A master equation for strongly interacting dipoles

TL;DR

This paper derives a gauge-invariant master equation for strongly interacting dipoles that explicitly includes Coulomb interactions, revealing separation-dependent corrections to energy transfer and emission spectra.

Contribution

The authors develop a new master equation in the Coulomb gauge that incorporates inter-dipole Coulomb energy within the system Hamiltonian, differing from standard approaches.

Findings

The master equation is gauge-invariant but depends on different S-matrix elements.

It predicts separation-dependent corrections to energy transfer and decay rates.

Application shows corrections to the emission spectrum of two dipoles.

Abstract

We consider a pair of dipoles for which direct electrostatic dipole-dipole interactions may be significantly larger than the coupling to transverse radiation. We derive a master equation using the Coulomb gauge, which naturally enables us to include the inter-dipole Coulomb energy within the system Hamiltonian rather than the interaction. In contrast, the standard master equation for a two- dipole system, which depends entirely on well-known gauge-invariant S-matrix elements, is usually derived using the multipolar gauge, wherein there is no explicit inter-dipole Coulomb interaction. We show using a generalised arbitrary-gauge light-matter Hamiltonian that this master equation is obtained in other gauges only if the inter-dipole Coulomb interaction is kept within the interaction Hamiltonian rather than the unperturbed part as in our derivation. Thus, our master equation, while still gauge-invariant, depends on different S-matrix elements, which give separation-dependent corrections to the standard matrix elements describing resonant energy transfer and collective decay. The two master equations coincide in the large separation limit where static couplings are negligible. We provide an application of our master equation by finding separation-dependent corrections to the natural emission spectrum of the two-dipole system.