Zeno physics in ultrastrong-coupling circuit QED

We study the Zeno and anti-Zeno effects in a superconducting qubit interacting strongly and ultrastrongly with a microwave resonator. Using a model of a frequently measured two-level system interacting with a quantized mode, we predict different behaviors and total control of the Zeno times depending on whether the rotating-wave approximation can be applied in the Jaynes-Cummings model. As an example, we show the dependence of our results with the properties of the initial field states.

Figure 1

Zeno effect in the strong-coupling regime, $g/\omega =0.05,$ for an initial state with an excited qubit and the resonator in a coherent state, $|\mathrm{e}\rangle \otimes |\alpha \rangle ,$ with $\left|\alpha \right|{}^2=9$ photons on average. The solid line is a plot of the collapses and revivals under free evolution in the RWA. The remaining lines correspond to the survival probability of the excited state under periodic measurements of the qubit, repeated at intervals of $\delta t=0.01\pi g^{-1}$. If the RWA is applied, the usual Zeno time is obtained [dashed (red) line]. If the RWA is not applied, we can either slow down the decay, setting the phase of the coherent state to $\theta =\pi /2$ [dash-dotted (green) line] or increasing it to $\theta =0$ [dotted (blue) line]. For sufficiently fast measurements, these numerical results and those obtained from the analytical expressions (9) and (10) are practically the same.

Figure 2

Zeno effect in the ultrastrong-coupling regime, when $g$, $\omega$, ${\omega }_0=1$ GHz. (top) Free evolution of the excited-state population in the qubit and (bottom) survival probability of the excited state for periodic measurements with $\delta t=0.1{\omega }^{-1}=0.1$ ns (each symbol represents a measurement). The inset shows the comparison between both plots. The solid line corresponds to an RWA model [solid (black) circles], while the other lines start from a coherent state with 9 photons and either $\theta =0$ [dashed line with solid (blue) squares] or $\theta =\pi /2$ [dash-dotted line with solid (red) triangles].

Figure 3

A phase qubit can be used to explore the Zeno effect: the escape rate of the unstable level $|\mathrm{e}\rangle$ is equivalent to a continuous measurement of that state.