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We report $^{115}$In nuclear magnetic resonance (NMR) measurements inCeCoIn$_5$ at low temperature ($T \approx 70$ mK) as a function of magneticfield ($H_0$) from 2 T to 13.5 T applied perpendicular to the $\hat c$-axis.NMR line shift reveals that below 10 T the spin susceptibility increases as$\sqrt{H_0}$. We associate this with an increase of the density of states dueto the Zeeman and Doppler-shifted quasiparticles extended outside the vortexcores in a d-wave superconductor. Above 10 T a new superconducting state isstabilized, possibly the modulated phase predicted by Fulde, Ferrell, Larkinand Ovchinnikov (FFLO). This phase is clearly identified by a strong and linearincrease of the NMR shift with the field, before a jump at the first ordertransition to the normal state.

Field Dependence of the Ground State in the Exotic Superconductor CeCoIn$_5$: a Nuclear Magnetic Resonance Investigation