Quantum fluctuations in Josephson junctions: a path integral approach

The mean-field dynamics of a bosonic Josephson junction can be studied comprehensively by means of a Lagrangian/action expressed in terms of the population imbalance z and the relative phase ϕ of the coupled condensates. When one dynamical variable dominates the other, an effective single-variable Lagrangian can be obtained by integrating out the less relevant collective coordinate. Extending previous results, we perform this reduction to derive a phase-only action (relevant for superconducting qubits in quantum computing) and an imbalance-only action. This procedure allows us to analytically derive, to first order in perturbation theory, the quantum effective action by means of a covariant background field method. The quantum-corrected Josephson frequency is also calculated in both the imbalance-only and phase-only frameworks and the results are compared.