## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Quantum interpolation for high-resolution sensing

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)

## Significance

Nanoscale magnetic resonance imaging enabled by quantum sensors is a promising path toward the outstanding goal of determining the structure of single biomolecules at room temperature. We develop a technique, which we name “quantum interpolation,” to improve the frequency resolution of these quantum sensors far beyond limitations set by the experimental controlling apparatus. The method relies on quantum interference to achieve high-fidelity interpolation of the quantum dynamics between hardware-allowed time samplings, thus allowing high-resolution sensing. We demonstrate over two orders of magnitude resolution gains, and discuss applications of our work to high-resolution nanoscale magnetic resonance imaging.

## Abstract

Recent advances in engineering and control of nanoscale quantum sensors have opened new paradigms in precision metrology. Unfortunately, hardware restrictions often limit the sensor performance. In nanoscale magnetic resonance probes, for instance, finite sampling times greatly limit the achievable sensitivity and spectral resolution. Here we introduce a technique for coherent quantum interpolation that can overcome these problems. Using a quantum sensor associated with the nitrogen vacancy center in diamond, we experimentally demonstrate that quantum interpolation can achieve spectroscopy of classical magnetic fields and individual quantum spins with orders of magnitude finer frequency resolution than conventionally possible. Not only is quantum interpolation an enabling technique to extract structural and chemical information from single biomolecules, but it can be directly applied to other quantum systems for superresolution quantum spectroscopy.

Precision metrology often needs to strike a compromise between signal contrast and resolution, because the hardware apparatus sets limits on the precision and sampling rate at which the data can be acquired. In some cases, classical interpolation techniques have become a standard tool to achieve a significantly higher resolution than the bare recorded data. For instance, the Hubble Space Telescope uses classical digital image processing algorithms like variable pixel linear reconstruction [Drizzle (1)] to construct a supersampled image from multiple low-resolution images captured at slightly different angles. Unfortunately, this classical interpolation method would fail for signals obtained from a quantum sensor, where the information is encoded in its quantum phase (2). Here we introduce a technique, which we call “quantum interpolation,” that can recover the intermediary quantum phase, by directly acting on the quantum probe dynamics, and effectively engineer an interpolated Hamiltonian. Crucially, by introducing an optimal interpolation construction, we can exploit otherwise deleterious quantum interferences to achieve high fidelity in the resulting quantum phase signal.

Quantum systems, such as trapped ions (3), superconducting qubits (4, 5), and spin defects (6, 7) have been shown to perform as excellent spectrum analyzers and lock-in detectors for both classical and quantum fields (8⇓–10). This sensing technique relies on modulation of the quantum probe during the interferometric detection of an external field. Such a modulation is typically achieved by a periodic sequence of

Our quantum interpolation technique can overcome these limitations in sensing resolution by capturing data points on a finer mesh than directly accessible due to experimental constraints, in analogy to classical interpolation. However, instead of interpolating the measured function values (which would contain no new information), the objective is to interpolate the ideal sensing evolution operator (propagator) in a coherent way. The key idea is presented in Fig. 1*A*. To achieve precision sensing at a desired frequency

## Quantum Interpolation

### Principles.

The building blocks of the quantum interpolated dynamics are propagators *A*, these operators *A*, *Inset*), we construct a propagator

Here *Optimal Construction*, can we achieve high-fidelity quantum interpolation, which would be otherwise limited by errors caused by the noncommutativity of the dynamics and further amplified when considering a large number of pulses.

Before giving more details of the optimal construction, we demonstrate the need and advantages of quantum interpolation (Fig. 1 *B* and *C*) by performing high spectral resolution magnetometry using the electronic spin of the nitrogen vacancy (NV) center in diamond (19) as a nanoscale probe (7, 10, 20). Using a conventional XY8-6 dynamical decoupling sequence (17, 18) to measure the ^{14}N nuclear spin of the NV center, we obtain a low-resolution signal where the expected sinc-like dip is barely resolved (Fig. 1*B*). Upon increasing the number of pulses to XY8-18, this narrow dip is completely lost. To enhance the signal resolution, we use the optimized interpolation sequence that completely mitigates the deleterious effects of timing resolution and reveals the folding back of the dip into a double peak due to strong interference between the NV and the ^{14}N spin (Fig. 1*C* and *SI Appendix*, *Interferometric Spin Sensing via the NV Center*). Thanks to quantum interpolation, the number of points that can now be sampled scales linearly with the number of pulses *N*, and the resolution still improves as

### Optimal Construction.

The ordering of the different pulse sequence blocks is a crucial step in achieving an interpolated propagator that remains the most faithful approximation of **1**, would lead to fast error accumulation and the failure of quantum interpolation. We tackle this problem by minimizing the deviation *D*) over the whole evolution. This procedure yields the optimal control sequence for any desired propagator, because we find that it also minimizes both the filter function error and the infidelity of the interpolated propagator, *SI Appendix*, *Optimal Quantum Interpolation Construction*).

## Experimental Realization

### Sensing Classical Fields.

To demonstrate the power of quantum interpolation, we perform high-resolution magnetometry of a classical single-tone AC magnetic field at the frequency *A*), we detect the spurious harmonic (21) of frequency *C*. Without quantum interpolation, we reach our experimental resolution limit after applying a sequence of only 64 *B* and *C*). Quantum interpolation 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 conventional dynamical decoupling sequences is manifest when the goal is to resolve signals with similar frequencies. Fig. 2*B* shows that our quantum sensor is able to easily detect a classic dual-tone perturbation, resolving fields that are separated by

A useful figure of merit to characterize the resolution enhancement of quantum interpolation, in analogy to band-pass filters, is the *Q* value of the sensing peak, *Q* value for conventional decoupling pulse sequences is set by the finite time resolution, *Q* can be linearly boosted with the pulse number to over 1,000 (Fig. 2*D*). Given typical NV coherence time (1 ms), *SI Appendix*, *Comparison with Other High-Resolution Sensing Techniques* for a detailed comparison with these methods).

### Sensing Quantum Systems.

Even more remarkably, the coherent construction of quantum interpolation ensures that one can measure not only classical signals, but also quantum systems [e.g., coupled spins (25)] with high spectral resolution. This result is nontrivial, because it implies that we are not only modulating the quantum sensor but also effectively engineering an interpolated Hamiltonian for the probed quantum system (26). Specifically, we consider a quantum probe coupled to the quantum system of interest via an interaction *A*) is given by

Sensing of the target quantum system is achieved via interference between the two evolution paths given by *SI Appendix*, *Quantum Interpolation: Theory & Practice*) because higher-order terms cancel out in the interference between the propagators

Consider, for example, the coupling of a quantum probe (the NV center) to two-level systems (nuclear spins *SI Appendix*, *Interferometric Spin Sensing via the NV Center*).

To experimentally demonstrate the high-precision sensing reached by quantum interpolation, we measure the ^{14}N nuclear spin via its coupling to the NV center electronic spin. Even if the ^{14}N is strongly coupled to the NV (^{14}N nuclear spin frequency is largely set by its quadrupolar interaction *B*). We used quantum interpolation to supersample the signal at 48 ps (a 41-fold resolution gain), revealing precise features of the spectral lineshape (Fig. 3), including the expected slight asymmetry in sidelobes (*SI Appendix*, *Interferometric Spin Sensing via the NV Center*). Detecting this distinct spectral feature confirms that quantum interpolation can, indeed, achieve a faithful measurement of the quantum signal, as we find an excellent match of the experimental data with the theoretical model, with the error being less than 3% for most interpolated points. The ability to probe the exact spectral lineshape provides far more information than just the signal peaks, especially when there could be overlapping peaks or environment-broadened linewidths.

## Conclusion and Outlook

These results have immediate and far-reaching consequences for nanoscale NV NMR (9, 10, 32), where our technique can map spin arrangements of a nearby single protein with a spatial resolution that dramatically improves with the number of pulses. The *Q* value provides an insightful way to quantify the resolution gains for these applications. With a ^{13}C chemical shifts of aldehyde and aromatic groups can now be measured (33). Beyond sensing nuclear spins, we envision quantum interpolation to have important applications in condensed matter, to sense high-frequency (hence high *Q*) 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 interpolation technique that achieves substantial gains in quantum sensing resolution. We demonstrated its advantages by performing high-frequency-resolution magnetometry of both classical fields and single spins using NV centers in diamond. The technique allows pushing spectral resolution limits to fully exploit the long coherence times of quantum probes under decoupling pulses. Quantum interpolation could also enhance the performance of other NV-based sensing technique. We experimentally demonstrated resolution gains of 112, and *Q*-value gains of over 1,000, although the ultimate limits of the technique can be at least an order of magnitude larger. Quantum interpolation thus turns quantum sensors into high-resolution and high-*Q* spectrum analyzers of classical and quantum fields. We expect quantum interpolation to be an enabling technique for nanoscale single-molecule spectroscopy at high magnetic fields (36), allowing the discrimination of chemical shifts and angstrom-resolution single-molecule structure.

## Acknowledgments

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

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: pcappell{at}mit.edu.

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 Editorial Board.

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

## References

- ↵.
- Fruchter A,
- Hook RN

- ↵
- ↵
- ↵.
- Yan F, et al.

- ↵
- ↵
- ↵
- ↵
- ↵.
- Shi F, et al.

- ↵.
- Lovchinsky I, et al.

- ↵.
- Cywinski L,
- Lutchyn RM,
- Nave CP,
- DasSarma S

- ↵
- ↵.
- Smith PES,
- Bensky G,
- Álvarez GA,
- Kurizki G,
- Frydman L

- ↵
- ↵.
- de Lange G,
- Wang ZH,
- Riste D,
- Dobrovitski VV,
- Hanson R

- ↵
- ↵.
- Staudacher T, et al.

- ↵.
- Mamin HJ, et al.

- ↵
- ↵
- ↵.
- Loretz M, et al.

- ↵.
- Laraoui A,
- Meriles CA

*ACS Nano*7(4):3403–3410. PMID: 23565720. - ↵.
- Belthangady C, et al.

- ↵
- ↵
- ↵.
- Ajoy A,
- Cappellaro P

- ↵
- ↵
- ↵
- ↵
- ↵.
- Chen M,
- Hirose M,
- Cappellaro P

- ↵.
- Ajoy A,
- Bissbort U,
- Lukin MD,
- Walsworth RL,
- Cappellaro P

- ↵.
- Ernst RR,
- Bodenhausen G,
- Wokaun A

- ↵.
- van der Sar T,
- Casola F,
- Walsworth R,
- Yacoby A

- ↵
- ↵.
- Hemmer P

## Citation Manager Formats

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.