The effects of system–environment correlations on heat transport and quantum entanglement via collision models

Abstract

By means of collision models, we study the effects of system–environment correlations (SECs) on the heat transport and quantum entanglement in both transient and steady-state regimes. In the considered models, the reservoirs are simulated through two chains of particles whose nearest-neighbor collisions induce the SECs. In the first model, the system is a qubit connecting two independent reservoirs with different temperatures. We show that the heat currents exhibit oscillations and even reversed flows from the cold reservoir to the hot one depending on intracollision strengths of reservoir particles. In the stationary regime, we observe a nonlinear relation between heat currents and intracollision strengths, which can be accounted for by the established steady-state SECs. In our second model, the system contains two interacting qubits and we show that the initial entanglement of the system either vanishes within finite steps of collisions or recovers from disappearing and retains nonzero value. The combined regions of relevant parameters that sustain steady-state entanglement are presented. We also present a method to enhance the steady-state entanglement by enlarging temperature differences of the two reservoirs.

Appendix

In this appendix, we present some explanations for the derivation of the steady-state as well as Eqs. (14)–(16) in the first model. We first assume a general form of the steady-state $\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}$ with matrix elements $\widetilde{\rho }_{kl}\equiv \left\langle \overline{k} \left| \widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}\right| \overline{l}\right\rangle$ ($k,l=1,2,\ldots ,8$) expressed in the ordered basis $\{\left| \overline{1}\right\rangle =\left| 000\right\rangle , \left| \overline{2}\right\rangle =\left| 001\right\rangle , \left| \overline{3}\right\rangle =\left| 010\right\rangle ,$ $\left| \overline{4}\right\rangle =\left| 011\right\rangle , \left| \overline{5}\right\rangle =\left| 100\right\rangle , \left| \overline{6}\right\rangle =\left| 101\right\rangle , \left| \overline{7}\right\rangle =\left| 110\right\rangle , \left| \overline{8}\right\rangle =\left| 111\right\rangle \}$ of $R_{m}^{(h)}$, S, and $R_{m}^{(c)}$. To derive the concrete form of the steady state, we should construct and solve 64 equations for these elements. Since the total systems of $R_{m}^{(h)}$, S, and $R_{m}^{(c)}$ have reached the stationary regime, their state $\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}$ should remain invariant under the successive collisions of $S-R_{m}^{(h)}$, $S-R_{m}^{(c)}$, $R_{m}^{(h)}-R_{m+1}^{(h)}$, and $R_{m}^{(c)}-R_{m+1}^{(c)}$ as given in Eqs. (11) and (12). Suppose the overall action of these collisions is $F(J_{h},J_{c},\varOmega )$, then we have $\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}=F(J_{h},J_{c},\varOmega )\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}F^{\dag }(J_{h},J_{c},\varOmega )$, from which we can derive the 64 equations for the elements $\widetilde{\rho }_{kl}$ and obtain their values via numerical calculations.

From the definition of heat current (13), we know that to calculate the released heat $\Delta \widetilde{Q}_{m}^{(h)}$ of the hot reservoir we should first derive the state $\widetilde{\rho ^{\prime }}_{R_{m}^{(h)}}$, i.e., the state of $R_{m}^{(h)}$ after collision with the system, as well as $\widetilde{\rho }_{R_{m}^{(h)}}$. Since we have assumed the general form of the steady-state $\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}$, $\widetilde{\rho }_{R_{m}^{(h)}}$ can be obtained straightforward by tracing $\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}$ over the degrees of freedom of S and $R^{(c)}_{m}$ and expressed in terms of the matrix elements $\widetilde{\rho }_{kl}$. Suppose the collision between S and $R^{(h)}_{m}$ is given by the action $f(J_{h})$, then we shall obtain $\widetilde{\rho }^{\prime }_{R_{m}^{(h)}}$ through tracing the transformed state $f(J_{h})\widetilde{\rho }_{R^{(h)}_{m}SR^{(c)}_{m}}f^{\dag }(J_{h})$ over the degrees of freedom of S and $R^{(c)}_{m}$. The form of $\widetilde{\rho }^{\prime }_{R_{m}^{(h)}}$ can thus be expressed by the matrix elements $\widetilde{\rho }_{kl}$ and $J_{h}$. In the total expression of $\Delta \widetilde{Q}_{m}^{(h)}$, we can easily find the terms contributed by the diagonal and off-diagonal elements of $\widetilde{\rho }_{kl}$. Therefore, we divide $\Delta \widetilde{Q}_{m}^{(h)}$ correspondingly into two parts as given in Eqs. (15) and (16).