Low-Frequency Quantum Sensing

Exquisite sensitivities are a prominent advantage of quantum sensors. Ramsey sequences allow precise measurement of direct current fields, while Hahn-echo-like sequences measure alternating current fields. However, the latter are restrained for use with high-frequency fields (above approximately $1$ kHz) due to finite coherence times, leaving less-sensitive noncoherent methods for the low-frequency range. In this paper, we propose to bridge the gap with a fitting-based algorithm with a frequency-independent sensitivity to coherently measure low-frequency fields. As the algorithm benefits from coherence-based measurements, its demonstration with a single nitrogen-vacancy center gives a sensitivity of $9.4$ nT ${\mathrm{Hz}}^{-0.5}$ for frequencies below about $0.6$ kHz down to near-constant fields. To inspect the potential in various scenarios, we apply the algorithm at a background field of tens of nTs, and we measure low-frequency signals via synchronization.

Figure 1

Algorithm. (a) Illustrative ODMR spectrum (dark red line at the top) with three valleys (indicated with red arrows) related to the energy differences between the $m_s=0$ and $m_s=-1$ states with hyperfine splitting due to the ${}^{14}\mathrm{N}$ nucleus. The inset shows the basic pulse sequence for FID measurements with delay $\tau$ between the MW $\pi /2$ pulses. (b) In the algorithm, a fixed-delay FID sequence is repeated during the period of the signal, measuring the yellow areas. Thus, the sequence functions as an oscilloscope with quantum measurements. (c) For a single sequence with the same fixed delay, by applying a dc field the spin rotates around the $z$ axis depending on the size of the area (here the dc field magnitude times the fixed delay), resulting in an oscillation of the readout (RO) signal (simulation results with blue crosses, fit with red line). Such a measurement can be used as calibration, for example for the amplitude $A$ (magenta dashed lines are the extrema) and the offset $O_S$ (black dashed line) from Eq. (1). (d) The final phase of the spin (analyzed simulation results with cyan circles) depends on the measured area during each subsequence. (e) By taking the shape of the areas into account while fitting, the parameters such as the ac field amplitude are retrieved (analyzed simulation results with cyan diamonds, resulting field with olive line). (f) With a single FID sequence the result would be ambiguous for larger fields given the rotational symmetry of the phase. However, the combination of multiple sequences allows some of this information to be retained (simulation results with blue dots, fit with orange line).

Figure 2

Exploratory measurements. (a) RO signal versus delay giving the FID measurement result (data with blue crosses, fit with red line) to determine $T_2^{\ast }$ (${1.05}_{-0.05}^{+0.05}$ ms). (b) Calibration measurement for a delay of $0.4$ ms (data with blue crosses, fit with red line). The horizontal black dashed line indicates the offset $O_S$, while the horizontal magenta dashed lines give the extrema due to amplitude $A$ [see Eq. (1)]. (c) Example measurement exhibiting the independence of phase and offset when measuring an ac field ($50$-Hz signal with an amplitude of $3.1$ nT, data with blue dots, fit with orange line). (d) Example measurement for a high dynamic range measurement, which is outside the linear regime (data with blue dots, fit with orange line). The amplitude is $100$ times the amplitude in (c) and (e). (e) Example measurement for a field of $1$ Hz with an amplitude of $3.1$ nT (data with blue dots, fit with orange line). The offset field of the N-$V$ center for the measurements in this figure is about $1.8$ mT.

Figure 3

Sensitivity. (a) Sensitivity versus frequency: data points with blue crosses (error bars indicate standard errors); calculated result with blue dashed line; fitted result for low frequency with horizontal red line (${\eta }_{\mathrm{ac}}={9.4}_{-0.2}^{+0.1}$ nT ${\mathrm{Hz}}^{-0.5}$); calculated result with Hahn-echo sequence for the current longest $T_2$ with magenta dashed line (adjusted from Ref. [3] with the parameters of the current experimental setup for fair comparison); calculated result for optimal pulsed ODMR [20] with orange dashed-dotted line; averaged result over low frequencies for Fourier-Lorentzian as a guide to the eye with purple dashed-dotted line; calculated result for dc sensing via cw ODMR as a guide to the eye with green dashed line at the top (for the current N-$V$ center and experimental setup, adjusted from Ref. [20]); threshold frequency with vertical cyan line ($f_{\mathrm{threshold}}=0.36$ kHz). Insets show two examples of the amplitude uncertainty versus measurement time $T_{\mathrm{meas}}$ results (data with blue diamonds, fits with olive lines), where fits give the sensitivities (left for $1$ Hz and right for $0.2$ kHz). Note the breaks on the vertical axis. Here, the offset field of the N-$V$ center is about $1.8$ mT. (b) The fixed delay applied in the subsequences for each frequency (blue crosses), with a maximum of $0.4$ ms (horizontal black dashed line). For high frequencies, this delay decreases since a minimum number of data points are required per period. (c) Number of subsequences $N$ (blue crosses, fit with red line) per period, thus data points, for each frequency, with a minimum of $4$ (horizontal black dashed line). The vertical olive dashed line is a guide to the eye at the maximum frequency where $N>4$.

Figure 4

Low-frequency sensing at low field. (a) Calibration measurement for a delay of $0.2$ ms around zero field (RO signal is readout signal, data with blue crosses, fit with red line). The cyan-shaded area indicates the range of fields that include the actual zero field based on FID measurements (Sec. V within the Supplemental Material [27]), while the extremum (vertical magenta line) in this region is when the additional field cancels the remaining background field exactly. The measurements in (b) and (c) are performed at a background field of $30$ nT (black dashed lines). (b) Example measurement for a field of $50$ Hz with an amplitude of $10$ nT at a background field of $30$ nT (data with blue dots, fit with orange line). (c) Uncertainty versus measurement time to find the sensitivity for the same signal as in (b) (data with blue diamonds, fit with olive line). (d) Near zero field, when the preparation MW pulse (first lower red rectangle after the laser pulse), here ${(\pi /2)}_x$, and the readout MW pulse, here ${(\pi /2)}_y$, are along different axes, the result depends on the spin state and roughly averages towards a field-independent value (black arrows). (e) When the readout MW pulse is along the same axis, here ${(\pi /2)}_x$, the readout result is the same (black line) for each spin state, enabling measurement of the field.

Figure 5

Synchronized low-frequency sensing. (a) Free nuclear precession RO signal versus time for part of the time axis (total length is $1.0$ s, repeated $10000$ times, data with blue dots, fit with orange line) measuring the response of a deionized water sample after applying a rf $\pi /2$ pulse. Here, the amplitude of the low-frequency signal ($7$ Hz) starts at $43.7$ nT (left side), and decays to $2.5$ nT (right side). (b) Free nuclear precession readout signal versus time for part of the time axis (total length is $1.0$ s, repeated $20000$ times, data with blue dots, fit with orange line) measuring an ethanol sample after applying a rf $\pi /2$ pulse. Here, the amplitudes of the low-frequency signal start at $9.6$ nT ($7$ Hz), $19.7$ nT ($14$ Hz), and $13.6$ nT ($21$ Hz), and decay to $0.4$ nT, $0.9$ nT, and $0.6$ nT, respectively. The offset field of the N-$V$ center for the measurements in this figure is about $0.5$ mT.