Influence of the thermal environment on entanglement dynamics in small rings of qubits

Numerical solutions of the stochastic Schrödinger equation given by quantum state diffusion approach to open quantum systems is used to study dynamics of nearest neighbor qubit pairs in systems of small number of qubits on rings with Heisenberg and transverse Ising interaction and under the influence of the thermal environment. In particular, the dependence of the pair entanglement dynamics on the temperature, number of qubits, the type of coupling, and the type of entanglement in the initial state was analyzed for systems of up to $N=10$ qubits. Periodic recurrence of relatively large values of the pair entanglement with dumping due to decoherence by the thermal noise is observed. It is concluded that the pair entanglement in rings with transverse Ising coupling and prepared in a separable initial state is the most resistant on the decoherence effects of the thermal noise, compared to the Heisenberg coupling or initial states with different types of entanglement.

Figure 1

$(1,2)$ (full black lines) and $(1+[N∕2],2+[N∕2\left]\right)$ (dotted gray lines) pair entanglement for the isolated rings of qubits with Heisenberg (c), (e) and transversal Ising (a), (b), (d), (f) coupling. $T$ represents the dimensionless time $T=\omega t$ and $E$ is the measure of entanglement Eq. (9). In (a) and (b) the initial state is $\mid W⟩$ and the number of qubits is (a) $N=7$ and (b) $N=8$. In (c) and (d) the initial state is $\mid \max ⟩$ and $N=7$. In (e) and (f) the initial state is $\mid \mathrm{sep}⟩$ and $N=7$.

Figure 2

Pair entanglement from $\mid W⟩$ initial state and $\overline{n}=0.5$. (a) Heisenberg coupling and $N=5$ (dotted gray), $N=6$ (gray thin), $N=7$ (gray thick), $N=8$ (black dotted), $N=9$ (black thin), and $N=10$ (black thick). In (b), (c), and (d) the thick black lines are for the Heisenberg coupling, the full gray lines are for the transverse Ising (1, 2) pair and in (b) gray dotted for $(1+[N∕2],2+[N∕2\left]\right)$ pair. (b) $N=6$; (c) $N=7$; (d) $N=8$. $T$ and $E$ are as in Fig. 1.

Figure 3

(Color online) $(1,2)$ pair entanglement dynamics from $\mid \max ⟩$ initial state for (a) and (b) $N=6$ and (c) and (d) $N=7$, for the (a) and (c) the Heisenberg coupling and for (b) and (d) the transverse Ising. The parameter $\overline{n}$ is $\overline{n}=0.5$ (dotted gray), $\overline{n}=1$ (black), and isolated system (green). $T$ and $E$ are as in Fig. 1.

Figure 4

(a) and (b) $(1,2)$ and (c) and (d) $(5,6)$ pair entanglement dynamics from $\mid \max ⟩$ initial state and $N=8$, for (a) and (c) the Heisenberg coupling and for (b) and (d) the transverse Ising, and $\overline{n}=0.5$ (black dotted) and isolated systems (black full). $T$ and $E$ are as in Fig. 1.

Figure 5

(Color online) $(1,2)$ pair entanglement dynamics from $\mid \mathrm{sep}⟩$ initial state for (a) and (b) $N=6$ and (c) and (d) $N=7$, for (a) and (c) the Heisenberg coupling and for (b) and (d) the transverse Ising. In all figures $\overline{n}=0.5$ (dotted), $\overline{n}=0.8$ (dashed), and isolated system (full). $T$ and $E$ are as in Fig. 1.

Figure 6

(Color online) (a) and (b) $(1,2)$ and (c) and (d) $(5,6)$ pair entanglement dynamics from $\mid \mathrm{sep}⟩$ initial state for $N=8$, for (a) and (c) the Heisenberg coupling and for (b) and (d) the transverse Ising, and $\overline{n}=0.5$ (dotted), $\overline{n}=1$ (dashed), and isolated system (full). $T$ and $E$ are as in Fig. 1.

Figure 7

Dependence of $(1,2)$ pair entanglement $E$ at $t={\tau }_N$ on the number of qubits $N$ for several fixed $\overline{n}$, for the transverse Ising coupling and $\mid \mathrm{sep}⟩$ initial state. (b) Isolated system (stars), $\overline{n}=0.5$ (circles), $\overline{n}=0.8$ (boxes), and $\overline{n}=1$ (down triangles). The lines serve only to guide the eye.