Mapping the Seven Phases of Anderson Localization
16 Jun 2026, Yanjiang
Scientists map all seven transport phases of Anderson localization within a single photonic Floquet lattice, including the elusive triple coexistence of extended, critical, and localized states.
In a laboratory at Southern University of Science and Technology in Shenzhen, a pulse of infrared light no longer than a few billionths of a second enters a loop of optical fiber that stretches for kilometres. Inside that loop, a carefully orchestrated sequence of polarisation splitters, wave plates, and fast modulators reshapes the pulse at every round trip—pushing and pulling its polarisation, delaying one half of the light, advancing the other—so that, step by step, the pulse dances across an artificial lattice of seventy sites. Single‑photon detectors listen in after every second orbit, recording where the light lands. The whole apparatus is, at its heart, a photonic Floquet lattice built by Yao Qin, Yucheng Wang, and their colleagues at the Shenzhen Institute for Quantum Science and Engineering. And with it, the team has achieved something that had long existed only on theorists’ blackboards: a complete, single‑system map of the seven distinct transport phases of Anderson localisation.
For more than six decades, the textbook picture of waves in a disordered medium has been starkly binary. If the disorder is weak, a quantum particle or a light wave travels freely—extended states, the physicists call them. Crank up the disorder beyond a threshold, and the wave becomes trapped: the states are localised, the transport stops. This dichotomy, formulated by Philip Anderson in 1958, has become one of the cornerstones of condensed‑matter physics. But modern theory, having long since outgrown the simple extended‑versus‑localised dialectic, paints a far richer landscape. In certain one‑dimensional systems—especially those with quasiperiodic rather than completely random disorder—a third, intermediate regime appears: critical states. These states are neither fully delocalised nor fully pinned. They occupy a strange middle ground where the wave has a fractal character, simultaneously spreading and staying put, and displaying no characteristic length scale at all. Furthermore, because a single system can harbour different transport behaviours at different energies, the three pure phases can coexist, producing a total of seven possible phases: extended (E), critical (C), localised (L), and the four coexistence phases E+L, C+L, E+C, and the elusive triple coexistence phase E+C+L.
“We observe all seven phases, including the elusive triply coexisting extended‑critical‑localized phase,” the authors write. And they do it inside a single recirculating optical‑loop platform, carefully tuning the hopping between lattice sites to sculpt the required quasiperiodic profile. The work appears in a preprint (arXiv:2606.14825) and constitutes the first experimental realisation of the complete Anderson‑localisation landscape.
The key to generating critical states—and to controlling how they mix with extended and localized ones—lies in what the team calls inhomogeneously distributed hopping zeros, or IDZs. In their photonic lattice, the hopping amplitude between adjacent sites is not uniform. It follows a quasiperiodic pattern controlled by two parameters, lambda₊ and lambda₋, that set the depth of modulation in the effective magnetic‑like hopping terms. When the modulation is strong enough, at certain positions the hopping amplitude becomes so small that, for all practical purposes, the light cannot jump between those sites. These are the IDZs. By strategically placing these zeros—and making sure they are not spaced evenly—the team breaks the lattice into regions where different transport personalities can live side by side. A broad region with no zeros might host extended states; a region hemmed in by zeros on both sides might host localised ones; and the boundaries, where zeros crowd close together, give birth to critical states that are confined yet oscillatory.
The architecture of the experiment is a one‑dimensional Floquet photonic lattice implemented in a recirculating optical‑loop setup. A 6‑nanosecond laser pulse at 1560 nm circulates through a 4‑km fibre loop that prevents temporal overlap between successive round trips. Two round trips make one complete Floquet cycle, during which the light’s polarisation (playing the role of a synthetic spin) undergoes a sequence of site‑dependent rotations and shifts. A crucial trick is that the loop is not a simple ring but contains an internal interferometer that splits the pulse into two time‑delayed components, mimicking nearest‑neighbour hopping both forward and backward. By programming the voltages on electro‑optic modulators, the researchers can imprint any desired hopping profile—including the quasiperiodic patterns that create IDZs—and observe the spatiotemporal evolution of the light over sixty Floquet cycles.
What they see is striking. When the parameters place the system in a purely extended phase, the light pulse spreads ballistically, its wavefront racing outward as a bright cone on the space‑time plot. In a purely localised phase, the pulse stays pinned near its starting position, its intensity barely changing after the first few cycles. In a purely critical phase, the light neither expands nor freezes: it oscillates within a tightly bounded region, its survival probability decaying in a peculiar, non‑exponential fashion that signals the absence of any characteristic diffusion scale. These three pure phases—E, C, L—are the building blocks. The team then dials the parameters to regions where the system should exhibit coexistence. In the E+L phase, for example, the spatiotemporal distribution splits into two clear sectors: a fast ballistic front from the extended component and a stationary core from the localized one. In the C+L phase, the critical oscillatory sector sits inside a confining box defined by the nearest hopping zeros, while a truly frozen L component lingers elsewhere. The most spectacular is the triple E+C+L phase: all three transport behaviours appear simultaneously in different parts of the same light pulse, their boundaries marked by the positions of the IDZs. The wavefront and survival probability diagnostics—extracted directly from the photon counts—match numerical simulations to a degree that confirms the designed phases are indeed present.
A natural question raised by earlier work on exact quantum critical states is whether approximate methods can faithfully reproduce the full phenomenology. Last year, a group led by Huang and colleagues realised exact quantum critical states in a photonic lattice by engineering genuine zero‑hopping points. In the present work, the IDZs are not mathematically exact zeros—they are merely extremely small—and the system is finite. Does that blur the critical behaviour? The data suggest that, for transport diagnostics such as wavefront velocity and survival probability, the answer is no. The critical signatures are robust enough to survive the approximations. However, the team does not report a full multifractal spectrum analysis, the gold‑standard diagnostic for true criticality. Whether the observed critical states would pass that rigorous test remains an open, and interesting, question. One could also worry about finite‑size effects and the overlap of the initial state with the desired spectral regions, but the close agreement between experiment and simulation across the entire phase diagram gives confidence that the essential physics is captured.
Beyond the proof of principle, the team goes further and tracks the phase transitions themselves. By fixing one control parameter and sweeping the other, they capture the dynamics as the system moves from one coexistence phase to another—say, from E+L through C+L to pure L, or from E+C through pure C to C+L. The spatiotemporal plots show extended sectors dying, critical oscillations appearing and then shrinking, and localised sectors growing. Quantitative diagnostics such as the survival probability and wavefront velocity change slope at the expected boundaries, offering an experimental measurement of mobility edges—the energies that separate extended from localised or critical states—in real time.
The achievement establishes a unified platform for exploring an entire class of disordered transport problems on a single table‑top. Because the hopping profile is entirely programmable, future experiments can, for example, investigate the multifractal properties of critical states directly, measure the full spectrum of fractal dimensions, or explore how interactions—by adding optical non‑linearities—modify the seven‑phase landscape. The loop architecture is also a natural testbed for studying the interplay of topology and disorder, because the Floquet lattice can host non‑trivial winding numbers that coexist with localisation.
For now, the team has delivered what no previous experiment could: a single, clean, and thoroughly mapped realisation of the complete Anderson‑localisation hierarchy. It is a map that will, one suspects, be pinned to the walls of many offices, a reminder that even in disorder, nature organises itself into a surprisingly orderly number of ways to travel—or to stay still.
— Yanjiang
Yanjiang is an online editor of LoomSci.com.
References
- Yao Qin et al., Experimental realization of the complete seven-phase Anderson-localization landscape, arXiv:2606.14825
- Huang et al., Experimental observation of exact quantum critical states, arXiv:2502.19185