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1. Band Dynamics of Multimode Resonant Nanophotonic Lattices with Adjustable Liquid Interfaces

   https://doi.org/10.3390/nano13162350  

   Razmjooei, Nasrin; Magnusson, Robert (August 2023, Nanomaterials)

   Subwavelength resonant lattices offer a wide range of fascinating spectral phenomena under broadside illumination. The resonance mechanism relies on the generation of lateral Bloch modes that are phase matched to evanescent diffraction orders. The spectral properties and the total number of resonance states are governed by the structure of leaky modes and the mode count. This study investigates the effect of interface modifications on the band dynamics and bound-state transitions in guided-mode resonant lattices. We provide photonic lattices comprising rectangular Si3N4 rods with a liquid film with an adjustable boundary. The band structures and band flips are examined through numerical simulations using the rigorous coupled-wave analysis (RCWA) method and analyzing the zero-order spectral reflectance as a function of the incident angle. The band structures and band flips are examined through numerical simulations, and the influences of the refractive index and the thickness of the oil layer on the band dynamics are investigated. The results reveal distinct resonance linewidths corresponding to different refractive indices of the oil layer. Furthermore, the effect of the oil thickness on the band dynamics is explored, demonstrating precise control over the number of propagating modes within the lattice structure. Theoretical simulations and experimental results are presented for a subwavelength silicon-nitride lattice combined with a liquid film featuring an adjustable boundary. The presence of a relatively thick liquid waveguiding region enables the emergence of additional modes, including the first four transverse-electric (TE) leaky modes, which produce observable resonance signatures. Through experimental manipulation of the basic lattice’s duty cycle, the four bands undergo quantifiable band transitions and closures. The experimental results obtained within the 1400–1600 nm spectral range exhibit reasonable agreement with the numerical analysis. These findings underscore the significant role played by the interface in shaping the band dynamics of the lattice structure, providing valuable insights into the design and optimization of photonic lattices with adjustable interfaces. 

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2. Extremal spectral gaps for periodic Schrödinger operators

   https://doi.org/10.1051/cocv/2018029  

   Kao, Chiu-Yen; Osting, Braxton (January 2019, ESAIM: Control, Optimisation and Calculus of Variations)

   The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m , we consider the problem of maximizing the gap-to-midgap ratio for the m th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m intervals. Optimal potentials are computed numerically using a rearrangement algorithm and are observed to be periodic. In two dimensions, we develop an efficient rearrangement method for this problem based on a semi-definite formulation and apply it to study properties of extremal potentials. We show that, provided a geometric assumption about the maximizer holds, a lattice of disks maximizes the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit parametrization of two-dimensional Bravais lattices, we also consider how the optimal value varies over all equal-volume lattices. 

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3. Transfer operator approach to 1d random band matrices

   https://doi.org/doi.org/10.1142/11060  

   Shcherbina, M.; Shcherbina, T. (April 2018, Proceedings of the International Congress of Mathematicians 2018)

   We discuss an application of the transfer operator approach to the analysis of the different spectral characteristics of 1d random band matrices (correlation functions of characteristic polynomials, density of states, spectral correlation functions). We show that when the bandwidth $$W$$ crosses the threshold $$W=N^{1/2}$$, the model has a kind of phase transition (crossover), whose nature can be explained by the spectral properties of the transfer operator. 

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4. Crystalline Quantum Circuits

   https://doi.org/10.1103/PRXQuantum.4.030313  

   Sommers, Grace M.; Huse, David A.; Gullans, Michael J. (July 2023, PRX Quantum)

   Random quantum circuits continue to inspire a wide range of applications in quantum information science and many-body quantum physics, while remaining analytically tractable through probabilistic methods. Motivated by an interest in deterministic circuits with similar applications, we construct classes of nonrandom unitary Clifford circuits by imposing translation invariance in both time and space. Further imposing dual unitarity, our circuits effectively become crystalline spacetime lattices whose vertices are swap or iswap two-qubit gates and whose edges may contain one-qubit gates. One can then require invariance under (subgroups of) the crystal’s point group. Working on the square and kagome lattices, we use the formalism of Clifford quantum cellular automata to describe operator spreading, entanglement generation, and recurrence times of these circuits. A full classification on the square lattice reveals, of particular interest, a “nonfractal good scrambling class” with dense operator spreading that generates codes with linear contiguous code distance and high performance under erasure errors at the end of the circuit. We also break unitarity by adding spacetime translation-invariant measurements and find a class of such circuits with fractal dynamics. 

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5. A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems

   https://doi.org/10.1090/cams/5  

   Lu, Jianfeng; Lu, Yulong (January 2022, Communications of the American Mathematical Society)

   This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schrödinger operator on a d d -dimensional hypercube with Neumann boundary condition. We prove that the convergence rate of the generalization error is independent of dimension d d , under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The latter is achieved by a fixed point argument based on the Krein-Rutman theorem. 

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