Preparation-protocol-dependent quantum Mpemba dynamics in a magnetically tunable graphene nanotorus qubit

Abstract: We investigate the quantum Mpemba effect in a graphene nanotorus qubit and show that its strength depends strongly on how the initial state is prepared. We compare two protocols that share the same final Hamiltonian, thermal bath, and Markovian Liouvillian: bare Gibbs preparation, in which the coherent drive is switched on only at the final quench, and driven steady-state preparation, in which the initial states are stationary states of the driven dynamics. Using the distance to the final stationary state and an integrated Mpemba parameter, we find broad regions of strong anomalous relaxation for bare Gibbs preparation, with $M_B$ approaching unity, whereas the driven steady-state protocol almost completely suppresses the effect. A Liouvillian-mode analysis reveals the mechanism: the bare Gibbs protocol typically yields a smaller hot-state weight in the slow active sector than the cold state, $R_s<1$, while the driven steady-state protocol predominantly gives $R_s>1$. Since the final Liouvillian spectrum is identical for both protocols at corresponding parameter points, the contrast originates from preparation-dependent modal weights rather than from changes in the decay-rate hierarchy. These results identify state preparation as a control parameter for anomalous quantum relaxation in a curved graphene qubit.