  Quantum dots attached to BCS superconducting leads exhibit a $0-\pi$ impurity
quantum phase transition, which can be experimentally controlled either by the
gate voltage or by the superconducting phase difference. For the pertinent
superconducting single-impurity Anderson model, we newly present two simple
analytical formulae describing the position of the phase boundary in parameter
space for the weakly correlated and Kondo regime, respectively. Furthermore, we
show that the two-level approximation provides an excellent description of the
low temperature physics of superconducting quantum dots near the phase
transition. We discuss reliability and mutual agreement of available finite
temperature numerical methods (Numerical Renormalization Group and Quantum
Monte Carlo) and suggest a novel approach for efficient determination of the
quantum phase boundary from measured finite temperature data. Our results
enable fast and efficient, yet reliable characterization and design of such
nanoscopic tunable Josephson junction devices.
