Effect of coherence of nonthermal reservoirs on heat transport in a microscopic collision model

We investigate the heat transport between two nonthermal reservoirs based on a microscopic collision model. We consider a bipartite system consisting of two identical subsystems, and each subsystem interacts with its own local reservoir, which consists of a large collection of initially uncorrelated ancillas. Then a heat transport is formed between two reservoirs by a sequence of pairwise collisions (intersubsystem and subsystem-local reservoir). In this paper we consider two kinds of the reservoir's initial states: the thermal state and the state with coherence whose diagonal elements are the same as that of the thermal state and the off-diagonal elements are nonzero. In this way, we define the effective temperature of the reservoir with coherence according to its diagonal elements. We find that for two reservoirs having coherence the direction of the steady current of heat is different for different phase differences between the two initial states of two reservoirs, especially the heat can transfer from the “cold reservoir” to the “hot reservoir” in the steady regime for particular phase difference. In the limit of the effective temperature difference between the two reservoirs $\mathrm{\Delta }T\to 0$, for most of the phase differences, the steady heat current increases with the increase of effective temperature until it reaches the high effective temperature limit, while for the thermal state or particular phase difference the steady heat current decreases with the increase of temperature at high temperatures, and in this case the conductance can be obtained.

Figure 1

Sketch of the protocol: A bipartite system $\mathcal{S}$ consists of subsystems ${\mathcal{S}}_A$ and ${\mathcal{S}}_B$; after intersubsystem interaction, ${\mathcal{S}}_A$ interacts with the $n\mathrm{th}$ ancilla (prepared in ${\eta }_n^h$), and ${\mathcal{S}}_B$ interacts with the $n\mathrm{th}$ ancilla (prepared in ${\eta }_n^c$); then this process is repeated and is directed to the $(n+1)\mathrm{th}$ ancilla. Thus the heat transport is established between the two reservoirs by a sequence of this process.

Figure 2

Heat current $J_h$ against the number of collisions $n$ for initial state (13) with $p=0.8$ and different relative phase ${\varphi }_h-{\varphi }_c=\{0,\pi /4,\pi /2,\pi ,5\pi /4,3\pi /2\}$. The dashed black line corresponds to $p=0$ in Eq. (13), i.e., the thermal state ${\rho }_{\beta }$. The inset is the magnified $J_h$ for the thermal state and state (13) with ${\varphi }_h-{\varphi }_c=0,\pi$. For all plots we choose the ground state of system ${|\psi \rangle }_S=|11\rangle ,\gamma =\pi /32,\delta =\pi /4,T_h=2T_c=2$ (i.e., ${\beta }_h=\frac{1}{2}{\beta }_c=\frac{1}{2}$), and $\omega =1$.

Figure 3

Steady heat current $J_h^{\mathrm{steady}}$ as a function of ${\varphi }_h-{\varphi }_c$ for initial state (13) with $p=0.8$ and $\delta =\pi /4$, and different interaction strength (a) $\gamma =\pi /2$, (b) $\gamma =\pi /4$, (c) $\gamma =\pi /12$, and (d) $\gamma =\pi /32$. For all plots we choose the ground state of system ${|\psi \rangle }_S=|11\rangle ,T_h=2T_c=2$ (i.e., ${\beta }_h=\frac{1}{2}{\beta }_c=\frac{1}{2}$), and $\omega =1$.

Figure 4

The conductance as a function of $T$ for state (13) with $p=0.8,{\varphi }_h-{\varphi }_c=\{0,\pi \}$; and $p=0$, i.e., thermal state ${\rho }_{\beta }$. $\gamma =\pi /32,\delta =\pi /4,\omega =1,T_h=T+\mathrm{\Delta }T/2$, and $T_c=T-\mathrm{\Delta }T/2$.

Figure 5

Steady heat currents as functions of $\mathrm{\Delta }T$ and $T$ for state (13) with $p=0.8,\gamma =\pi /32,\delta =\pi /4;T_h=T+\mathrm{\Delta }T/2$ and $T_c=T-\mathrm{\Delta }T/2$, and $\omega =1$. (a) ${\varphi }_h-{\varphi }_c=\pi /4$; (b) ${\varphi }_h-{\varphi }_c=5\pi /4$.