Application of an inverse Holstein-Primakoff transformation to the Jaynes-Cummings model

TL;DR

This paper introduces a modified Holstein-Primakoff transformation that maps bosonic operators to an $SU(2)$ algebra, enabling efficient derivation of classical Hamiltonians for quantum systems like the Jaynes-Cummings model.

Contribution

It presents a novel modification of the Holstein-Primakoff transformation and applies Lieb's classical limit prescription to derive classical Hamiltonians from quantum models.

Findings

Efficient classical Hamiltonian derivation for the Jaynes-Cummings model

New $SU(2)$-based bosonic operator representation

Simplified analysis of quantum-classical correspondence

Abstract

A modification of the Holstein-Primakoff transformation is proposed such that creation and annihilation operators for a bosonic field are rewritten as operators of a $SU(2)$ algebra. Once it is applied to a quantum Hamiltonian, a subsequent application of the prescription by Lieb to obtain the classical limit for spin operators allows one to write efficiently a classical Hamiltonian for the system. This process is illustrated for the $M$-atom Jaynes-Cummings model.