Smallest quantum thermal machine: The effect of strong coupling and distributed thermal tasks

The functions of the smallest self-contained thermal machine consisting of a single qutrit are studied when the weak internal coupling assumption is relaxed. It is shown that in the presence of one target to be cooled the strong coupling is not beneficial to the refrigeration. The reason is explained by examining the effect of the strong coupling on the contributions of all eigenstates transitions to the heat current of the related thermal reservoir. When acting simultaneously on two targets, the machine can be manipulated to implement distributed tasks on them, such as cooling one target and meanwhile heating another one, by adjusting the coupling strengths between the machine with the two targets. In particular, we show that the machine can realize temperature reversal for the two qubits, namely, the qubit that is coupled to the high temperature reservoir is refrigerated to a temperature below that of the qubit contacting with the low temperature reservoir.

Figure 1

(a) Schematic diagram of the physical model that consists of a machine (qutrit) acts on one target (qubit). The machine is affected by two thermal reservoirs, i.e., a reservoir $R_h$ with high temperature $T_h$ and a reservoir $R_r$ with room temperature $T_r$, through transitions of $\left|1\right.〉\leftrightarrow \left|2\right.〉$ and $\left|1\right.〉\leftrightarrow \left|3\right.〉$, respectively, while the qubit is exposed to a cold reservoir $R_c$ with temperature $T_c$. The machine acts on the target via the resonant interaction between the transition of $\left|2\right.〉\leftrightarrow \left|3\right.〉$ and $\left|g\right.〉\leftrightarrow \left|e\right.〉$. (b) The levels of the bare states and eigenstates of the machine and the qubit. The bare states $\left|2e\right.〉$ and $\left|3g\right.〉$ are degenerate and constitute the eigenstates $\left|{\lambda }_4\right.〉$ and $\left|{\lambda }_5\right.〉$ through superposition. The transitions among the six eigenstates induced by the the reservoirs $R_r, R_h$, and $R_c$ are identified with orange, red, and blue colors, respectively.

Figure 2

(a) Heat currents $Q_c, Q_r$, and $Q_h$ and (b) the temperature $T_A$ of the target qubit as a function of the temperature $T_h$ of the hot reservoir in the weak coupling regime for $\mathrm{\Omega }=0.001{\epsilon }_3$ and ${\gamma }_c={\gamma }_r={\gamma }_h=0.001{\epsilon }_3$. The other parameters are set as ${\epsilon }_2=2, {\epsilon }_3=10, T_c=10$, and $T_r=30$. The horizontal lines at zero in (a) and at 10 in (b) are added to guide the eye for the heat crossover and refrigeration, respectively.

Figure 3

(a) The heat currents $Q_c$ and (b) the temperature $T_A$ as a function of the temperature $T_h$ of the hot reservoir for different machine-qubit coupling strengths $\mathrm{\Omega }$. The other parameters are set as ${\gamma }_c={\gamma }_r={\gamma }_h=0.001{\epsilon }_3, {\epsilon }_2=2, {\epsilon }_3=10, T_c=10$, and $T_r=30$. The horizontal lines at zero in (a) and at 10 in (b) are added to guide the eye for the heat crossover and refrigeration, respectively.

Figure 4

The contributions of all the associated transitions, i.e., $\left|{\lambda }_1\right.〉\leftrightarrow \left|{\lambda }_2\right.〉, \left|{\lambda }_3\right.〉\leftrightarrow \left|{\lambda }_4\right.〉, \left|{\lambda }_3\right.〉\leftrightarrow \left|{\lambda }_5\right.〉, \left|{\lambda }_4\right.〉\leftrightarrow \left|{\lambda }_6\right.〉, \left|{\lambda }_5\right.〉\leftrightarrow \left|{\lambda }_6\right.〉$, to the heat current of the cold reservoir $R_c$, namely, $[{\mathrm{\Gamma }}_{i{i}^{\prime }}^{\left(c\right)}{\overline{\rho }}_{i^{\prime }{i}^{\prime }}-{\mathrm{\Gamma }}_{i^{\prime }i}^{\left(c\right)}{\overline{\rho }}_{ii}]{\omega }_{i^{\prime }i}$ with $(i,i^{\prime })=(2,1),(4,3),(5,3),(6,4)$, and $(6,5)$, as a function of $\mathrm{\Omega }$. The red solid line denotes $Q_c$, i.e., the sum of all the contributions. The inset shows the transitions of the eigenstates: For the small $\mathrm{\Omega }$ the net effects of all the transitions are excitations via heat absorptions from $R_c$, while with increasing $\mathrm{\Omega }$ the net effects of three transitions become decaying and thus release heat to $R_c$. The other parameters are set as ${\gamma }_c={\gamma }_r={\gamma }_h=0.001{\epsilon }_3, {\epsilon }_2=2, {\epsilon }_3=10, T_c=10, T_r=30$, and $T_h=150$.

Figure 5

Schematic diagram of our second model which is the same as that in Fig. 1 but with two target qubits $A$ and $B$. The two qubits $A$ and $B$ are exposed to independent cold reservoirs $R_{c1}$ with temperature $T_{c1}$ and $R_{c2}$ with temperature $T_{c2}$, respectively. The machine acts on the targets still via the resonant interaction between the transition of $\left|2\right.〉\leftrightarrow \left|3\right.〉$ and the qubits.

Figure 6

The temperatures $T_A$ and $T_B$ of qubits $A$ and $B$ as a function of the temperature $T_h$ of the hot reservoir for different intracouplings between the machine and the two qubits with ${\mathrm{\Omega }}_1=3, {\mathrm{\Omega }}_2=0.8$. The other parameters are set as ${\gamma }_{c1}={\gamma }_{c2}={\gamma }_r={\gamma }_h=0.01, {\epsilon }_2=7, {\epsilon }_3=15, T_{c1}=T_{c2}=10$, and $T_r=20$.

Figure 7

Contour plots of $T_A-T_{c1}$ (a) and $T_B-T_{c2}$ (b) versus ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ for $T_{c1}=10>T_{c2}=9$. The other parameters are set as $T_r=19, T_h=150, {\epsilon }_2=7, {\epsilon }_3=15$, and ${\gamma }_{c1}={\gamma }_{c2}={\gamma }_r={\gamma }_h=0.01$.

Figure 8

The dependences of temperatures $T_A$ and $T_B$ on ${\mathrm{\Omega }}_1$ for given values of ${\mathrm{\Omega }}_2$ (a)–(c) and on ${\mathrm{\Omega }}_2$ for given values of ${\mathrm{\Omega }}_1$ (d)–(f). The parameters are the same as that in Fig. 7.

Figure 9

Contour plots of $T_A-T_B$ versus ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ for $T_{c1}=10>T_{c2}=9$. The parameters are the same as that in Fig. 7.