Lindblad formalism based on fermion-to-qubit mapping for nonequilibrium open quantum systems

We present an alternative form of master equation, applicable to the analysis of nonequilibrium dynamics of fermionic open quantum systems. The formalism considers a general scenario, composed by a multipartite quantum system in contact with several reservoirs, each one with a specific chemical potential and in thermal equilibrium. With the help of Jordan-Wigner transformation, we perform a fermion-to-qubit mapping to derive a set of Lindblad superoperators that can be straightforwardly used on a wide range of physical setups. To illustrate our approach, we explore the effect of a charge sensor, acting as a probe, over the dynamics of electrons on coupled quantum molecules. The probe consists of a quantum dot attached to source and drain leads, that allows a current flow. The dynamics of populations, entanglement degree, and purity show how the probe is behind the sudden deaths and rebirths of entanglement, at short times. Then, the evolution leads the system to an asymptotic state being a statistical mixture. Those are signatures that the probe induces dephasing, a process that destroys the coherence of the quantum system.

Figure 1

Illustration of a generic setup composed of a multipartite system $\mathcal{S}$ interacting with $M$ reservoirs.

Figure 2

Scheme of the coupled quantum molecules with a probe: the first (second) molecule is composed by dots 1 and 2 (3 and 4). Tunneling is allowed inside the molecules, described by parameter ${\mathrm{\Delta }}_{ij}$. The electrostatic interactions between electrons inside the molecules, with parameter $U_{a\left(b\right)}$, provide the coupling between the molecules. The probe consists of a narrow channel provided by dot 5, attached to a source (S) and a drain (D), coupled capacitive with $U_{\mathrm{p}}$ as illustrated.

Figure 3

Time evolution of cross-populations, Eqs. (20), for the states belonging to the subspace ${\mathcal{M}}_{2QB}$, considering probe coupling $U_{\mathrm{p}}=3J$ and ${\mathrm{\Gamma }}_{55}^{\left(5\right)}=J$. (a) $P_{|1001\rangle }$ (solid black line) and $P_{|0110\rangle }$ (solid light brown line). (b) $P_{|1010\rangle }$ (black line with symbols) and $P_{|0101\rangle }$ (light brown line with symbols). Inset: Time evolution of $P_{|1001\rangle }$ and $P_{|0110\rangle }$ cross-populations when the probe is turned off ($U_{\mathrm{p}}=0$). Physical parameters are those for creation of Bell states: ${\mathrm{\Delta }}_{12}={\mathrm{\Delta }}_{34}=\mathrm{\Delta }=\sqrt{3}/8J,{\varepsilon }_i^{\left(\mathcal{M}\right)}=0,U_a=(3/4)J,U_b=(1/4)J$.

Figure 4

Time evolution of the populations of the quantum dots in coupled quantum molecules, for the same physical parameters as in Fig. 3. (a) $\langle {\widehat{N}}_1\rangle$ (solid black lines) and $\langle {\widehat{N}}_2\rangle$ (solid light brown lines), corresponding to the molecule coupled with the probe. We also plot the behavior of both $\langle {\widehat{N}}_1\rangle$ (dashed black line) and $\langle {\widehat{N}}_2\rangle$ (dashed light brown line) when the probe is turned off ($U_{\mathrm{p}}=0$). (b) $\langle {\widehat{N}}_3\rangle$ (light brown line with symbols) and $\langle {\widehat{N}}_4\rangle$ (black line with symbols), corresponding to the second molecule.

Figure 5

Dynamics of (a) concurrence ($C$) and (b) linear entropy ($S$) for several values of coupling $U_{\mathrm{p}}$, for the same choice of physical parameters used in Fig. 3: $U_{\mathrm{p}}=0$ (dashed light brown lines), $U_{\mathrm{p}}=J$ (solid light brown lines), $U_{\mathrm{p}}=2J$ (dashed black lines), and $U_{\mathrm{p}}=3J$ (solid black lines).