Dissipative quantum entanglement dynamics of two and three qubits due to the dynamical Lamb effect

A theoretical framework to investigate the time evolution of the quantum entanglement due to the dynamical Lamb effect between two and three superconducting qubits coupled to a coplanar waveguide in the presence of different sources of dissipation is developed. Guidelines on how to proceed in the $N$-qubit case are also given. We quantitatively analyze the case of single switching of the coupling and absence of dissipation within a perturbative approach and show that it is a good approximation to the case of periodic switching of the coupling for high frequencies of switching. The same systems are analyzed for the general case of periodic switching of the coupling at any frequency and in the presence of dissipation via numerical calculations. Different measures of entanglement compatible with mixed states are adopted. It is demonstrated that the different measures show a different level of detail of the latter. The concurrence and the negativity are obtained in the two-qubit case; the three-$\pi$ and the negativity are obtained in the three-qubit case. It is shown that time-dependent Greenberger-Horne-Zeilinger states can be created even in the presence of dissipation. To maximize the quantum entanglement between the qubits, the effects of tuning several parameters of the system are investigated.

Figure 1

Comparison of the time evolution of different entanglement measures of two qubits between the perturbative calculations (“Perturbation”) and the numerical calculations (“Averaged”) for the case of averaged qubit-cavity coupling and the numerical calculations (“Switching”) for the case of high-frequency switching of the coupling. (a),(c),(e) The time evolution of the concurrence, the mutual information, and the negativity, respectively, for the case of averaged coupling. (b),(d),(f) The comparison of the time evolution of the same quantities including the case of fast switching of the coupling.

Figure 2

Time evolution of different measures of entanglement for frequencies of switching of the coupling ${\varpi }_s$ in the range ${\varpi }_s\in \left[{\omega }_0,4{\omega }_0\right]$. (a) Concurrence and (b) negativity.

Figure 3

Time evolution of different measures of entanglement for frequencies of cavity photons ${\omega }_c$ in the range ${\omega }_c\in \left[\frac{{\omega }_0}{2},\frac{3{\omega }_0}{2}\right]$. (a) Concurrence and (b) negativity.

Figure 4

Time evolution of different measures of entanglement for the decay rate $\kappa$ of the cavity photons in the range $\kappa \in \left[0,2\pi \times 1.6\text{GHz}\right]$. (a) Concurrence and (b) negativity.

Figure 5

Comparison of the time evolution of (a),(b) negativity and (c),(d) three-$\pi$ between the analytical (“Perturbation”) and the numerical (“Averaged”) calculations for the case of averaged qubit-cavity coupling, and the numerical calculations (“Switching”) for the case of fast oscillatory coupling. (a),(c) The comparison for the case of average decoupling only; (b),(d) the results obtained for the case of fast switching coupling.

Figure 6

Time evolution of different measures of entanglement for frequencies of switching of the coupling ${\varpi }_s$ in the range ${\varpi }_s\in \left[{\omega }_0,4{\omega }_0\right]$ for three qubits with equal transition frequency ${\omega }_0$. (a) Negativity and (b) three-$\pi$.

Figure 7

Time evolution of (a) negativity of the first qubit, (b) negativity of the second qubit, (c) negativity of the third qubit, and (d) three-$\pi$ when the frequency of switching of the coupling is tuned over the range ${\varpi }_s\in \left[{\omega }_0,4{\omega }_0\right]$ for qubits with different transition frequencies.