Quantum thermodynamic uncertainty relation and macroscopic superconducting coherence

Abstract

Stability and efficiency are mutually exclusive in a thermodynamic process, e.g. in a thermal machine. Any effort to reduce the fluctuations of a certain output quantity is necessarily accompanied by an increase of entropy production, therefore lowering its efficiency. This interplay is beautifully captured by the so called Thermodynamic Uncertainty Relations (TURs) which set a lower bound on the relative uncertainty of a current for a given rate of entropy production. Their status in hybrid normal-superconducting (N-S) devices has remained unsettled. We show that, in the subgap regime, departures from the normal quantum TUR are governed by {\it macroscopic} superconducting coherence quantified by the pair amplitude, and that introducing a dephasing probe suppresses this coherence and restores the bound. We further derive a hybrid quantum TUR that is general for two-terminal N-S junctions in the Andreev regime: the inequality is never violated, is saturated only at vanishing current, and is related to the normal quantum bound under the replacement (e to 2e). For N-S quantum dot and Cooper-pair-splitter systems we compute current and noise and show that deviations from the normal bound track the pair amplitude on the central region. The results establish a direct link between superconducting macroscopic coherence and nonequilibrium fluctuations and supply a general bound for the Andreev regime.

Figures (6)

• Figure 1: A central region is coupled to a superconducting (S) lead with tunneling rate $\Gamma_{\rm S}$, and to two normal leads $\eta={\rm L,R}$ with tunneling rates $\Gamma_{\rm \eta}$, chemical potential $\mu_{\rm \eta}$ and kept at temperature $T$. a) The central region corresponds to a quantum dot (with energy level $\epsilon$) and one of the normal leads is considered as a probe coupled to the dot by a tunneling rate $\Gamma_{\rm P}$ and with a chemical potential $\mu_{\rm P}$ that is adjusted so that the current flowing in P is zero. b) The central region is a double quantum dot. The superconducting lead is also coupled to the dots through the nonlocal tunneling rate $\Gamma_{\mathrm{C}}$.
• Figure 2: Fano factor as a function of (a) $\epsilon$ and (b) $\mu_{\rm N}$ and the ratio $\Gamma_{\rm S}/\Gamma_{\rm N}$ . The red area shows the region in which the quantum bound $\mathcal{B_{\rm qu}}$ is broken, and the yellow area corresponds to violations of the classical bound $\mathcal{B_{\rm cl}}$. Parameters (when fixed): $\Gamma_{\rm N} = 0.6 k_BT$, $\mu_{\rm N} = k_BT$ and $\epsilon = 0$.
• Figure 3: The value of $\Gamma_{\rm P}$ for which the decoherence is strong enough that the quantum (blue) and classical (orange) TUR stop being violated as a function of $\epsilon$. Parameters: $\Gamma_{\rm N}=0.6k_B T$, $\Gamma_{\rm S} = \sqrt{\frac{5}{3}}\Gamma_{\rm N}$ and $\mu_{\rm N}=k_B T$.
• Figure 4: Regions of violation of the classical (yellow) and quantum (red) TURs in the CPS system. The plotted quantities correspond to the left-hand side of Eq. \ref{['eq:ine']} where the respective inequality is violated. The top panel displays the dependence on the tunneling rates $\Gamma_{\mathrm{S}}/\Gamma_{\mathrm{N}}$ and $\Gamma_{\mathrm{C}}/\Gamma_{\mathrm{N}}$, while the bottom panels show the dependence on the energy levels $\varepsilon_{\eta}$ for the three representative points in the top panel where the violations are maximal. Simultaneous LAR and CAR processes result in the maximal violation of the TURs. The black dashed line marks the saturation of the quantum bound. Parameters (when fixed): $\Gamma_{\mathrm{N}} = 0.6\, k_{\mathrm{B}}T$, $\varepsilon_{\eta} = 0$, and $\mu_{\mathrm{N}} = k_{\mathrm{B}}T$.
• Figure 5: Regions of violation of the classical (yellow) and quantum (red) TURs in the CPS system. The plotted quantities correspond to the left-hand side of Eq. (1) where the corresponding inequality is violated. The dependence on the energy level $\varepsilon \equiv \varepsilon_L = -\varepsilon_R$ and the tunneling rates $\Gamma_{\mathrm{S}}/\Gamma_{\mathrm{N}} = \Gamma_{\mathrm{C}}/\Gamma_{\mathrm{N}}$ is displayed in the top panel, while the dependence on the normalized chemical potential $\mu_{\mathrm{N}}/(k_{\mathrm{B}}T)$ is shown in the bottom panel. In the top panel, with $\mu_{\mathrm{N}} = k_{\mathrm{B}}T$ fixed, the largest departures from both bounds occur for nearly symmetric couplings, $\Gamma_{\mathrm{S}} = \Gamma_{\mathrm{C}} \approx \Gamma_{\mathrm{N}}$, at $|\varepsilon| \simeq \sqrt{5/12},\Gamma_{\mathrm{N}}$. In the bottom panel, fixing $\Gamma_{\mathrm{S}} = \Gamma_{\mathrm{C}} = \Gamma_{\mathrm{N}}$, the classical TUR reaches a minimum of $-0.3$ around $|\mu_{\mathrm{N}}|/(k_{\mathrm{B}}T) = 3.4$, while the quantum TUR attains $-0.075$ near $|\mu_{\mathrm{N}}|/(k_{\mathrm{B}}T) = 2$ (both at $|\varepsilon| \simeq \sqrt{5/12},\Gamma_{\mathrm{N}}$). Parameters (when fixed): $\Gamma_{\mathrm{N}} = 0.6\,k_{\mathrm{B}}T$, $\mu_{\mathrm{N}} = k_{\mathrm{B}}T$, and $\Gamma_{\mathrm{S}} = \Gamma_{\mathrm{C}} = \Gamma_{\mathrm{N}}$.