Krylov Complexity for Jacobi Coherent States

TL;DR

This paper extends Krylov complexity methods to Jacobi group coherent states, enabling analysis of squeezed states in quantum optics through augmented Lanczos algorithms and benchmarking their numerical computation.

Contribution

It develops a computational framework for Krylov complexity in Jacobi group coherent states, combining analytical and numerical techniques.

Findings

Lanczos algorithm can be augmented for Jacobi group states.

Benchmarking scheme for Lanczos coefficients is established.

Framework generalizes to higher-dimensional Jacobi groups.

Abstract

We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group $H_{1}$, and the symplectic group, $Sp(2,\mathbb{R})\simeq SU(1,1)$. Such coherent states are physically realized as squeezed states in, for example, quantum optics. With the Krylov basis for both the $SU(1,1)$ and Heisenberg-Weyl groups being well understood, their semi-direct product is also partially analytically tractable. We exploit this to benchmark a scheme to numerically compute the Lanczos coefficients which, in principle, generalizes to the more general Jacobi group $H_{n}\rtimes Sp(2n,\mathbb{R})$.