  A generalized Bose-Hubbard model in a two-mode approximation is applied to
study the rotational dynamics of a direct-current atomtronic quantum
interference device. Modified values of on-site interaction and pair-tunneling
parameters of the Hamiltonian, derived from the small-oscillation periods of
the Josephson modes, are shown to provide an excellent agreement to the
Gross-Pitaevskii simulation results for the whole rotational frequency range,
reaching also the critical values of imbalance and current. This amounts to a
full validation of the semiclassical approximation of the modified Hamiltonian,
whose quantization is employed to investigate the quantum features of the
stationary states. Focusing on the frequency interval where the potential
energy presents two minima, it is shown that the central frequency, at which
such minima are symmetric, yields an atom number parity-protected qubit with a
maximum entanglement of both persistent-current states, similar to those of
superconducting circuits threaded by a half-quantum of applied flux. Such a
parity protection scheme survives within a small interval around the central
frequency, which sets the minimum rotational frequency precision that should be
required to implement the qubit. It is found that such a maximum admissible
error in the frequency determination turns out to be inversely proportional to
the qubit quality factor that measures the gap between the qubit energy levels
and the following levels. It is shown that the chemical potential or condensate
particle number could be employed as suitable control parameters to achieve the
best trade-off between such qubit characteristics.
