Efficiency of optimal control for noisy spin qubits in diamond

Hendry M. Lim; Genko T. Genov; Roberto Sailer; Alfaiz Fahrurrachman; Muhammad A. Majidi; Fedor Jelezko; Ressa S. Said

doi:10.1103/3hnz-bysr

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Efficiency of optimal control for noisy spin qubits in diamond

Hendry M. Lim, Genko T. Genov, Roberto Sailer, Alfaiz Fahrurrachman, Muhammad A. Majidi, Fedor Jelezko, Ressa S. Said

2025Physical Review Applied 24, 054064CC BY 4.0LCP000019

Phys. Rev. Appl. 24, 054064 (2025) · DOI: 10.1103/3hnz-bysr

License: CC BY 4.0.

Abstract

Decoherence is a major challenge for quantum technologies, whose negative effects can be mitigated by quantum optimal control. The decoherence dynamics significantly varies depending on the character- istics of the system's environment, consequently affecting the optimization outcomes. In this work, we investigate the optimization of a negatively charged nitrogen-vacancy (NV−) center qubit-spin-inversion control pulse, and observe two error-compensation mechanisms, depending on how long the correlation time is compared to the pulse duration. Additionally, we analyze the effects of control space constraints and optimization options to identify a strategy to improve the optimization performance. Finally, we present experimental realizations of the optimized pulses, exhibiting the applicability of the procedure. Our work serves as a generic yet essential guide to implementing optimal control in the presence of realistic noise.

Figures

200 panels with data across 9 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

panel (2): Lower sub-plot of the print: sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for $\delta$-noise correlation times $\tau_\delta = 100$, 10, 1, 0.1 and 0.01 $\mu$s, with no amplitude noise, and the dotted ideal curve. Bands are plus and minus one standard deviation over repeated runs. The upper sub-plot and the inset are not hosted. data.csv

Fig. 2

panel (a): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid) against evolution time $t$ ($\mu$s) at $\tau_\delta = 100$ $\mu$s for case 1.a.i. ($\varphi(t) = \pi/2$, $N_c = 5$, $\beta_{\max} = 3$), with the dash-dotted narrow pulse. Dotted lines mark the $\pm 5$ amplitude limits. The printed $J_{\mathrm{opt}} = 0.016 \pm 0.001$ and $J_{\mathrm{narrow}} = 0.097 \pm 0.003$ are not drawn. data.csv
panel (b): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid) against evolution time $t$ ($\mu$s) at $\tau_\delta = 10$ $\mu$s for case 1.a.i. ($\varphi(t) = \pi/2$, $N_c = 5$, $\beta_{\max} = 3$), with the dash-dotted narrow pulse. Dotted lines mark the $\pm 5$ amplitude limits. The printed $J_{\mathrm{opt}} = 0.022 \pm 0.002$ and $J_{\mathrm{narrow}} = 0.096 \pm 0.003$ are not drawn. data.csv
panel (c): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid) against evolution time $t$ ($\mu$s) at $\tau_\delta = 1$ $\mu$s for case 1.a.i. ($\varphi(t) = \pi/2$, $N_c = 5$, $\beta_{\max} = 3$), with the dash-dotted narrow pulse. Dotted lines mark the $\pm 5$ amplitude limits. The printed $J_{\mathrm{opt}} = 0.053 \pm 0.002$ and $J_{\mathrm{narrow}} = 0.098 \pm 0.003$ are not drawn. data.csv
panel (d): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid) against evolution time $t$ ($\mu$s) at $\tau_\delta = 0.1$ $\mu$s for case 1.a.i. ($\varphi(t) = \pi/2$, $N_c = 5$, $\beta_{\max} = 3$), with the dash-dotted narrow pulse. Dotted lines mark the $\pm 5$ amplitude limits. The printed $J_{\mathrm{opt}} = 0.124 \pm 0.003$ and $J_{\mathrm{narrow}} = 0.102 \pm 0.003$ are not drawn. data.csv
panel (e): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid) against evolution time $t$ ($\mu$s) at $\tau_\delta = 0.01$ $\mu$s for case 1.a.i. ($\varphi(t) = \pi/2$, $N_c = 5$, $\beta_{\max} = 3$), with the dash-dotted narrow pulse. Dotted lines mark the $\pm 5$ amplitude limits. The printed $J_{\mathrm{opt}} = 0.138 \pm 0.004$ and $J_{\mathrm{narrow}} = 0.119 \pm 0.004$ are not drawn. data.csv

Fig. 3

panel (a): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) at $\tau_\delta = 100$ $\mu$s for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$. Dotted lines mark the $\pm 5$ amplitude limits. The printed annotation, $J_{\mathrm{opt}} = 0.016 \pm 0.001$, is not drawn. data.csv
panel (b): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) at $\tau_\delta = 100$ $\mu$s for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$. Dotted lines mark the $\pm 5$ amplitude limits. The printed annotation, $J_{\mathrm{opt}} = 0.016 \pm 0.002$, is not drawn. data.csv
panel (c): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) at $\tau_\delta = 100$ $\mu$s for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$. Dotted lines mark the $\pm 5$ amplitude limits. The printed annotation, $J_{\mathrm{opt}} = 0.023 \pm 0.002$, is not drawn. data.csv
panel (d): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) at $\tau_\delta = 100$ $\mu$s for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$. Dotted lines mark the $\pm 5$ amplitude limits. The printed annotation, $J_{\mathrm{opt}} = 0.045 \pm 0.002$, is not drawn. data.csv
panel (e): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) at $\tau_\delta = 100$ $\mu$s for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$. Dotted lines mark the $\pm 5$ amplitude limits. The printed annotation, $J_{\mathrm{opt}} = 0.040 \pm 0.002$, is not drawn. data.csv

Fig. 4

panel (a): Experimentally (Exp., solid) and numerically (Num., dashed) generated optimized pulse shape $f_y(t)$ against evolution time $t$ ($\mu$s), for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark 0 and $\pm 5$. The printed title, $J_{\mathrm{exp}}/J_{\mathrm{num}} = 1.067$, is not drawn. data.csv
panel (b): Experimentally (Exp., solid) and numerically (Num., dashed) generated optimized pulse shape $f_y(t)$ against evolution time $t$ ($\mu$s), for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark 0 and $\pm 5$. The printed title, $J_{\mathrm{exp}}/J_{\mathrm{num}} = 1.003$, is not drawn. The printed caption lists this case under (c). data.csv
panel (c): Experimentally (Exp., solid) and numerically (Num., dashed) generated optimized pulse shape $f_y(t)$ against evolution time $t$ ($\mu$s), for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark 0 and $\pm 5$. The printed title, $J_{\mathrm{exp}}/J_{\mathrm{num}} = 0.966$, is not drawn. The printed caption lists this case under (b). data.csv
panel (d): Experimentally (Exp., solid) and numerically (Num., dashed) generated optimized pulse shape $f_y(t)$ against evolution time $t$ ($\mu$s), for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark 0 and $\pm 5$. The printed title, $J_{\mathrm{exp}}/J_{\mathrm{num}} = 0.995$, is not drawn. data.csv

Fig. 5

panel (a): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) under a rectangular control pulse of short duration and small rotation phase, for three noise conditions: $\delta$ only (dashed), $\epsilon$ only (dash-dotted) and Both (solid). Shaded bands are plus and minus one standard deviation. Dotted lines mark $\pm 1$. data.csv
panel (b): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) under a rectangular control pulse of short duration and large rotation phase, for three noise conditions: $\delta$ only (dashed), $\epsilon$ only (dash-dotted) and Both (solid). Shaded bands are plus and minus one standard deviation. Dotted lines mark $\pm 1$. data.csv
panel (c): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) under a rectangular control pulse of long duration and small rotation phase, for three noise conditions: $\delta$ only (dashed), $\epsilon$ only (dash-dotted) and Both (solid). Shaded bands are plus and minus one standard deviation. Dotted lines mark $\pm 1$. data.csv
panel (d): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) under a rectangular control pulse of long duration and large rotation phase, for three noise conditions: $\delta$ only (dashed), $\epsilon$ only (dash-dotted) and Both (solid). Shaded bands are plus and minus one standard deviation. Dotted lines mark $\pm 1$. data.csv

Fig. 6

panel (1): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.016 \pm 0.001$, is not drawn. data.csv
panel (2): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.022 \pm 0.002$, is not drawn. data.csv
panel (3): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.053 \pm 0.002$, is not drawn. data.csv
panel (4): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.124 \pm 0.003$, is not drawn. data.csv
panel (5): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.138 \pm 0.004$, is not drawn. data.csv
panel (6): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.040 \pm 0.002$, is not drawn. data.csv
panel (7): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.024 \pm 0.001$, is not drawn. data.csv
panel (8): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.056 \pm 0.002$, is not drawn. data.csv
panel (9): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.103 \pm 0.003$, is not drawn. data.csv
panel (10): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.004$, is not drawn. data.csv
panel (11): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.045 \pm 0.002$, is not drawn. data.csv
panel (12): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.048 \pm 0.001$, is not drawn. data.csv
panel (13): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.063 \pm 0.002$, is not drawn. data.csv
panel (14): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.134 \pm 0.003$, is not drawn. data.csv
panel (15): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.150 \pm 0.004$, is not drawn. data.csv
panel (16): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.044 \pm 0.001$, is not drawn. data.csv
panel (17): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.031 \pm 0.001$, is not drawn. data.csv
panel (18): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.045 \pm 0.001$, is not drawn. data.csv
panel (19): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.100 \pm 0.003$, is not drawn. data.csv
panel (20): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.144 \pm 0.004$, is not drawn. data.csv
panel (21): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.016 \pm 0.002$, is not drawn. data.csv
panel (22): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.024 \pm 0.001$, is not drawn. data.csv
panel (23): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (24): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.113 \pm 0.003$, is not drawn. data.csv
panel (25): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.139 \pm 0.004$, is not drawn. data.csv
panel (26): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.015 \pm 0.001$, is not drawn. data.csv
panel (27): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.023 \pm 0.001$, is not drawn. data.csv
panel (28): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.054 \pm 0.002$, is not drawn. data.csv
panel (29): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.105 \pm 0.003$, is not drawn. data.csv
panel (30): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.003$, is not drawn. data.csv
panel (31): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.083 \pm 0.002$, is not drawn. data.csv
panel (32): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.026 \pm 0.001$, is not drawn. data.csv
panel (33): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (34): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.131 \pm 0.003$, is not drawn. data.csv
panel (35): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.153 \pm 0.004$, is not drawn. data.csv
panel (36): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.030 \pm 0.001$, is not drawn. data.csv
panel (37): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.058 \pm 0.002$, is not drawn. data.csv
panel (38): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.056 \pm 0.001$, is not drawn. data.csv
panel (39): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.101 \pm 0.003$, is not drawn. data.csv
panel (40): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.142 \pm 0.004$, is not drawn. data.csv
panel (41): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.023 \pm 0.002$, is not drawn. data.csv
panel (42): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.030 \pm 0.002$, is not drawn. data.csv
panel (43): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.084 \pm 0.003$, is not drawn. data.csv
panel (44): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.110 \pm 0.003$, is not drawn. data.csv
panel (45): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.147 \pm 0.005$, is not drawn. data.csv
panel (46): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.040 \pm 0.003$, is not drawn. data.csv
panel (47): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.057 \pm 0.003$, is not drawn. data.csv
panel (48): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.126 \pm 0.003$, is not drawn. data.csv
panel (49): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.108 \pm 0.003$, is not drawn. data.csv
panel (50): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.140 \pm 0.004$, is not drawn. data.csv
panel (51): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.079 \pm 0.003$, is not drawn. data.csv
panel (52): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (53): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.174 \pm 0.005$, is not drawn. data.csv
panel (54): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.208 \pm 0.004$, is not drawn. data.csv
panel (55): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.141 \pm 0.004$, is not drawn. data.csv
panel (56): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 100$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.114 \pm 0.002$, is not drawn. data.csv
panel (57): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 10$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.087 \pm 0.002$, is not drawn. data.csv
panel (58): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.083 \pm 0.003$, is not drawn. data.csv
panel (59): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.003$, is not drawn. data.csv
panel (60): Optimized modulation components $f_x$ (dashed) and $f_y$ (solid), as $f_j(t)$, against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s. Dotted lines mark the $\pm 5$ amplitude limits. The printed title, $J_{\mathrm{opt}} = 0.174 \pm 0.004$, is not drawn. data.csv

Fig. 7

panel (1): Tile plot of the optimized cost function $J_{\mathrm{opt}}$ for the twelve optimization option cases, 1.a.i. to 3.b.ii. (rows), and five correlation times $\tau_\delta = 100$, 10, 1, 0.1 and 0.01 $\mu$s (columns). Each tile prints its value plus and minus its standard deviation; darker tiles are lower $J_{\mathrm{opt}}$. data.csv

Fig. 8

panel (1): Cost function $J$ against optimization iteration (iter) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.016 \pm 0.001$, is not drawn. data.csv
panel (2): Cost function $J$ against optimization iteration (iter) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.022 \pm 0.002$, is not drawn. data.csv
panel (3): Cost function $J$ against optimization iteration (iter) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.053 \pm 0.002$, is not drawn. data.csv
panel (4): Cost function $J$ against optimization iteration (iter) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.124 \pm 0.003$, is not drawn. data.csv
panel (5): Cost function $J$ against optimization iteration (iter) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.138 \pm 0.004$, is not drawn. data.csv
panel (6): Cost function $J$ against optimization iteration (iter) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.040 \pm 0.002$, is not drawn. data.csv
panel (7): Cost function $J$ against optimization iteration (iter) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.024 \pm 0.001$, is not drawn. data.csv
panel (8): Cost function $J$ against optimization iteration (iter) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.056 \pm 0.002$, is not drawn. data.csv
panel (9): Cost function $J$ against optimization iteration (iter) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.103 \pm 0.003$, is not drawn. data.csv
panel (10): Cost function $J$ against optimization iteration (iter) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.004$, is not drawn. data.csv
panel (11): Cost function $J$ against optimization iteration (iter) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.045 \pm 0.002$, is not drawn. data.csv
panel (12): Cost function $J$ against optimization iteration (iter) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.048 \pm 0.001$, is not drawn. data.csv
panel (13): Cost function $J$ against optimization iteration (iter) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.063 \pm 0.002$, is not drawn. data.csv
panel (14): Cost function $J$ against optimization iteration (iter) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.134 \pm 0.003$, is not drawn. data.csv
panel (15): Cost function $J$ against optimization iteration (iter) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.150 \pm 0.004$, is not drawn. data.csv
panel (16): Cost function $J$ against optimization iteration (iter) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.044 \pm 0.001$, is not drawn. data.csv
panel (17): Cost function $J$ against optimization iteration (iter) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.031 \pm 0.001$, is not drawn. data.csv
panel (18): Cost function $J$ against optimization iteration (iter) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.045 \pm 0.001$, is not drawn. data.csv
panel (19): Cost function $J$ against optimization iteration (iter) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.100 \pm 0.003$, is not drawn. data.csv
panel (20): Cost function $J$ against optimization iteration (iter) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.144 \pm 0.004$, is not drawn. data.csv
panel (21): Cost function $J$ against optimization iteration (iter) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.016 \pm 0.002$, is not drawn. data.csv
panel (22): Cost function $J$ against optimization iteration (iter) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.024 \pm 0.001$, is not drawn. data.csv
panel (23): Cost function $J$ against optimization iteration (iter) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (24): Cost function $J$ against optimization iteration (iter) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.113 \pm 0.003$, is not drawn. data.csv
panel (25): Cost function $J$ against optimization iteration (iter) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.139 \pm 0.004$, is not drawn. data.csv
panel (26): Cost function $J$ against optimization iteration (iter) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.015 \pm 0.001$, is not drawn. data.csv
panel (27): Cost function $J$ against optimization iteration (iter) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.023 \pm 0.001$, is not drawn. data.csv
panel (28): Cost function $J$ against optimization iteration (iter) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.054 \pm 0.002$, is not drawn. data.csv
panel (29): Cost function $J$ against optimization iteration (iter) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.105 \pm 0.003$, is not drawn. data.csv
panel (30): Cost function $J$ against optimization iteration (iter) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.003$, is not drawn. data.csv
panel (31): Cost function $J$ against optimization iteration (iter) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.083 \pm 0.002$, is not drawn. data.csv
panel (32): Cost function $J$ against optimization iteration (iter) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.026 \pm 0.001$, is not drawn. data.csv
panel (33): Cost function $J$ against optimization iteration (iter) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (34): Cost function $J$ against optimization iteration (iter) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.131 \pm 0.003$, is not drawn. data.csv
panel (35): Cost function $J$ against optimization iteration (iter) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.153 \pm 0.004$, is not drawn. data.csv
panel (36): Cost function $J$ against optimization iteration (iter) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.030 \pm 0.001$, is not drawn. data.csv
panel (37): Cost function $J$ against optimization iteration (iter) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.058 \pm 0.002$, is not drawn. data.csv
panel (38): Cost function $J$ against optimization iteration (iter) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.056 \pm 0.001$, is not drawn. data.csv
panel (39): Cost function $J$ against optimization iteration (iter) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.101 \pm 0.003$, is not drawn. data.csv
panel (40): Cost function $J$ against optimization iteration (iter) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.142 \pm 0.004$, is not drawn. data.csv
panel (41): Cost function $J$ against optimization iteration (iter) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.023 \pm 0.002$, is not drawn. data.csv
panel (42): Cost function $J$ against optimization iteration (iter) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.030 \pm 0.002$, is not drawn. data.csv
panel (43): Cost function $J$ against optimization iteration (iter) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.084 \pm 0.003$, is not drawn. data.csv
panel (44): Cost function $J$ against optimization iteration (iter) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.110 \pm 0.003$, is not drawn. data.csv
panel (45): Cost function $J$ against optimization iteration (iter) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.147 \pm 0.005$, is not drawn. data.csv
panel (46): Cost function $J$ against optimization iteration (iter) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.040 \pm 0.003$, is not drawn. data.csv
panel (47): Cost function $J$ against optimization iteration (iter) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.057 \pm 0.003$, is not drawn. data.csv
panel (48): Cost function $J$ against optimization iteration (iter) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.126 \pm 0.003$, is not drawn. data.csv
panel (49): Cost function $J$ against optimization iteration (iter) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.108 \pm 0.003$, is not drawn. data.csv
panel (50): Cost function $J$ against optimization iteration (iter) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.140 \pm 0.004$, is not drawn. data.csv
panel (51): Cost function $J$ against optimization iteration (iter) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.079 \pm 0.003$, is not drawn. data.csv
panel (52): Cost function $J$ against optimization iteration (iter) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (53): Cost function $J$ against optimization iteration (iter) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.174 \pm 0.005$, is not drawn. data.csv
panel (54): Cost function $J$ against optimization iteration (iter) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.208 \pm 0.004$, is not drawn. data.csv
panel (55): Cost function $J$ against optimization iteration (iter) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.141 \pm 0.004$, is not drawn. data.csv
panel (56): Cost function $J$ against optimization iteration (iter) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 100$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.114 \pm 0.002$, is not drawn. data.csv
panel (57): Cost function $J$ against optimization iteration (iter) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 10$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.087 \pm 0.002$, is not drawn. data.csv
panel (58): Cost function $J$ against optimization iteration (iter) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.083 \pm 0.003$, is not drawn. data.csv
panel (59): Cost function $J$ against optimization iteration (iter) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.003$, is not drawn. data.csv
panel (60): Cost function $J$ against optimization iteration (iter) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s. The printed title, $J_{\mathrm{opt}} = 0.174 \pm 0.004$, is not drawn. data.csv

Fig. 9

panel (1): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.016 \pm 0.001$, is not drawn. data.csv
panel (2): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.022 \pm 0.002$, is not drawn. data.csv
panel (3): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.053 \pm 0.002$, is not drawn. data.csv
panel (4): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.124 \pm 0.003$, is not drawn. data.csv
panel (5): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.138 \pm 0.004$, is not drawn. data.csv
panel (6): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.040 \pm 0.002$, is not drawn. data.csv
panel (7): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.024 \pm 0.001$, is not drawn. data.csv
panel (8): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.056 \pm 0.002$, is not drawn. data.csv
panel (9): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.103 \pm 0.003$, is not drawn. data.csv
panel (10): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.004$, is not drawn. data.csv
panel (11): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.045 \pm 0.002$, is not drawn. data.csv
panel (12): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.048 \pm 0.001$, is not drawn. data.csv
panel (13): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.063 \pm 0.002$, is not drawn. data.csv
panel (14): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.134 \pm 0.003$, is not drawn. data.csv
panel (15): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.150 \pm 0.004$, is not drawn. data.csv
panel (16): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.044 \pm 0.001$, is not drawn. data.csv
panel (17): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.031 \pm 0.001$, is not drawn. data.csv
panel (18): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.045 \pm 0.001$, is not drawn. data.csv
panel (19): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.100 \pm 0.003$, is not drawn. data.csv
panel (20): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 1.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\pi/2, 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.144 \pm 0.004$, is not drawn. data.csv
panel (21): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.016 \pm 0.002$, is not drawn. data.csv
panel (22): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.024 \pm 0.001$, is not drawn. data.csv
panel (23): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (24): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.113 \pm 0.003$, is not drawn. data.csv
panel (25): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.139 \pm 0.004$, is not drawn. data.csv
panel (26): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.015 \pm 0.001$, is not drawn. data.csv
panel (27): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.023 \pm 0.001$, is not drawn. data.csv
panel (28): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.054 \pm 0.002$, is not drawn. data.csv
panel (29): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.105 \pm 0.003$, is not drawn. data.csv
panel (30): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.003$, is not drawn. data.csv
panel (31): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.083 \pm 0.002$, is not drawn. data.csv
panel (32): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.026 \pm 0.001$, is not drawn. data.csv
panel (33): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (34): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.131 \pm 0.003$, is not drawn. data.csv
panel (35): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.153 \pm 0.004$, is not drawn. data.csv
panel (36): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.030 \pm 0.001$, is not drawn. data.csv
panel (37): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.058 \pm 0.002$, is not drawn. data.csv
panel (38): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.056 \pm 0.001$, is not drawn. data.csv
panel (39): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.101 \pm 0.003$, is not drawn. data.csv
panel (40): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 2.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\mathrm{const.}, 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.142 \pm 0.004$, is not drawn. data.csv
panel (41): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.023 \pm 0.002$, is not drawn. data.csv
panel (42): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.030 \pm 0.002$, is not drawn. data.csv
panel (43): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.084 \pm 0.003$, is not drawn. data.csv
panel (44): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.110 \pm 0.003$, is not drawn. data.csv
panel (45): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 3]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.147 \pm 0.005$, is not drawn. data.csv
panel (46): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.040 \pm 0.003$, is not drawn. data.csv
panel (47): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.057 \pm 0.003$, is not drawn. data.csv
panel (48): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.126 \pm 0.003$, is not drawn. data.csv
panel (49): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.108 \pm 0.003$, is not drawn. data.csv
panel (50): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.a.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 5, 8]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.140 \pm 0.004$, is not drawn. data.csv
panel (51): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.079 \pm 0.003$, is not drawn. data.csv
panel (52): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.055 \pm 0.002$, is not drawn. data.csv
panel (53): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.174 \pm 0.005$, is not drawn. data.csv
panel (54): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.208 \pm 0.004$, is not drawn. data.csv
panel (55): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.i., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 3]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.141 \pm 0.004$, is not drawn. data.csv
panel (56): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 100$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.114 \pm 0.002$, is not drawn. data.csv
panel (57): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 10$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.087 \pm 0.002$, is not drawn. data.csv
panel (58): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.083 \pm 0.003$, is not drawn. data.csv
panel (59): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 0.1$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.132 \pm 0.003$, is not drawn. data.csv
panel (60): Sample-averaged population inversion $\overline{\langle\sigma_z\rangle}$ against evolution time $t$ ($\mu$s) for case 3.b.ii., $[\varphi(t), N_c, \beta_{\max}] = [\varphi(t), 10, 8]$, at $\tau_\delta = 0.01$ $\mu$s, under the Optimized (solid) and Unoptimized (dotted) pulse. Black dotted lines mark $\pm 1$. The printed title, $J_{\mathrm{opt}} = 0.174 \pm 0.004$, is not drawn. data.csv

Cite

Hendry M. Lim, Genko T. Genov, Roberto Sailer, Alfaiz Fahrurrachman, Muhammad A. Majidi, Fedor Jelezko, Ressa S. Said. Efficiency of optimal control for noisy spin qubits in diamond. Phys. Rev. Appl. 24, 054064 (2025). https://doi.org/10.1103/3hnz-bysr

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