Signatures of the $A^2$ term in ultrastrongly-coupled oscillators

TL;DR

This paper investigates the effects of the $A^2$ term in the Hamiltonian of ultrastrongly coupled bosonic matter-cavity systems, revealing equal polariton populations and proposing experimental tests.

Contribution

It introduces a detailed analysis of the $A^2$ term's signatures in ultrastrong coupling, including a microscopic Hamiltonian refinement and experimental predictions.

Findings

Polariton populations are always equal under the Thomas-Reiche-Kuhn sum rule.

The vacuum state contains a nonzero polariton population in the ultrastrong regime.

Experimental signatures can be observed via cavity radiation after rapid coupling switching.

Abstract

We study a bosonic matter excitation coupled to a single-mode cavity field via electric dipole. Counter-rotating and $A^2$ terms are included in the interaction model, ${\mathbf A}$ being the vector potential of the cavity field. In the ultrastrong coupling regime the vacuum of the bare modes is no longer the ground state of the Hamiltonian and contains a nonzero population of polaritons, the true normal modes of the system. If the parameters of the model satisfy the Thomas-Reiche-Kuhn sum rule, we find that the two polaritons are always equally populated. We show how this prediction could be tested in a quenching experiment, by rapidly switching on the coupling and analyzing the radiation emitted by the cavity. A refinement of the model based on a microscopic minimal coupling Hamiltonian is also provided, and its consequences on our results are characterized analytically.