s3c.2.pdf quantum information and measurement (qim) v...
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S3C.2.pdf Quantum Information and Measurement (QIM) V:Quantum Technologies © OSA 2019
Optimal control of diamond spin qubits forquantum sensing in noisy environments
Francesco Poggiali1,2, Santiago Hernandez-Gomez1,2, Paola Cappellaro1,3, NicoleFabbri1,2
1LENS European Laboratory for Non linear Spectroscopy, Universita degli Studi di Firenze, I-50019 SestoFiorentino, Italy
2CNR-INO Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche, I-50019 Sesto Fiorentino, Italy3Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA
02139, [email protected]
Abstract: We devise a robust quantum sensing scheme based on optimal control. We ex-perimentally demonstrate sensitivity enhancement of diamond spin-qubits sensors to mea-sure ultraweak time-varying magnetic fields in noisy environments.
OCIS codes: 270.2500, 270.5585.
1. Introduction
Optimal control theory [1] provides valuable tools for steering the dynamics of a quantum system and performingdesired operations, with high fidelity and speed. However, quantum sensing poses peculiar challenges to control.The information on the unitary dynamics of the system is not a priori accessible, due to the presence of unknownexternal field to be measured. Thus, fidelity cannot be a suitable figure of merit to optimize control. Moreover, thegoal of sensing requires both maximizing the effect of the field to be measured, and protecting the sensor fromunwanted external noise, two tasks that are inherently contrasting each other.
We devise an optimal control scheme that exploits sensitivity as a novel optimization metric [2]. We exper-imentally implement this control paradigm using a Nitrogen-Vacancy (NV) center in diamond as a magneticfield sensor. Advantageously, sensitivity includes both the unknown external field to be measured, and decoher-ence induced by environmental noise, which we spectroscopically characterize [3]. Our optimal control schemedemonstrates superior performances in complex sensing tasks, such as measuring multi-chromatic ac fields andarbitrary waveforms.
2. Discussion
The electronic spin of the Nitrogen-Vacancy center has emerged as a prominent quantum sensor thanks to com-bination of sensitivity, spatial resolution, and ultralong coherence. In the presence of a static bias magnetic field,we exploit the Sz = 0,−1 ground-state spin projections as an effective two-level system. The spin qubit is cou-pled both to the external field b(t) to be measured (inducing unitary but unknown dynamics), and to noise β (t),inducing decoherence. We assume β (t) to be described by a stochastic variable with power spectral density S(ω)in frequency domain. Control is applied to the spin qubit via pulsed dynamical decoupling (DD) [4], consisting inseries of π pulses that repeatedly flip the spin, thus reversing its evolution. A Ramsey interferometer embeddingthe control sequences measures b(t).
The considered sensing task consists in measuring the amplitude of an ultraweak spectrally characterized mag-netic field with multiple frequency components. Common equispaced DD pulse sequences show high frequencyselectivity that can be exploited to measure monochromatic ac magnetic fields [6], but for this reason they mayexhibit suboptimal performances when probing a multitone target field due to attenuation of some frequencycomponents. We have thus designed a direct and fast search method that looks for the optimal pulse sequenceminimizing the cost function η , that is, sensitivity. Note that sensitivity η = eχn(t)
√t/|φn(t)|, depends on the
applied control both via the decoherence functional χn(t), and via the spin phase φn(t) accumulated during theevolution time t under the unknown field. The search is performed in a multidimensional parameter space. Weanalyze which optimization parameters (e.g., total sensing time, π-pulse positions, signal phase, signal triggertime) provide the largest improvement without requiring excessive computational resources. An example of op-timization for a multitone magnetic field is given in Fig. 1 a, showing a remarkable enhancement of sensitivitycompared to standard control when measuring multitone ac fields at fixed interrogation time.
To effectively suppress the effect of unwanted noise, superimposed to the field to be measured, the knowledge ofits sources is critical. We thus demonstrate a noise spectroscopy protocol [3] that complements our optimal control
S3C.2.pdf Quantum Information and Measurement (QIM) V:Quantum Technologies © OSA 2019
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Fig. 1. (a) Inverse of sensitivity, measured in the presence of a sample multitone target field b(t)under equispaced DD sequences (dots), and under optimized control (diamonds). The curves repre-sent the theoretical predictions for the equispaced sequence (green) and the optimized one (blue).The inset shows the time dependece of b(t) and subsets of the two pulse distributions (comprising50 pulses in both the cases). (b) Noise spectrum (in blue) and coupling with nearby nuclei (in red),due to Carbon nuclear spins inside diamond. (c) Time evolution of spin coherence under differentnon-equispaced AXY8 pulse sequences. Dots are experimental data with statistical error. Red linesare the predicted coherence simulated using the measured environment, with no free parameters.
strategy. In the absence of any target field, we characterize the nuclear spin environment of the NV qubit insidediamond, comprising both coherently coupled quantum systems, and a larger quantum bath that can be modeled asa classical stochastic field β (t) (Fig. 1 b). We overcome the limitation due to the relatively short coherence timesby using the higher-order harmonics of DD sequence to provide noise attenuation, and avoid signal saturationfrom the strong noise peak. Remarkably, in the weak coupling regime of the qubit to the spin environment (highbias field), the obtained environment model shows predictive behavior (Fig. 1 c) under different different kinds oftime-dependent control, beyond monochromatic filters.
3. Conclusions
We demonstrated a robust and efficient control method adapted for quantum sensing applications. A precise mod-eling of the NV spin environment can be used to further improve environment noise suppression, specificallytailored to the measured noise spectrum.
References
1. S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Koeckenberger, R. Kosloff, I. Kuprov, B. Luy, S.Schirmer, T. Schulte-Herbrueggen, D. Sugny, and F.K. Wilhelm, “Training Schrdinger’s Cat: QuantumOptimal Control”, Eur. Phys. J. D 69, 279 (2015).
2. F. Poggiali, P. Cappellaro, N. Fabbri, “Optimal Control for One-Qubit Quantum Sensing”, Physical ReviewX 8, 021059 (2018).
3. S. Hernandez-Gomez, F. Poggiali, P. Cappellaro, N. Fabbri, “Noise spectroscopy of a quantum-classicalenvironment with a diamond qubit”, arXiv:1808.08222.
4. L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998).5. T. Yuge, S. Sasaki, and Y. Hirayama, Phys. Rev. Lett. 107, 170504 (2011).6. J.M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P.R. Hemmer, A. Yacoby, R. Walsworth,
and M. D. Lukin, High-Sensitivity Diamond Magnetometer with Nanoscale Resolution, Nat. Phys. 4, 810(2008).