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Quantum interpolation for high-resolution sensingAshok Ajoya,b, Yi-Xiang Liua,b, Kasturi Sahaa,b, Luca Marsegliaa,b, Jean-Christophe Jaskulaa,b, Ulf Bissborta,b,c,and Paola Cappellaroa,b,1

aResearch Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139; bDepartment of Nuclear Science & Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139; and cSingapore University of Technology and Design, 487372 Singapore

Edited by Renbao Liu, The Chinese University of Hong Kong, Hong Kong, and accepted by Editorial Board Member Evelyn L. Hu November 28, 2016(received for review July 2, 2016)

Recent advances in engineering and control of nanoscale quan-tum sensors have opened new paradigms in precision metrology.Unfortunately, hardware restrictions often limit the sensor per-formance. In nanoscale magnetic resonance probes, for instance,finite sampling times greatly limit the achievable sensitivity andspectral resolution. Here we introduce a technique for coher-ent quantum interpolation that can overcome these problems.Using a quantum sensor associated with the nitrogen vacancycenter in diamond, we experimentally demonstrate that quan-tum interpolation can achieve spectroscopy of classical magneticfields and individual quantum spins with orders of magnitudefiner frequency resolution than conventionally possible. Not onlyis quantum interpolation an enabling technique to extract struc-tural and chemical information from single biomolecules, but itcan be directly applied to other quantum systems for superreso-lution quantum spectroscopy.

quantum sensing | quantum control | nanoscale NMR | NV centers

Precision metrology often needs to strike a compromisebetween signal contrast and resolution, because the hard-

ware apparatus sets limits on the precision and sampling rate atwhich the data can be acquired. In some cases, classical inter-polation techniques have become a standard tool to achieve asignificantly higher resolution than the bare recorded data. Forinstance, the Hubble Space Telescope uses classical digital imageprocessing algorithms like variable pixel linear reconstruction[Drizzle (1)] to construct a supersampled image from multi-ple low-resolution images captured at slightly different angles.Unfortunately, this classical interpolation method would failfor signals obtained from a quantum sensor, where the infor-mation is encoded in its quantum phase (2). Here we intro-duce a technique, which we call “quantum interpolation,” thatcan recover the intermediary quantum phase, by directly act-ing on the quantum probe dynamics, and effectively engineer aninterpolated Hamiltonian. Crucially, by introducing an optimalinterpolation construction, we can exploit otherwise deleteriousquantum interferences to achieve high fidelity in the resultingquantum phase signal.

Quantum systems, such as trapped ions (3), superconductingqubits (4, 5), and spin defects (6, 7) have been shown to per-form as excellent spectrum analyzers and lock-in detectors forboth classical and quantum fields (8–10). This sensing techniquerelies on modulation of the quantum probe during the interfero-metric detection of an external field. Such a modulation is typ-ically achieved by a periodic sequence of π-pulses that invertthe sign of the coupling of the external field to the quantumprobe, leading to an effective time-dependent modulation f (t)of the field (11–13). These sequences, more frequently used fordynamical decoupling (14, 15), can be described by sharp band-pass filter functions obtained from the Fourier transform of f (t).This description lies at the basis of their application for precisionspectroscopy, as the filter is well approximated by modified sincfunctions, F (ντ,N )≈ sin2(4πNντ)

sin2(4πντ), where 2τ is the time inter-

val between π-pulses, and N is the number of pulses. The filterpassband is centered at ν= 1/(4τ), its rejection (signal contrast)

increases with the number of pulses as N 2, and the band-passbandwidth (frequency resolution) decreases as ∆ν= 1/(4N τ),tremendously improving frequency resolution with increasingpulse numbers. Unfortunately, this high resolution can only beobtained if the experimental apparatus allows a correspondinglyfine time sampling ∆τ , with a precision 1/(4N ν). In practice, thisis an extremely serious limitation, because conventional hard-ware sampling bounds are quickly saturated, leading to losses inboth signal contrast and spectral resolution.

Our quantum interpolation technique can overcome these lim-itations in sensing resolution by capturing data points on a finermesh than directly accessible due to experimental constraints,in analogy to classical interpolation. However, instead of inter-polating the measured function values (which would contain nonew information), the objective is to interpolate the ideal sens-ing evolution operator (propagator) in a coherent way. The keyidea is presented in Fig. 1A. To achieve precision sensing ata desired frequency ν= 1/4τ , we use control sequences withdifferent π-pulse separations, τk = k∆τ and τk+1 = (k + 1)∆τ ,where ∆τ is the minimum timing step allowed by the hard-ware (and k is an integer). By combining different numbers ofthese building blocks, we can achieve an effective evolution forthe desired time N τ = (N − p)τk + pτk+1 (where 0≤ p≤N isan integer). This result would be trivial if the effective Hamil-tonian H during the pulsed evolution were constant, because(e iHτk )

N−p(e iHτk+1)

p= e iHN τ . This simple prescription, how-

ever, hides the subtleties of our quantum interpolation scheme;indeed, the high-frequency resolution of dynamical decouplingsensing schemes arises exactly from the fact that the effectiveHamiltonian does depend on the pulse timing. Then, extremecare must be taken in building the interpolated dynamics by a

Significance

Nanoscale magnetic resonance imaging enabled by quantumsensors is a promising path toward the outstanding goal ofdetermining the structure of single biomolecules at room tem-perature. We develop a technique, which we name “quantuminterpolation,” to improve the frequency resolution of thesequantum sensors far beyond limitations set by the experimen-tal controlling apparatus. The method relies on quantum inter-ference to achieve high-fidelity interpolation of the quantumdynamics between hardware-allowed time samplings, thusallowing high-resolution sensing. We demonstrate over twoorders of magnitude resolution gains, and discuss applicationsof our work to high-resolution nanoscale magnetic resonanceimaging.

Author contributions: A.A. and P.C. designed research; A.A., Y.-X.L., K.S., L.M., J.-C.J., U.B.,and P.C. performed research; A.A., Y.-X.L., K.S., L.M., J.-C.J., U.B., and P.C. analyzed data;and A.A., Y.-X.L., K.S., L.M., J.-C.J., U.B., and P.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. R.L. is a Guest Editor invited by the EditorialBoard.1To whom correspondence should be addressed. Email: pcappell@mit.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1610835114/-/DCSupplemental.

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A B C D

Fig. 1. Quantum Interpolation scheme. (A) Conceptual picture of quantum interpolation. The set of useful sensing propagators (UN(τk+p/N), blue line)constitute a one-dimensional manifold in the 4D operator space, shown here in the dominant 3D subspace (see SI Appendix, Quantum Interpolation:Theory & Practice, Geometric Representation). Conventionally, the unitary evolution of a quantum sensor can only be probed at discrete intervals UN(τk)(black cubes), with τk = k∆τ , and classical signal interpolation would miss an accurate description (dashed black line). Quantum interpolation faithfullyapproximates the evolution (UN(τk+p/N), green spheres) by coherent combination of pulse sequences [Insets, CPMG (16) sequences], allowing for samplingat arbitrary small time intervals. (B and C) (Top) NMR signal from a single 14N spin associated with the NV quantum sensor. (B) Sensing with conventionalsequences limited to ∆τ = 2 ns. (C) Quantum interpolation, improving the resolution to 110 ps. (Bottom) Quantum interpolation can reveal details of thesignal (the folding and reflection of the central peak leading to the double peak), as expected from the theoretical line shape at large pulse numbers (see SIAppendix, Interferometric Spin Sensing via the NV Center). (D) Filter function description of quantum interpolation. (Top) Time domain filter function f(t)for the desired (dashed green lines) and interpolated pulse sequence (solid blue lines) for the simplest case of a half-time interpolation with total sequencetime T . The deviation between these filters is the error function ε (shaded regions in Middle) that needs to be minimized for an optimal interpolationconstruction. (Bottom) Frequency domain representation of both filter functions and the Fourier Transform (FT) of their difference.

suitable ordering of the sequence blocks that will engineer thecorrect interpolated Hamiltonian.

Quantum InterpolationPrinciples. The building blocks of the quantum interpolateddynamics are propagators U(τk ) describing the quantum probeevolution under a control sequence unit composed of π-pulsesseparated by a time τk = k∆τ . Experimentally, thus, we combinewell-known control sequences such as the XY8 sequence (16–18). As shown in Fig. 1A, these operators U(τk ) can be thought

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Fig. 2. High-resolution sensing and spectroscopy. (A) Detection of the spurious harmonic of an AC magnetic field via quantum-interpolated XY16sequences. The incoherent external magnetic field is generated by an AC current at fAC = 2.5 MHz through a 20-µm wire located in the vicinity of theNV center. Our hardware limitation (∆τ = 1 ns) translates in a frequency resolution of ∆fAC = 35.3 kHz, and would cause a severe suppression of thedetected signal as its linewidth decreases linearly with the number of π-pulses. In the rightmost image, quantum interpolation enables supersampling at8.9 ps (an effective boost of 112), which still permits to resolve clearly a linewidth of 2.5 kHz. (B) Detection of incoherent AC magnetic fields with twodistinct frequencies. Quantum interpolation with a maximum of 672 π-pulses allows for a resolution gain of a factor 72 and faithfully reconstructs the ACfields, even if the two frequencies are not resolved by regular XY16 sequences with our timing resolution. (C) Linewidth of the detected AC magnetometrysignal (from A) with regular sampling (blue) and supersampling (green). The error bars are residuals to a Gaussian fit. (D) Sensing quality factor Q = f/∆fextracted from B. Conventional dynamical decoupling sequences can only achieve Q≤ 100. This limit can be surpassed with quantum interpolation, scalinglinearly with number of pulses, to reach Q ≈ 1,000.

of as a discrete subset of all of the possible propagators that onemight want to access for quantum sensing. Combining sequenceswith different timings (Fig. 1A, Inset), we construct a propagatorU (τk+p/N ) that approximates the desired interpolated dynam-ics, U(τk+p/N ),

UN (τk+p/N )≡P

{N−p∏m=1

U(τk )

p∏n=1

U(τk+1)

}≈ UN (τk+p/N ).

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Here P describes a permutation of the propagators (the pulsesequence order). Only by optimizing P , as explained in OptimalConstruction, can we achieve high-fidelity quantum interpolation,which would be otherwise limited by errors caused by the non-commutativity of the dynamics and further amplified when con-sidering a large number of pulses.

Before giving more details of the optimal construction, wedemonstrate the need and advantages of quantum interpolation(Fig. 1 B and C) by performing high spectral resolution magne-tometry using the electronic spin of the nitrogen vacancy (NV)center in diamond (19) as a nanoscale probe (7, 10, 20). Usinga conventional XY8-6 dynamical decoupling sequence (17, 18)to measure the 14N nuclear spin of the NV center, we obtain alow-resolution signal where the expected sinc-like dip is barelyresolved (Fig. 1B). Upon increasing the number of pulses toXY8-18, this narrow dip is completely lost. To enhance the sig-nal resolution, we use the optimized interpolation sequence thatcompletely mitigates the deleterious effects of timing resolutionand reveals the folding back of the dip into a double peak dueto strong interference between the NV and the 14N spin (Fig. 1Cand SI Appendix, Interferometric Spin Sensing via the NV Center).Thanks to quantum interpolation, the number of points that cannow be sampled scales linearly with the number of pulses N , andthe resolution still improves as 1/N . The sensing resolution isthus determined only by the quantum probe coherence time T2

(simultaneously extended due to dynamical decoupling) and thenumber of pulses that can be reliably applied.

Optimal Construction. The ordering of the different pulse seq-uence blocks is a crucial step in achieving an interpolatedpropagator that remains the most faithful approximation ofUN (τk+p/N ), even at large N . For instance, a naive construction,P = 1 in Eq. 1, would lead to fast error accumulation and thefailure of quantum interpolation. We tackle this problem by min-imizing the deviation ε(t) = |fU (t)− fU (t)| of the filter functionin the time domain (shaded regions in Fig. 1D) over the wholeevolution. This procedure yields the optimal control sequencefor any desired propagator, because we find that it also minimizesboth the filter function error and the infidelity of the interpolatedpropagator, UN (τk+p/N ), with the ideal one, UN (τk+p/N ). Intu-itively, the optimal construction compensates the error at eachdecoupling sequence block, to achieve a constant error that doesnot depend on the number N of pulses and scales as ∆τ , allow-ing access to a large number of interpolated points. We showanalytically and numerically that the errors for any interpolatedpropagator are approximately equal, and bounded by the error ofU 2(τk+1/2) = U2(τk+1/2) +O(∆τ2) (see SI Appendix, OptimalQuantum Interpolation Construction).

Experimental RealizationSensing Classical Fields. To demonstrate the power of quan-tum interpolation, we perform high-resolution magnetometryof a classical single-tone AC magnetic field at the frequencyfAC = 2.5 MHz. By applying optimally ordered quantum interpo-lated sequences (Fig. 2A), we detect the spurious harmonic (21)of frequency 2fAC. As the number of π-pulses is increased, thefilter function associated with the equivalent XY-N sequences,and, accordingly, the measured signal, becomes narrower. Thesignal linewidths are not affected by the finite time resolution, ashighlighted in Fig. 2C. Without quantum interpolation, we reachour experimental resolution limit after applying a sequence ofonly 64 π-pulses (XY8-8 sequence; Fig. 1 B and C). Quantuminterpolation enables AC magnetometry far beyond this limit:We obtain an improvement by a factor 112 in timing resolution,corresponding to a sampling time of 8.9 ps.

The advantage of quantum interpolation over conventionaldynamical decoupling sequences is manifest when the goal is to

resolve signals with similar frequencies. Fig. 2B shows that ourquantum sensor is able to easily detect a classic dual-tone pertur-bation, resolving fields that are separated by ∆f = 6.2 kHz, farbelow the limit set by our native 1-ns hardware time resolution.

A useful figure of merit to characterize the resolutionenhancement of quantum interpolation, in analogy to band-passfilters, is the Q value of the sensing peak, Q = f /∆f . The Q valuefor conventional decoupling pulse sequences is set by the finitetime resolution, Q = 1/(2f ∆τ). Quantum interpolation lifts thisconstraint, allowing Q ≈ 2N /π, limited only by the coherencetime T2, Nmax≤T2/(2τ), and pulse errors. Our experimentsillustrate that the effective sensing Q can be linearly boostedwith the pulse number to over 1,000 (Fig. 2D). Given typicalNV coherence time (1 ms), π-pulse length (50 ns), and timingresolution (1 ns), an impressive increase of about 104 over thehardware limits is achievable. Quantum interpolation can alsoenhance alternative high-resolution sensing techniques like cor-relation spectroscopy (22), and Hartmann–Hahn sensing (23, 24)(see also SI Appendix, Comparison with Other High-ResolutionSensing Techniques for a detailed comparison with thesemethods).

Sensing Quantum Systems. Even more remarkably, the coher-ent construction of quantum interpolation ensures that onecan measure not only classical signals, but also quantum sys-tems [e.g., coupled spins (25)] with high spectral resolution.This result is nontrivial, because it implies that we are notonly modulating the quantum sensor but also effectively engi-neering an interpolated Hamiltonian for the probed quantumsystem (26). Specifically, we consider a quantum probe cou-pled to the quantum system of interest via an interaction H =|0〉〈0|H0 + |1〉〈1|H1. Here H0,1 is the target system’s Hamil-tonian (of dimension D), which depends on which eigenstate|0〉, |1〉 the probe is in. Then, the propagator under a π-pulse

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Fig. 3. High-resolution spin detection. A modified XY8-12 sequenceenables an effective sampling at 48 ps (a 41-fold resolution gain with respectto the hardware-imposed ∆τ = 2 ns). The lineshape of the 14N NMR signaldisplays a slight asymmetry in the signal sidelobes, an expected feature (seeSI Appendix, Interferometric Spin Sensing via the NV Center) of the NMRsignal under the XY8 sequence (solid line). The agreement with the theory-fitted curve is very good, reflected by the relative residual SD being 3%.

Ajoy et al. PNAS Early Edition | 3 of 5

train (with timings as in the XY8 sequence; Fig. 1A) is given byUN (τ) = |0〉〈0|UN

0 (τ) + |1〉〈1|UN1 (τ), with

UN0,1(τ) = (e−iH0,1τe−iH1,02τe−iH0,1τ )

N. [2]

Sensing of the target quantum system is achieved via interfer-ence between the two evolution paths given by UN

0,1(τ), whichresults in the quantum probe signal S =

[1 + Tr(UN

0 UN†1 )/D

]/2

(27, 28). The interference is enhanced by increasing the num-ber of pulses N , and by a careful choice of the time τ , mak-ing it susceptible once again to finite timing resolution. Quan-tum interpolation can overcome this limitation, engineering anypropagator UN

0,1(τk+p/N )≈UN0,1(τk+p/N ) by suitably combining

UN−p0,1 (τk ) and Up

0,1(τk+1). It is somewhat surprising that thisconstruction would work: When considering a large numberof pulses, one would expect that the noncommutativity of thepropagators and the nonconvergence of the perturbative Baker–Campbell–Hausdorff expansion would amplify the discrepancybetween ideal and interpolated propagator. Fortunately, the con-struction developed for classical fields still keeps the error small(as we show in SI Appendix, Quantum Interpolation: Theory &Practice) because higher-order terms cancel out in the interfer-ence between the propagators U0,1.

Consider, for example, the coupling of a quantum probe (theNV center) to two-level systems (nuclear spins 1/2). NV cen-ters implanted a few nanometers below the diamond surfacehave recently emerged as the prime technology toward the long-standing goal of obtaining high spatial resolution structure ofsingle molecules in their natural environment, by performingnanoscale NMR spectroscopy (10, 17, 18, 29, 30). The outstand-ing key challenge is resolving the spectral features (and hencepositions) of densely packed networks of spins in such molecules.Frequency differences, as small as a few Hertz, arise from chem-ical shifts and the coupling to the NV. The Hamiltonian of thej th spin in the molecule is given byH(j)

0 =ωLIzj ; H(j)1 =ωLIzj +∑

ν A(j)zν Iνj , where ωL is the Larmor frequency of the spins, and

A(j)zν are the components of the coupling to the NV center. Then,

the U0,1 propagators are composed of nuclear spin rotations con-ditioned on the NV state. For weakly coupled spins, the max-imum interference signal arises at τ =π/

(2ωL + A

(j)zz

), when

the propagators correspond to rotations around two nonparallelaxes separated by an angle α= arctan

[A

(j)zx /(ωL +A

(j)zz /2)

]. The

angle between the nuclear spin rotation axes in the two NV man-ifolds is amplified with every subsequent application of π-pulse,giving rise to a signal contrast that grows with N 2. The destruc-tive interference is also amplified away from the sensing peak(27, 28), leading to a sinc linewidth that falls as 1/(4N τ), similarto the results obtained using the semiclassical filter picture (seeSI Appendix, Interferometric Spin Sensing via the NV Center).

To experimentally demonstrate the high-precision sensingreached by quantum interpolation, we measure the 14N nuclearspin via its coupling to the NV center electronic spin. Even ifthe 14N is strongly coupled to the NV (Azz =− 2.16 MHz), itusually does not give rise to an interferometric signal becauseof its transverse coupling Azx = 0. However, a small perpen-dicular field B⊥= 0.62 G generates an effective transverse

coupling γeB⊥Axx/(∆0− γeBz ), with Axx =− 2.62 MHz (31)and γe = 2.8 MHz/G as the NV gyromagnetic ratio. This effectbecomes sizable at a longitudinal magnetic field Bz = 955.7 Gthat almost compensates the NV zero-field splitting ∆0 = 2.87GHz. The 14N nuclear spin frequency is largely set by itsquadrupolar interaction P =− 4.95 MHz, a high frequencybeyond our timing resolution (Fig. 1B). We used quantum inter-polation to supersample the signal at 48 ps (a 41-fold resolu-tion gain), revealing precise features of the spectral lineshape(Fig. 3), including the expected slight asymmetry in sidelobes (SIAppendix, Interferometric Spin Sensing via the NV Center). Detect-ing this distinct spectral feature confirms that quantum interpo-lation can, indeed, achieve a faithful measurement of the quan-tum signal, as we find an excellent match of the experimentaldata with the theoretical model, with the error being less than3% for most interpolated points. The ability to probe the exactspectral lineshape provides far more information than just thesignal peaks, especially when there could be overlapping peaksor environment-broadened linewidths.

Conclusion and OutlookThese results have immediate and far-reaching consequences fornanoscale NV NMR (9, 10, 32), where our technique can mapspin arrangements of a nearby single protein with a spatial res-olution that dramatically improves with the number of pulses.The Q value provides an insightful way to quantify the resolu-tion gains for these applications. With a Q ≈ 104 that is cur-rently achievable, 13C chemical shifts of aldehyde and aromaticgroups can now be measured (33). Beyond sensing nuclear spins,we envision quantum interpolation to have important applica-tions in condensed matter, to sense high-frequency (hence highQ) signals (34), such as those arising from the excitation of spin-wave modes in magnetic materials like yittrium iron garnett (35).

In conclusion, we have developed a quantum interpolationtechnique that achieves substantial gains in quantum sensing res-olution. We demonstrated its advantages by performing high-frequency-resolution magnetometry of both classical fields andsingle spins using NV centers in diamond. The technique allowspushing spectral resolution limits to fully exploit the long coher-ence times of quantum probes under decoupling pulses. Quan-tum interpolation could also enhance the performance of otherNV-based sensing technique. We experimentally demonstratedresolution gains of 112, and Q-value gains of over 1,000, althoughthe ultimate limits of the technique can be at least an order ofmagnitude larger. Quantum interpolation thus turns quantumsensors into high-resolution and high-Q spectrum analyzers ofclassical and quantum fields. We expect quantum interpolationto be an enabling technique for nanoscale single-molecule spec-troscopy at high magnetic fields (36), allowing the discrimina-tion of chemical shifts and angstrom-resolution single-moleculestructure.

ACKNOWLEDGMENTS. We thank E. Bauch, F. Casola, D. Glenn, F. Jelezko,S. Lloyd, M. Lukin, E. Rosenfeld, and R. Walsworth for stimulating discus-sions and careful reading of the manuscript. This work was supported, inpart, by the US Army Research Office through Grants W911NF-11-1-0400 andW911NF-15-1-0548 and by the NSF Grant PHY0551153 (Center for UltracoldAtoms).

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