Steady-state entanglement and coherence of two coupled qubits in equilibrium and nonequilibrium environments

We analytically and numerically investigate the steady-state entanglement and coherence of two coupled qubits, each interacting with a local boson or fermion reservoir, based on the Bloch-Redfield master equation beyond the secular approximation. We find that there is nonvanishing steady-state coherence in the nonequilibrium scenario, which grows monotonically with the nonequilibrium condition quantified by the temperature difference or chemical potential difference of the two baths. The steady-state entanglement, in general, is a nonmonotonic function of the nonequilibrium condition as well as the bath parameters in the equilibrium setting. We also discover that weak interqubit coupling and high base temperature or chemical potential of the baths can strongly suppress the steady-state entanglement and coherence, regardless of the strength of the nonequilibrium condition. On the other hand, the energy detuning of the two qubits, when used in a compensatory way with the nonequilibrium condition, can lead to significant enhancement of the steady-state entanglement in some parameter regimes. In addition, the qubits typically have a stronger steady-state entanglement when coupled to fermion baths exchanging particles with the system than boson baths exchanging energy with the system, under similar conditions. We also identify a close connection between the energy current flowing through the system and the steady-state coherence. Preliminary investigations suggest that these results are insensitive to the form of the reservoir spectral densities in the Markovian regime. Feasible experimental realization of measuring the steady-state entanglement and coherence is discussed for the coupled qubit system in nonequilibrium environments. Our findings offer some general guidelines for optimizing the steady-state entanglement and coherence in the coupled qubit system and may find potential applications in quantum information technology.

Figure 1

Schematic diagram of the physical model under consideration. The coupled qubit system, with respective energy spacings ${\omega }_1$ and ${\omega }_2$, are coupled to each other and immersed in their individual reservoirs.

Figure 2

The levels of the four eigenstates $|E_i\rangle$ for $i=1,2,3,4$ of the coupled qubit system. The red and green lines represent two groups of energy-level transitions induced by the system-reservoir interaction (see the text for detail).

Figure 3

(a) The steady-state concurrence for the boson reservoir case. (b) The steady-state coherence in the bare-state representation for the boson reservoir case. The parameters are set as ${\omega }_1={\omega }_2=10,J_1=J_2=1$.

Figure 4

(a) The steady-state concurrence for the fermion reservoir case. (b) The steady-state coherence in the bare-state representation for the fermion reservoir case. The parameters are set as ${\omega }_1={\omega }_2=10,J_1=J_2=1,T_1=T_2=1.5$.

Figure 5

The steady-state concurrence, coherence in the eigenstate and bare-state representations, populations $a={\rho }_{11}$ and $d={\rho }_{22}$ as well as $\sqrt{ad}$, for the coupled qubit system interacting with individual boson reservoirs at different temperatures. The parameters are set as ${\omega }_1={\omega }_2=10,J_1=J_2=1,\lambda =6$.

Figure 6

The steady-state concurrence (a), the coherence in the eigenstate representation (b), and the coherence in the bare-state representation (c) for the coupled qubit system immersed in individual fermion reservoirs with different chemical potentials. The parameters are set as ${\omega }_1={\omega }_2=10,J_1=J_2=1,\lambda =6,T_1=T_2=1.5$.

Figure 7

The entanglement phase diagrams of the system of the two coupled qubits immersed in boson baths (a) and fermion baths (b). The parameters are set as $J_1=J_2=1$, $\lambda =6$, $\overline{\omega }=10$ in panels (a) and (b); $\overline{T}=3$ in panel (a); and $T_1=T_2=1.5$, $\overline{\mu }=4$ in panel (b).

Figure 8

The steady-state coherence and entanglement for boson reservoirs [(a), (b)] and fermion reservoirs [(c), (d)]. The parameters are set as ${\omega }_1={\omega }_2=10,{\omega }_{c1}={\omega }_{c2}=40$; [(a), (b)] $T_1=2,T_2=T_1+\mathrm{\Delta }T$; [(c), (d)] $T_1=T_2=1.5,{\mu }_1=4,{\mu }_2={\mu }_1+\mathrm{\Delta }\mu$.

Figure 9

The energy current in the steady state for boson reservoirs (a) and fermion reservoirs (b) for different inter qubit coupling strengthes. The parameters are set as ${\omega }_1={\omega }_2=10,J_1=J_2=1$; (a) $T_1=2,T_2=T_1+\mathrm{\Delta }T$; (b) $T_1=T_2=1.5,{\mu }_1=4,{\mu }_2={\mu }_1+\mathrm{\Delta }\mu$.