lenses ($\text{L}_1$-$\text{L}_2$) in a telescopic configuration and a pinhole (Ph). Its polarization is adjusted with a half-wave plate (HWP) and a quarter-wave plate (QWP), which corresponds to setting the coin input state. It then propagates through a sequence of dichroic LC metasurfaces and uniform plates, whose optical actions realize the operations of the NH QW dynamics. The output modes can be resolved in the focal plane of a lens, where they form a line of separated Gaussian spots, from which the experimental probability distribution can be extracted. (b) Representative experimental and theoretical QW distributions corresponding to a protocol with ${\delta=\pi}$, average dichroism ${\bar{\eta}=0.57}$, and horizontal input polarization, $\ket{H}=(\ket{L}+\ket{R})/\sqrt 2$. Experimental results (left) are compared to numerical simulations of the implemented QW (center) and of the unitary QW having the same value of $\delta$ but $\eta=0$ (right).
• Figure 3: Experimental results. Evolution of a localized walker in a NH QW up to five steps. Experimental data and numerical simulations refer to horizontal polarization as the input coin state, with different levels of average dichroism: (a) ${\bar{\eta}=0.57}$, (b) $\bar{\eta}=0.40$, (c) $\bar{\eta}=0.13$. The average similarity obtained at each step is reported within each panel. The dashed horizontal lines represent the mean distribution for the experimental data and the shadowed region above each bar corresponds to one standard deviation.
• Figure 4: Walker variance in a NH QW. Variance of the walker probability distribution as a function of the number of steps. The input polarization is $\ket{H}$. Experimental data (top) are compared with theoretical prediction (bottom). The shadowed region above each bar corresponds to one standard deviation. The pink dashed curve is obtained as the average best fit of the variance for the different configurations at each step and serves as a guide for the eye.
• Figure S1: Experimental results. Evolution of a localized walker in a NH QW up to five steps. Experimental data and numerical simulations refer to horizontal polarization as the input coin state, with different levels of average dichroism: (a) ${\bar{\eta}=0.48}$, (b) $\bar{\eta}=0.31$, (c) $\bar{\eta}=0.23$. The average similarity obtained at each step is reported within each panel. The dashed horizontal lines represent the mean distribution for the experimental data and the shadowed region above each bar corresponds to one standard deviation.