Permutational symmetry for identical multi-level systems: a second quantized approach

TL;DR

This paper introduces a second quantized framework that leverages permutational symmetry to simplify the dynamics of multiple identical multi-level quantum systems, significantly reducing computational complexity.

Contribution

It presents a novel second quantized approach to exploit permutational symmetry in multi-level systems, enabling efficient modeling of collective dynamics.

Findings

Dimensionality reduction in system dynamics

Efficient mapping to bosonic modes

Application to Lindblad dynamics with collective dissipation

Abstract

We develop a framework that provides a straightforward approach to fully exploit the permutational symmetry of identical multi-level systems. By taking into account the permutational symmetry, we outline a simple scheme that allows to map the dynamics of $N$ identical $d$-level systems to the dynamics of $d$ bosonic modes with $N$ particles, achieving an exponential reduction on the dimensionality of the problem in a simple and straightforward way. In particular, we consider the Lindblad dynamics of several identical multi-level systems interacting with a common subsystem under the action of collective dissipation terms.