Critical Unstable Qubits: an Application to $B^0\bar{B}^0$-Meson System

Abstract: We extend our previous work on a novel class of unstable qubits which we have identified recently and called them Critical Unstable Qubits (CUQs). The characteristic property of CUQs is that the energy-level and decay-width vectors, ${\bf E}$ and ${\bf \Gamma}$, are orthogonal to one another, and the key parameter $r = |{\bf \Gamma}|/|2{\bf E}|$ is less than 1. Most remarkably, CUQs exhibit two atypical behaviours: (i) they display coherence-decoherence oscillations in a co-decaying frame of the system described by a unit Bloch vector ${\bf b}$, and (ii) the unit Bloch vector ${\bf b}$ describing a pure CUQ sweeps out unequal areas during equal intervals of time, while rotating about the vector ${\bf E}$. The latter anharmonic phenomenon emerges beyond the usual oscillatory pattern due to the energy-level difference of the two-level quantum system, which governs an ordinary qubit. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of non-sinusoidal oscillation of a CUQ. We apply the results of our formalism to the $B^0\bar{B}^0$-meson system and derive, for the first time, generic upper limits on these new observables.