Nonvanishing effect of detuning errors in dynamical-decoupling-based quantum sensing experiments

J. E. Lang, T. Madhavan, J.-P. Tetienne, D. A. Broadway, L. T. Hall, T. Teraji, T. S. Monteiro, A. Stacey, and L. C. L. Hollenberg

Phys. Rev. A 99, 012110 – Published 9 January 2019

Abstract

Characteristic dips appear in the coherence traces of a probe qubit when dynamical decoupling (DD) is applied in synchrony with the precession of target nuclear spins, forming the basis for nanoscale nuclear magnetic resonance (NMR). The frequency of the microwave control pulses is chosen to match the qubit transition but this can be detuned from resonance by experimental errors, hyperfine coupling intrinsic to the qubit, or inhomogeneous broadening. The detuning acts as an additional static field which is generally assumed to be completely removed in Hahn echo and DD experiments. Here we demonstrate that this is not the case in the presence of finite pulse-durations, where a detuning can drastically alter the coherence response of the probe qubit, with important implications for sensing applications. Using the electronic spin associated with a nitrogen-vacancy center in diamond as a test qubit system, we analytically and experimentally study the qubit coherence response under CPMG and XY8 dynamical decoupling control schemes in the presence of finite pulse-durations and static detunings. Most striking is the splitting of the NMR resonance under CPMG, whereas under XY8 the amplitude of the NMR signal is modulated. Our work shows that the detuning error must not be neglected when extracting data from quantum sensor coherence traces.

Received 10 September 2018

DOI:https://doi.org/10.1103/PhysRevA.99.012110

Physics Subject Headings (PhySH)

Quantum coherence & coherence measures Quantum metrology Semiclassical physics

General Physics

Authors & Affiliations

J. E. Lang^1,*, T. Madhavan², J.-P. Tetienne^2,†, D. A. Broadway^2,3, L. T. Hall², T. Teraji⁴, T. S. Monteiro¹, A. Stacey^2,3, and L. C. L. Hollenberg^2,3

¹Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom
²School of Physics, The University of Melbourne, Victoria 3010, Australia
³Centre for Quantum Computation and Communication Technology, School of Physics, The University of Melbourne, Victoria 3010, Australia
⁴National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan

^*jacob.lang.14@ucl.ac.uk
^†jtetienne@unimelb.edu.au

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Vol. 99, Iss. 1 — January 2019

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Figure 1
(a) The two dynamical decoupling pulse sequences considered in this work to detect an oscillating (ac) magnetic signal generated, for instance, by an ensemble of nuclear spins. The blue and red pulses correspond to the $0^{\circ}$ and $90^{\circ}$ phase shifts of the microwave driving field, respectively. (b) Schematic of the experiment: a single NV center below the diamond surface, with a semi-infinite sample of nuclear spins above the surface. (c) Transition frequencies of the NV electronic spin as a function of the axial magnetic field $B_{0}$ . The hyperfine structure due to the ${}^{15}N$ nuclear spin of the NV center ( $m_{I} = \pm \frac{1}{2}$ ) is shown, with the shading of the lines denoting the amount of nuclear spin polarization under optical pumping; in particular, near the ESLAC at $B \approx 500$ G the nuclear spin is almost fully polarized into the $m_{I} = - \frac{1}{2}$ state [42].
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Figure 2
(a) Modulation functions $(f_{x}, f_{y}, f_{z})$ in the time domain for the CPMG sequence with finite-duration pulses for zero and nonzero detuning errors. The simulation here has a pulse spacing of $τ = 240$ ns, a pulse width of $t_{p} = 40$ ns, and detunings of $Δ = 0$ (top graph) and $Δ / 2 π = 1.5$ MHz (bottom). These values were chosen to be representative of the experiment performed in Fig. 3. (b) Filter function, $| \tilde{f} {(ω) |}^{2}$ , for the CPMG sequences shown in panel (a) but with a total of $N = 128$ pulses. For zero detuning there is a central peak at $ω_{DD} / 2 π = 2.08$ MHz which splits at nonzero detuning. (c) Scan of the filter function shown in panel (b) as a function of the detuning strength $Δ$ . (d) Scan of the filter function shown in panel (b) as a function of the pulse width $t_{p}$ . The green solid lines indicate the equivalent slices in each map and these also correspond to the filter function shown in panel (b).
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Figure 3
(a,b) NMR spectra obtained using the CPMG sequence for various values of the detuning $Δ$ , with a $π$ -pulse duration of $t_{p} = 40$ ns, a number of $π$ pulses $N = 336$ , and under a magnetic field $B \approx 500$ G. The different spectra are vertically offset for clarity. The solid lines are double-Lorentzian fits used to estimate the positions of the two NMR dips, $τ_{1}$ and $τ_{2}$ . (b) Positions of the two NMR dips extracted from panel (a), $τ_{1}$ and $τ_{2}$ , as a function of the detuning $Δ$ . (c) Positions of the two NMR dips as a function of the $π$ -pulse duration $t_{p}$ for a fixed detuning of $Δ / 2 π = + 1.5$ MHz. Here the number of pulses is $N = 256$ [shallower NV compared to panels (a) and (b)]. In panels (b) and (c), the blue dots are the experimental data (blue lines are a guide to the eye) while the red lines are the theoretical data.
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Figure 4
(a) Filter function $| \tilde{f} {(ω) |}^{2}$ for the XY8 sequence for various values of the detuning $Δ$ , with $N = 376$ , $t_{p} = 40$ ns, and $τ = 235$ ns. (b,c) NMR spectra recorded from the same NV as in Fig. 3 using the XY8 sequence with $t_{p} = 40$ ns and $N = 376$ , for positive (b) and negative (c) detunings. (d) Decoherence curves recorded for a different NV under the XY8 sequence, with $N = 128$ , $t_{p} = 40$ ns, and a detuning of $Δ = 0$ (black line) or $Δ / 2 π = + 3$ MHz (blue). Also shown for comparison is the case of the CPMG sequence with $Δ / 2 π = + 3$ MHz (red). (e) Parameters $d_{NV}$ and $T_{2 n}^{*}$ werwe obtained by fitting the model of Ref. [33] to the spectra shown in panels (b) and (c), for detunings between $- 2$ and +2 MHz. The lines are a guide to the eye. The vertical error bars are the standard error from the fit.
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Figure 5
Modulation functions $(f_{x}, f_{y}, f_{z})$ in the time domain for the XY8 sequence with finite-duration pulses for zero and nonzero detuning errors. The simulation here has a pulse spacing of $τ = 240$ ns, a pulse width of $t_{p} = 40$ ns, and detunings of $Δ = 0$ (top graph) and $Δ / 2 π = 1.5$ MHz (bottom graph)—matching the parameters for the CPMG sequence in Fig. 2.
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