Observing the Non-Locality of a Single Massive Particle

The state of a single particle after it passes through a beam splitter (BS) is expressed as a superposition of two states, each of which corresponds to the particle traveling along a path, e.g., |left>+|right>.However, if we represent the same state in the number basis it is written as |01>+|10>, which looks like one of the maximally entangled Bell states.Then, does the single particle state after a BS exhibit the quantum non-locality?Can the Bell inequalities be violated?Particularly for single massive particles, the Bell test seems tricky because we cannot rotate a measurement basis to the form |0>+|1> due to the particle-number superselection rule.Here, we show a scheme to convert the mode entanglement of a massive particle to a Bell-testable form.The conversion can be done using a BEC as a resource, but we also consider the case where no resource is provided initially.