In this article we explore the requirements for enabling high quality optically detected magnetic resonance (ODMR) spectroscopy in a conventional gradient force optical tweezers using nanodiamonds containing nitrogen-vacancy (NV$^{-}$) centres. We find that modulation of the infrared (1064 nm) trapping laser during spectroscopic measurements substantially improves the ODMR contrast compared with continuous wave trapping. The work is significant as it allows trapping and quantum sensing protocols to be performed in conditions where signal contrast is substantially reduced. We demonstrate the utility of the technique by resolving NV$^{-}$ spin projections within the ODMR spectrum. Manipulating the orientation of the nanodiamond via the trapping laser polarisation, we observe changes in spectral features. Theoretical modelling then allows us to infer the crystallographic orientation of the NV$^{-}$. This is an essential capability for future magnetic sensing applications of optically trapped nanodiamonds.
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