Limits on Spectral Resolution Measurements by Quantum Probes

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 timescale 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.

Figure 1

Resolution problem. (a) When the two signals are close enough (blue and orange), their sum looks like one Lorentzian (green) making it impossible to differentiate between two sources and one broad source. Experimental scheme. (b) Experimental scheme for a set of frequencies. A set of $n$ measurements is made during the time that the phase of the signal is stable, where the time of each measurement is $\tau$ and is limited by the coherence time of the probe ($T_2$). The sets are repeated $N$ times to obtain more statistics, i.e., $N$ uncorrelated sets. The estimation of the frequencies is done by calculating the likelihood function and locating its maximum value [16, 17, 18]; i.e., $\max \mathcal{L}(\{{\delta }_k,{\mathrm{\Omega }}_k,{\phi }_k^{(\ell )}{\}}_{k,\ell })$, where $k$ is the index of the frequencies ($\delta$) and $\ell$ designates the phases $(\phi )$, and $\mathrm{\Omega }$ is the Rabi frequency of the signal.

Figure 2

Resolving the experimental data. (a) Two frequencies with a frequency difference below the DFT limit, ${\delta }_1=250[2\pi \mathrm{Hz}]$ and ${\delta }_2=251.6[2\pi \mathrm{Hz}]$ (${\delta }_2-{\delta }_1=0.4[2\pi T_{\phi }^{-1}]$) were resolved. The Rabi frequency of the signal was $\mathrm{\Omega }\approx 12[2\pi \mathrm{kHz}]$. The inset, and the blue line over the histogram, show the DFT of 33 measurement sets. The figure depicts a histogram of the estimators from $2^{10}$ iterations of MLE, each over the 33 datasets randomly chosen from the total of 880 datasets. The average estimators are $⟨{\tilde{\delta }}_1⟩=250.22\pm 0.45[2\pi \mathrm{Hz}]$, and $⟨{\tilde{\delta }}_2⟩=251.68\pm 0.40[2\pi \mathrm{Hz}]$ (the errors represent the SD), i.e., over $2.4\sigma$ apart. (b) The result of the same procedure, for data containing only a single frequency. The average of the difference is $⟨|{\tilde{\delta }}_1-{\tilde{\delta }}_2|⟩=0.51\pm 0.53[2\pi \mathrm{Hz}]$. (c) The histogram of $|{\delta }_2-{\delta }_1|$ taken from the data in (a) (purple) and (b) (green). Note that the $100[\mathrm{Hz}]$ component in the spectrum (and harmonics) is noise due to fluctuations in the room’s fluorescent lights.

Figure 3

SD of estimators from global search vs the Cramér-Rao bound. The SD of the estimators are plotted in black circles with error bars calculated from the SD of the statistics, and the root of the mean square error is plotted in green asterisks. For comparison the Cramér-Rao bound of ${\delta }_2-{\delta }_1$ calculated from the probability (red), and the average probability over the amplitudes (blue)—the scattering is due to the randomness of the phases. The dashed black line indicates the $2-3\sigma$ certainty regime. The signal is generated with two frequencies of ${\delta }_1=e^4[2\pi T_{\phi }^{-1}]$ and ${\delta }_2={\delta }_1+\mathrm{\Delta }\delta$, with phases randomly taken from a uniform distribution $\phi \sim U(0,2\pi )$ and amplitudes randomly taken from a Rayleigh distribution $\mathrm{\Omega }\sim R(\sqrt{2/\pi }⟨\mathrm{\Omega }⟩)$.