Temperature and voltage measurement in quantum systems far from equilibrium

We show that a local measurement of temperature and voltage for a quantum system in steady state, arbitrarily far from equilibrium, with arbitrary interactions within the system, is unique when it exists. This is interpreted as a consequence of the second law of thermodynamics. We further derive a necessary and sufficient condition for the existence of a solution. In this regard, we find that a positive temperature solution exists whenever there is no net population inversion. However, when there is a net population inversion, we may characterize the system with a unique negative temperature. Voltage and temperature measurements are treated on an equal footing: They are simultaneously measured in a noninvasive manner, via a weakly coupled thermoelectric probe, defined by requiring vanishing charge and heat dissipation into the probe. Our results strongly suggest that a local temperature measurement without a simultaneous local voltage measurement, or vice versa, is a misleading characterization of the state of a nonequilibrium quantum electron system. These results provide a firm mathematical foundation for voltage and temperature measurements far from equilibrium.

Figure 1

Illustration of the measurement setup: The quantum conductor represented below is in a nonequilibrium steady state. A weakly coupled scanning tunneling probe noninvasively measures the local voltage $\left({\mu }_p\right)$ and local temperature $\left(T_p\right)$ simultaneously: By requiring both a vanishing net charge exchange $\left(I_p^{\left(0\right)}=0\right)$ and a vanishing net heat exchange $\left(I_p^{\left(1\right)}=0\right)$ with the system. The nonequilibrium steady state has been prepared, in this particular illustration, via the electrical and thermal bias of the strongly coupled reservoirs (1 and 2). The measurement method itself is completely general and does not depend upon (a) how such a nonequilibrium steady state is prepared, (b) how far from equilibrium the quantum electron system is driven, and (c) the nature of interactions within that system.

Figure 2

Illustration of Lemma 1: The contour $PQ$ shown in magenta cuts the number current contour $I_p^{\left(0\right)}=0$ (or any $I_p^{\left(0\right)}=\text{const.})$ exactly once. The contour line from $P$ to $Q$ is at a constant temperature $\left(T_p=\text{const.}\right)$, and illustrates the Clausius statement: The number current is monotonically decreasing along $PQ$. The system and bias conditions are detailed in Sec. 5d.

Figure 3

Illustration of Lemma 2: The contour $PQ$ shown in magenta cuts $I_p^{\left(1\right)}=0$ (or any $I_p^{\left(1\right)}=\text{const.})$ exactly once. Contour $PQ$ is defined along constant voltage ${\mu }_p=\text{const.}$, and illustrates the Clausius statement: The heat current is monotonically decreasing along $PQ$. The system and bias conditions are detailed in Sec. 5d.

Figure 4

Left panel: Illustration of Theorem 2 for positive temperatures. The contour $PQ$ along $I_p^{\left(0\right)}=0$ (shown in blue) cuts the contour $I_p^{\left(1\right)}=0$ (shown in red) exactly once. Contour $PQ$ illustrates a certain statement of the second law of thermodynamics: The heat current is monotonically decreasing along $PQ$ (thus implying uniqueness). Right panel: The local spectrum sampled by the probe $\overline{A}\left(\omega \right)$ (black), the nonequilibrium distribution function $f_s\left(\omega \right)$ (red), and the probe Fermi-Dirac distribution $f_p\left(\omega \right)$ (blue) corresponding to the unique solution in the left panel. The resonances in the spectrum $\overline{A}\left(\omega \right)$ correspond to the eigenstates of the closed two-level Hamiltonian (see Sec. 5d) ${\epsilon }_{\pm }=\pm 1$ shown in magenta. The Fermi-Dirac distribution is monotonically decreasing with energy, and corresponds to a situation with positive temperature (no net population inversion). The necessary and sufficient condition for the existence of a positive temperature solution is stated in Theorem 3.

Figure 5

Left panel: Illustration of Theorem 2 for negative temperatures. The contour $PQ$ along $I_p^{\left(0\right)}=0$ (shown in blue) cuts the contour $I_p^{\left(1\right)}=0$ (shown in red) exactly once. Contour $PQ$ illustrates a certain statement of the second law of thermodynamics: The heat current is monotonically decreasing along $PQ$ (thus implying uniqueness). Right panel: The local spectrum sampled by the probe $\overline{A}\left(\omega \right)$ (black, and nearly unchanged from Fig. 4), the nonequilibrium distribution function $f_s\left(\omega \right)$ (red), and the probe Fermi-Dirac distribution $f_p\left(\omega \right)$ (blue) which corresponds to the unique solution (shown in the left panel). The resonances in the spectrum $\overline{A}\left(\omega \right)$ correspond to the eigenstates of the closed two-level Hamiltonian (see Sec. 5d) ${\epsilon }_{\pm }=\pm 1$ shown in magenta. The system has a net population inversion, satisfying the conditions of Corollary 3.1, and the probe Fermi-Dirac distribution is monotonically increasing with energy, corresponding to a negative temperature.

Figure 6

Schematic diagram of a two-level system coupled to two electron reservoirs under bias. (a) Bias condition not leading to population inversion. (b) Bias condition leading to population inversion due to direct injection into excited state.