Interferometric Readout of Momentum-Space Topology in a Programmable Dissipative Photonic Circuit

AI summary

Overview

Research area: Non-Hermitian topological photonics, programmable photonic integrated circuits, and quantum simulation of synthetic momentum-space models.

Technical level: Advanced. The paper assumes familiarity with non-Hermitian Hamiltonians, Zak phases, Chern numbers, exceptional points, and unitary dilation.

Scope: The paper reports an experimental method, implemented on a programmable photonic integrated circuit, that reads out momentum-space topological invariants directly from the phase of a non-Hermitian bulk evolution, even when strong dissipation suppresses the underlying amplitudes.

What This Paper Is About

Topological invariants such as the Zak phase and the Chern number are defined directly in momentum space, but in photonics they are usually inferred indirectly from edge or interface states, or from real-space wavepacket dynamics. Those approaches require carefully terminated finite structures, fixed geometries, or long-time bulk averages. The paper's goal is to measure the momentum-space invariant itself, point by point, on reconfigurable hardware, and to do so in the strongly dissipative regime where the amplitudes carrying the topological phase are heavily suppressed.

Key Contributions

1. Interferometric readout of a non-Hermitian bulk evolution. The authors embed a non-unitary propagator into a larger unitary through Sz.-Nagy dilation and recover the relative phase of the two signal amplitudes using four phase-shifted intensity measurements at (\phi = 0, \pi/2, \pi, 3\pi/2). The resulting interferometric coherence is (C(k,t) = [I_0 - I_\pi] + i[I_{3\pi/2} - I_{\pi/2}]).

2. Direct extraction of the Zak phase in the non-Hermitian SSH model. The method distinguishes trivial and non-trivial Zak phases by integrating (\partial_k \arg C) across the Brillouin zone, with no real-space boundary or spectrally resolved edge mode required.

3. An interferometric signature of an exceptional point. The authors define a coherence winding (q(\alpha)) by integrating (d\arg C) around a closed contour in the ((\mathrm{Re},C, \mathrm{Im},C)) plane and measure (q = 1) for counter-clockwise and (q = -1) for clockwise traversal.

4. Extension to a synthetic torus and a first Chern number. A non-Hermitian Rice-Mele pump is programmed over ((k,\theta)), and the first Chern number is extracted from the winding of the Zak phase, giving (\mathrm{Ch}_1 = 0) and (\mathrm{Ch}_1 = 1) for trivial and non-trivial cycles.

Main Findings

• Zak phase distinguishes the two phases. For the non-trivial case ((j_1,j_2,\gamma) = (0.5,1.5,4)), the accumulated Zak phase reaches (1.02\pi) at (k=\pi); for the trivial case ((1.5,0.5,4)), it reaches (-0.00148\pi). Across the transition at fixed (j_1 = 0.5), the average values are (-0.0254\pi) and (1.04\pi) on the two sides, quantized to (0) and (\pi). These data are taken at (t=2) with (N_k = 25) momentum points over (k \in [0,\pi]), i.e. 100 unitaries per parameter set.

• Readout fidelity. For the representative well-conditioned interval (t \in [1,2]), the root-mean-square errors relative to simulation are (0.0613\pi) for the phase and (0.0194) for the magnitude. Deviations concentrate where (|C(k,t)|) is small and the phase is ill-conditioned.

• Exceptional-point winding. Both simulation and experiment give (q=1) for counter-clockwise and (q=-1) for clockwise encirclement of the EP-induced coherence zero. The authors emphasize that this integer (q) is distinct from the conventional eigenvalue vorticity of a second-order EP, which yields a half-integer winding (\pm 1/2) upon encircling the EP in parameter space.

• First Chern number from a Rice-Mele pump. For the non-trivial cycle with ((j_0,\delta,\Delta_c,\Delta_0,\gamma,t) = (2,0.7,0,0.8,2,0.25)) and input state ((1,-i)^T/\sqrt{2}), the phase and magnitude RMSEs are (0.0208\pi) and (0.0302). The momentum-averaged magnitudes have RMSEs of (0.00769) and (0.00864) for the non-trivial and trivial cycles. The reconstructed pump trajectory gives (\mathrm{Ch}_1 = 1) and (\mathrm{Ch}_1 = 0), with fitted loops separated from the gap closing by margins of (0.805) (non-trivial) and (0.175) (trivial).

• Dissipative regime is handled explicitly. For (|f(k)| < \gamma/2), the two bands are distinguished not by (\mathrm{Re},E_\pm(k)) but by (\mathrm{Im},E_\pm(k)), i.e. by their decay rates. The dilation plus interferometry recovers phase information in this regime.

• Resource scaling. An (N)-band model requires (N) signal and (N) ancillary modes. The two-dimensional data use (N_k = N_\theta = 25) uniformly spaced points over (k,\theta \in [0,2\pi)), i.e. 2500 unitaries per cycle; panels showing (k \in [0,\pi]) evaluate RMSEs over that half-range.

Methodology in Plain English

Each momentum point is realized as a separately programmed unitary on the same photonic chip, rather than as a physical lattice with edges. Because the target non-Hermitian evolution does not conserve probability, it cannot be run directly on unitary optics: the propagator (e^{-iH(k)t}) is first rescaled by its largest singular value to make it a contraction, then embedded into a larger unitary acting on a signal subspace ((A,B)) plus an ancilla subspace ((\tilde A,\tilde B)). The chip implements a compiled (4\times4) transformation (U(k,t,\phi) = U_{\mathrm{int}}(\phi)U_{\mathrm{dil}}(k,t)).

After the dilated evolution, the signal amplitudes are mixed against four phase-shifted references, and the four output intensities are combined into the complex coherence (C(k,t)). Its argument gives the relative phase of the (A) and (B) amplitudes; its modulus tells how strong the readout is. Repeating at successive (k) points traces the phase across the Brillouin zone, and integrating that phase gives the Zak phase. For the two-dimensional case, the same procedure is repeated over a pump parameter (\theta), and the winding of the Zak phase over one cycle gives the first Chern number. In that case, candidate Rice-Mele loops are fitted to the measured coherence before the invariant is computed.

Why This Matters

The work shifts momentum-space topology from an inferred property to a directly measured one on reconfigurable hardware. It avoids the need for finite structures with well-separated edge modes, avoids the discrete momentum sampling and finite-size sensitivity of real-space dynamical methods such as the mean chiral displacement, and remains viable where dissipation is strong enough to suppress the amplitudes that carry the topological phase. It also connects exceptional-point physics to an experimentally accessible coherence signal rather than to direct eigenvalue spectroscopy.

Potential applications, which the paper does not itself demonstrate:

• Programmable topological photonic devices whose response can be reconfigured across a phase diagram without fabricating a new sample.
• Metrology and sensing built on exceptional-point-enhanced or phase-sensitive interferometric readouts.
• Optical communications and signal processing using reconfigurable integrated photonic meshes with designed non-Hermitian response.
• Quantum and classical simulation of non-Hermitian symmetry classes and topological invariants that are inaccessible to fixed-geometry platforms.

Industry relevance: the method relies on programmable photonic integrated circuits of the kind pursued by integrated-photonics foundries, and on calibration of imperfect devices. It points toward reusable hardware for exploring topological parameter spaces, rather than one-off samples per model.

Future Directions

• Scaling to multi-band models. The paper states that an (N)-band model needs (N) signal and (N) ancillary modes; demonstrating larger (N) experimentally is a natural next step.
• Other non-Hermitian symmetry classes and invariants. The authors argue that the platform's reconfigurability suits experimental investigation of invariants that fixed-geometry platforms cannot reach.
• Calibration and error mitigation. The finite fidelity of the programmed unitary is identified as the relevant imperfection; a calibrated model of an imperfect PIC is proposed to account for the actual device response.
• Higher-dimensional synthetic spaces. The Rice-Mele demonstration establishes a two-dimensional synthetic torus; extending the approach to richer pump cycles and higher-dimensional parameter spaces remains open. The paper does not report device footprint, material platform, or a detailed experimental noise budget.

Target Audience

Researchers in topological photonics, non-Hermitian physics, and integrated photonics; experimentalists building programmable optical processors for simulating quantum and classical Hamiltonians; and graduate students or advanced readers interested in direct momentum-space measurement of topological invariants. Readers seeking introductory background on non-Hermitian topology or Sz.-Nagy dilation will need supplementary material, since the paper assumes that background.

Authors’ abstract

Topology under non-Hermitian dynamics is encoded in the phase of the bulk evolution, yet strong dissipation suppresses the amplitudes carrying it. We resolve this using a programmable photonic integrated circuit that implements such dynamics in synthetic momentum space through unitary dilation, with the phase recovered by phase-shifted interferometry. For the non-Hermitian Su-Schrieffer-Heeger model, the method distinguishes trivial and non-trivial Zak phases and yields a coherence winding $q=\pm 1$ induced by an exceptional point. Extending to a synthetic torus via a Rice-Mele pump gives the first Chern number $\mathrm{Ch}_1=0,1$ for trivial and non-trivial cycles, showing a programmable route to momentum-space topology under strongly dissipative non-Hermitian dynamics.