Thermal transistor and thermometer based on Coulomb-coupled conductors

We study a three-terminal setup consisting of a single-level quantum dot capacitively coupled to a quantum point contact. The point contact connects to a source and drain reservoir while the quantum dot is coupled to a single base reservoir. This setup has been used to implement a noninvasive, nanoscale thermometer for the bath reservoir by detecting the current in the quantum point contact. Here, we demonstrate that the device can also be operated as a thermal transistor where the average (charge and heat) current through the quantum point contact is controlled via the temperature of the base reservoir. We characterize the performances of this device both as a transistor and a thermometer and derive the operating condition maximizing their respective sensitivities. The present analysis is useful for the control of charge and heat flow and high precision thermometry at the nanoscale.

Figure 1

(a) The QPC (beige region) and a single electron quantum dot connected to three terminals. Source and drain reservoirs have chemical potentials ${\mu }_{\text{L}},{\mu }_{\text{R}}$ and temperatures ${\mathrm{\Theta }}_{\text{L}},{\mathrm{\Theta }}_{\text{R}}$. The base reservoir, with temperature $\mathrm{\Theta }$ and electrochemical potential $\mu$, is tunnel coupled to the quantum dot with tunneling rates $W_{\mathrm{on}/\mathrm{off}}$. The capacitance $C$ mediates the dependence of the current through the QPC, $I_{\alpha }$, upon the charge state $\alpha$ of the quantum dot, (b) $\alpha =0$ or (c) $\alpha =1$.

Figure 2

The normalized current $({\langle I\rangle }_{\mathrm{\Theta }}-\langle I_1\rangle )/\mathrm{\Delta }I$ and the normalized differential sensitivity $\left(\right|\varepsilon |/{\mathrm{\Delta }I\left)\right|d\langle I\rangle }_{\mathrm{\Theta }}/d\mathrm{\Theta }|=\xi (\varepsilon /\mathrm{\Theta })$ versus the normalized temperature $\mathrm{\Theta }/\left|\varepsilon \right|$. The blue and the red lines are for the cases $\varepsilon >0$ and $\varepsilon <0$, respectively.

Figure 3

QPC transmission in the regime where the different dot occupations correspond to fully closed and open channels. The requirement to reach this regime is to have $({\mu }_{\text{R}}-U_0)\gg \hslash \omega$ and $(U_1-{\mu }_{\text{L}})\gg \hslash \omega$.

Figure 4

A typical cycle of a metrological process consists of the following steps: (i) The probe (blue rectangle) is initially prepared in some known state. (ii) The probe interacts with some physical process (magenta rectangle) that depends on the true value of the estimation parameter of interest. The interaction is turned off before moving to the next step. (iii) The interaction between the probe and the measuring apparatus is turned on to perform a noiseless projective measurement. When the measurement is done, the interaction is turned off and the measurement outcome is sent to the computer for later processing.

Figure 5

Comparison of analytics of the standard, Eq. (20), and the always-on protocol, Eq. (29), with Monte Carlo (MC) simulations. We ignore the shot noise in the QPC for both protocols. We take $\varepsilon /\mathrm{\Theta }$ to be the optimal value 2.4 for both protocols. In general, if our prior knowledge about $\mathrm{\Theta }$ is very loose and $\varepsilon$ can be far detuned from its optimal value, then the optimal sensitivity shown here can be achieved by adaptive measurements.

Figure 6

The normalized standard deviation of the thermometer against the normalized temperature when $\left|\varepsilon \right|$ is kept fixed. The blue and red lines are plotted according to Eqs. (19) and (28) respectively, where $c=10$ in Eq. (19). We see that for both protocols, the normalized standard deviation $\sqrt{\mathrm{Var}\left(\widehat{\mathrm{\Theta }}\right)}/\mathrm{\Theta }$ scales as $exp\left(\right|\varepsilon |/\mathrm{\Theta })$ at low temperature and $\mathrm{\Theta }/\left|\varepsilon \right|$ at high temperature.