A photonic thermalization gap in disordered lattices

📝 Abstract

The formation of gaps – forbidden ranges in the values of a physical parameter – is a ubiquitous feature of a variety of physical systems: from energy bandgaps of electrons in periodic lattices and their analogs in photonic, phononic, and plasmonic systems to pseudo energy gaps in aperiodic quasicrystals. Here, we report on a thermalization' gap for light propagating in finite disordered structures characterized by disorder-immune chiral symmetry -- the appearance of the eigenvalues and eigenvectors in skew-symmetric pairs. In this class of systems, the span of sub- thermal photon statistics is inaccessible to input coherent light, which -- once the steady state is reached -- always emerges with super-thermal statistics no matter how small the disorder level. We formulate an independent constraint that must be satisfied by the input field for the chiral symmetry to be activated’ and the gap to be observed. This unique feature enables a new form of photon-statistics interferometry: the deterministic tuning of photon statistics – from sub-thermal to super-thermal – in a compact device, without changing the disorder level, via controlled excitation-symmetry-breaking realized by sculpting the amplitude or phase of the input coherent field.

💡 Analysis

The formation of gaps – forbidden ranges in the values of a physical parameter – is a ubiquitous feature of a variety of physical systems: from energy bandgaps of electrons in periodic lattices and their analogs in photonic, phononic, and plasmonic systems to pseudo energy gaps in aperiodic quasicrystals. Here, we report on a thermalization' gap for light propagating in finite disordered structures characterized by disorder-immune chiral symmetry -- the appearance of the eigenvalues and eigenvectors in skew-symmetric pairs. In this class of systems, the span of sub- thermal photon statistics is inaccessible to input coherent light, which -- once the steady state is reached -- always emerges with super-thermal statistics no matter how small the disorder level. We formulate an independent constraint that must be satisfied by the input field for the chiral symmetry to be activated’ and the gap to be observed. This unique feature enables a new form of photon-statistics interferometry: the deterministic tuning of photon statistics – from sub-thermal to super-thermal – in a compact device, without changing the disorder level, via controlled excitation-symmetry-breaking realized by sculpting the amplitude or phase of the input coherent field.

📄 Content

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A photonic thermalization gap in disordered lattices H. Esat Kondakci, Ayman F. Abouraddy and Bahaa E. A. Saleh CREOL, The College of Optics & Photonics, University of Central Florida, Orlando, Florida 32816, USA

The formation of gaps – forbidden ranges in the values of a physical parameter – is a ubiquitous feature of a variety of physical systems: from energy bandgaps of electrons in periodic lattices1 and their analogs in photonic2, phononic3, and plasmonic4 systems to pseudo energy gaps in aperiodic quasicrystals.5 Here, we report on a ‘thermalization’ gap for light propagating in finite disordered structures characterized by disorder-immune chiral symmetry6 – the appearance of the eigenvalues and eigenvectors in skew-symmetric pairs. In this class of systems, the span of sub- thermal photon statistics is inaccessible to input coherent light, which – once the steady state is reached – always emerges with super-thermal statistics no matter how small the disorder level. We formulate an independent constraint that must be satisfied by the input field for the chiral symmetry to be ‘activated’ and the gap to be observed. This unique feature enables a new form of photon-statistics interferometry: the deterministic tuning of photon statistics – from sub-thermal to super-thermal – in a compact device, without changing the disorder level, via controlled excitation-symmetry-breaking realized by sculpting the amplitude or phase of the input coherent field.

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For electrons in crystals, lattice symmetries play a critical role in establishing energy gaps, and introducing disorder typically diminishes their role [1]. One exception lies in certain disorder- immune symmetries that emerge in random matrix theory [7], such as chiral [6] and particle-hole symmetric ensembles [8], which play decisive roles in diverse areas of physics ranging from superconductivity [9] to quantum chromodynamics [10]. A hallmark of disorder-immune chiral symmetry [7,11] is that the system Hamiltonian can be transformed into a block off-diagonal matrix representation, corresponding to separate bipartite sublattices [12]. We elucidate this concept in the context of the one-dimensional (1D) disordered tight-binding lattice models depicted in Fig. 1a,b, one of which maintains chiral symmetry. In Fig. 1a, coupling between lattice sites is fixed 𝐶𝐶𝑥𝑥,𝑥𝑥+1 = 𝐶𝐶̅ (𝑥𝑥 is the site index) while their energies 𝛽𝛽𝑥𝑥 are randomly perturbed – so-called diagonal disorder [13]. Alternatively, these energies may be held fixed 𝛽𝛽𝑥𝑥= 𝛽𝛽̅, while the couplings are perturbed – so called off-diagonal disorder (Fig. 1b) [14]. Only in the latter case can the Hamiltonian be cast in block off-diagonal form by dividing the Hilbert space into subspaces of even- and odd-indexed lattice sites. Such lattices provide a setting for studying disorder-immune chiral symmetry, which is also realized in two-dimensional (2D) lattices such as square, hexagonal [11,12], and even-sited ring lattices and certain Penrose tilings [15] under conditions of off-diagonal disorder. A Hamiltonian endowed with chiral symmetry features eigenvalues and eigenvectors that occur in skew-symmetric pairs in every realization of the disorder. Chiral ensembles raise fundamental questions regarding the impact on transport of the interplay between disorder and symmetry. Specifically, can one detect unambiguous traces of the underlying symmetry in the statistics of the transported wave even at maximal lattice disorder?
In investigating these questions, optics provides a particularly useful platform to explore random matrix theory and the ramifications of disorder-immune symmetries. Since the randomness of the lattice influences the propagating light wave, the emerging random light must be described using the tools of statistical optics and optical coherence theory. Starting from the Helmholtz wave equation for monochromatic light, we obtain a Schrödinger-like equation in the paraxial limit [16,17],

𝑖𝑖 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕+ 1 2𝑘𝑘o𝑛𝑛o ∇T 2𝐴𝐴+ 𝑘𝑘oΔ𝑛𝑛(𝑥𝑥, 𝑦𝑦)𝐴𝐴= 0;

(1) where the axial propagation coordinate 𝑧𝑧 plays the role of time, ∇T is the transverse Laplacian along the 𝑥𝑥 and 𝑦𝑦 coordinates, 𝐴𝐴(𝑥𝑥, 𝑦𝑦, 𝑧𝑧) is the slowly varying optical field envelope, 𝑘𝑘o is the free-space wave number, and Δ𝑛𝑛(𝑥𝑥, 𝑦𝑦) is an axially invariant perturbation to an averaged refractive index 𝑛𝑛o [17]. If the index features continuous but localized perturbations forming a disordered lattice, then the system is akin to a set of parallel coupled waveguides. Assuming the perturbations allow for single-mode waveguides with nearest-neighbor-only coupling, then the above equation can be mapped onto a set of coupled equations for the complex amplitudes 𝐸𝐸𝑥𝑥 at the lattice sites (see Supplementary) [18],

−𝑖𝑖 𝑑𝑑𝐸𝐸𝑥𝑥 𝑑𝑑𝑑𝑑= 𝛽𝛽𝑥𝑥𝐸𝐸𝑥𝑥+ 𝐶𝐶𝑥𝑥,𝑥𝑥+1𝐸𝐸𝑥𝑥+1 + 𝐶𝐶𝑥𝑥,𝑥𝑥−1𝐸𝐸𝑥𝑥−1.

(2) 3

In a finite-width lattice of 2𝑁𝑁+ 1 waveguides, labeled 𝑥𝑥= −𝑁𝑁, … , 𝑁𝑁, 𝛽𝛽𝑥𝑥 is the

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