Quantum sensing with arbitrary frequency resolution

TL;DR

This paper presents a quantum sensing method that achieves arbitrary frequency resolution using quantum lock-in detection with nitrogen vacancy centers, enabling highly sensitive magnetic field measurements over a broad bandwidth.

Contribution

The authors introduce a novel quantum sensing approach with frequency resolution limited only by an external clock, demonstrated using NV centers for magnetic field detection.

Findings

Achieved 70 μHz frequency resolution over MHz bandwidth.

Demonstrated high sensitivity with SNR > 10,000:1 in one hour.

Applicable to magnetic resonance and quantum simulation.

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 uHz over a MHz bandwidth. The continuous sampling further guarantees an excellent sensitivity, reaching a signal-to-noise ratio in excess of 10,000:1 for a 170 nT test signal measured during a one-hour interval. Our technique has applications in magnetic resonance spectroscopy, quantum simulation, and sensitive signal detection.

Equations

$${\varphi }_k=\frac{2t_{\mathrm{a}}}{\pi }x(t_k),$$

$$p_k=\frac{1}{2}(1-\sin {\varphi }_k)\approx \frac{1}{2}(1-{\varphi }_k)$$

$$y_k=\mathrm{Pois}\left[C(1-ϵ\mathrm{Bn}[p_k])\right],$$

$$\mathrm{SNR}=\frac{Y_j}{{\sigma }_Y}.$$

$$Y_j\approx \frac{1}{16}{N}^2{C}^2{ϵ}^2{\varphi }_{\max }^2,$$

$$\mathrm{SNR}=\frac{\frac{1}{16}N{C}^2{ϵ}^2{\varphi }_{\max }^2}{\frac{1}{4}{C}^2{ϵ}^2+C(1-\frac{ϵ}{2})}.$$

$$\left(\begin{array}{c}{\vec{Y}}_1\\ {\vec{Y}}_2\\ ⋮\\ {\vec{Y}}_p\end{array}\right)=\left(\begin{array}{c}{\Phi }_1\\ {\Phi }_2\\ ⋮\\ {\Phi }_p\end{array}\right)\vec{X},$$

Full text

Quantum sensors with new capabilities are driving the field of precision metrology Giovannetti et al. (2011); Degen et al. (2016). In particular, spin qubits associated with crystal defects in diamond Doherty et al. (2013) and other materials Pla et al. (2012); Widmann et al. (2015); Kolesov et al. (2012) have emerged as excellent probes with nanometer spatial resolution Balasubramanian et al. (2008); Maletinsky et al. (2012). 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 techniques from quantum control 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 Cywinski et al. (2008); Kotler et al. (2011); Lange et al. (2011); Alvarez and Suter (2011). Other, more recent techniques include dressed-state approaches Yan et al. (2013); Loretz et al. (2013), Floquet spectroscopy Lang et al. (2015), and correlative measurements Laraoui et al. (2010, 2013). Crucially, the spectral resolution of all of these techniques is limited by the state life time of the qubit probe. For nitrogen vacancy centers in diamond, reported spectral resolutions are a few Hz at best, even when assisted by a long-lived quantum memory Zaiser et al. (2016); Rosskopf et al. (2016); Pfender et al. (2016).

We introduce a simple concept where 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 Kotler et al. (2011); Lange et al. (2011) 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 Donoho (2006). The periodic sampling further guarantees that the signal-to-noise ratio (SNR) increases in proportion to the measurement time. We demonstrate our method by recording signal traces for up to $4\,\mathrm{hours}$, reaching a frequency resolution of $70\,\mathrm{\,\mu{\rm Hz}}$ and a precision of $260\,\mathrm{nHz}$ with $\mathrm{SNR}>10^{4}$.

Our experimental demonstration makes use of a spin qubit formed by the negatively-charged nitrogen vacancy (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 Maletinsky et al. (2012) and addresses a broad range of potential applications in physics, materials science and biology Rondin et al. (2014); Schirhagl et al. (2014).

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

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 $\pi$ pulses with inter-pulse spacing $\tau$. This defines the lock-in detection frequency $f_{\mathrm{LI}}=m/(2\tau)$ where $m=1,3,5,...$ is the harmonic order. The frequency bandwidth of the lock-in is approximately $f_{\mathrm{LI}}\pm 1/(2t_{\mathrm{a}})$ Lange et al. (2011); Degen et al. (2016). For an a.c. signal $x(t)=\Omega\cos(2\pi f_{\mathrm{ac}}t)$ with a frequency $f_{\mathrm{ac}}\approx f_{\mathrm{LI}}$ within this bandwidth, the quantum phase accumulated after time $t_{\mathrm{a}}$ is

[TABLE]

where $t_{k}$ marks the start of the lock-in measurement and $\Omega$ is the signal amplitude in units of angular frequency (Supplementary Text 1). Crucially, although the quantum phase is accumulated over an extended time interval $[t_{k},t_{k}+t_{\mathrm{a}}]$, its value reflects the instantaneous value of $x(t)$ at time $t=t_{k}$. To read out the quantum phase, the quantum state is measured in the $Y$ basis yielding a probability

[TABLE]

to find the system pointing along the $-Y$ direction. The approximation is for small $|\phi_{k}|\ll\pi/2$ within the sensor’s linear range Kotler et al. (2013). Optical readout finally converts the projected state into a photon number $y_{k}$. Because state projection and optical readout are stochastic processes, $y_{k}$ is a random variable,

[TABLE]

where Bn is a Bernoulli process which takes the value 1 with a probability of $p_{k}$ and the value 0 with probability $1-p_{k}$, and Pois is a Poisson process that reflects the photon shot noise. $C$ is a variable readout gain and $\epsilon$ is the optical contrast.

By collecting a time trace of $N$ measurement outputs $\{y_{k}\}_{k=1}^{N}$ at sampling times $t_{k}=kt_{\mathrm{s}}$, we can sample the signal $x(t)$ at a sub-Nyquist rate $f_{\mathrm{s}}=1/t_{\mathrm{s}}$. 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 $\delta f=f_{\mathrm{s}}/N$ 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 sub-levels of single NV centers located in diamond nanopillar waveguides (see Materials and Methods). At a bias field of $457\,\mathrm{mT}$ applied along the NV symmetry axis, the transition frequency between these states is $9916\,\mathrm{MHz}$. We initialized the qubit using a 532 nm laser pulse and a microwave $\pi/2$ pulse, and detected the qubit state using a phase-shifted $\pi/2$ pulse followed by optical readout. We used an indirect readout scheme where the final qubit state was first stored in the 15N nuclear spin ($I=1/2$), serving as a memory qubit Rosskopf et al. (2016), and we then repetitively read out the 15N spin state by a nuclear quantum non-demolition (QND) measurement Jiang et al. (2009); Neumann et al. (2010). By varying the number of QND measurements $n$ we could adjust the readout gain $C$ between ca. $0-230$ photons. The optical contrast was $\epsilon\approx 0.35$.

As a first illustration of the continuous sampling technique, Figs. 2A,B show a time trace and spectrum of an amplitude modulated (AM) magnetic test signal with carrier frequency $f_{\mathrm{c}}=601.2547\,\mathrm{kHz}$ and modulation frequency $f_{\mathrm{AM}}=10\,\mathrm{mHz}$. The test signal had an amplitude of approximately $170\,\mathrm{nT}$, corresponding to $\Omega=2\pi\times 4.7\,\mathrm{kHz}$, and was generated by passing an a.c. current through a nearby wire. The signal contained three components at frequencies $f_{\mathrm{c}}$ and $f_{\mathrm{c}}\pm f_{\mathrm{AM}}$ with a power ratio of 1:4:1. The frequency resolution $\delta f$ of the spectrum, obtained from a time trace of one hour duration, was $\delta f=1/T=278\,\mathrm{\,\mu{\rm Hz}}$ (Fig. 2B, right inset). Because our signal was undersampled, the abscissa in Fig. 2B indicates the detuning from $f_{\mathrm{c}}$ rather than the absolute frequency. The observed $\,\mu{\rm Hz}$ frequency resolution and the consistent amplitude ratio between carrier and side peaks illustrate the capabilities of our method.

While our strategy allows for an arbitrary frequency resolution $\delta f$, we are more interested in how precisely we can determine a signal’s linewidth and center frequency in the experiment. Fig. 2D depicts the decrease in the fitted linewidth parameter $\gamma$ with increasing measurement time $T$ for four signals. Signals (i-iii) were produced by amplitude modulation similar to Fig. 2B and had zero intrinsic linewidth, $\gamma_{\mathrm{int}}=0$. The linewidth parameter for these signals scaled with $\gamma\propto T^{-1}$, 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 (FM) with Gaussian noise in order to mimic a non-zero intrinsic linewidth $\gamma_{\mathrm{int}}>0$. The linewidth parameter for signal (iv) initially also decreased as $T^{-1}$ but leveled out as $\gamma$ approached $\gamma_{\mathrm{int}}$. Fig. 2D further shows the fit errors in the peak frequencies for all signals. Here, a $T^{-1.5}$ scaling was observed that reduced to $T^{-0.5}$ once the intrinsic linewidth became significant, 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. In order to quantify the sensitivity, we compared the peak amplitude $Y_{j}$ with the standard deviation $\sigma_{Y}$ of the noise floor in the power spectrum (see Fig 2B). This defines a power signal-to-noise ratio,

[TABLE]

Assuming that the entire signal power is concentrated in a single Fourier component $Y_{j}$ of the spectrum (i.e., that the linewidth of the signal is smaller than $\delta f$), it follows from Eq. 2 and 3 that

[TABLE]

where $\phi_{\mathrm{max}}=2t_{\mathrm{a}}\Omega/\pi$ 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 $\sigma_{Y}$ is the sum of two contributions, one from quantum projection noise with variance $\frac{1}{4}NC^{2}\epsilon^{2}$ and one from optical shot noise with variance $NC(1-\frac{\epsilon}{2})$. The SNR becomes

[TABLE]

Because $N=Tf_{\mathrm{s}}$, the SNR improves proportional to the duration of the time record $T$.

To further optimize the SNR, we adjusted the phase amplitude $\phi_{\mathrm{max}}$ and the readout gain $C$. We achieved this by varying the sensing time $t_{\mathrm{a}}$ and the readout time $t_{\mathrm{r}}$. First, we increased the sensing time $t_{\mathrm{a}}$ so that the quantum phase covered the full linear range of the lock-in, typically $\phi_{\mathrm{max}}\sim 0.5$. Although larger $\phi_{\mathrm{max}}$ are possible, the response of the lock-in becomes non-linear Kotler et al. (2013) 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, 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 $C(n)\approx n\times 0.105\,\mathrm{photons}$ and $t_{\mathrm{r}}\approx n\times 2.32\,\mathrm{\,\mu{\rm s}}$. Fig. 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 $n\approx 260$, which corresponded to the threshold gain $C_{\mathrm{thresh}}=4/\epsilon^{2}-2/\epsilon\approx 27$ where shot noise and quantum projection noise are balanced Degen et al. (2016). Increasing the gain beyond $C_{\mathrm{thresh}}$ 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 Neumann et al. (2010).

Thus far, all 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 $f_{\mathrm{c}}\sim 601\,\mathrm{kHz}$ was sampled at $f_{\mathrm{s}}=0.237\,\mathrm{kHz}$, which is about $5\cdot 10^{3}$ times slower than the Nyquist rate. We now discuss a strategy that overcomes this limitation using compressive sampling. We implemented this strategy by recording a set of time traces with slightly different sampling rates $f_{\mathrm{s}}$.

Compressive sampling (CS) exploits our prior knowledge about the sparsity of the wideband spectrum Donoho (2006); Candes et al. (2006). Suppose the vector $\vec{X}$ holds the Fourier components of the desired wideband spectrum sampled at or above the Nyquist rate, and the vectors $\vec{Y}_{i}$ represent a small set of undersampled spectra with $i=1,2,\dots,p$. We can express our measured undersampled spectra $\vec{Y}_{i}$ by the linear system

[TABLE]

where $\Phi_{i}$ are sampling matrices folding the wideband spectrum into the bandwidths of the undersampled spectra Sun et al. (2012). To reconstruct the wideband spectrum, we solve Eq. (7) for $\vec{X}$. Although the linear system is highly underdetermined, a solution can be found if $\vec{X}$ is sparse (is significantly non-zero only for a few frequencies) and the $\Phi_{i}$ 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 $f_{\mathrm{s}}$ via the delay time $t_{\mathrm{d}}$. To ensure incoherence between the sampling matrices, we randomized our choices of $t_{\mathrm{d}}$. Fig. 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 1,200 kHz. Although the SNR of the reconstructed spectrum is reduced due to 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 Mamin et al. (2013); Staudacher et al. (2013); Loretz et al. (2014), where 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 Cai et al. (2013). 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 Mehring (2012); Reif (2012). Ultrahigh resolution NMR is able to resolve couplings of a few mHz Appelt et al. (2006) under favorable conditions and achieves $<20\,\mathrm{Hz}$ linewidths even for 1H in dense solid samples Reif (2012). 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 $\propto T$ scaling of the SNR.

We thank T. Rosskopf, K. Chang, A. Retzker and F. Jelezko for experimental support and useful discussions. This work was supported by the Swiss NSF Project Grant $200021\_137520$, by the Swiss NSF NCCR QSIT, and by the FP7-611143 DIADEMS programme of the European Commission.

Note added to the arXiv version: This arXiv preprint corresponds to the manuscript resubmitted to Science Boss et al. (2017) with some further literature references included. We also acknowledge similar work by Jelezko and coworkers Schmitt et al. (2017) and Walsworth and coworkers Bucher et al. (2017).