Qubit-oscillator system: An analytical treatment of the ultra-strong coupling regime

TL;DR

This paper provides an analytical approach to studying a qubit coupled to an oscillator in the ultrastrong coupling regime, extending beyond the Jaynes-Cummings model and exploring complex energy spectra and dynamics.

Contribution

It introduces a higher-order Van Vleck perturbation theory method to analyze the qubit-oscillator system in the ultrastrong coupling regime, including finite bias cases.

Findings

Analytical treatment valid in the ultrastrong coupling limit

Multiple contributing frequencies in the system's dynamics

Non-trivial dressing of the tunneling matrix element affects frequency suppression

Abstract

We examine a two-level system coupled to a quantum oscillator, typically representing experiments in cavity and circuit quantum electrodynamics. We show how such a system can be treated analytically in the ultrastrong coupling limit, where the ratio $g/\Omega$ between coupling strength and oscillator frequency approaches unity and goes beyond. In this regime the Jaynes-Cummings model is known to fail, because counter-rotating terms have to be taken into account. By using Van Vleck perturbation theory to higher orders in the qubit tunneling matrix element $\Delta$ we are able to enlarge the regime of applicability of existing analytical treatments, including in particular also the finite bias case. We present a detailed discussion on the energy spectrum of the system and on the dynamics of the qubit for an oscillator at low temperature. We consider the coupling strength $g$ to all orders, and the validity of our approach is even enhanced in the ultrastrong coupling regime. Looking at the Fourier spectrum of the population difference, we find that many frequencies are contributing to the dynamics. They are gathered into groups whose spacing depends on the qubit-oscillator detuning. Furthermore, the dynamics is not governed anymore by a vacuum Rabi splitting which scales linearly with $g$, but by a non-trivial dressing of the tunneling matrix element, which can be used to suppress specific frequencies through a variation of the coupling.