Bell-inequality test for spatial-mode entanglement of a single massive particle

Abstract

Experiments violating Bell inequalities have formed our belief that the world at its smallest is genuinely non-local. While many non-locality experiments use the first quantised picture, the physics of fields of indistinguishable particles is captured most conveniently by second quantisation. This implies the possibility of non-local correlations, such as entanglement, between modes of the field. In this paper, we propose an experimental scheme that tests the theoretically predicted entanglement between modes in space occupied by massive bosons. A successful test of this scheme would not only indicate that mode entanglement is as genuine as particle entanglement, despite the particle number superselection rule, but would also prove that this superselection rule can be overcome.

Sub–shot-noise sensitivities without entanglement

“…Recently, particle-number nonconserving superpositions have attracted renewed interest in the context of cold atoms. It was suggested that when the two modes are coupled to a third one containing a Bose-Einstein condensate that forms a "coherent" reservoir, such superpositions might become possible and might have applications for quantum communication [33,34]. Below we will argue that this kind of "naturally occurring" entanglement is very useful for precision measurements.…”

“…For laser light in a coherent state with N 1 or even the most squeezed states currently available [54,55], the fluctuations of N still satisfy σ (N )/N 1, and the average photon number is therefore representative of the photon number in any realization. A state of the form (33) with N K/2 maximizes the photon-number fluctuations, however, and obviously the average value of N is never realized (only N = 0 or N = K are). Nevertheless, the scaling with N is highly relevant practically, as it corresponds to the mean energy in the state, which is indeed what makes producing the state costly.…”

Bell inequality for pairs of particle-number-superselection-rule restricted states

“…However, to reliably confirm spatial-mode entanglement of a single massive particle in an experiment, one would have to ensure that no additional * l.heaney1@physics.ox.ac.uk entanglement entered the system via the particle reservoir (see [23,24] for further discussion of this point) and even if this were guaranteed, such a coupling to the reservoir is likely to be difficult to implement under realistic conditions. Moreover, both in the photon experiment [15] and in theoretical proposals [9,21], a local postselection of the measurement outcomes was required to see a violation of the CHSH inequality, which meant that these tests only strictly probed for entanglement (i.e., the Bell inequality was used as an entanglement witness [25]), and not the more stringent property of nonlocality.…”

Spatial Entanglement of Formation in Thermal States of One or Two Identical Non-Interacting Massive Bosons

“…However, schemes to detect single particle and/or spatial entanglement unambiguously continue to be devised and attempted, e.g. [21][22][23][24][25], but are so far probably not free of loopholes.…”