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

Abstract

We present an exact analytic study of unitary coherence dynamics in a minimal two qubit superconducting system. By deriving the full time evolution operator and propagating Bell state initial conditions, we obtain closed form time dependent pure state density matrices and an explicit analytic expression for the $C_{l_1}$ norm of coherence. Two of the Bell states are shown to be invariant under the model dynamics with constant coherence, while the other two exhibit controlled, parameter dependent coherence oscillations. The oscillatory behaviour is governed by two distinct frequency scales that map directly onto the circuit coupling and tunnelling parameters, allowing 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 optimisation and provide a clear foundation for incorporating dissipation and for experimental validation.