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Quantum sensing with arbitrary frequency resolution

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Science  26 May 2017:
Vol. 356, Issue 6340, pp. 837-840
DOI: 10.1126/science.aam7009

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.

Science, this issue p. 832, p. 837; see also p. 802

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 104 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 (46) 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) (911). 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 > 104.

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 ts. Each sampling instance consists of three periods, including a quantum lock-in measurement of duration ta, qubit state readout during tr, and an additional delay time td to accommodate for experimental overhead and adjust the sampling rate. The sampling period is then ts = ta + tr + td.

Fig. 1 Basic concept of continuous sampling.

The qubit sensor stroboscopically probes an ac signal x(t) in intervals of the sampling period ts. Each sampling instance k consists of sensor initialization (green), a phase measurement using quantum lock-in detection (red pulses), and sensor readout (yellow). A sensor output yk is proportional to the quantum phase ϕk and to the instantaneous value of x(tk) at time stamp tk (blue dots). A time trace {yk} of sensor outputs therefore contains the undersampled signal x(tk) (gray oscillation). In our experiment, the sensor qubit was implemented by the electronic spin of a NV center in diamond. A laser pulse was used for initialization and nuclear-spin–assisted optical detection for readout. Quantum lock-in detection was implemented by a CPMG sequence with interpulse spacing τ and either 16 or 32 π pulses (materials and methods) (30).

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 fLI = m/(2τ), where m = 1,3,5,... is the harmonic order. The frequency bandwidth of the lock-in is approximately fLI ± 1/(2ta) (2, 10). For an ac signal x(t) = Ωcos(2πfact) with afrequency facfLI within this bandwidth, the quantum phase accumulated after time ta isEmbedded Image(1)where tk 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 [tk, tk + ta], its value reflects the instantaneous value of x(t) at time t = tk. To read out the quantum phase, the quantum state is measured in the Y basis, yielding a probabilityEmbedded Image(2)to find the system pointing along the −Y direction. The approximation is for small Embedded Image within the sensor’s linear range (20). Optical readout finally converts the projected state into a photon number yk. Because state projection and optical readout are stochastic processes, yk is a random variableEmbedded Image(3)where Bn is a Bernoulli process, which takes the value 1 with a probability of pk and the value 0 with probability 1 − pk, and Pois is a Poisson process that reflects the photon shot noise. C is a variable readout gain and Embedded Image is the optical contrast.

By collecting a time trace of N measurement outputs Embedded Image at sampling times tk = kts, we can sample the signal x(t) at a sub-Nyquist rate fs = 1/ts. 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 δf = fs/N that is arbitrarily fine.

To implement our continuous sampling technique, we used the qubit formed by the mS = 0 and mS = –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 15N nuclear spin (I = 1/2), serving as a memory qubit, and we then repetitively read out the 15N 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 Embedded Image.

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 fc = 601.2547 kHz and modulation frequency fAM = 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 fc and fc ± fAM, with a power ratio of 1:4:1. The frequency resolution δf of the spectrum, obtained from a time trace of 1-hour duration, was δf = 1/T = 278 μHz (Fig. 2B, right inset). Because our signal was undersampled, the abscissa in Fig. 2B indicates the detuning from fc 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.

Fig. 2 Continuous sampling of ac magnetic signals.

(A) A 1.26-s excerpt of a 1-hour time trace sampled at ts = 4.2 ms. The vertical axis shows photon counts. The telegraph-like behavior results from the stochastic quantum state projection. (B) Fourier spectrum (power) of the full 1-hour time trace. Three peaks are visible that correspond to the central and side peaks of the amplitude-modulated signal. The horizontal axis indicates the detuning from the carrier frequency. The insets show the noise floor (left inset) and the frequency resolution (right inset) of the spectrum. a.u., arbitrary units. (C) Fitted linewidth parameter γ as a function of total measurement time T for four signals. Data points (i) to (iii) originate from coherent signals and data points (iv) from an incoherent signal with an artificial line broadening of γint = 0.76 mHz. Original spectra are presented in fig. S3. (D) Uncertainty in the fitted peak frequencies for the different peaks. Solid and dashed lines are guides to the eye.

Though our strategy allows for an arbitrary frequency resolution δf, 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, γint = 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 γint > 0. The linewidth parameter for signal (iv) initially also decreased with the T−1 scaling but leveled out as γ approached γint. 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 Yj with the standard deviation σY of the noise floor in the power spectrum (Fig. 2B). This defines a power SNREmbedded Image(4)Assuming that the entire signal power is concentrated in a single Fourier component Yj of the spectrum (i.e., that the linewidth of the signal is smaller than δf), it follows from Eqs. 2 and 3 thatEmbedded Image(5)where ϕmax = 2taΩ/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 σY is the sum of two contributions: one from quantum projection noise with variance Embedded Image and one from optical shot noise with variance Embedded Image. The SNR becomesEmbedded Image(6)Because N = Tfs, the SNR improves proportional to the duration of the time record T.

To further optimize the SNR, we adjusted the phase amplitude ϕmax and the readout gain C. We achieved this by varying the sensing time ta and the readout time tr. First, we increased the sensing time ta so that the quantum phase covered the full linear range of the lock-in, typically ϕmax ~ 0.5. Although larger ϕmax 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 trn × 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 Embedded Image, where shot noise and quantum projection noise are balanced (2). Increasing the gain beyond Cthresh 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).

Fig. 3 Signal-to-noise ratio (SNR) as a function of the readout gain C.

The readout gain is adjusted via the number of repetitive QND measurements n of the nuclear memory qubit. Blue dots are the experimental data and represent the mean SNR of six time traces with N = 3.85 × 105 samples. The dashed line represents the ideal SNR predicted by theory (Eq. 6). The solid line, in addition, takes the depolarization of the nuclear 15N memory qubit into account (supplementary text 3). The filled blue region indicates the standard error of the fit. From our data, we extract a 15N depolarization rate Γ ≈ 1.4 × 10−4 per readout. The corresponding optimum SNR (n = 260) for a 1-hour measurement interval is 1.2 × 104 (materials and methods) (30).

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 fc ∼ 601 kHz was sampled at fs = 0.237 kHz, which is about 5 × 103 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 fs.

Compressive sampling (CS) exploits our prior knowledge about the sparsity of the wideband spectrum (17). Suppose the vector Embedded Image holds the Fourier components of the desired wideband spectrum sampled at or above the Nyquist rate, and the vectors Embedded Image represent a small set of p undersampled spectra with i = 1,2,...,p. We can express our measured undersampled spectra Embedded Image by the linear systemEmbedded Image(7)where Φi 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 Embedded Image. Although the linear system is highly underdetermined, a solution can be found if Embedded Image is sparse (i.e., if it is substantially nonzero for only a few frequencies) and the Φi are mutually incoherent.

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 fs via the delay time td. To ensure incoherence between the sampling matrices, we randomized our choices of td. 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.

Fig. 4 Wideband spectral reconstruction based on compressive sampling.

(A) Undersampled spectra obtained from time records with i = 1,...,7 random chosen sampling rates fs,i. The data represent records of duration T = 2 s that were averaged for 1.5 hours. (B and C) Reconstructed wideband spectrum containing l = 7 tones in two subbands. The insets show the individual tones with frequencies f(i) = fc – 15 Hz, f(ii) = fc, f(iii) = fc + 15 Hz, and f(iv) = fc + 1000 Hz (where fc = 400.75 kHz), as well as Embedded Image, Embedded Image, and Embedded Image (where Embedded Image). The noise reflects the incomplete image rejection of the reconstruction procedure.

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 1H 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 (3140)

References and Notes

  1. See supplementary materials.
  2. 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.
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