## Enhancing quantum sensing

The quantum properties of the nitrogen vacancy (NV) defect in diamond can be used as an atomic compass needle that is sensitive to tiny variations in magnetic field. Schmitt *et al.* and Boss *et al.* successfully enhanced this sensitivity by several orders of magnitude (see the Perspective by Jordan). They applied a sequence of pulses to the NV center, the timing of which was set by and compared with a highly stable oscillator. This allowed them to measure the frequency of an oscillating magnetic field (megahertz bandwidth) with submillihertz resolution. Such enhanced precision measurement could be applied, for example, to improve nuclear magnetic resonance-based imaging protocols of single molecules.

## Abstract

Quantum sensing takes advantage of well-controlled quantum systems for performing measurements with high sensitivity and precision. We have implemented a concept for quantum sensing with arbitrary frequency resolution, independent of the qubit probe and limited only by the stability of an external synchronization clock. Our concept makes use of quantum lock-in detection to continuously probe a signal of interest. Using the electronic spin of a single nitrogen-vacancy center in diamond, we demonstrate detection of oscillating magnetic fields with a frequency resolution of 70 microhertz over a megahertz bandwidth. The continuous sampling further guarantees an enhanced sensitivity, reaching a signal-to-noise ratio in excess of 10^{4} for a 170-nanotesla test signal measured during a 1-hour interval. Our technique has applications in magnetic resonance spectroscopy, quantum simulation, and sensitive signal detection.

Quantum sensors with new capabilities are driving the field of precision metrology (*1*, *2*). In particular, spin qubits associated with crystal defects in diamond (*3*) and other materials (*4*–*6*) have emerged as highly sensitive probes with nanometer spatial resolution (*7*, *8*). Because the defect spins are well isolated from the environment, they can be controlled with high fidelity, allowing researchers to implement sophisticated quantum manipulation protocols.

A particularly important sensing task is the spectral decomposition of time-varying signals into their frequency components. Quantum metrology employs quantum control techniques to reach this goal. For example, dynamical decoupling methods—originally developed for protecting qubits from decoherence—have been adapted for detecting alternating signals with narrow bandwidth and high signal-to-noise ratio (SNR) (*9*–*11*). Other, more recent techniques include dressed-state approaches (*12*, *13*), Floquet spectroscopy (*14*), and correlative measurements (*15*). Crucially, the spectral resolution of all of these techniques is limited by the state lifetime of the qubit probe. For nitrogen-vacancy (NV) centers in diamond, reported spectral resolutions are a few hertz at best, even when assisted by a long-lived quantum memory (*16*).

We introduce a simple concept in which the frequency estimation is solely limited by the stability of an external, classical reference clock and the total available measurement time. Our method takes advantage of the quantum lock-in amplifier (*9*, *10*), which is used to stroboscopically sample the signal of interest. Although the acquired signal is highly undersampled, we show that the original wideband spectrum can be recovered by compressive sampling methods (*17*). The periodic sampling further guarantees that the SNR increases in proportion to the measurement time. We demonstrate our method by recording signal traces for up to 4 hours, reaching a frequency resolution of 70 μHz and a precision of 260 nHz with SNR *>* 10^{4}.

Our experimental demonstration makes use of a spin qubit formed by the negatively charged NV center in diamond. The NV center is a suitable object for our demonstration because it can be efficiently initialized, manipulated, and read out at room temperature by optical and microwave pulses. Furthermore, its sensing technology is well developed (*8*) and addresses a broad range of potential applications in physics, materials science, and biology (*18*, *19*).

Our approach (Fig. 1) relies on periodic sampling of a signal *x*(*t*) in intervals of a sampling period *t*_{s}. Each sampling instance consists of three periods, including a quantum lock-in measurement of duration *t*_{a}, qubit state readout during *t*_{r}, and an additional delay time *t*_{d} to accommodate for experimental overhead and adjust the sampling rate. The sampling period is then *t*_{s} = *t*_{a} + *t*_{r} + *t*_{d}.

To implement the quantum lock-in measurement, we use a Carr-Purcell-Meiboom-Gill (CPMG) decoupling sequence (red pulses in Fig. 1). Specifically, we initialize the qubit to the +*X* state of the *X* basis and modulate it by a series of π pulses with interpulse spacing τ. This defines the lock-in detection frequency *f*_{LI} = *m*/(2τ), where *m* = 1,3,5,*...* is the harmonic order. The frequency bandwidth of the lock-in is approximately *f*_{LI} ± 1/(2*t*_{a}) (*2*, *10*). For an ac signal *x*(*t*) = Ωcos(2π*f*_{ac}*t*) with afrequency *f*_{ac} ≈ *f*_{LI} within this bandwidth, the quantum phase accumulated after time *t*_{a} is(1)where *t _{k}* marks the start of the lock-in measurement, and Ω is the signal amplitude in units of angular frequency (supplementary text 1). Crucially, although the quantum phase is accumulated over an extended time interval [

*t*,

_{k}*t*+

_{k}*t*

_{a}], its value reflects the instantaneous value of

*x*(

*t*) at time

*t*=

*t*. To read out the quantum phase, the quantum state is measured in the

_{k}*Y*basis, yielding a probability(2)to find the system pointing along the −

*Y*direction. The approximation is for small within the sensor’s linear range (

*20*). Optical readout finally converts the projected state into a photon number

*y*. Because state projection and optical readout are stochastic processes,

_{k}*y*is a random variable(3)where Bn is a Bernoulli process, which takes the value 1 with a probability of

_{k}*p*and the value 0 with probability 1 −

_{k}*p*, and Pois is a Poisson process that reflects the photon shot noise.

_{k}*C*is a variable readout gain and is the optical contrast.

By collecting a time trace of *N* measurement outputs at sampling times *t _{k}* =

*kt*

_{s}, we can sample the signal

*x*(

*t*) at a sub-Nyquist rate

*f*

_{s}= 1/

*t*

_{s}. Hence, a Fourier transform of the time trace reveals a discrete undersampled spectrum of

*x*(

*t*). Crucially, the number of samples

*N*can be made as large as desired, allowing for a frequency resolution δ

*f*=

*f*

_{s}/

*N*that is arbitrarily fine.

To implement our continuous sampling technique, we used the qubit formed by the *m*_{S} = 0 and *m*_{S} = –1 spin sublevels of single NV centers located in diamond nanopillar waveguides (materials and methods). At a bias field of 457 mT applied along the NV symmetry axis, the transition frequency between these states is 9916 MHz. We initialized the qubit using a 532-nm laser pulse and a microwave π/2 pulse and detected the qubit state using a phase-shifted π/2 pulse followed by optical readout. We used an indirect readout scheme in which the final qubit state was first stored in the ^{15}N nuclear spin (*I* = 1/2), serving as a memory qubit, and we then repetitively read out the ^{15}N spin state by a nuclear quantum nondemolition (QND) measurement (*21*, *22*). By varying the number of QND measurements *n*, we could adjust the readout gain *C* between 0 and ~230 photons. The optical contrast was .

As a first illustration of the continuous sampling technique, Fig. 2, A and B, show a time trace and spectrum of an amplitude-modulated (AM) magnetic test signal with carrier frequency *f*_{c} = 601.2547 kHz and modulation frequency *f*_{AM} = 10 mHz. The test signal had an amplitude of ~170 nT, corresponding to Ω = 2π × 4.7 kHz, and was generated by passing an ac current through a nearby wire. The signal contained three components at frequencies *f*_{c} and *f*_{c} ± *f*_{AM}, with a power ratio of 1:4:1. The frequency resolution δ*f* of the spectrum, obtained from a time trace of 1-hour duration, was δ*f* = 1/*T* = 278 μHz (Fig. 2B, right inset). Because our signal was undersampled, the abscissa in Fig. 2B indicates the detuning from *f*_{c} rather than the absolute frequency. The observed microhertz frequency resolution and the consistent amplitude ratio between the carrier and side peaks illustrate the capabilities of our method.

Though our strategy allows for an arbitrary frequency resolution δ*f*, we are more interested in how precisely we can determine a signal’s linewidth and center frequency in the experiment. Figure 2C depicts the decrease in the fitted linewidth parameter γ with increasing measurement time *T* for four signals. Signals (i) to (iii) were produced by amplitude modulation, as in Fig. 2B, and had zero intrinsic linewidth, γ_{int} = 0. The linewidth parameter for these signals scaled with γ ∝ *T*^{−1}, which represents the Fourier transform limit of the continuous sampling method. The *T*^{−1} scaling is expected to continue until the phase noise in the reference clock or the frequency jitter in the signal generator become dominating. Signal (iv), on the other hand, was artificially broadened by frequency modulation with Gaussian noise to mimic a nonzero intrinsic linewidth γ_{int} *>* 0. The linewidth parameter for signal (iv) initially also decreased with the *T*^{−1} scaling but leveled out as γ approached γ_{int}. Figure 2D further shows the fit errors in the peak frequencies for all signals. Here an observed *T*^{−1.5} scaling reduced to *T*^{−0.5} once the intrinsic linewidth became dominating, as expected for the scaling of the spectral amplitude variance (supplementary text 2).

We next examined how the sensitivity of the sensor can be optimized. To quantify the sensitivity, we compared the peak amplitude *Y _{j}* with the standard deviation σ

*of the noise floor in the power spectrum (Fig. 2B). This defines a power SNR(4)Assuming that the entire signal power is concentrated in a single Fourier component*

_{Y}*Y*of the spectrum (i.e., that the linewidth of the signal is smaller than δ

_{j}*f*), it follows from Eqs. 2 and 3 that(5)where ϕ

_{max}= 2

*t*

_{a}Ω/π is the signal amplitude expressed in units of the accumulated phase (supplementary text 3). The approximation is again for small signals within the linear response of the lock-in (Eq. 2). The noise σ

*is the sum of two contributions: one from quantum projection noise with variance and one from optical shot noise with variance . The SNR becomes(6)Because*

_{Y}*N*=

*Tf*

_{s}, the SNR improves proportional to the duration of the time record

*T*.

To further optimize the SNR, we adjusted the phase amplitude ϕ_{max} and the readout gain *C*. We achieved this by varying the sensing time *t*_{a} and the readout time *t*_{r}. First, we increased the sensing time *t*_{a} so that the quantum phase covered the full linear range of the lock-in, typically ϕ_{max} ~ 0.5. Although larger ϕ_{max} values are possible, the response of the lock-in becomes nonlinear (*20*), and harmonics are generated in the spectrum. This complicates the interpretation while providing little further improvement in the SNR (supplementary text 1 and figs. S4 and S5). Next, we turned up the gain *C* until sensor readout became dominated by quantum projection noise. In our experiment, we could adjust *C* by varying the number *n* of QND measurements of the nuclear memory qubit, where *C*(*n*) ≈ *n* × 0.105 photons and *t*_{r} ≈ *n* × 2.32 μs. Figure 3 plots the SNR for signal (ii) in Fig. 2C as a function of *n*. The SNR increased rapidly for small *n* until it saturated around *n* ≈ 260, which corresponded to the threshold gain , where shot noise and quantum projection noise are balanced (*2*). Increasing the gain beyond *C*_{thresh} only marginally improved the SNR and eventually even degraded it. The degradation at very high gains was due to the imperfection of the nuclear quantum memory, which became depolarized under optical illumination (*22*).

Thus far, all of our measurements reported relative rather than absolute signal frequencies. The measurement of absolute signal frequencies is hindered by the large undersampling. For example, in Fig. 2B, a signal of frequency *f*_{c} ∼ 601 kHz was sampled at *f*_{s} = 0.237 kHz, which is about 5 × 10^{3} times slower than the Nyquist rate. We now discuss a strategy that makes use of compressive sampling to overcome this limitation. We implemented this strategy by recording a set of time traces with slightly different sampling rates *f*_{s}.

Compressive sampling (CS) exploits our prior knowledge about the sparsity of the wideband spectrum (*17*). Suppose the vector holds the Fourier components of the desired wideband spectrum sampled at or above the Nyquist rate, and the vectors represent a small set of *p* undersampled spectra with *i* = 1,2,*...*,*p*. We can express our measured undersampled spectra by the linear system(7)where Φ* _{i}* are sampling matrices folding the wideband spectrum into the bandwidths of the undersampled spectra (

*23*). To reconstruct the wideband spectrum, we solve Eq. 7 for . Although the linear system is highly underdetermined, a solution can be found if is sparse (i.e., if it is substantially nonzero for only a few frequencies) and the Φ

*are mutually incoherent.*

_{i}To demonstrate wideband spectral reconstruction, we implemented a CS scheme to recover *l* = 7 tones from a set of *p* = 7 undersampled spectra (supplementary text 4). We adjusted the sampling frequencies *f*_{s} via the delay time *t*_{d}. To ensure incoherence between the sampling matrices, we randomized our choices of *t*_{d}. Figure 4 shows the undersampled spectra together with the reconstructed wideband spectrum. The tones in these spectra were contained in two 20-kHz-wide frequency bands, one centered at the first harmonic of the lock-in filter function at 400 kHz and one at the third harmonic at 1200 kHz. Although the SNR of the reconstructed spectrum is reduced because of incomplete image rejection, the experiment clearly demonstrates that the absolute peak frequencies can be unambiguously recovered. The image rejection can be improved by increasing the number of spectra *p*.

Our experiments demonstrate that a quantum sensor can achieve a frequency resolution far beyond its intrinsic state lifetime, limited only by the stability of an external synchronization clock. Looking forward, quantum sensing with arbitrary frequency resolution has important applications in sensitive magnetic and electric field detection. A high spectral resolution is, for example, essential for nanoscale nuclear magnetic resonance (NMR) imaging experiments (*24*, *25*), in which minute spectral shifts can be used to infer atomic positions, internuclear distance vectors, and molecular connectivity. Spectral addressability is also important for operating large-scale quantum registers in solid-state quantum simulators (*26*). Although NMR spectra are often broadened by internuclear interactions, a rich repertoire of line narrowing and isotope dilution techniques exists for refining the spectral resolution (*27*, *28*). Ultrahigh-resolution NMR is able to resolve couplings of a few millihertz (*29*) under favorable conditions and achieves *<*20-Hz linewidths, even for ^{1}H in dense solid samples (*28*). Finally, continuous sampling can provide sensitivity gains when measuring weak, modulated signals. This is because of the high duty cycle achieved by continuously probing the signal during the measurement time *T*, combined with the favorable ∝ *T* scaling of the SNR.

## Supplementary Materials

www.sciencemag.org/content/356/6340/837/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S5

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵See supplementary materials.
- ↵
- ↵
**Acknowledgments:**We thank T. Rosskopf, K. Chang, A. Retzker, and F. Jelezko for experimental support and useful discussions. This work was supported by the Swiss National Science Foundation (SNSF) Project grant 200021 137520, the National Competence Center in Research on Quantum Science and Information Technology (NCCR QSIT), and the FP7-611143 DIADEMS program of the European Commission.