The limits of frequency resolution in nano NMR experiments have been discussed extensively in recent years. It is believed that there is a crucial difference between the ability to resolve a few frequencies and the precision of estimating a single one. Whereas the efficiency of single frequency estimation gradually increases with the square root of the number of measurements, the ability to resolve two frequencies is limited by the specific time scale of the signal and cannot be compensated for by extra measurements. Here we show theoretically and demonstrate experimentally that the relationship between these quantities is more subtle and both are only limited by the Cramér-Rao bound of a single frequency estimation.We consider the problem of spectral resolution; i.e., differentiating between two close frequency components of a signal. In the nano NMR setting this can be formulated as follows. A time dependent signal is coupled to a two-level system by a term such as H = f (t)σ z , where σ z is a Pauli matrix, with the aim to assess the spectral content of f (t). This problem has been extensively examined in the past few years via NV centers in diamond [1][2][3][4][5][6][7][8][9][10][11]. The limit of resolution of the frequency spectrum of signals is believed to be set by the linewidth of the power spectrum [1][2][3] where the liquid state is dominated by diffusion [9,10,12,13] . The main problem is illustrated in Fig. 1a, where two signals that are close enough manifest a power spectrum which is similar to that of a single broad frequency. This intuition that resolution is limited by the line-width is based on the Rayleigh criterion from optics [17,18] where an analogy is drawn between the wavelength and the line-width. This notion is one of the main pillars of spectroscopy. Here, we challenge this concept.The traditional method of spectroscopy with quantum sensors uses dynamical decoupling pulses for a certain duration, since the fluorescence as a function of the dynamical decoupling frequency reflects the spectrum of the signal [19,20]. Thus when implementing this method two frequencies are only resolvable if the difference between them is larger than, roughly, T −1 2 , where T 2 is the coherence time of the probe. This was believed to impose a fundamental limit on frequency resolution. However, it was realized that by transferring the quantum phase of the sensor to the state population that survives up to longer T 1 relaxation times [13,21], resolution could be improved. Moreover, by using a hybrid quantum system where an additional long-lived qubit acts as a more stable clock [4,5,10,[22][23][24] the limit could be extended to the coherence time of the ancilla qubit. Recently it was realized that the quantum memory could be replaced by a classical one [1][2][3]25]. In these contributions it was shown that although the efficiency of estimating a single frequency improves with the number of measurements, the resolution limit is set by the specific time scale of the scheme.

No abstract