Abstract
Essential to the functionality of qubitbased sensors are control protocols, which shape their response in frequency space. However, in common control routines outofband spectral leakage complicates interpretation of the sensor’s signal. In this work, we leverage discrete prolate spheroidal sequences (a.k.a. Slepian sequences) to synthesize provably optimal narrowband controls ideally suited to spectral estimation of a qubit’s noisy environment. Experiments with trapped ions demonstrate how spectral leakage may be reduced by orders of magnitude over conventional controls when a near resonant driving field is modulated by Slepians, and how the desired narrowband sensitivity may be tuned using concepts from RF engineering. We demonstrate that classical multitaper techniques for spectral analysis can be ported to the quantum domain and combined with Bayesian estimation tools to experimentally reconstruct complex noise spectra. We then deploy these techniques to identify previously immeasurable frequencyresolved amplitude noise in our qubit’s microwave synthesis chain.
Introduction
Industrial, metrological, and medical applications provide a strong pull for advanced nanoscale sensing techniques^{1}, exploiting the exquisite sensitivity of quantum coherent systems to their environments^{2,3,4}. Qubits naturally exhibit broadband coupling to their environments, but the application of a temporally modulated driving field can alter their frequency response in a desired way. For instance, application of modulation which periodically flips the qubit’s state has allowed for a narrowband spectral response^{5}, which may be tuned by adjusting the interpulse spacing or extending the sequence duration. This general approach to “dynamical decoupling noise spectroscopy” has seen broad adoption in quantum information^{6,7,8,9,10,11}, as well as in nanoscale diamond sensors for biomedical and physicsbased applications^{12,13,14,15}. However, control implemented in this form suffers from the significant complication of spectral leakage, where signals outside of a target sensing frequency band can contribute to the sensor’s response (Fig. 1b), and if not properly accounted for, can lead to bias when estimating the spectral density of a signal from experimental data^{16,17}.
In an ideal scenario, for frequency domain spectral estimation applications, the chosen control protocol would be sensitive only within a userdetermined band. Pulsed dynamical decoupling is often employed because the leading component of the filter transfer function describing the modulated sensor’s performance is narrowly peaked^{5,7}. Examination of the control propagator describing the time–domain response of a qubit subject to this control, however, reveals that the effective squarewaveform of the control propagator (Fig. 1c) leads to the appearance of an infinite chain of harmonics in the Fourier domain (Fig. 1e). These outofband harmonics can then contribute bias in noise spectroscopy protocols.
The problem of spectral leakage is well known in classical signal processing and has led to the development of time–bandwidthoptimized functions for use in spectral estimation. The discrete prolate spheroidal sequences (DPSS)^{18} are an orthogonal set of discrete time functions that maximize a signal’s energy within a predefined frequency band (Fig. 1d)^{19}. The DPSS form the basis of the multitaper method of spectral analysis^{20}, which is employed in estimation problems across a wide range of physical, computational, and biomedical disciplines^{21,22}. Additionally, DPSS have also been suggested in magnetic resonance imaging to avoid outofband excitation (socalled Gibbs artifact^{23}) and have recently enabled the design of optimal control algorithms for unitary quantum dynamics, that incorporate bandwidth constraints^{24}. This strong base of demonstrations motivates our use of DPSSmodulated pulses for quantum sensing.
In this work, we adapt DPSS functions to the problem of spectral leakage in quantum control for qubitbased sensors. We introduce the concept of continuously amplitudemodulated control waveforms defined by DPSS functions as effective window functions, and we demonstrate that such controls afford suppression of spectral leakage in the quantum setting. Experiments with trapped atomic ions are used to reconstruct the filter transfer functions and show orders of magnitude improvement in suppression of outofband signals relative to conventional square waveforms (e.g., pulsed dynamical decoupling or instantaneous phaseflips under driven rotary echo^{25,26}). We then present a series of RF engineeringinspired techniques to shift the band center of the control’s frequency response and combine this with tomographic measurements to allow disambiguation of sensor response in a cluttered background. Two techniques for spectral reconstruction are presented, both inspired by the original classical multitaper method, and we compare their performance for experimentally reconstructed spectra. Experimental results show that our new noninverting quantum multitaper performs similarly to more computationally intensive Bayesian estimation routines, at the expense of frequency resolution in the reconstruction. Finally, we employ these techniques to characterize otherwise inaccessible properties of our qubit drive system and provide frequencyresolved characterization of system noise and nonlinearities with calibrated sensitivity to 0.001 dB.
Results
The DPSS functions
For a time–domain sequence consisting of N elements, characterized by sampling interval Δt, and halfbandwidth parameter W ∈ (0, 1/2), the DPSS may be defined as real solutions to the eigenvalue problem
where \(v_m^{(k)}(N,W)\) is the nth element of the kth order DPSS for k, n ∈ {0, 1, …, N − 1}. The discrete Fourier transform of \(v_m^{(k)}(N,W)\) into the (angular) frequency domain [−π/Δt, π/Δt] is the discrete prolate spheroidal wavefunction, U ^{(k)}(N, W;ω), which is spectrally concentrated in [−ω _{ B }, ω _{ B }] ≡ [−2πW/Δt, 2πW/Δt].
The eigenvalue λ _{ k }(N, W) directly quantifies the spectral concentration of U ^{(k)}(N, W;ω), i.e., the fraction of spectral power within the target band compared to the total spectral power, as \(\lambda _k(N,W) = \frac{{{\int}_{  \omega _B}^{\omega _B} {\rm d}\omega U^{(k)}(N,W;\omega )^2}}{{{\int}_{  \pi /{\mathrm{\Delta }}t}^{\pi /{\mathrm{\Delta }}t} {\rm d}\omega U^{(k)}(N,W;\omega )^2}}.\) DPSS are optimal in the sense that, among all sequences with fixed N and W, they are the ones that maximize the above ratio. Spectral concentration is highest for k = 0, and decreases with increasing k; the DPSS of order \(k < 2\left\lfloor {NW} \right\rfloor  1\) have ≥70% of their spectral weight within the target band (Fig. 1e; Supplementary Table 1). Physically, the time–bandwidth product NW controls the fraction of the desired pulse energy within the pulse time τ = NΔt. Increasing NW enables concentration to be maintained for higher k, but this has the disadvantage of extending the target band in the frequency domain. In practice, setting NW = k + 1 is usually a satisfactory compromise between these factors.
DPSS filter transfer function reconstruction
To characterize the frequency response of a qubitbased sensor undergoing an arbitrary control protocol in the presence of multiaxis classical noise, we rely on the filter transfer function formalism^{5,27,28,29,30,31}, which we also outline in the Supplementary Note 4. The qubit sensor’s response to its environment under the application of control is given approximately by the average measured fidelity of the operation, denoted here as \({\cal F}_{{\mathrm{av}}}\). This is captured, in the weaknoise limit (Methods), by the spectral overlap of the noise power spectral density in multiple quadratures, S _{ i }(ω), with a transfer function describing the control, F _{ i }(ω), as \({\cal F}_{{\mathrm{av}}} \approx 1  {\mathrm{exp}}\left[ {  \pi ^{  1}\mathop {\sum}\nolimits_{i = \Omega ,z} {\int} {\mathrm{d}}\omega F_i(\omega )S_i(\omega )} \right].\) Here, the sum is taken over noise contributions in the amplitude quadrature, proportional to the applied control, i.e., the qubit Rabi frequency ∝Ωσ _{ x }, and in the dephasing quadrature, ∝σ _{ z }. The presence of a signal in the sensor’s target band, defined by the applied control modulation, will be manifested as a reduction in the fidelity of the operation implemented. The latter may be the identity, for odd k, or, in general, another nontrivial quantum state transformation (Fig. 1a).
Analytic calculation of F _{Ω}(ω) both for flattop modulation (commonly associated with dynamical decoupling protocols and here a rotary spin echo, Fig. 1c) and for piecewiseconstant modulation defined by the DPSSmodulated pulses, reveals the superior spectral concentration of the latter (Fig. 1e). While the main lobe of F _{Ω}(ω) is broader inside the target band (blue shading) for DPSS modulation as compared to the rotary spin echo, leakage outside the target band is significantly suppressed. For the rotary spin echo, spectral leakage increases outofband sensitivity by 30–80 dB relative to the DPSS, quantified by the value L _{ k } (Fig. 1e).
We perform experiments to directly test the spectral response of a driven qubitbased sensor using trapped ^{171}Yb^{+} ions, where the qubit is realized through the hyperfine splitting of the ^{1}S_{1/2} ground state with a transition frequency ~12.6 GHz. We can modulate the amplitude and phase of the driving microwave field arbitrarily using a vector signal generator, providing full control of the qubit state on the Bloch sphere. We employ projective measurements of the qubit state in the zbasis and average over experiments to identify deviations from ideal control operations, which constitute our signal of interest, P(↑_{ z }). Details of the experimental system appear in refs. ^{32,33} and in the Supplementary Note 1.
We verify the spectral properties of DPSSmodulated qubit sensors by performing frequencyselective system identification to map out the effective spectral response of the driven qubit (Fig. 2a). A small singlefrequency perturbation β _{Ω}(t) = α cos(ω _{sid} t + φ) is added to the applied control envelope of the driving field, producing \({\mathrm{\Omega }}(t) \mapsto {\mathrm{\Omega }}(t)(1 + \beta _{\mathrm{\Omega }}(t))\). Since this results in S _{Ω}(ω) ∝ δ(ω − ω _{sid}), by scanning the tunable modulation frequency ω _{sid} and averaging over phase φ for fixed modulation depth α, we effectively reconstruct the filter transfer function of the control, F _{Ω}(ω), through measurements of the average fidelity metric \({\cal F}_{{\mathrm{av}}}\). In this setting, we obtain \({\cal F}_{{\mathrm{av}}}\) directly from the projective measurements via \(P\left( { \uparrow _z} \right) = 1  {\cal F}_{{\mathrm{av}}} \approx 1  {\mathrm{exp}}\left[ {  \pi ^{  1}{\int} {\mathrm{d}}\omega F_{\mathrm{\Omega }}(\omega )S_{\mathrm{\Omega }}(\omega )} \right]\), where we assume negligible dephasing. Experimental reconstruction of the qubits’ spectral response under DPSSmodulated pulses for k = 1 shows good agreement with the analytically calculated fidelity, in addition to the expected broadening as NW is increased (Fig. 2b).
Using the same technique, we can experimentally compare the frequency response of qubits driven by DPSSmodulated pulses to their flattop counterparts, as shown in Fig. 2c–e. These experiments demonstrate the superior spectral concentration in the target band (shaded region); measurements on qubits subject to DPSSmodulated pulses show no detectable sensitivity to perturbations (given by β _{Ω}(t)) outside of the target band, whereas flattop pulses exhibit significant outofband harmonics (marked by arrows). Such sensor responses outside of the target band constitute a source of spectral leakage in sensing applications.
Extending DPSS control capabilities
In order to implement DPSSmodulated pulses for spectral reconstruction applications, we apply additional analog modulation techniques designed to shift the band center from zero to a userdefined frequency ω _{ s } ^{34} (Methods). We employ two modulation protocols: cosinusoidal (COS) modulation shifts both positive and negative frequency components by ω _{ s }, while singlesideband (SSB) modulation shifts the band center by ω _{ s } and suppresses either the positive or negative frequencies, thereby reducing the bandwidth by one half. Experiments using both techniques demonstrate maintenance of the critical spectral concentration of the DPSSmodulated pulses within the shifted bands (Fig. 3a, b). Further details are included in Supplementary Note 6.
Quantum sensing applications also require the ability to disambiguate changes in the measured operational fidelity due to target signals within a single quadrature, e.g., Ωσ _{ x }, from alternate “interfering” sources, which may be manifested similarly in projective measurements. For instance, the presence of a Hamiltonian term ∝σ _{ z } during a driven operation ∝σ _{ x } will reduce the measured fidelity of the driven operation in a manner similar to the presence of noise only proportional to the control^{30}. Consequently, a single measurement is insufficient to determine which process is at play. To detect and compensate for such effects, we use tomographic reconstruction^{35}, by preparing the qubit state along the three Cartesian directions, applying DPSS control, and performing independent sequential measurements in the corresponding bases (Fig. 3c). In our experiments, we simultaneously apply a target signal ∝σ _{ x } as above, and an additional white dephasing term ∝σ _{ z }, which contributes to the sensor’s overall response in a way that obfuscates the measurement of the target. We then isolate the target signal’s contribution by combining three projective measurements as \({\cal S}\) ≡ (1 + P(↑_{ x }) − P(↑_{ y }) − P(↑_{ z }))/2, as derived in the Supplementary Note 3. Reconstructed values of \({\cal S}\) (red markers, Fig. 3c) reproduce the results expected without any σ _{ z }terms well (red line), successfully correcting for a vertical offset that would otherwise bias a spectral estimate.
Multitaper spectral reconstruction with DPSS controls
With demonstrations of the relevant bandlimited properties of qubits subject to DPSSmodulated pulses as well as essential bandshifting techniques complete, we move on to demonstrate our spectral reconstruction capabilities. As a sample application, we reconstruct an engineered amplitude noise spectrum (Fig. 3d). We employ four different DPSSmodulated pulses with k = 1, 3, 5, 7, and NW = 7, bandshifted by SSB at nine different shift frequencies, ω _{ s }, each resulting in an individual estimate or “data taper”. The spacing of the modulation frequencies was chosen to be about 1/2 of the bandwidth of the filter functions, which yields measurements with sensitivity in overlapping bands. The various estimates are combined to produce a reconstruction of the target noise spectrum, which we accomplish this using two distinct techniques. While both are inspired by Thomson’s multitaper approach^{20}, they also differ in important ways.
In its original form, the multitaper method aims to estimate the spectrum of a stationary Gaussian process from a finite set of discrete time samples. In this technique, DPSS waveforms are used to window the time–domain data in postprocessing, producing a set of estimates of the spectrum in the target band. While each “singletaper” estimate is, in itself, an estimate of the spectrum χ ^{2}distributed about the true value, the key idea of multitaper estimation is to combine different estimates into a weightedsum estimator with superior statistical properties. Thanks to the orthogonality of the DPSS, this procedure results in a χ ^{2}distribution with a greater number of degrees of freedom, ensuring consistency and increasing variance efficiency^{20,21}. In order to offset the introduction of outofband bias from higherorder DPSS, the final estimate is determined through an adaptive weighting procedure, designed to favor the lowestorder estimates with the best spectral concentration in the band.
The first reconstruction technique we employ is closest in spirit to the original multitaper, with one crucial distinction; by applying DPSS amplitude modulation to the quantum sensor we are, in effect, windowing the noise process before any measurements are made. This stands in contrast to the manner in which classical multitaper estimates are determined by postprocessing a set of discrete time samples. Measured fidelities determine preliminary spectral estimates at the center of each band, which are then weighted according to Thomson’s adaptive procedure to obtain a final set of estimates. In our second approach, we combine the use of multiple DPSS tapers with Bayesian estimation techniques. Each band, corresponding to a shift frequency, ω _{ s }, is subdivided into a set of smaller segments. For each band, solving a linear inversion determines the Bayesian maximum a posteriori estimate of the spectrum in each segment, which serves as a preliminary estimate. As each segment is contained in multiple bands, the preliminary estimates are weighted by their Fisher information to determine a final estimate of the spectrum in each segment. Additional details on both reconstruction methods are given in the Supplementary Notes 8 and 9.
In our experiments, the Bayesian reconstruction in Fig. 3d uses the multitaper as a prior, and offers slightly improved resolution of the highfrequency cutoff. Both procedures produce spectral estimates which quantitatively match the applied spectrum (within resolution limits), and accurately identify the presence of a highfrequency cutoff in the noise. While the Bayesian approach relies on linear inversion and is thus computationally less efficient and stable than the adaptive multitaper, the flexibility in the choice of the model spectrum to be used as a prior, as well as the inband segmentation, allow for improved resolution and the possibility to handle complex (e.g., “mixed”, consisting of both smooth and line components) spectra, in principle. Advantages of Bayesian spectral estimation are further highlighted in^{36}.
Characterization of native system noise
We conclude by using DPSSmodulated pulses to obtain frequencyresolved information about native noise and nonlinearities in our control system at the end of the synthesis chain (which includes the vector signal generator, an amplifier, cabling, and a waveguidetocoax converter). For this experiment, we use a single ion and perform DPSSmodulated pulses with k = 0, producing a net rotation equivalent to ~400π rotations, ideally enacting \({\Bbb I}\). We calibrate sensitivity to noise by first applying a singlefrequency modulation at the shifted band center frequency, ω _{sid} = ω _{ s }, and averaging over phase (“x” markers, Fig. 4a). We compare this value against interleaved measurements conducted without applied noise to determine the minimum sensitivity achieving SNR ~1. These measurements demonstrate our ability to detect bandlimited amplitude noise with modulation depth as low as ~0.001 dB. In measurements taken at different values of ω _{ s }, we observe a reproducible deviation from the ideal operation over much of the scan range, with a distinct feature around 20 kHz. We confirm that the measured signals are a manifestation of a frequencydependent response in the synthesis chain rather than, e.g., extrinsic decoherence, by adding an initial small amplitudeoffset pulse to a bandshifted DPSSmodulated pulse and scanning over the magnitude of that offset. Investigations into the source of this behavior are ongoing at the time this manuscript is prepared. Ultimately, this approach provides information that we believe is otherwise inaccessible via independent characterization of hardware components in our system.
Discussion
The demonstrations presented here indicate that appropriately crafted quantum control protocols for qubitbased sensors have the ability to overcome significant technical limitations in contemporary quantum sensing experiments. These protocols can be applied to any qubitbased sensor in which arbitrary phase and amplitude modulation of the driving field is possible and spectral concentration is desired. It is noteworthy that by reducing the need to account for highfrequency harmonics in the Fourier response of the modulation pattern, the relatively simple and computationally efficient adaptive multitaper approach to spectrum reconstruction performs similarly to the more complex Bayesian estimation procedure under the conditions we tested. While a full comparative study of both single and multitaper spectral reconstruction techniques using flattop vs. DPSSmodulated pulses is beyond our current scope, preliminary simulations reinforce the utility of Slepian filters in mitigating leakageinduced artifacts in reconstruction. A detailed analysis will be the subject of an upcoming manuscript, along with developing mathematical bounds for spectral leakage and performing a quantitative assessment of the impact of leakage as a function of the target spectrum. Future experiments will also involve the extension of DPSSmodulated control to sensing of additive dephasing noise and multiqubit settings, in order to provide an expanded toolkit of bandlimited controls for quantum sensors.
Methods
Analog modulation techniques
Carrier waves are commonly modulated in radioengineering to multiplex signals within a certain frequency band. We use the same approach to shift the sensitivity of our control pulses in the frequency domain. We employ two of these techniques, COS modulation (also commonly known as amplitude modulation), and singlesideband modulation. By multiplying the time–domain control pulses with a cosine function, so that \({\mathrm{\Omega }}_{{\mathrm{mod}},n}^{{\mathrm{COS}}} \equiv v_n^{(k)}(N,W){\mathrm{cos}}(n\omega _s{\mathrm{\Delta }}t)\), we convolve the original transfer function with two delta function at ±ω _{ s }. In a standard Fourier transform, the positive frequency component is reflected about the y axis. The negative frequency component becomes visible when ω _{ s } is greater than the bandwidth of the pulse such that one copy of the positive and negative sidebands appear in the positive frequency domain. To recover the original appearance of the DPSS filter functions, we may alternatively use singlesideband modulation. In this case, the filter will either be at a higher or lower frequency than ω _{ s }, depending on the sign in \({\mathrm{\Omega }}_{{\mathrm{mod}},n}^{{\mathrm{SSB}}} \equiv v_n^{(k)}(N,W){\mathrm{cos}}(n\omega _s{\mathrm{\Delta }}t)\) ± \({\cal H}\left[ {v_n^{(k)}(N,W)} \right]{\mathrm{sin}}(n\omega _s{\mathrm{\Delta }}t)\). We precalculate the waveform numerically using the Hilbert transform, \({\cal H}\left[ {v_n^{(k)}(N,W)} \right]\), and apply it directly from our microwave source.
Weaknoise limit
The regime in which the firstorder fidelity approximation we employ is valid is the weaknoise limit. This requires that the smallness parameter, as defined in refs. ^{28,30}, \(\xi = \left[ {{\int}_0^\tau \mathrm{d}t\left\ {\beta (t)} \right\} \right]^{1/2} < 1\). For the case of system identification, where β _{Ω}(t) = α cos(ω _{sid} t), as well as for pure amplitude noise, β(t) = β _{Ω}(t)Ω(t), we calculate \(\xi = \alpha A{\mathrm{/}}2\sqrt 2\), where A is the pulse area. For pulses where A = π, the upper bound for the weaknoise limit is at α ≈ 0.9.
Data availability
Data used in figures and computer scripts used to produce DPSSmodulated pulses and perform the spectral reconstruction are available at https://github.com/qclsydney/researchsupplements.
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Acknowledgements
We acknowledge K. Shi, C. Edmunds, and A. Milne for contributions to the experimental system and C. Granade and C. Ferrie for useful discussions on Bayesian estimation. This work was partially supported by the ARC Centre of Excellence for Engineered Quantum Systems CE110001013, the US Army Research Office under Contract W911NF12R0012, IARPA via Department of Interior National Business Center contract number 201212050800010, and a private grant from H. & A. Harley.
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V.M.F. and S.M. developed experimental hardware, built the experimental control system, and obtained the presented data with theoretical techniques developed by L.M.N., D.L., and L.V. Numerical simulations and data analysis were performed by V.M.F., S.M., L.M.N., and L.V. M.J.B. conceived the experiment and led development of the experimental system. W.de.F. provided initial numerical simulations. V.M.F., S.M., L.M.N., L.V., and M.J.B. wrote the manuscript.
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Frey, V.M., Mavadia, S., Norris, L.M. et al. Application of optimal bandlimited control protocols to quantum noise sensing. Nat Commun 8, 2189 (2017). https://doi.org/10.1038/s41467017022982
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