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We demonstrate the trajectory measurement of the single neutral atomsdeterministically using a high-finesse optical micro-cavity. Single atomstrongly couples to the high-order transverse vacuum TEM_{10} mode, instead ofthe usual TEM_{00} mode, and the parameter of the system is (g_{10},\kappa,\gamma )=2\pi \times (20.5,2.6,2.6)MHz. The atoms simply fall down freely fromthe magneto-optic trap into the cavity modes and the trajectories of the singleatoms are linear. The transmission spectrums of atoms passing through the TEM10mode are detected by a single photon counting modules and well fitted. Thanksto the tilted cavity transverse TEM10 mode, which is inclined to the verticaldirection about 45 degrees and it helps us, for the first time, to eliminatethe degenerate trajectory of the single atom falling through the cavity and getthe unique atom trajectory. Atom position with high precision of 0.1{\mu}m inthe off-axis direction (axis y) is obtained, and the spatial resolution of5.6{\mu}m is achieved in time of 10{\mu}s along the vertical direction (axisx). The average velocity of the atoms is also measured from the atom transits,which determines the temperature of the atoms in magneto-optic trap, 186{\mu}K{\pm} 19{\mu}K.

Elimination of degenerate trajectory of single atom strongly coupled to the tilted cavity TEM10 mode