Dynamics and phonon-induced decoherence of Andreev level qubit

We present a detailed theory for the Andreev level qubit, a system consisting of a highly transmissive quantum point contact embedded in a superconducting loop. The two-level Hamiltonian for Andreev levels interacting with quantum phase fluctuations is derived by using a path integral method. We also derive a kinetic equation describing qubit decoherence due to interaction of the Andreev levels with acoustic phonons. The collision terms are nonlinear due to the fermionic nature of the Andreev states, leading to slow nonexponential relaxation and dephasing of the qubit at temperatures smaller than the qubit level spacing.