Leggett-Garg inequalities for a quantum top affected by classical noise

The violation of the Leggett-Garg inequality is studied for a quantum top (with angular momentum J(z) of integer or half-integer size), being driven by classical Gaussian white noise. The form of a longitudinal (J(z)) or a transverse (J(x)) coupling of noise to the angular momentum affects both (i) to what extent the Leggett-Garg inequality is violated and (ii) how this violation is influenced by the size j of the spinning top and direction of a coupling (transverse or longitudinal). We introduce j-independent method, using two-dimensional invariant subspace of the system Hilbert space, which allows us to find out strict analytical solution for a noise-free system and with longitudinal coupling and to extract from the whole dynamics effects purely induced by a noise. It is demonstrated that in the semi-classical limit of a large angular momentum j and for the transverse coupling, the Leggett-Garg inequalities become more strongly violated as compared to the deep quantum regime of small j.