Non-Markovian qubit dynamics in a circuit-QED setup

We consider a circuit-QED setup that allows the induction and control of non-Markovian dynamics of a qubit. Non-Markovianity is enforced over the qubit by means of its direct coupling to a bosonic mode which is controllably coupled to another qubit-mode system. We show that this configuration can be achieved in a circuit-QED setup consisting of two initially independent superconducting circuits, each formed by one charge qubit and one transmission-line resonator, which are put in interaction by coupling the resonators to a current-biased Josephson junction. We solve this problem exactly and then proceed with a thorough investigation of the emergent non-Markovianity in the dynamics of the qubits. Our study might serve the context for the first experimental assessment of non-Markovianity in a multielement solid-state device.

Figure 1

Sketch of the elements comprising the circuit-QED model used in this work. Charge qubit $A$ is coupled to transmission line resonator $a$, while charge qubit $B$ is coupled to transmission line resonator $b$. These are Cooper pair boxes with tunable Josephson coupling. The coupling between the resonators is achieved by dispersively coupling a CBJJ qubit $C$ to both resonators.

Figure 2

We depict the basic idea behind the investigation of non-Markovianity in this work. Qubit $A$ is subjected to Markovian dephasing $\gamma$ and is coupled to other subsystems directly through $g$ and indirectly thorough $\lambda$. Provided that $g=0$, the time evolution of qubit $A$ will certainly be Markovian due solely to the $\gamma$ environment. Once $g$ is turned on, non-Markovian features may appear. The situation becomes even more interesting by introducing the intermode coupling.

Figure 3

Dynamics of $\sigma \left(t\right)$ for various values of the dephasing rates and $\lambda =0$. We have taken $\gamma /2\pi =0.3$ MHz (solid line), $1.0$ MHz (dashed line), and $1.5$ MHz (dot-dashed line). As for the other parameters, we used $g/2\pi =50.0$ MHz and $\omega /2\pi =10.0$ GHz.

Figure 4

Dynamics of $\sigma \left(t\right)$ as a function of time and dephasing. We used $g/2\pi =50.0$ MHz and $\omega /2\pi =10.0$ GHz.

Figure 5

Dynamics of $\sigma \left(t\right)$ for various mode-mode coupling constants $\lambda$. We considered $\lambda /2\pi =10$ MHz (solid line) and $\lambda /2\pi =50$ MHz (dashed line). For the other parameters we used $\gamma /2\pi =0.3$ MHz, $g/2\pi =50.0$ MHz, and $\omega /2\pi =10.0$ GHz.

Figure 6

Dynamics of ${\sigma }_{\mathrm{mix}}\left(t\right)$ obtained considering the initial state in Eq. (22) for various amplitudes $\alpha$ and $\beta$. We took $\alpha =\beta =0$ (solid line), $\alpha =\beta =1$ (dashed line), and $\alpha =\beta =2$ (dot-dashed line). For the other parameters we used $\gamma /2\pi =0.3$ MHz, $g/2\pi =50.0$ MHz, and $\omega /2\pi =10.0$ GHz.

Figure 7

Dynamics of ${\sigma }_{\mathrm{mix}}\left(t\right)$ obtained considering the initial state in Eq. (22) for various dephasing rates and decoupled modes $\lambda =0$. We considered $\gamma /2\pi =0.3$ MHz (solid line), $\gamma /2\pi =5.0$ MHz (dashed line), $\gamma /2\pi =10.0$ MHz (dot-dashed line). For the other parameters we used $g/2\pi =50.0$ MHz and $\omega /2\pi =10.0$ GHz.