  To obtain Bell statistics from hybrid states composed of finite- and
infinite-dimensional systems, we propose a hybrid measurement scheme, in which
the continuous mode is measured using the generalized pseudospin operators,
while the finite (two)-dimensional system is measured in the usual Pauli basis.
Maximizing the Bell expression with these measurements leads to the violations
of local realism which is referred to as hybrid nonlocality. We demonstrate the
utility of our strategy in a realistic setting of cavity quantum
electrodynamics, where an atom interacts with a single mode of an
electromagnetic field under the Jaynes-Cummings Hamiltonian. We dynamically
compute the quenched averaged value of hybrid nonlocality in imperfect
situations by incorporating disorder in the atom-cavity coupling strength. In
the disordered case, we introduce two kinds of measurement scenarios to
determine the Bell statistics -- in one situation, experimentalists can tune
the optimal settings according to the interaction strength while such
controlled power is absent in the other case. In contrast to the oscillatory
behavior observed in the ordered case, the quenched averaged violation
saturates to a finite value in some parameter regimes in the former case,
thereby highlighting an advantage of disordered systems. We also examine the
connection between Wigner negativity and hybrid nonlocality.
