Non-Markovian entanglement evolution of two uncoupled qubits

Two noninteracting qubits, initially prepared in an entangled state, are coupled to their own independent environments and evolve under their influence. The reduced non-Markovian dynamics of two qubits is exact for arbitrary model parameters. Necessary and sufficient conditions for nonvanishing entanglement are formulated for both zero and nonzero temperatures and arbitrary time. It is shown that (i) entanglement dynamics can effectively be controlled by a finite quantum system coupled to one of the qubits and (ii) dynamical symmetry of the controlling quantum system can influence significantly entanglement of the qubits and results in its nonmonotonic decay.

Figure 1

(Color online) Time evolution for negativity from all four Bell states (15). The first qubit is coupled to the sub-Ohmic $\left(\mu =-0.1\right)$, Ohmic $\left(\mu =0\right)$, or super-Ohmic $\left(\mu =0.1\right)$ bath and the second qubit evolves freely, i.e., $H_{\pm }=\pm 1$. The cutoff frequency ${\omega }_c={10}^3$, the coupling constant $\lambda =0.1$, temperature $T=0$, and the imperfect preparation parameter $p=0$, cf. Eq. (14). Time is given in units of $t_0$.

Figure 2

(Color online) Negativity of the qubit-qubit system. The single-mode control is determined by (26, 27) with ${\gamma }_{\pm }=1$. The initial state of the controlling quantum system is the number state $|\Omega ⟩=|N⟩$ with $N=0,1,5$. The reservoir $R_1$ is super-Ohmic at $T=0$. Remaining parameters are the same as in Fig. 1.

Figure 3

(Color online) Negativity of the qubit-qubit system. The two-mode control is determined by (32, 33) with ${\gamma }_1={\gamma }_2=1$. The initial state of the controlling quantum system is the number state given by Eqs. (35) or (36) with $N=1$. The reservoir $R_1$ is super-Ohmic at $T=0$. Remaining parameters are the same as in Fig. 1.

Figure 4

(Color online) The same as in Fig. 3 for selected values of $N$ of the initial state (35) or (36) of the controlling quantum system.

Figure 5

(Color online) The same as in Fig. 3 for selected values of $z$ of the initially entangled coherent state (37) of the controlling quantum system.