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To obtain Bell statistics from hybrid states composed of finite- andinfinite-dimensional systems, we propose a hybrid measurement scheme, in whichthe 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 violationsof local realism which is referred to as hybrid nonlocality. We demonstrate theutility of our strategy in a realistic setting of cavity quantumelectrodynamics, where an atom interacts with a single mode of anelectromagnetic field under the Jaynes-Cummings Hamiltonian. We dynamicallycompute the quenched averaged value of hybrid nonlocality in imperfectsituations by incorporating disorder in the atom-cavity coupling strength. Inthe disordered case, we introduce two kinds of measurement scenarios todetermine the Bell statistics -- in one situation, experimentalists can tunethe optimal settings according to the interaction strength while suchcontrolled power is absent in the other case. In contrast to the oscillatorybehavior observed in the ordered case, the quenched averaged violationsaturates to a finite value in some parameter regimes in the former case,thereby highlighting an advantage of disordered systems. We also examine theconnection between Wigner negativity and hybrid nonlocality.

Hybrid nonlocality via atom photon interactions with and without impurities