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Spatiotemporal Topological Phase Transition in non-Hermitian Photonic System

This paper experimentally demonstrates a unified spatiotemporal topological phase transition in a static non-Hermitian photonic crystal by engineering a waveguide-assisted SSH model that bridges energy and momentum band topologies, enabling real-time control over band evolution through spatial translation.

Original authors: Zimeng Zeng, Zhuoyang Li, Jiayao Liu, Zelong He, Zhaona Wang

Published 2026-02-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zimeng Zeng, Zhuoyang Li, Jiayao Liu, Zelong He, Zhaona Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where light doesn't just travel through space, but also dances through time. Usually, scientists study how light behaves in space (like in a crystal made of glass) or how it behaves in time (like a material that changes its properties every second). These are two different rulebooks.

This paper introduces a clever trick to combine these two rulebooks into one, but with a twist: they do it using a completely static object (something that doesn't move or change over time) by using a concept called "loss" (energy disappearing).

Here is the breakdown of their discovery using simple analogies:

1. The Problem: Two Separate Worlds

Think of Spatial Crystals as a hallway with a repeating pattern of doors. Light moves through them, and the "gaps" in the pattern determine which colors of light can pass. This is about Energy.
Think of Temporal Crystals as a hallway where the doors open and close in a rhythmic pattern over time. Light moves through them, and the "gaps" determine how the light's momentum changes. This is about Time.

For a long time, scientists studied these separately. To study the "Time" version, you usually need a machine that physically changes the material incredibly fast (like a strobe light flashing at optical speeds), which is extremely hard to build.

2. The Solution: The "Lossy" Waveguide

The researchers built a special "hallway" for light made of a waveguide (a pipe for light) and a grating (a comb-like structure).

The Trick: Instead of trying to make the material change over time, they introduced loss (they made some parts of the hallway absorb light, like a sponge soaking up water).
The Analogy: Imagine a musical instrument. If you play a note perfectly, it rings clearly (Energy band). If you put your hand on the string to dampen it (Loss), the sound changes completely, and the rules of how the note behaves shift.
By carefully tuning how much light is "soaked up" (loss) versus how much the light waves connect to each other (coupling), they created a system that behaves as if it were changing in time, even though the object is sitting still.

3. The Map: A Unified "Spatiotemporal" Landscape

The team created a 2D map (a phase diagram) that acts like a GPS for light.

The Axes: One side of the map is "Loss," and the other is "Coupling."
The Zones:
- Blue Zones (Energy): Here, the light behaves like it's in a normal crystal (spatial).
- Purple Zones (Momentum/Time): Here, the light behaves like it's in a time-crystal (temporal).
- The Border: There is a special line where the rules flip. Crossing this line is like stepping from a world where time flows normally into a world where time behaves strangely.

They found four distinct "countries" on this map. You can travel from one to another just by changing the amount of loss or coupling, without ever needing to move the object or change it over time.

4. The Experiment: The "Gradient" Slide

To prove this works, they didn't just simulate it; they built it.

The Gradient: They fabricated a single piece of material where the "comb teeth" get slightly different shapes from one end to the other.
- One end: Has a lot of "loss" and weak connection.
- The other end: Has little "loss" and strong connection.
- The Middle: Is the transition zone.
The Walk: They shined a laser on the material and slowly moved the laser spot from one end to the other.
The Result: As the laser moved across the static material, the light's behavior changed continuously. It started in a "Spatial" mode, crossed a mysterious "Time-like" gap (where the light split in a specific way), and ended up in a different "Spatial" mode.

5. The "Exceptional Points" (The Magic Corners)

In the middle of their experiment, they found something called Exceptional Points (EPs).

Analogy: Imagine two roads merging into one. At the merge point, the two roads become indistinguishable. If you go past that point, the roads split again, but they might look different than before.
In their experiment, at these specific points, the light waves merged perfectly and then split apart in a way that proved they had crossed a topological boundary. They saw this happen multiple times in their single sample.

Summary

The paper claims to have built a static platform (a fixed piece of hardware) that acts as a universal translator between "Space" physics and "Time" physics. By using loss as a control knob, they created a complete map of how light can transition between these two states. They proved this by sliding a laser across a specially designed, gradually changing material and watching the light's "personality" shift in real-time, effectively simulating the complex physics of time crystals without needing any moving parts or fast-switching electronics.

Technical Summary: Spatiotemporal Topological Phase Transition in a Non-Hermitian Photonic System

Problem Statement
While energy-band topology in spatial photonic crystals (PCs) and momentum-band topology in temporal crystals have been established as distinct frameworks for probing topological phases, their unification within a static platform remains unexplored. Conventional approaches to spatiotemporal topology rely on physical temporal modulation of material parameters, a requirement that is extremely demanding at optical frequencies. Although recent efforts have utilized non-Hermiticity in spatial PCs to simulate temporal topology via PT-symmetry breaking, these studies often remain tethered to time-dependent frameworks or fail to provide a global phase diagram. A fundamental challenge persists: is it possible to achieve a complete spatiotemporal topological transition and a predictive phase diagram using a completely static Hamiltonian without temporal modulation?

Methodology
The authors propose and realize a static platform based on a waveguide-assisted non-Hermitian Su-Schrieffer-Heeger (SSH) model.

Theoretical Framework: The system is modeled as a 1D non-Hermitian SSH chain where the duty cycle (DC) and etching depth (ED) of a grating structure control the relative loss ( $\eta$ ) and coupling strength ( $\lambda$ ). This creates a two-dimensional parameter space. The authors derive effective Hamiltonians for both small-momentum regions (energy-band regime) and frequency-dependent regions (momentum-band regime).
Topological Analysis: Recognizing that conventional topological invariants (e.g., winding numbers in the complex energy plane or Zak phases in a generalized Brillouin zone) are inapplicable due to the complex-momentum spectrum and the lack of temporal periodicity, the authors employ a biorthogonal parallel-transport method. This approach tracks the holonomy of Bloch eigenstates along an open path in the frequency parameter to identify topological phases.
Experimental Implementation: A one-dimensional grating-waveguide coupled structure is fabricated using dual-beam interference lithography with a tailored development process to control DC and ED.
- Static Observation: Angle-resolved reflection spectroscopy is used to observe Bloch momentum-band gaps and Exceptional Points (EPs). A spectral-derivative method is developed to reconstruct band structures from transmission data obscured by non-Hermitian mode broadening.
- Gradient Implementation: To realize a continuous transition, a spatially graded PC is fabricated where a Gaussian-shaped interference envelope creates a smooth gradient in DC and ED. This projects the abstract $(\lambda, \eta)$ parameter space onto real-space coordinates.

Key Results

Unified Phase Diagram: The study constructs a complete spatiotemporal topological phase diagram in the $(\lambda, \eta)$ parameter space. This diagram reveals four distinct topological phases: two energy-band phases (trivial and non-trivial) and two momentum-band phases (Phase I and Phase II). These phases are separated by a boundary defined by a Dirac cone condition, with the point $(1, 0)$ serving as a four-phase transition point.
Static Spatiotemporal Transition: The system demonstrates that PT-symmetry breaking (driven by loss) opens a momentum-band gap, while coupling strength governs energy-band topology. The interplay allows for a continuous transition between energy-band and momentum-band regimes without temporal modulation.
Experimental Verification of Momentum Gaps: Angle-resolved reflection spectra confirm the existence of Bloch momentum-band gaps centered at $k=0$ and the presence of EP pairs where bands merge. The spectral-derivative method successfully reconstructs band structures from transmission data, showing excellent agreement with the non-Hermitian SSH model.
Continuous Transition in Gradient PCs: By scanning across the fabricated gradient sample, the authors experimentally trace a path that transitions from a topologically trivial energy-band phase, through a momentum-band gap region, to a non-trivial energy-band phase. This spatial scan effectively maps onto the theoretical pathways (e.g., paths ①-③) in the global phase diagram. The transition is accompanied by a band inversion of symmetry-protected bound states in the continuum (SP-BIC).

Significance and Claims
The paper claims to bridge the gap between spatial and temporal topological physics by demonstrating that a static, non-Hermitian photonic system can serve as a synthetic spacetime arena.

Unification: It unifies energy-band and momentum-band topologies within a single, static parameter space, moving beyond the isolated study of spatial or temporal crystals.
New Characterization: It introduces a viable theoretical approach (biorthogonal parallel transport) for analyzing topological invariants in non-chiral, non-Hermitian systems where standard methods fail.
Experimental Platform: It establishes a versatile, bias-free platform for exploring synthetic spacetime physics. The work demonstrates that real-time control over band topology evolution pathways can be achieved simply by translating a probe across a static, spatially graded sample.
Future Outlook: The authors suggest this approach opens new avenues for controlling light via non-Hermitian band engineering, offering a more accessible and cost-effective static platform for generating EPs and Bloch momentum-band gaps compared to conventional dynamic optical systems. The findings are presented as a step toward robust signal control and nonreciprocal meta-devices operating in static, loss-engineered environments.