Quantum Correlation and Coherence in Dissipative Two SC-Qubit Systems Interacting with a Coherent SC-Cavity

Abstract

An analytical solution of the master equation is obtained for a coherent SC-cavity interacts with two SC-qubits. With this solution, the effect of an intrinsic decoherence on the growth of coherence loss (mixture) and the entanglement degradation is shown. Due to the intrinsic decoherence, the oscillatory behavior of the qubit-inversion is deteriorated, and its amplitude decreases. Where its stationary state refers to that the energy is stored more in the SC-qubits. The growth of the mixture is investigated for the entire system and its sub-systems of the SC-cavity and two qubits. The entanglement is studied for two different partitions, between the SC-cavity and the two qubits as well as between the two qubits. It is found that the generation of the entanglement and mixture and their stationary values depend on the initial coherence intensity and the intrinsic decoherence rate. The stationary entanglement and mixture, and the phenomena of sudden appearance and disappearance of the entanglement could be controlled by adjusting the intrinsic noise rate and the initial coherence intensity.

Appendix

Here, the (3) can be rewritten as:

$$ \frac{d}{dt}\hat{\rho}=\hat{L}\hat{\rho }, $$

(16)

where the super-operator $\hat{L}$ is given by:

$$ \hat{L}\ast =-i\left[\hat{H},\ast \right]-\frac{1}{2\gamma}\left[{\hat{H}}^2\ast -2\hat{H}\ast \hat{H}+\ast {\hat{H}}^2\right]. $$

(17)

Therefore,

$$ \hat{\rho}(t)={e}^{\hat{L}t}\hat{\rho}(0). $$

(18)

Then, we write the initial density matrix in terms the used eigenstates $\mid {\psi}_i^m\Big\rangle$ of (6) as:

$$ {\displaystyle \begin{array}{rcl}\rho (0)& =& \sum \limits_{m,n=0}{\eta}_{m,n}\left[{a}_m\sqrt{2}|{\psi}_1^m\Big\rangle +{b}_m\left(|{\psi}_2^m\Big\rangle +|{\psi}_3^m\Big\rangle \right)\right]\\ {}& & \times \left[{a}_n\sqrt{2}\right\langle {\psi}_1^n\mid +{b}_n\left(\right\langle {\psi}_2^n\mid +\left\langle {\psi}_3^n\mid \right)\Big].\end{array}} $$

(19)

With the super-operator $\hat{L}$, the time evolution of the density matrices $\mid {\psi}_i^m\left\rangle \right\langle {\psi}_j^n\mid$ is given by

$$ {\displaystyle \begin{array}{rcl}{e}^{\hat{L}t}\mid {\psi}_i^m\left\rangle \right\langle {\psi}_j^n\mid & =& {\beta}_{ij}^{mn}\mid {\psi}_i^m\left\rangle \right\langle {\psi}_j^n\mid \\ {}& =& {e}^{\left[-,i,\left({\varepsilon}_i^m,-,{\varepsilon}_j^n\right),t,-,\frac{1}{2\gamma },{\left({\varepsilon}_i^m,-,{\varepsilon}_j^n\right)}^2,t\right]}\mid {\psi}_i^m\left\rangle \right\langle {\psi}_j^n\mid .\end{array}} $$

(20)

By using the (18–20), we get the final density matrix, $\hat{\rho}(t)$, as in the (5). After that, we write the final density matrix, $\hat{\rho}(t)$ in the space states {|v₁〉 = |e_Ae_B,n〉,|v₂〉 = |e_Ag_B,n + 1〉,|v₃〉 = |g_Ae_B,n + 1〉,|v₄〉 = |g_Ag_B,n + 2〉} by using (6).