Non-Stationary Decoherence in Superconducting Qubits: Memory Multi-Fractional Brownian Motion and a Time-Dependent Quantum Brownian Motion Extension

Building upon our prior work [1], we present a unified stochastic drift model (SdM) for superconducting charge qubits based on memory multi-fractional Brownian motion (mmFBM). The classical sector employs a time-dependent Hurst exponent H(t) and adaptive memory kernel K(t,s), capturing non-stationary 1/f^beta noise and long-range temporal correlations inaccessible to conventional models. The quantum extension is formulated via a time-dependent Caldeira--Leggett environment with spectral density J(omega;t) = eta(t) omega_c^{1-s(t)} omega^{s(t)} exp-omega/omega_c, where s(t) = 2H(t)-1, consistently reproducing beta(t) = 2H(t)-1. Four central results emerge: (1) relaxation and noise amplitudes act independently on energy decay; (2) time-varying H(t) matches experimental 1/f spectra more accurately than any constant exponent; (3) adaptive kernel dynamics preserve correlations without artificial damping; and (4) simulations predict coherence times T1 5.00 x 10⁶ ns, T2 4.18 x 10⁵ ns consistent with theory when charge noise dominates. The qubit exhibits stretched-exponential Ramsey and echo decay, non-Markovian dephasing, and a temperature-driven quantum-to-classical crossover. We derive the effective time-local Lindblad master equation, establish the classical mmFBM limit at high temperatures, and provide experimentally testable scaling relations. The non-exponential decay patterns reveal fundamental limitations of Markovian approaches, and the framework guides the design of noise-resilient qubit architectures.