Analytic $C_{\ell_1}$ norm of Coherence Evolution for Bell States under a Two-Qubit Superconducting Hamiltonian

Abstract

We present the first exact analytic study of unitary coherence dynamics in a minimal two qubit superconducting system with zz-coupling and transverse Josephson tunneling. By deriving the full time evolution operator without approximations and propagating all four Bell state initial conditions, we obtain novel closed form time dependent pure state density matrices and an explicit analytic expression for the C_(l_1 ) norm of coherence that, to our knowledge, has not been previously reported in the literature. Two of the Bell states are shown to be invariant under the model dynamics with constant coherence, whereas the other two exhibit controlled, parameter dependent coherence oscillations. The oscillatory behavior is governed by two distinct frequency scales that directly map onto the circuit coupling and tunneling parameters, allowing for predictable tuning of amplitude and periodicity. Numerical visualizations clarify operating regimes for transient enhancement or suppression of coherence. These results deliver compact, analytically tractable tools for parameter optimization and provide a clear foundation for incorporating dissipation and experimental validation.