Tolerance in the Ramsey interference of a trapped nanodiamond

The scheme recently proposed in [M. Scala et al., Phys Rev Lett 111, 180403 (2013)], where a gravity-dependent phase shift is induced on the spin of a nitrogen-vacancy (NV) center in a trapped nanodiamond by the interaction between its magnetic moment and the quantized motion of the particle, provides a way to detect spatial quantum superpositions by means of spin measurements only. Here, the effect of unwanted coupling with other motional degrees of freedom is considered and we show that it does not affect the validity of the scheme. Both this coupling and the additional error source due to misalignment between the quantization axis of the NV center spin and the trapping axis are shown not to change the qualitative behavior of the system, so that a proof-of- principle experiment can be neatly performed. Our analysis, which shows that the scheme retains the important features of not requiring ground state cooling and of being resistant to thermal fluctuations, can be useful for the several schemes which have been proposed recently for testing macroscopic superpositions in trapped microsystems.