We report on Fourier spectroscopy experiments performed with near-surface nitrogen-vacancy centers in a diamond chip. By detecting the free precession of nuclear spins rather than applying a multipulse quantum sensing protocol, we are able to unambiguously identify the NMR species devoid of harmonics. We further show that by engineering different Hamiltonians during free precession, the hyperfine coupling parameters as well as the nuclear Larmor frequency can be selectively measured with high precision (here 5 digits). The protocols can be combined to demonstrate two-dimensional Fourier spectroscopy. The technique will be useful for mapping nuclear coordinates in molecules en route to imaging their atomic structure.Nitrogen-vacancy (NV) centers in diamond have opened exciting perspectives for the ultrasensitive detection of nuclear magnetic resonance (NMR), with possible applications to molecular structure imaging and chemical nanoanalytics [1][2][3]. NMR signals are detected by placing an analyte on a diamond chip engineered with a surface layer of NV centers, and measuring the weak magnetic dipole fields of nuclei via optically detected magnetic resonance [4,5]. Examples of the rapid recent progress in NV-NMR include the detection of small numbers of nuclei within voxels of a few (nm) 3 [6,7], the detection of multiple nuclear isotopes [8,9] and naturally occurring adsorption layers [6], the observation of surface diffusion and molecular motion [10,11], scanning imaging with < 20 nm spatial resolution [9,12], and the spatial mapping of up to 8 internal 13 C nuclei [13]. One of the far goals of NV-NMR is the detection and three-dimensional localization of individual nuclei in single molecules deterministically placed on the diamond chip [1,13,14].Sensitive detection of nuclear magnetic signals is possible with multipulse sequences that consist of a series of π pulses (see Fig. 1a). These sequences act like a narrow-band lock-in amplifier [15] whose demodulation frequency f = 1/(2τ ) is set by the delay time τ between the pulses [14,16,17]. By varying τ a frequency spectrum of the magnetic field can be recorded. Multipulse spectroscopy of NMR signals has been reported for many nuclear isotopes, including 1 H, 13 C, 14 N, 15 N, 19 F, and possibly 29 Si and 31 P [3, 6-9, 12, 18]. These experiments have, however, also revealed some important shortcomings of the method, including a modest spectral resolution [2,19] and ambiguities in peak assignments due to signal harmonics [18]. The fundamental reason for both effects is the indirect way nuclear spin signals are detected via their influence on the electronic spin.A more natural way for measuring NMR signals is to observe the free nuclear precession in the absence of microwave or radio-frequency pulses, reminiscent of the "free induction decay" in conventional NMR Fourier spectroscopy. The free nuclear precession can be detected by performing two consecutive nuclear spin measurements and incrementing the duration t 1 between the measurements. Mamin et al. [2] have ...