Robust fermionic-mode entanglement of a nanoelectronic system in non-Markovian environments

A maximal steady-state fermionic entanglement of a nanoelectronic system is generated in finite temperature non-Markovian environments. The fermionic entanglement dynamics is presented by connecting the exact solution of the system with an appropriate definition of fermionic entanglement. We prove that the two understandings of the dissipationless non-Markovian dynamics, namely, the bound state and the modified Laplace transformation, are completely equivalent. For comparison, the steady-state entanglement is also studied in the wide-band limit and Born-Markovian approximation. When the environments have a finite band structure, we find that the system presents various kinds of relaxation processes. The final states can be thermal or thermal-like states, quantum memory states, and oscillating quantum memory states. Our study provides an analytical way to explore the non-Markovian entanglement dynamics of identical fermions in a realistic setting, i.e., finite-temperature reservoirs with a cutoff spectrum.

Figure 1

Density plot of steady-state entanglement ${\overline{E}}^s$ versus the on-site energy ${\epsilon }_l$ and the chemical potential ${\mu }_l$. In (a) and (b), we keep ${\mu }_1={\mu }_2=5\Gamma$, but in (c) and (d), we take the complete symmetric condition where ${\epsilon }_1={\epsilon }_2=\epsilon$ and ${\mu }_1={\mu }_2=\mu$. The other parameters are ${\Gamma }_1={\Gamma }_2=\Gamma$, $k_B{T}_1=k_B{T}_2=0.5\Gamma$, and $G=0.5\Gamma$.

Figure 2

Entanglement evolution of the DQDs with Lorentzian spectral density, where (a): $k_{B}T=G=0.5\Gamma$, (b): $d=2\Gamma$, $G=0.5\Gamma ,$ and (c): $k_{B}T=d=0.5\Gamma$. Insets show the asymptotic dynamics of $\overline{E}$ with different bandwidths and temperatures. The initial state is ${\widehat{a}}_1^{†}\left|\right|\mathrm{vac}\rangle \rangle$. The other parameters are $\epsilon =\mu =2\Gamma$. Here we consider the complete symmetric condition, so all the subscripts are omitted.

Figure 3

Steady-state entanglement ${\overline{E}}^s$ versus the interdot tunnel coupling $G$. The DQDs are set to be in resonance with $\epsilon =\mu =2\Gamma$ and the temperature $k_{B}T=0.5\Gamma$.

Figure 4

Density plot of steady-state entanglement ${\overline{E}}^s$ versus the on-site energy ${\epsilon }_l$. The corresponding results of the wide-band limit (WBL) and Born-Markovian approximation (BM) are plotted in (a) and (b), respectively. We keep ${\mu }_1={\mu }_2=5\Gamma$, and the other parameters are the same as given in Fig. 1.

Figure 5

Schematic plots of the root structure of Eq. (16), i.e., $det[\omega \mathbf{I}-\mathbf{M}-\mathit{\Sigma }(\omega \left)\right]=0$. The purple-shaded regimes on the real axis $\omega$ correspond to the bandwidth region $\mathbf{J}\left(\omega \right)\ne 0$. The green-dashed curve is the parabola ${\omega }^2-({\epsilon }_1+{\epsilon }_2)\omega +{\epsilon }_1{\epsilon }_2-G^2$.