Controlling heat flows among three reservoirs asymmetrically coupled to two two-level systems

We study heat flows among three thermal reservoirs via two two-level systems (TLSs). Two reservoirs are coupled to one TLS and the third reservoir to the second TLS. The two TLSs are also coupled to each other, thus bridging the third reservoir with the two other reservoirs. We show that the magnitudes and directions of the reservoirs' heat currents can be controlled by varying the various damping rates of the two TLSs due to coupling with the corresponding reservoirs. First, it is shown that by changing the damping rate due to one reservoir, magnitudes of heat currents of the other two reservoirs can behave in completely different manners, namely, although one may be enhanced, the other may instead be suppressed, and vice versa. Second, the sign of the heat current of one reservoir may change (i.e., crossover from heat absorption to heat release, or vice versa) if a damping rate or the coupling strength between the two TLSs is swept through a critical value, which depends on the temperature settings for the three reservoirs. Due to the asymmetric couplings of the two TLSs to the three reservoirs, the thermal rectification occurs without introducing any additional asymmetry to the systems.

Figure 1

(a) Schematic diagram of the physical model under consideration. Two TLSs $A$ and $B$ are coupled to each other with a strength $g$. The TLS $A$ is coupled simultaneously to two heat reservoirs $R_A^{\left(1\right)}$ and $R_A^{\left(2\right)}$ with temperatures $T_A^{\left(1\right)}$ and $T_A^{\left(2\right)}$, respectively, while $B$ is coupled to a single heat reservoir $R_B$ with temperature $T_B$. (b) The levels of the four eigenstates $\left|{\lambda }_i\right.〉 \left(i=1,2,3,4\right)$ of Hamiltonian ${\widehat{H}}_S$, given by Eq. (1), for the two coupled TLSs. In the presence of the baths, there exist bath-induced exciting and damping processes among the four eigenstates. The effective transition rates are given by ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2,{\mathrm{\Gamma }}_3$, and ${\mathrm{\Gamma }}_4$ in Eq. (12).

Figure 2

Heat currents $Q_A^{\left(1\right)}$ (dashed line), $Q_A^{\left(2\right)}$ (dash-dotted line), and $Q_B$ (solid line) as a function of the scaled damping rates ${\mathrm{\Gamma }}_A^{\left(2\right)}/{\mathrm{\Gamma }}_A^{\left(1\right)}$. The insets show the variations of $|Q_A^{\left(1\right)}|,|Q_A^{\left(2\right)}|$, and $|Q_B|$. The temperatures are set as (a) $T_A^{\left(1\right)}=T_A^{\left(2\right)}={\mathrm{\Gamma }}_A^{\left(1\right)}<T_B=10{\mathrm{\Gamma }}_A^{\left(1\right)}$ and (b) $T_A^{\left(1\right)}=T_A^{\left(2\right)}=10{\mathrm{\Gamma }}_A^{\left(1\right)}>T_B={\mathrm{\Gamma }}_A^{\left(1\right)}$. The other parameters are set as ${\omega }_A=10{\mathrm{\Gamma }}_A^{\left(1\right)},{\omega }_B=9{\mathrm{\Gamma }}_A^{\left(1\right)},g={\mathrm{\Gamma }}_A^{\left(1\right)}$, and ${\mathrm{\Gamma }}_B={\mathrm{\Gamma }}_A^{\left(1\right)}$.

Figure 3

Population variations with the scaled damping rate ${\mathrm{\Gamma }}_A^{\left(2\right)}/{\mathrm{\Gamma }}_A^{\left(1\right)}$ for the same parameters used in Figs. 2 and 2.

Figure 4

Heat currents $Q_A^{\left(1\right)}$ (dashed line), $Q_A^{\left(2\right)}$ (dash-dotted line), and $Q_B$ (solid line) as a function of the scaled damping rates ${\mathrm{\Gamma }}_B/{\mathrm{\Gamma }}_A^{\left(2\right)}$. The temperatures are chosen as (a) $T_A^{\left(1\right)}={\mathrm{\Gamma }}_A^{\left(2\right)},T_A^{\left(2\right)}=6{\mathrm{\Gamma }}_A^{\left(2\right)},T_B=10{\mathrm{\Gamma }}_A^{\left(2\right)}$ and (b) $T_A^{\left(1\right)}=10{\mathrm{\Gamma }}_A^{\left(2\right)},T_A^{\left(2\right)}=6{\mathrm{\Gamma }}_A^{\left(2\right)},T_B={\mathrm{\Gamma }}_A^{\left(2\right)}$. The other parameters are chosen as ${\mathrm{\Gamma }}_A^{\left(1\right)}=0.5{\mathrm{\Gamma }}_A^{\left(2\right)},g=4{\mathrm{\Gamma }}_A^{\left(2\right)},{\omega }_A=10{\mathrm{\Gamma }}_A^{\left(2\right)}$, and ${\omega }_B=9{\mathrm{\Gamma }}_A^{\left(2\right)}$. The horizontal line at zero is added to guide the eye for the heat crossover.

Figure 5

(a) $Q_A^{\left(2\right)}$ vs ${\mathrm{\Gamma }}_B/{\mathrm{\Gamma }}_A^{\left(2\right)}$ with various temperatures $T_A^{\left(2\right)}$ between $T_A^{\left(1\right)}$ and $T_B$ for $T_A^{\left(1\right)}={\mathrm{\Gamma }}_A^{\left(2\right)}$ and $T_B=10{\mathrm{\Gamma }}_A^2$. (b) Same as in (a), with various temperatures $T_A^{\left(2\right)}$ between $T_B$ and $T_A^{\left(1\right)}$ for $T_B={\mathrm{\Gamma }}_A^{\left(2\right)}$ and $T_A^{\left(1\right)}=10{\mathrm{\Gamma }}_A^{\left(2\right)}$. The other parameters are chosen as in Fig. 4. The horizontal line at zero is added to guide the eye for the heat crossover.

Figure 6

Heat currents $Q_A^{\left(1\right)}$ (dashed line), $Q_A^{\left(2\right)}$ (dash-dotted line), and $Q_B$ (solid line) as a function of the scaled coupling strength $g/\mathrm{\Gamma }$ of TLSs $A$ and $B$ for different temperature settings. The other parameters are set as ${\omega }_A=10\mathrm{\Gamma },{\omega }_B=9\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_A^{\left(1\right)}={\mathrm{\Gamma }}_A^{\left(2\right)}={\mathrm{\Gamma }}_B=\mathrm{\Gamma }$. The horizontal line at zero is added to guide the eye for the heat crossover.

Figure 7

(a) $Q_A^{\left(2\right)}$ vs $g/\mathrm{\Gamma }$ with different values of temperatures $T_A^{\left(2\right)}$ for $T_A^{\left(1\right)}=\mathrm{\Gamma }<T_A^{\left(2\right)}<T_B=10\mathrm{\Gamma }$; (b) $Q_A^{\left(1\right)}$ vs $g/\mathrm{\Gamma }$ with different values of temperatures $T_A^{\left(1\right)}$ for $T_B=\mathrm{\Gamma }<T_A^{\left(1\right)}<T_A^{\left(2\right)}=10\mathrm{\Gamma }$. The other parameters used here are the same as in Fig. 6. The horizontal line at zero is added to guide the eye for the heat crossover.

Figure 8

Rectification as a function of scaled coupling strength $g/\mathrm{\Gamma }$ for different lower temperature $T_{<}$ with a given higher temperature $T_{>}=10\mathrm{\Gamma }$. Two nonresonant TLSs are considered in (a) with ${\omega }_A=10\mathrm{\Gamma }$ and ${\omega }_B=4\mathrm{\Gamma }$, while two identical TLSs are considered in (b) with ${\omega }_A={\omega }_B=10\mathrm{\Gamma }$. The horizontal line at zero in (b) denotes the rectification in the two-reservoir case, namely, each of two identical TLSs being coupled to a single reservoir.