Josephson Current Through a Lateral Quantum Dot Connected to Two Superconducting Leads by Ballistic Contacts

We study the Josephson current -- phase relationship in a chaotic quantum dot coupled to superconductors by ballistic contacts. In this regime, strong proximity effect induces a superconductivity in the quantum dot that leads to a significant modification in the electron density of states and formation of the supercurrent between the leads. The magnitude of the supercurrent depends on the phase difference of superconducting order parameter in the leads and shows anharmonic dependence on the phase difference. We find that when the Thouless energy on the dot exceeds the superconducting energy gap, the second harmonic of the supercurrent becomes comparable in magnitude to the first harmonic. On the technical level, we use the Keldysh formalism in the framework of the circuit theory to compute dependence of the Josephson energy and current on the superconducting phases in the leads. We discuss how these quantities change as a function of the Thouless energy and the superconducting gap in the leads, and discuss anharmonicity of gatemon qubits with quantum dot Josephson junctions.