Can single photon excitation of two spatially separated modes lead to a violation of Bell inequality via homodyne measurements

We reconsider the all-optical homodyne-measurement based experimental schemes that aim to reveal Bell nonclassicality of a single photon, often termed `nonlocality'. We focus on the schemes put forward by Tan, Walls and Collett (TWC, 1991) and Hardy (1994). In the light of our previous work the Tan, Walls and Collett setup can be described by a precise local hidden variable model, hence the claimed nonclassicality of this proposal is apparent, whereas the nonclassicality proof proposed by Hardy is impeccable. In this work we resolve the following problem: which feature of the Hardy's approach is crucial for its successful confirmation of nonclassicality. The scheme of Hardy differs from the Tan, Walls and Collett setup in two aspects. (i) It introduces a superposition of a single photon excitation with vacuum as the initial state of one of the input modes of a 50-50 beamsplitter, which creates the superposition state of two separable (exit) modes under investigation. (ii) In the final measurements Hardy's proposal utilises a varying strengths of the local oscillator fields, whereas in the TWC case they are constant. In fact the local oscillators in Hardy's scheme are either on or off (the local setting is specified by the presence or absence of the local auxiliary field). We show that it is the varying strength of the local oscillators, from setting to setting, which is the crucial feature enabling violation of local realism in the Hardy setup, whereas it is not necessary to use initial superposition of a single photon excitation with vacuum as the initial state of the input mode. Neither one needs to operate in the fully on/off detection scheme. Despite the failure of the Tan, Walls and Collett scheme in proving Bell nonclassicality, we show that their scheme can serve as an entanglement indicator.