Analytical comparison of the first- and second-order resonances for implementation of the dynamical Casimir effect in nonstationary circuit QED

TL;DR

This paper analytically and numerically compares first- and second-order resonances in nonstationary circuit QED, revealing that second-order resonances can induce similar phenomena at halved frequencies but with lower transition rates, impacting dynamical Casimir effect applications.

Contribution

It provides a detailed analytical and numerical analysis of second-order resonances in nonstationary circuit QED, highlighting their potential and limitations for dynamical Casimir effect implementation.

Findings

Second-order resonances occur at halved frequencies of first-order ones.

Transition rates for second-order resonances are significantly smaller.

Modulation of coupling strength is less effective at second-order resonances.

Abstract

We investigate analytically and numerically the nonstationary circuit QED setup in which $N$ independent qubits interact with a single mode of the Electromagnetic field confined in a resonator. We consider the harmonic time modulation of some parameter (atomic transition frequency or the atom-field coupling strength) and derive the unitary dynamics up to the second order in the modulation depth for $N=1$ and $N\gg 1$. It is shown that all the resonant phenomena that occur for modulation frequencies $\sim 2\omega _{0}$ (where $\omega _{0}$ is the cavity frequency) also occur for the halved frequencies. However, in the latter case the associated transition rates are significantly smaller and the modulation of the coupling strength is less effective. The transition rates are evaluated explicitly and the prospects of employing the second-order resonances in the phenomena related to the dynamical Casimir effect are examined.