The paper by C. W. J. Beenakker explores the Josephson effect in a junction coupled to a gapless electron reservoir in the normal state. The author extends the scattering theory of the Josephson effect to include this coupling, leading to a reduction in the supercurrent carried by an Andreev level at zero temperature. Specifically, the reduction factor is given by $(2/\pi) \arctan(2 \varepsilon_A \tau / \hbar)$, where $\varepsilon_A$ is the energy of the Andreev level and $\tau$ is the quasiparticle escape rate from the reservoir.
The study is motivated by the thermodynamic properties of quantum dots coupled to superconductors, where the Josephson effect can be affected by dissipative electromagnetic environments. The main contribution is the derivation of closed-form expressions for the supercurrent-phase relationship in the presence of a spatially uniform coupling rate to the electron reservoir. This system is a simple example of a "non-Hermitian Josephson junction," a topic of current interest.
The paper discusses three applications: a quantum dot Josephson junction, a point contact Josephson junction, and a long SNS junction. For each application, the author provides detailed mathematical derivations and plots to illustrate the effects of the electron reservoir on the supercurrent. The results are compared with recent work on non-Hermitian Josephson junctions, particularly the zero-temperature current-phase relation derived by Shen, Lu, Lado, and Trif, which aligns with the findings presented in this paper.
In conclusion, the coupling to a gapless electron reservoir reduces the supercurrent through a Josephson junction, with the reduction factor depending on the energy of the Andreev level and the quasiparticle escape rate. This effect is significant in both weakly coupled quantum dot junctions and short point contacts, while a more complex dependence is observed in long junctions where states above the gap contribute to the supercurrent.The paper by C. W. J. Beenakker explores the Josephson effect in a junction coupled to a gapless electron reservoir in the normal state. The author extends the scattering theory of the Josephson effect to include this coupling, leading to a reduction in the supercurrent carried by an Andreev level at zero temperature. Specifically, the reduction factor is given by $(2/\pi) \arctan(2 \varepsilon_A \tau / \hbar)$, where $\varepsilon_A$ is the energy of the Andreev level and $\tau$ is the quasiparticle escape rate from the reservoir.
The study is motivated by the thermodynamic properties of quantum dots coupled to superconductors, where the Josephson effect can be affected by dissipative electromagnetic environments. The main contribution is the derivation of closed-form expressions for the supercurrent-phase relationship in the presence of a spatially uniform coupling rate to the electron reservoir. This system is a simple example of a "non-Hermitian Josephson junction," a topic of current interest.
The paper discusses three applications: a quantum dot Josephson junction, a point contact Josephson junction, and a long SNS junction. For each application, the author provides detailed mathematical derivations and plots to illustrate the effects of the electron reservoir on the supercurrent. The results are compared with recent work on non-Hermitian Josephson junctions, particularly the zero-temperature current-phase relation derived by Shen, Lu, Lado, and Trif, which aligns with the findings presented in this paper.
In conclusion, the coupling to a gapless electron reservoir reduces the supercurrent through a Josephson junction, with the reduction factor depending on the energy of the Andreev level and the quasiparticle escape rate. This effect is significant in both weakly coupled quantum dot junctions and short point contacts, while a more complex dependence is observed in long junctions where states above the gap contribute to the supercurrent.