Symplectic Split-Operator Propagators from Tridiagonalized Multi-Mode Bosonic Hilbert Spaces for Bose-Hubbard Hamiltonians

TL;DR

This paper introduces a method to efficiently diagonalize and simulate large bosonic multimode systems, like optomechanical and Bose-Hubbard models, using tridiagonalization and symplectic split-operator propagators, enabling larger basis sizes.

Contribution

The authors develop a tridiagonalization approach for bosonic systems that allows exact diagonalization with sparse algorithms, improving simulation capacity for complex quantum models.

Findings

Exact tridiagonalization of bosonic Hamiltonians achieved

Efficient diagonalization with $D imes D$ matrices and sparse algorithms

Implementation of symplectic split-operator propagators with minimal basis re-indexing

Abstract

In this methods paper, we show how to tridia\-go\-nalize two families of bosonic multimode systems: optomechanical and Bose-Hubbard hamiltonians. Using tools from number theory, we devise a rendering of these systems in the form of exact $D \times D$ tridiagonal symmetric matrices with real-valued entries. Such matrices can subsequently be exactly diagonalized using specialized sparse-matrix algorithms that need on the order of $D \ln(D)$ steps. This makes it possible to describe systems with much larger numbers of basis states than available to date. It also allows for efficient diagonal representation of large, accurate, symplectic split-operator propagators for which we moreover show that the required basis changes can be implemented by simple re-indexing, at marginal computational cost.