Darlington pair of quantum thermal transistors

Recent progress in manipulating individual quantum systems has led to the development of quantum thermal transistors, which control the thermal conductivity between two of its terminals according to an input signal on its third terminal. With several models for individual thermal transistors already developed, the next natural step is to investigate how multiple thermal transistors can interconnect to build useful composite devices. In electronics literature, the Darlington pair is a two-transistor configuration commonly used to construct electronic amplifiers with higher gain than is possible with a single transistor. We create the electronic Darlington pair's thermal equivalent using two individual thermal transistors in a similar configuration. Unlike previously studied models, multitransistor configurations like this contain internal transistor interconnections whose temperatures cannot be biased externally but are determined by the individual transistors' internal dynamics. We refine previous models to incorporate these transistor-transistor interactions and introduce an intermediate thermal bath to facilitate the thermal energy exchange between the Darlington pair transistors. We investigate temperature-based and optical field-based control strategies of the Darlington pair in terms of both steady-state and transient thermal flow characteristics through numerical simulations. Under both control strategies, the thermal Darlington pair's steady-state performance exhibits superior thermal amplification, sensitivity, and thermodynamic efficiency than an equivalent single thermal transistor. Our results closely mirror those expected from the corresponding electronic Darlington pair. Hence, we envision that we may readily adapt the intermediate bath formalism we developed in this work to translate a wide variety of useful electronic multitransistor configurations to their thermal equivalents.

Figure 1

The system consists of two quantum thermal transistors [17] $S_1$ and $S_2$ interconnected in a Darlington configuration. Each thermal bath $B_P$ (where $P=\{H,N,I,C\}$) is characterized by its temperature $T_P$. The external optical field $F$ on $S_1$ allows for optical control of the heat flows if necessary [25]. The energy flow rates between the baths $B_P$ (or optical field $F$) and the quantum transistors are represented by the $J\mathrm{s}$ in the directions specified. (Inset) Block diagram of an electronic Darlington pair built using BJTs.

Figure 2

Steady-state heat flows to and from $B_I$ obtained from individual transistor models when $T_I$ is artificially held constant at different values. (Top) Temperature-controlled configuration for control parameter $T_N=0.4T_H$. (Bottom) Optically controlled configuration for control parameter ${\mathrm{\Omega }}_F=0.3\kappa \mathrm{\Delta }$. All other parameters are as given in Sec. 3a.

Figure 3

Steady-state temperature $T_I$ of the intermediate bath $B_I$ for the full control parameter range. (Top) Temperature-controlled configuration while varying $T_N$ within the range $[T_C,T_H]$. (Bottom) Optically controlled configuration while varying ${\mathrm{\Omega }}_F$ within the range $[0,0.7\kappa \mathrm{\Delta }]$. All other parameters are as given in Sec. 3a.

Figure 4

(Solid) Overall steady-state energy flow rates for the external thermal baths $B_H$, $B_N$, and $B_C$ of the temperature-controlled configuration of the Darlington pair system for $T_N=[T_C,T_H]$ range. (Dashed) For comparison, the same figures if the Darlington pair is replaced with a single temperature-controlled thermal transistor [17]. All simulation parameters are as given in Sec. 3a.

Figure 5

(Solid) Thermal flow amplification factor ${\beta }_{\text{Te}}$ and thermal flow sensitivity factor ${\eta }_{\text{Te}}^P$ for the temperature-controlled configuration of the Darlington pair system for $T_N=[T_C,T_H]$ range. (Dashed) For comparison, the same figures if the Darlington pair is replaced with a single temperature-controlled thermal transistor [17]. All simulation parameters are as given in Sec. 3a.

Figure 6

(Solid) Overall steady-state energy flow rates for the external optical field $F$ and the thermal baths $B_H$, $B_N$, and $B_C$ of the optically controlled configuration of the Darlington pair system for ${\mathrm{\Omega }}_F=[0,0.7\kappa \mathrm{\Delta }]$ range. (Dashed) For comparison, the same figures if the Darlington pair is replaced with a single optically controlled thermal gate [25]. All simulation parameters are as given in Sec. 3a.

Figure 7

(Solid) Thermal flow amplification factor ${\beta }_{\text{Op}}$ and thermal flow sensitivity factor ${\eta }_{\text{Op}}^P$ for the optically controlled configuration of the Darlington pair system for ${\mathrm{\Omega }}_F=[0,0.7\kappa \mathrm{\Delta }]$ range. (Dashed) For comparison, the same figures if the Darlington pair is replaced with a single optically controlled thermal gate [25]. All simulation parameters are as given in Sec. 3a.

Figure 8

Transient characteristics of the optically controlled Darlington pair (solid) and its corresponding single transistor (dashed) when optical Rabi frequency is suddenly increased from ${\mathrm{\Omega }}_F=0$ to ${\mathrm{\Omega }}_F=0.5\kappa \mathrm{\Delta }$. (Top) $C_I=20k_{\text{B}}$. (Bottom) $C_I=40k_{\text{B}}$. All other simulation parameters are as given in Sec. 3a.

Figure 9

The state and transition rate diagrams [25] of the two individual thermal transistors in the steady state of the temperature-controlled Darlington pair. The control bath temperature is set at $T_N=0.4T_H$, corresponding to a partially conducting state of the Dalington pair. The intermediate bath temperature at the steady state is found to be $T_I\approx 0.7T_H$. All other simulation parameters are as given in Sec. 3a.