Stochastic model of noise for a quantum thermal transistor

Our focus in this investigation lies in developing a noise model for a quantum thermal transistor model inspired by its electronic counterpart, with the primary aim of establishing a platform for constructing analogous models. Previous studies on coupled two-level systems-based thermal transistors were focused on their average energy exchange. In this paper, we shift our attention to exploring the stochastic behavior of such thermal transistors due to the disturbances caused to their environment, such as continuous measurements. In the literature, the master equation for the transistor model is derived using the reduced dynamics method. This way, it masks the study of the stochastic nature of the energy flows in the system due to disturbances to the environment. In this paper, we describe a quantum trajectory under measurement theory whose ensemble average unravels the master equation for a quantum thermal transistor. This allows us to analyze the fluctuations and noise levels in the transistor model with greater detail. Then, we produce a numerical solution for the transistor dynamics based on Euler-Maruyama approximation. This helps to establish a model for the thermal transistor, drawing parallels to the small-signal/noise model in an electronic transistor. We define two parameters, thermal conductance and output thermal resistance, to describe the small signal-like model for the thermal transistor. Through these investigations, we seek to gain insights that can help design advanced heat management devices at the quantum level.

Figure 1

Transistor arrangement with three TLSs as terminals interacting with reservoirs $B_L, B_M$, and $B_R$, each with temperatures $T_L,T_M$, and $T_R$. The interactions of baths with TLSs are shown in solid lines. The three detectors that monitor the baths are also shown. The three energy flows at steady state, $J_L, J_M$, and $J_R$, are marked in the diagram.

Figure 2

A thermal transistor representing noise from baths as $\left(\frac{{\kappa }_L}{\hslash }\right), \left(\frac{{\kappa }_M}{\hslash }\right)$, and $\left(\frac{{\kappa }_R}{\hslash }\right)$, and the fluctuating energy flows as $j_L\left(t\right), j_M\left(t\right)$, and $j_R\left(t\right)$ due to continuous monitoring of baths.

Figure 3

Transistor dynamics from (a) reduced density matrix approach and (b) using the quantum trajectory method.

Figure 4

(a) The stochastic energy flows $J_L\left(t\right), J_M\left(t\right)$, and $J_R\left(t\right)$ from the baths for 100 realizations, keeping $\zeta =0.02$, and (b) average energy flows ${\overline{J}}_L, {\overline{J}}_M$, and ${\overline{J}}_R$.

Figure 5

The average energy flows ${\overline{J}}_{\text{P}}$ (solid lines) at steady state with no monitoring, and the energy flows at a steady state from the stochastic model $J_{\text{P}}$ (dotted lines) considering 100 realizations, and $\zeta =0.02$ varying with the control bath temperature $T_M$. Note that the variation of $J_L$ is approximately linear up to $T_M=0.15T$, which prompts us to use this gradient to define a new parameter for the model as thermal conductance(c).

Figure 6

The amplification factor variation with the original model and changing the noise levels in three cases at a fixed ${\kappa }_{\text{P}}=0.01$, by varying $\zeta$ at 0, 0.02, and 0.09. The amplification factor usually remains a constant within the selected transistor operating range $0.05T\le T_M\le 0.1T$ for lower levels of noise and then it starts to decrease for $0.1T<T_M\le 0.15T$ with a slightly higher rate when there is conditioning.

Figure 7

Thermal efficiency variation with the original model and changing the noise levels in three cases at a fixed ${\kappa }_{\text{P}}=0.01$, by varying $\zeta$ at 0, 0.02, and 0.09. This efficiency factor does not have significant changes in the selected conditioning strengths. It has an increase over our selected transistor operating region $0.05T\le T_M\le 0.15T$, varying between 10 and 130, diverging at $0.16T$ when $J_M$ hits a minimum. Then, it starts to decrease as $T_M$ gets closer to $T_L$.

Figure 8

Transistor rectification factor (a) between terminals $M$ and $L$ ($R_{M\to L}$) and (b) between terminals $M$ and $R$ ($R_{M\to R}$). We observe that there is not perfect but some rectification between the input terminal $M$ and $L/R$ terminals for the transistor operating region. This behavior indicates the challenging nature of achieving a quantum thermal transistor with characteristics exactly similar to those found in NPN and PNP diode junctions. Thus, the thermal transistor is not exactly an electronic transistor but it can have similar characteristics showing some rectification.

Figure 9

Thermal sensitivity variation is visualized by changing the control bath temperature. We notice that ${\eta }_{TM}^L$ and ${\eta }_{TM}^L$ remain nearly constant within the region $0.05T\le T_M\le 0.15T$ and ${\eta }_{TM}^M\approx 0$. These parameter variations are helpful for establishing a small-signal model in this region.

Figure 10

The diagram expresses the variation of the thermal conductance $c$ between hot and cold baths with the input temperature $T_M$, keeping temperatures $T_L$ and $T_R$ constant when $\zeta =0.02$, and ${\kappa }_{\text{P}}=0.01$.

Figure 11

The diagram expresses the variation in thermal resistance $r_e$ between hot $T_L$ and cold $T_R$ baths at $\zeta =0.02$ and ${\kappa }_{\mathrm{P}}=0.01$, with constant input temperatures $T_M=0.05T,0.1T,0.15T$. Note that as we increase the control temperature $T_M$ the resistance for the heat flow reduces further.

Figure 12

The diagram expresses the parameters $c$ and $r_e$. This is the small-signal equivalent of the quantum thermal transistor.

Figure 13

The visualization of the detection process.