Observation of Non-Hermitian Spectral Deformation in Complex Momentum Space

TL;DR

This work demonstrates the experimental observation of non-Hermitian spectral deformation in complex momentum space by engineering a non-Hermitian SSH-like lattice with long-range couplings in a synthetic OAM dimension inside a degenerate optical cavity. The authors encode complex momentum via $K=k-i\mu$ with $\beta=e^{-iK}$ and perform complex-momentum–resolved spectroscopy using a phase-only SLM to project onto left non-Bloch states, enabling direct mapping of $E_s(\beta)$ as $\mu$ varies. They extract open-boundary spectra, GBZ via $\mu_{GBZ}$, exceptional points, and the Ronkin-function landscape, validating the non-Bloch band theory and its spectral geometry. The results establish a versatile photonic platform for probing non-Hermitian physics in complex momentum space and point to extensions to higher-dimensional Amoeba formulations and generalized Brillouin zones. The work provides a direct experimental bridge between spectral deformation, GBZ topology, and non-Bloch invariants with potential broad impact on non-Hermitian quantum systems.

Abstract

Open systems feature a variety of phenomena that arise from non-Hermitian physics. Recent theoretical studies have offered much insights into these phenomena through the non-Bloch band theory, though many of the theory's key features are experimentally elusive. For instance, the correspondence between complex momenta and non-Hermitian bands, while central to non-Bloch band theory, has so far defied direct experimental observation. Here we experimentally study the non-Hermitian spectral deformation in complex-momentum space, by implementing a non-Hermitian lattice with long-range couplings in the synthetic orbital-angular-momentum (OAM) dimension of photons inside a degenerate cavity. Encoding the complex momenta in the phase and amplitude modulations of the OAM modes, and devising a complex-momentum-resolved projective detection, we reconstruct the spectral deformation in momentum space, where the eigenspectrum on the complex plane morphs through distinct geometries. This enables us to experimentally extract key information of the system under the non-Bloch band theory, including exceptional points in the complex-momentum space, the open-boundary spectra, and the generalized Brillouin zone. Our work demonstrates a versatile platform for exploring non-Hermitian physics and non-Bloch band theory, and opens the avenue for direct experimental investigation of non-Bloch features in the complex-momentum space.

Figures (5)

• Figure 1: Spectral geometry and deformation of a non-Hermitian SSH model.a. Schematic of the non-Hermitian SSH lattice with long-range couplings. The gray dashed box indicates a unit cell, and the red and blue spheres denote the sublattice sites. b. Evolution of the complex energy spectrum $E(k - i\mu)$ with varying imaginary momentum component $\mu$ for $\delta_2=0$. Blue loops represent eigenspectra under the periodic boundary condition (PBC), with $\mu=0$. Gray lines indicate those under the open boundary condition (OBC), with $\mu_{\text{GBZ}}=-0.23$. Red loops mark the emergence of exceptional points (EPs) at $\mu=-0.48$, where the line gap closes. The arrows indicate the spectral winding direction as $k$ increases from $0$ to $2\pi$. c. The Wasserstein metric $G_w(\mu) = \partial_\mu A(\mu) / 2\pi$, where $A(\mu)$ is the area enclosed by the eigenspectrum. The colored points mark the parameters for the corresponding spectra in b. d. Spectral geometry with long-range couplings. Self-intersections of the spectrum trace the eigenspectrum under OBC (gray dashed), with the corresponding complex momenta moving along the GBZ. Inset (magnified view): self-intersections, which connect the interiors of loops with winding number $w \neq 0$.
• Figure 2: Experimental implementation and complex-momentum-resolved measurements.a. Schematic of the experimental setup. Two mirrors form a degenerate optical cavity, with the red path indicating the trajectory of the probe light. Colored optical elements inside the cavity introduce controllable mode couplings. A spatial light modulator (SLM) followed by a single-mode fiber (SMF) are used for projective detection. Projection with arbitrary non-Bloch states $\langle\beta^{L}|$ are realized by loading the corresponding phase pattern onto the SLM. Inset: For a complete eigenspectra measurement, we scan $\beta$ along circular trajectories with different radius ($\mu$) on the complex plane, and record the output light intensity. WP: wave plate; PPBS: partial polarization beam splitter; FC: fiber coupler; PD: photodetector. b. Illustration of the non-Hermitian SSH model encoded in different OAM modes. The colored annuli represent the intensity and phase profiles of the OAM modes. Red and blue regions denote components with left- and right-circular polarizations, respectively. The arrows indicate mode couplings, whose colors correspond to the optical elements in a that generate these couplings.
• Figure 3: Measurement of the spectral deformation without long-range couplings.a, c, e, and g. Transmission spectra measured at $\mu = 0, -0.1, -0.23,$ and $-0.48$ under the parameters $(\delta_1, \delta_2, \eta, \gamma) = (0.31\pi, 0, 0.25\pi, 0.057\pi)$. The top panel shows the experimental results, and the bottom panel displays the corresponding theoretical predictions. The color scale is normalized to [0, 1]. b, d, f, and h. The dots represent the complex eigenenergies extracted from a, c, e, and g, and the solid curves denote the corresponding theoretical calculations. Here $\mu_{\text{GBZ}}=-0.23$ resides on the GBZ, and f corresponds to the eigenspectrum under OBC.
• Figure 4: Ronkin function measurement.a Ronkin-function landscape from experimental data (left) and theoretical calculations (right) under the parameters $(\delta_1, \delta_2, \eta, \gamma) = (0.31\pi, 0, 0.25\pi, 0.057\pi)$. b Ronkin function with fixed $E = 0$ and $E = 0.74\pi$, respectively. Experimental data are shown as dots, and theoretical curves are shown as solid lines.
• Figure 5: Measurement of the spectral deformation with long-range couplings.a, c, e, and g. Experimental (top) and theoretical (bottom) transmission spectra at $\mu = 0, -0.03, -0.06,$ and $-0.09$, respectively, with $(\delta_1, \delta_2, \eta, \gamma) = (0.13\pi, 0.5\pi, -0.125\pi, 0.036\pi)$. b, d, f, and h. Dots represent the complex eigenenergies extracted from panels a, c, e, and g; solid curves show the corresponding theoretical results. The thick gray curves denote the eigenspectra under OBC.