Non-classical correlations in two quantum dots coupled in a coherent resonator field under decoherence

Abstract

An analytical description is obtained for two excitons; each exciton is in one of two distant quantum dots embedded in a nano-mechanical resonator which is initially prepared in a superposition of coherent states with intrinsic decoherence. We use a particular method based on the dressed states of the model Hamiltonian. The robustness of the generated non-classical correlations is investigated via the measurement-induced non-locality and geometric quantum discord, compared with the log-negativity. The three measures present generated different correlations that depend on the initial coherence states and their intensities, and intrinsic decoherence. It is witnessed that the phenomena of sudden appearance and disappearance of entanglement are occurring and repeated at chosen intervals of time; they can disappear due to the intrinsic decoherence. The correlations were observed to attain highest robustness under initial coherent state, with decoherence parameter. Quantum correlation functions survive in the stationary states, and the amount of stationary correlations could be controlled by adjusting the values of the intrinsic decoherence, the initial state of the nano-mechanical resonator and its coherence intensity.

Appendix

By using the Eqs. (6) and (7), the final density matrix of the Eq. (2) is given by

$$\begin{aligned} \hat{\rho } (t)= & {} \mathrm{e}^{\hat{L}t}\hat{\rho }(0)=\sum _{m,n=0}\sum ^{3}_{i,j=0} \beta _{m}\beta _{n} \tilde{\alpha }_{ij}|\Psi ^{m}_{i}\rangle \langle \Psi ^{n}_{j}|. \end{aligned}$$

Where $\tilde{\alpha }_{mn}=\alpha _{mn}l^{mn}_{ij}$. $\alpha _{11}=2a^{m}_{1}a^{n}_{1}$, $\alpha _{12}=\alpha _{13}=a^{m}_{1} a^{n}_{2}\sqrt{2}$, $\alpha _{21}=\alpha _{31}=a^{m}_{2} a^{n}_{1}\sqrt{2}$ and $\alpha _{22}= \alpha _{23}= \alpha _{32}= \alpha _{33}=a^{m}_{2} a^{n}_{2}$. By using the canonical transform of Eq. (5), the analytical solution of Eq. (2) is given by Eq. (8). Here, we find the coefficients $h^{mn}_{ij}$ by collecting the coefficients of each one of the basic states of the two qubits and field, $\{ |i\rangle (i=1-4)\}$. They are given by

$$\begin{aligned} h^{mn}_{11}= & {} 2a^{m}_{1}a^{n}_{1}\tilde{\alpha }_{11} +a^{m}_{1}a^{n}_{2}\sqrt{2}(\tilde{\alpha }_{12}+\tilde{\alpha }_{13}) +a^{m}_{2}a^{n}_{1}\sqrt{2}(\tilde{\alpha }_{21}+\tilde{\alpha }_{31}) \\&+\,a^{m}_{2}a^{n}_{2}(\tilde{\alpha }_{22}+\tilde{\alpha }_{23}+\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{12}= & {} h^{mn}_{13}=\frac{1}{\sqrt{2}}a^{m}_{1}(-\tilde{\alpha }_{12}+\tilde{\alpha }_{13}) +\frac{1}{2}a^{m}_{2}(-\tilde{\alpha }_{22}+\tilde{\alpha }_{23}-\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{14}= & {} -2a^{m}_{1}a^{n}_{2}\tilde{\alpha }_{11} +a^{m}_{1}a^{n}_{1}\sqrt{2}(\tilde{\alpha }_{12}+\tilde{\alpha }_{13}) -a^{m}_{2}a^{n}_{2}\sqrt{2}(\tilde{\alpha }_{21}+\tilde{\alpha }_{31}) \\&+\,a^{m}_{2}a^{n}_{1}(\tilde{\alpha }_{22}+\tilde{\alpha }_{23}+\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{21}= & {} h^{mn}_{31}=\frac{1}{\sqrt{2}}a^{n}_{1}(-\tilde{\alpha }_{21}+\tilde{\alpha }_{31}) +\frac{1}{2} a^{n}_{2}(-\tilde{\alpha }_{22}-\tilde{\alpha }_{23}+\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{22}= & {} h^{mn}_{23}=h^{mn}_{32}=h^{mn}_{33}=\frac{1}{4}(\tilde{\alpha }_{22}-\tilde{\alpha }_{23} -\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{24}= & {} h^{mn}_{34}=\frac{1}{\sqrt{2}}a^{n}_{2}(\tilde{\alpha }_{21}-\tilde{\alpha }_{31}) +\frac{1}{2}a^{n}_{1}(-\tilde{\alpha }_{22}-\tilde{\alpha }_{23} +\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{41}= & {} -2a^{m}_{2}a^{n}_{1}\tilde{\alpha }_{11} -a^{m}_{2}a^{n}_{2}\sqrt{2}(\tilde{\alpha }_{12}+\tilde{\alpha }_{13}) +a^{m}_{1}a^{n}_{1}\sqrt{2}(\tilde{\alpha }_{21}+\tilde{\alpha }_{31}) \\&+\,a^{m}_{1}a^{n}_{2}(\tilde{\alpha }_{22}+\tilde{\alpha }_{23}+\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{42}= & {} h^{mn}_{43}=\frac{1}{\sqrt{2}}a^{m}_{2}(\tilde{\alpha }_{12}-\tilde{\alpha }_{13}) +\frac{1}{2}a^{m}_{1}(-\tilde{\alpha }_{22}+\tilde{\alpha }_{23}-\tilde{\alpha }_{32}+\tilde{\alpha }_{33}), \\ h^{mn}_{44}= & {} 2a^{m}_{2}a^{n}_{2}\tilde{\alpha }_{11} -a^{m}_{2}a^{n}_{1}\sqrt{2}(\tilde{\alpha }_{12}+\tilde{\alpha }_{13}) -a^{m}_{1}a^{n}_{2}\sqrt{2}(\tilde{\alpha }_{21}+\tilde{\alpha }_{31}) \\&+\,a^{m}_{1}a^{n}_{1}(\tilde{\alpha }_{22}+\tilde{\alpha }_{23}+\tilde{\alpha }_{32}+\tilde{\alpha }_{33}). \end{aligned}$$