The experimental verification of quantum fluctuation relations for driven open quantum system is currently a challenge, due to the conceptual and operative difficulty of distinguishing work and heat. The nitrogen-vacancy (NV) center in diamond has been recently proposed as a controlled test bed to study fluctuation relations in the presence of an engineered dissipative channel, in absence of work (Hernández-Gómez et al 2020 Phys. Rev. Res. 2 023327). Here, we extend those studies to exploring the validity of quantum fluctuation relations in a driven-dissipative scenario, where the spin exchanges energy both with its surroundings because of a thermal gradient, and with an external work source. We experimentally prove the validity of the quantum fluctuation relations in the presence of cyclic driving in two cases, when the spin exchanges energy with an effective infinite-temperature reservoir, and when the total work vanishes at stroboscopic times—although the power delivered to the NV center is non-null. Our results represent the first experimental study of quantum fluctuation relation in driven open quantum systems.

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Precise frequency measurements are important in applications ranging from navigation and imaging to computation and communication. Here we outline the optimal quantum strategies for frequency discrimination and estimation in the context of quantum spectroscopy, and we compare the effectiveness of different readout strategies. Using a single NV center in diamond, we implement the optimal frequency discrimination protocol to discriminate two frequencies separated by 2 kHz with a single 44 μs measurement, a factor of ten below the Fourier limit. For frequency estimation, we achieve a frequency sensitivity of 1.6 µHz/Hz2 for a 1.7 µT amplitude signal, which is within a factor of 2 from the quantum limit. Our results are foundational for discrimination and estimation problems in nanoscale nuclear magnetic resonance spectroscopy.

Diffusion noise is a major source of spectral line broadening in liquid state nano-scale nuclear magnetic resonance with shallow nitrogen-vacancy centres, whose main consequence is a limited spectral resolution. This limitation arises by virtue of the widely accepted assumption that nuclear spin signal correlations decay exponentially in nano-NMR. However, a more accurate analysis of diffusion shows that correlations survive for a longer time due to a power-law scaling, yielding the possibility for improved resolution and altering our understanding of diffusion at the nano-scale. Nevertheless, such behaviour remains to be demonstrated in experiments. Using three different experimental setups and disparate measurement techniques, we present overwhelming evidence of power-law decay of correlations. These result in sharp-peaked spectral lines, for which diffusion broadening need not be a limitation to resolution.