Entanglement between qubits induced by a common environment with a gap

We study a system of two qubits interacting with a common environment, described by a two-spin boson model. We demonstrate two competing roles of the environment: inducing entanglement between the two qubits and making them decoherent. For the environment of a single harmonic oscillator, if its frequency is commensurate with the induced two-qubit coupling strength, the two qubits could be maximally entangled and the environment could be separable. In the case of the environment of a bosonic bath, the gap of its spectral density function is essential to generate entanglement between two qubits at equilibrium and for it to be used as a quantum data bus.

Figure 1

Two circles refer to two qubits and an oval is an environment. The coupling between qubits and the common environment (solid arrow) induces the indirect interaction between two qubits (dashed arrow).

Figure 2

(Color online) Concurrence $C$ (red solid), $C_{\text{ideal}}$ (black thick solid), entropy $2S∕3$ (blue dashed), and the overlap $\exp [-{\Gamma }_R\left(t\right)]$ (cyan dashed-dotted) as a function of time for $\omega ∕\lambda =4\sqrt{3}$ (a), 5 (b), and 1 (c). (d) Maximum concurrence $C_{\max }$, maximum entropy $S_{\max }$, average concurrence $C_{\mathrm{avg}}$, and average entropy $S_{\mathrm{avg}}$ as a function of $n={(\omega ∕4\lambda )}^2$ for $0⩽\theta t<\pi ∕2$.

Figure 3

(Color online) (a) $\exp [-{\Gamma }_R\left(t\right)]$ as a function of ${\omega }_{c}t$ and (b) its log scale plot. Legends (1)–(6) refer to $({\omega }_0∕{\omega }_c,\alpha )=(0,0.25)$, (0.01,0.25), (0.1,0.25), (0,0.5), (0.01,0.5), and (0.1,0.5), respectively. Inset of (a): the spectral density function $J\left(\omega \right)$ with a gap ${\omega }_0$. (c) Concurrence $C$ (solid), entropy $2S∕3$ (dotted), and $\exp [-{\Gamma }_R\left(t\right)]$ (dashed) as a function of $\theta t$.

Figure 4

(Color online). (a) Maximum concurrence of the two qubits. (b) Entropy of two qubits at equilibrium as a function of $\alpha$ and ${\omega }_0∕{\omega }_c$. (c) $\exp [-{\Gamma }_R\left(\infty \right)]$ as a function of temperature $T$ and ${\omega }_0∕{\omega }_c$ for $\alpha =0.25$.