Steady-state entanglement enhanced by a dissipative ancilla

We investigate how to enhance entanglement in the steady state of interacting two-level systems. The steady state is reached by spontaneous decay of the individual systems. When we additionally couple them to a dissipative two-level ancilla with variable eigenfrequency and coupling strength, we observe a considerable enhancement effect in the entanglement of this steady state. Moreover, we see that the increased entanglement is directly connected to the selection of certain excited states via the environment disturbing the ancilla. This effect could be used in dissipative state preparation schemes, and as a test-bed for decoherence models.

Figure 1

We are interested in the steady-state entanglement between systems $A$ and $B$ and especially how this entanglement depends on the properties of the dissipative ancilla $C$. The model consists of three two-level systems $A,B$, and $C$ with eigenfrequencies $\omega$ and ${\omega }_C$ that are coupled symmetrically, where $J$ and $J_C$ are the coupling strengths. Each of the systems is located in a heat bath characterized by the decoherence parameters $\gamma$ and ${\gamma }_C$, respectively.

Figure 2

For $J=+0.62$ and ${\gamma }_C=\gamma ={10}^{-3}$, the steady-state entanglement between systems $A$ and $B$, as measured by the negativity $N$, decays with increased absolute value of coupling $J_C$ to the dissipative ancilla $C$. The entanglement is maximal for $J_C=0$ and arbitrary ${\omega }_C$, which means the dissipative ancilla cannot enhance the bipartite entanglement $N$.

Figure 3

For $J=-0.62$ and ${\gamma }_C=\gamma ={10}^{-3}$ the entanglement between systems $A$ and $B$, as measured by the negativity $N$, does not simply decrease when we increase the interaction strength $J_C$ with the dissipative ancilla $C$. There is a distinctive maximum $N=0.180$ (black dot) of entanglement for an appropriately chosen eigenfrequency ${\omega }_C=0.55$ and coupling strength $J_C=0.01$, also visualized in the inset.

Figure 4

The value of bipartite entanglement $N$ in dependence on the decoherence rate ${\gamma }_C$. The parameters ${\omega }_C=0.55,J_C=0.01,J=-0.62$, and $\gamma ={10}^{-3}$ are kept constant. Only for values of ${\gamma }_C>0.64$ does the enhancement effect cease to exist and the entanglement in the uncoupled system of $N=0.155$ (black dashed line) is superior.

Figure 5

We show the dynamics of the fidelities $F_n\left(t\right)$, starting from the initial state in Eq. (8). In particular, we compare the parameter sets from Eqs. (11) and (13) in Figs. 5 and 5, respectively. We observe in both that after a transient evolution an equilibrium is reached where the mixture of the steady state is dominated by the individual ground state $|E_0\rangle$. The insets illustrate that by changing the sign of $J$ and choosing the optimized decoherence parameter ${\gamma }_C$, the mixture is mainly altered by the suppressing state $|E_2\rangle$ at the expense of state $|E_0\rangle$. This explains the change in entanglement with respect to subsystem $AB$.

Figure 6

The value of bipartite entanglement $N$ in dependence on the decoherence rate ${\gamma }_C$. The parameters ${\omega }_C=-0.74,J_C=0.01,J=-0.31$, and $\gamma ={10}^{-3}$ are kept constant. As seen before in Fig. 4, the bath of ancilla $C$ is crucial to obtaining an enhancement effect.

Figure 7

The dynamics of the fidelities $F_n\left(t\right)$, starting from the initial state in Eq. (8) with the parameters from Eq. (17). After a transient evolution an equilibrium is reached where the mixture of the steady state is now dominated by the excited state $|E_4\rangle$. This is in stark contrast to the situation observed in Fig. 5 and explains the significantly different entanglement properties.