Cross correlations between charge noise and critical-current fluctuations in a four-level tunable Josephson system

A protocol to test cross correlations between charge and critical-current noise in the small superconducting contacts of an asymmetric Cooper-pair transistor coupled to a phase qubit is presented. The superconducting circuit behaves as a tunable four-level quantum system that can be prepared in two different configurations where cross-correlation terms are, respectively, absent or present and therefore, in principle, detectable. The measurements of the cross correlations are performed either through the escape probability of the dc superconducting quantum interference device or a quantum-state tomography of a few elements of the reduced density matrix of the four-level quantum system.