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We show violations of Leggett-Garg inequalities to be possible forsingle-mode cat-states evolving dynamically in the presence of a nonlinearquantum interaction arising from, for instance, a Kerr medium. In order toprove the results, we derive a generalised version of the Leggett-Garginequality involving different cat-states at different times. The violationsdemonstrate failure of the premise of macro-realism as defined by Leggett andGarg, provided extra assumptions associated with experimental tests are valid.With the additional assumption of stationarity, violations of the Leggett-Garginequality are predicted for the multi-component cat-states observed in theBose-Einstein condensate and superconducting circuit experiments of Greiner etal. (Nature 419, 51, (2002)) and Kirchmair et al. (Nature, 495, 205 (2013)).The violations demonstrate a mesoscopic quantum coherence, by negating that thesystem can be in a classical mixture of mesoscopically distinct coherentstates. Higher orders of nonlinearity are also studied and shown to give strongviolation of Leggett-Garg inequalities.

Leggett-Garg tests of macro-realism for dynamical cat-states evolving in a nonlinear medium