Josephson effect in a junction coupled to an electron reservoir Open Access

The thermodynamic properties of a quantum dot are affected by the coupling to a superconductor. The Josephson effect is a striking example, and a current will flow through the quantum dot in equilibrium if the quantum dot forms a weak link between two superconductors.¹ This supercurrent $I(ϕ)$ depends periodically on the phase difference $ϕ$ of the pair potential in the superconductors. A dissipative electromagnetic environment may degrade the supercurrent via phase fluctuations.^2–4 Here, we investigate an altogether different decay mechanism, the coupling of the quantum dot to a gapless electron reservoir (see Fig. 1).

The reservoir is a source of dephasing because quasiparticles that enter it from the quantum dot return without any phase coherence. Such a dephasing mechanism for persistent currents was introduced by Büttiker⁵ and applied to superconducting circuits by several authors.^6–13 Our main advance is that we obtain closed-form expressions for the supercurrent–phase relationship, for the case of a spatially uniform coupling rate to the electron reservoir.

Because the effective Hamiltonian of the quantum dot becomes non-Hermitian when we open it up to the reservoir, such a system is a simple example of a “non-Hermitian Josephson junction,” a topic of current interest.^14–17 Our explicit expressions support Ref. 17 in the debate over the proper generalization of the current–phase relationship $I∝dE/dϕ$ to complex eigenvalues E of the effective Hamiltonian.

We open up the Josephson junction by weakly coupling it to a gapless electron reservoir in the normal state. Quasiparticles enter the reservoir at a rate $1/τ≡2γ/ℏ$⁠, assumed to be spatially uniform in the junction region. This can be modeled by coupling to the reservoir via a spatially extended tunnel barrier.^20,21 The reservoir is closed, and it does not drain any current, but particles that enter it return to the junction without any phase coherence, so they no longer contribute to the supercurrent.

The full scattering matrix of the normal region, including the scattering channels to the electron reservoir, is unitary, with $SN(ε+iγ)$ a sub-unitary submatrix for the scattering channels to the superconductors. Similarly, the Andreev reflection matrix $RA(ε+iγ)$ is now sub-unitary also for $ε<Δ0$⁠.

As a first application, we consider the case that the two superconductors are coupled via a quantum dot containing a single resonant level, at energy $εR$ relative to the Fermi level. We ignore Coulomb blockade effects, which will be less significant in the open system.

The second application is a point contact junction of length L short compared to the superconducting coherence length $ξ0=ℏvF/Δ0$⁠. In this short-junction regime, the energy dependence of $SN$ can be neglected relative to the energy dependence of $RA$⁠, and we may evaluate $SN$ at the Fermi level. We assume that time reversal symmetry is preserved; hence, $σySN∗σy=SN†$⁠.

For the third and final application, we consider a superconductor–normal-metal–superconductor (SNS) junction of length $L≫ξ0$⁠. Both states above and below $Δ0$ then contribute to the supercurrent.

The coupling of the Josephson junction to the electron reservoir pushes the poles of its scattering matrix into the lower half of the complex plane, down to $E=±εA−iγ$⁠. These poles can be considered as the complex eigenvalues of an effective non-Hermitian Hamiltonian $H$⁠. The open Josephson junction thus provides a physical realization of the non-Hermitian Josephson effect studied in Refs. 14–17. Different generalizations have been proposed for the relation $I∝dE/dϕ$ when E is complex. Let us compare these with our findings.

In summary, we have calculated how the coupling to a gapless electron reservoir in the normal state reduces the supercurrent through a Josephson junction. A simple answer is obtained for the model where the escape rate $1/τ≡2γ/ℏ$ of quasiparticles into the reservoir is spatially uniform. At zero temperature, the reduction factor for a given Andreev level equals $(2/π)arctan(2εAτ/ℏ)$⁠. This applies to a weakly coupled quantum dot Josephson junction, or to a short point contact (length L smaller than the coherence length $ξ0$⁠). A more complicated $τ$-dependence is obtained in a long junction (⁠ $L≫ξ0$⁠), when also states above the gap contribute to the supercurrent.

I have benefited from discussions with A. R. Akhmerov, Yu. V. Nazarov, P.-X. Shen, and T. Vakhtel. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Advanced Grant No. 832256).

The author has no conflicts to disclose.

C. W. J. Beenakker: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.