Quantum Thermal Machine as a Thermometer

We propose the use of a quantum thermal machine for low-temperature thermometry. A hot thermal reservoir coupled to the machine allows for simultaneously cooling the sample while determining its temperature without knowing the model-dependent coupling constants. In its most simple form, the proposed scheme works for all thermal machines that perform at Otto efficiency and can reach Carnot efficiency. We consider a circuit QED implementation that allows for precise thermometry down to $\sim 15\mathrm{mK}$ with realistic parameters. Based on the quantum Fisher information, this is close to the optimal achievable performance. This implementation demonstrates that our proposal is particularly promising in systems where thermalization between different components of an experimental setup cannot be guaranteed.

Figure 1

Sketch of the thermometer. In order to measure the temperature of the cold bath, a quantum thermal machine is operated as a refrigerator, inducing a heat current from the cold bath to the hot bath. This requires energy from the work source $W$. The hot temperature is then increased until the machine reaches the Carnot point where its power consumption and the heat flows vanish (i.e., $P=J_c=J_h=0$). For machines that perform with a well-known efficiency (e.g., the Otto efficiency $\eta =1-{\mathrm{\Omega }}_c/{\mathrm{\Omega }}_h$), the cold temperature can then be deduced from the hot temperature through the relation $\eta ={\eta }_C=1-T_c/T_h$, without the need of knowing any model-dependent coupling constants. In this way, a low precision measurement of a hot temperature is converted into a high precision measurement of a low temperature. Operating the thermal machine as a refrigerator avoids any heating of the cold bath.

Figure 2

Circuit QED implementation of the thermometer. Two harmonic oscillators with frequencies ${\mathrm{\Omega }}_h$ and ${\mathrm{\Omega }}_c$ are coupled to thermal baths at temperatures $T_c$ and $T_h$, respectively, and to each other through a Josephson junction. The external bias voltage $V$ ensures that the cold bath is being cooled while determining $T_c$.

Figure 3

Performance of a circuit QED implementation. (a) Charge current as a function of the hot temperature $T_h$ for fixed $T_c=15\mathrm{mK}$. The refrigerator is initiated at $T_h$ below the Carnot point (dotted circle), leading to cooling of the cold bath. $T_h$ is then increased until the current vanishes (solid circle). This point can only be determined up to a certain error $\mathrm{\Delta }I$ that induces an uncertainty in the final $T_h$ (shaded area). (b) Error in the temperature estimation, Eq. (2). Measurement errors of $\mathrm{\Delta }I=0.3\mathrm{pA}$ and $\mathrm{\Delta }{T}_h=10\mathrm{mK}$ lead to $\mathrm{\Delta }{T}_c<2\mathrm{mK}$ down to temperatures of 15 mK. Blue (solid) lines are numeric solutions; green (dashed) lines are obtained analytically using a simplified model [49]. Parameters: ${\mathrm{\Omega }}_h=2\pi \times 8.5\mathrm{GHz}$, ${\mathrm{\Omega }}_c=2\pi \times 1\mathrm{GHz}$, ${\kappa }_h={\kappa }_c=2\pi \times 0.06\mathrm{GHz}$, $E_J=2\pi \times 0.2\mathrm{GHz}$, ${\lambda }_h={\lambda }_c=0.3$.

Figure 4

Single-shot precision from a current measurement (blue solid) and from the QFI (blue dashed) at the Carnot point, for the simple model with oscillator frequencies and couplings given in the caption of Fig. 3, and fitting parameter $g=E_J/8$.