Observation of complete delocalization in disordered photonic lattices

Technical Summary: Observation of Complete Delocalization in Disordered Photonic Lattices

Problem Statement
In quantum transport, the nature of eigenstates dictates system behavior. While perfectly ordered lattices support extended Bloch modes enabling ballistic transmission, the introduction of disorder typically leads to Anderson localization (AL), where states become localized and diffusion ceases. Although exceptions to AL exist in systems with specific parameter correlations (e.g., random-dimer models) or quasiperiodicity, these generally result in extended states only at discrete energy eigenvalues. Theoretical proposals have suggested the possibility of engineering bands of extended states across the entire energy spectrum in disordered systems, but experimental verification of complete delocalization (perfect transmission) across the full allowed energy range in a completely disordered lattice has not been previously reported.

Methodology
The authors investigate a disordered diamond-dot (DD) chain using a tight-binding formalism.

• Theoretical Framework: The system is modeled as a 1D chain of diamond plaquettes with on-site potentials $\varepsilon_n = \varepsilon_0$ and nearest-neighbor hopping amplitudes $t$ and $\bar{t}$. A magnetic flux $\Phi$ is introduced into each diamond plaquette. Using a real-space renormalization technique, the DD chain is mapped to an effective 1D model. The authors analyze the transfer matrices to identify conditions under which the system commutes, leading to a continuous energy spectrum of extended states.
• Experimental Implementation: The study utilizes femtosecond (fs) laser-writing technology to fabricate photonic lattices consisting of optical waveguides. This technique allows for precise control over the lattice geometry, specifically the dimensions of the diamond plaquettes ($d_x = d_y$).
• Parameters: The critical geometrical parameter is defined as the hopping ratio $\Delta \equiv \bar{t}/t$. The experiments are conducted with two magnetic flux configurations: zero flux ($\Phi = 0$) and effective $\pi$ flux ($\Phi = \pi$), the latter achieved by inserting dipolar P-waveguides to induce negative coupling.
• Validation: Theoretical predictions are validated through numerical simulations of the dynamical model and direct experimental observation of light transmission probabilities and intensity output profiles.

Key Contributions and Results

1. Analytical Proof of Complete Delocalization: The authors analytically demonstrate that when the hopping ratio satisfies the critical condition $\Delta_c = 1/\sqrt{2}$ at zero flux ($\Phi = 0$), the transfer matrices of the disordered system commute. This condition renders the completely disordered DD chain effectively equivalent to a perfectly ordered 1D lattice. Consequently, the system exhibits a continuous energy spectrum where all single-particle states are perfectly extended Bloch modes, resulting in perfect transmission ($T=1$) for the entire allowed range of energies, independent of the specific disorder realization.
2. Experimental Observation of Ballistic Transmission: In photonic lattices fabricated with varying diamond geometries, the authors observed a transition from localized to ballistic transport.
   • For geometries deviating from the critical condition (e.g., $d_x = 20$ $\mu$m, where $\Delta < \Delta_c$), light transport was suppressed due to destructive interference and Anderson-like localization.
   • At the critical geometry ($d_x \approx 13$ $\mu$m, satisfying $\Delta \approx \Delta_c$), the system became transparent. Experimental data showed transmission probabilities reaching approximately 91% for a single plaquette and ~80% for lattices containing 9 to 32 randomly distributed plaquettes. This confirms the persistence of delocalization across the full energy band in a disordered medium.
3. Aharonov-Bohm (AB) Caging: The study further demonstrates that introducing an effective magnetic flux of $\Phi = \pi$ induces extreme localization (AB caging) regardless of the lattice geometry. Under this condition, the effective hopping becomes zero, leading to a flat-band regime with compact eigenstates and near-zero transmission ($T \sim 1\%$). This effect arises from destructive interference between the upper and lower channels of the diamond plaquettes.

Significance
The paper reports the first direct experimental observation of the complete absence of Anderson localization and the occurrence of perfect ballistic transmission in a completely disordered lattice system across the entire energy spectrum. The work establishes that specific geometrical conditions can create a "transparent window" that overrides the effects of disorder. Furthermore, it highlights the tunability of the system, showing that the same disordered lattice can be switched between a perfectly conducting state (at $\Delta_c, \Phi=0$) and a perfectly insulating state (at $\Phi=\pi$) via magnetic flux control. These findings provide a platform for controlling energy transmission from ballistic to zero transmission in disordered systems, with potential relevance to optical transmission control and wave physics phenomena involving transport and localization.