Quantum derivation of Manley Rowe type relations

TL;DR

This paper derives quantum analogs of Manley-Rowe relations for time-dependent harmonic oscillators using the Ermakov Lewis invariant, linking quantum number and phase operators with classical power transfer equations.

Contribution

It introduces a quantum derivation of Manley-Rowe type relations based on the Ermakov Lewis invariant, connecting quantum operators with classical nonlinear optical equations.

Findings

Quantum number and phase operators are related through the invariant.

Derived relations are equivalent to classical power density equations.

Applicable to photon number excitations in nonlinear optics.

Abstract

The Ermakov Lewis quantum invariant for the time dependent harmonic oscillator is expressed in terms of number and phase operators. The identification of these variables is made in accordance with the correspondence principle and the amplitude and phase representation of the classical orthogonal functions invariant. The relationship between the number and phase operators is established through this invariant as the system evolves from one frequency to another. In the specific case where the excitations represent the photon number, these relations are equivalent to the power density transport equations derived in nonlinear optical processes.