Quantum sensing in the fractional Fourier domain

TL;DR

The paper tackles extracting time-varying spectral information with quantum sensors by extending measurements beyond time- or frequency-domain limits to the fractional Fourier domain (FRFT). It introduces a theoretical framework using FRFT-domain filters $h_j^{(\alpha)}(t)$ controlled by $q = \cot\alpha$, linking the accumulated phase $\Phi$ to the FRFT of the stimulus, and validates this with an NV ensemble. Experimental results show that matching the FRFT chirp rate to the signal yields substantial gains in spectral estimation and detection, achieving up to roughly two orders-of-magnitude improvement in mean-squared error at large $|q|$, with Bayesian analyses supporting the advantage. This work broadens quantum sensing capabilities to nontrivial time-frequency geometries, with potential impact on nanoscale NMR, radar-like sensing, and nonstationary-signal metrology, and suggests avenues for extensions to stochastic signals and adaptive sampling.

Abstract

Certain quantum sensing protocols rely on qubits that are initialized, coherently driven in the presence of a stimulus to be measured, then read out. Most widely employed pulse sequences used to drive sensing qubits act locally in either the time or frequency domain. We introduce a generalized set of sequences that effect a measurement in any fractional Fourier domain, i.e. along a linear trajectory of arbitrary angle through the time-frequency plane. Using an ensemble of nitrogen-vacancy centers we experimentally demonstrate advantages in sensing signals with time-varying spectra.

Figures (4)

• Figure 1: Three examples of ordinary DD filters defined in Eq. (\ref{['eq_hdef']}), as represented in the (a) time domain, (b), frequency domain, (c) and FRFT domain of order $\alpha = \pi/4$. (d) Filled blobs are bounded by the half-max contours of the Wigner representations of these same three filters. Black line is parallel to the $u_{\alpha=\pi/4}$-axis. Units are arbitrary.
• Figure 2: Same explanation as for Fig. \ref{['fig:fourier filter set']}, except for filters with $\alpha = \pi/4$.
• Figure 3: Experimental spectra of AC magnetic field synthesized according to Eq. (\ref{['eq_AWG_signal']}) for various $q_1$ and $f_1 \in \{1.2, 1.3\}$ MHz, as recorded by an NV ensemble driven with either (a) $q$-matched FRFT sequences, or (b) ordinary unchirped DD sequences. Both show good agreement with calculated spectra in (c) and (d), respectively.
• Figure 4: Statistical analysis of experimental data based on $q$-matched FRFT (solid) and unchirped DD (dashed) measurements. (a) Mean-squared error of least-squares estimates of $f_1$ (black). For each $q_1$, 10 measurements were recorded with ground truth $f_1 = 1.2$ MHz and 10 with $f_1 = 1.3$ MHz. Black dots and error bars mark the means and standard deviations, respectively, of 1000 bootstrapped resamples. Red lines depict Bayesian CRBs as described in the text. Blue lines depict the CRBs of the sampling-frequency-adapted measurement described in the text. (b) Experimentally realized MAP error rates (black) for tests of binary hypotheses $f_1 = 1.2$ MHz and $f_1 = 1.3$ MHz. If no errors were recorded across the 20 trials the data point is plotted at $10^{-31}$. Red lines depict the Bayes error rates for this binary hypothesis test.