Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reduced projection method for photonic moiré lattices

Zixuan Gao, Zhenli Xu, Zhiguo Yang

TL;DR

This work introduces a reduced projection method (RPM) for solving quasiperiodic Schrödinger eigenvalue problems arising in photonic moiré lattices. By leveraging a higher-dimensional periodic parent function and a projection matrix, the authors prove fast decay of generalized Fourier coefficients along the projection direction and develop a reduced basis technique with rigorous error estimates. The RPM dramatically lowers the degrees of freedom and computational cost compared to the classical projection method, validated by 1D and 2D numerical experiments showing exponential convergence and substantial speedups. The approach enables efficient simulation of high-dimensional quasiperiodic systems and offers a pathway to tackle more complex moiré lattices and related quasiperiodic problems.

Abstract

This paper presents a reduced projection method for the solution of quasiperiodic Schrödinger eigenvalue problems for photonic moiré lattices. Using the properties of the Schrödinger operator in higher-dimensional space via a projection matrix, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate along a fixed direction associated with the projection matrix. An efficient reduction strategy of the basis space is then proposed to reduce the degrees of freedom significantly. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small portion of the degrees of freedom is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of photonic moiré lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.

Reduced projection method for photonic moiré lattices

TL;DR

This work introduces a reduced projection method (RPM) for solving quasiperiodic Schrödinger eigenvalue problems arising in photonic moiré lattices. By leveraging a higher-dimensional periodic parent function and a projection matrix, the authors prove fast decay of generalized Fourier coefficients along the projection direction and develop a reduced basis technique with rigorous error estimates. The RPM dramatically lowers the degrees of freedom and computational cost compared to the classical projection method, validated by 1D and 2D numerical experiments showing exponential convergence and substantial speedups. The approach enables efficient simulation of high-dimensional quasiperiodic systems and offers a pathway to tackle more complex moiré lattices and related quasiperiodic problems.

Abstract

This paper presents a reduced projection method for the solution of quasiperiodic Schrödinger eigenvalue problems for photonic moiré lattices. Using the properties of the Schrödinger operator in higher-dimensional space via a projection matrix, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate along a fixed direction associated with the projection matrix. An efficient reduction strategy of the basis space is then proposed to reduce the degrees of freedom significantly. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small portion of the degrees of freedom is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of photonic moiré lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.
Paper Structure (10 sections, 7 theorems, 78 equations, 8 figures, 3 tables, 1 algorithm)

This paper contains 10 sections, 7 theorems, 78 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries on quasiperiodic functions
  3. Reduced projection method
  4. Decay rate of the generalized Fourier coefficients
  5. Numerical scheme
  6. Solution Algorithm
  7. Numerical examples
  8. 1D example
  9. 2D example
  10. Conclusions

Key Result

lemma thmcounterlemma

Let $m\in\mathbb Z^+$, suppose $F(\bm{x})\in H^m_{\rm per}(\mathbb [0,T]^n)$, then

Figures (8)

  • Figure 1: The generalized Fourier coefficient modulus of eigenfunctions for the 1D quasiperiodic potential. (a) The generalized Fourier coefficient modulus of eigenfunctions as function of $\bm q$. (b) The error $\hbox{Err}(D)$ as function of $D$ for spectrum 0.5945 and 0.6297. In both panels, $E_0=1$ and $N=180$.
  • Figure 2: The DOF (a) and the condition number (b) as function of $D$ using the RPM in one dimension with $N=180$. Correspondingly, the DOF of the PM is $N^2=32400$.
  • Figure 3: Absolute error of eigenvalues $\varepsilon$ and the $L^2$-error of the first eigenfunction $\delta$. (a,b): Error as function of $N$ for different $D$. (c,d): Error as function of $D$ for different $N$.
  • Figure 4: Error of the normalized first eigenfunctions obtained by the RPM in interval $[0, 1]$ for different $E_0$. In each panel, $D=25$ and three different $N$ are calculated.
  • Figure 5: The generalized Fourier coefficients (modulus) of eigenfunctions in the $\bm{q}$ space for $E_0=1$ and $N=30$. Results are present by logarithms with base $10$.
  • ...and 3 more figures

Theorems & Definitions (12)

  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 1
  • theorem 2
  • proof
  • lemma thmcounterlemma
  • theorem 3
  • proof
  • remark thmcounterremark
  • ...and 2 more