Thermodynamic uncertainty relations in superconducting junctions

Abstract

Quantum conductors attached to metallic reservoirs have been demonstrated to overcome the thermodynamic uncertainty relation (TUR), a trade-off relation between the amount of dissipation and the absence of charge and heat current fluctuations. Here, we report large TUR violations when superconducting reservoirs replace metallic ones. The coexistence of different transport processes, namely (multiple) Andreev reflection, where electrons and their retro-reflected holes create Cooper pairs, in addition to the normal quasiparticle transport is identified as the source for such TUR breakdowns. The large TUR violation is a remarkable advantage for building low dissipative and highly stable quantum thermal machines.

Figures (2)

• Figure 1: (a) Single-QP tunneling in which a QP tunnels into the empty density of states of the superconductor transferring one electron. The applied voltage has to exceed the SC gap, $V \geq \Delta$. (b) AR in which an electron is reflected as a hole inside the superconducting gap transferring two electrons. The voltage is arbitrary, $V\geq 0$. (c) QP tunneling between two SCs at voltages above $2\Delta$. (d) AR from a voltage biased superconductor to another superconductor transferring two electrons. The voltage has to exceed the SC gap, $V \geq \Delta$. (e) First MAR, where an incoming electron is reflected as a hole and that hole is retro-reflected as another electron, transferring a total of three electrons. The voltage has to exceed $V \geq 2\Delta/3$. (f) Higher order MAR involves a process that transfers four electrons with onset voltage $V \geq 2\Delta/4$. The voltage thresholds correspond to the case of zero temperature.
• Figure 2: (a) Critical transmission $\tau_c$ for breaking TUR for a NS junction as a function of the bias voltage $V/\Delta$ and the temperature $k_{\rm B}T/\Delta$. For areas with no color, TUR is not broken. (b) The minimal TUR breaking coefficient $\mathcal{F}_{\rm min}$ as a function of the dimensionless temperature $k_{\rm B}T/\Delta$ and the transmission $\tau$ for a NS junction. A negative coefficient indicates broken TUR. The black solid line indicates the phase boundary. The dotted line indicates the approximate phase boundary from the coefficient $C_{\rm neq}$ [see Eq. (\ref{['CneqNS']})]. The inset shows a zoom-in into the region where TUR is broken. (c) Same as in (a) for an SS junction. Dotted lines indicate special voltages of onsets of different transport processes, namely $V/\Delta = 1, 2/3, 1/2$ for AR, MAR and 4th order MAR. Grey area shows a parameter realm that was not analyzed. (d) Same as in (b) for an SS junction with the phase boundary as a solid line. The inset shows a zoom-in into the region where TUR is broken.