[1] Viola L, Knill E and Lloyd S 1999 Phys. Rev. Lett. 82 2417 [2] Suter D and Álvarez G A 2016 Rev. Mod. Phys. 88 041001 [3] Biercuk M J, Uys H, VanDevender A P, Shiga N, Itano W M and Bollinger J J 2009 Phys. Rev. A 79 062324 [4] Kotler S, Akerman N, Glickman Y, Keselman A and Ozeri R 2011 Nature 473 61 [5] Bohnet J G, Sawyer B C, Britton J W, Wall M L, Rey A M, Foss-Feig M and Bollinger J J 2016 Science 352 1297 [6] Kaewuam R, Tan T R, Arnold K J, Chanu S R, Zhang Z and Barrett M D 2020 Phys. Rev. Lett. 124 083202 [7] Wang P, Luan C Y, Qiao M, Um M, Zhang J, Wang Y, Yuan X, Gu M, Zhang J and Kim K 2021 Nat. Commun. 12 233 [8] Petta J, Johnson A, Taylor J, Laird E, Yacoby A, Lukin M, Marcus C, Hanson M and Gossard A 2005 Science 309 2180 [9] Liu R B, Yao W and Sham L J 2007 New J. Phys. 9 226 [10] Cappellaro P, Jiang L, Hodges J S and Lukin M D 2009 Phys. Rev. Lett. 102 210502 [11] de Lange G, Wang Z H, Risté D, Dobrovitski V V and Hanson R 2010 Science 330 60 [12] Medford J, Cywiński L, Barthel C, Marcus C M, Hanson M P and Gossard A C 2012 Phys. Rev. Lett. 108 086802 [13] Rong X, Geng J, Wang Z, Zhang Q, Ju C, Shi F, Duan C K and Du J 2014 Phys. Rev. Lett. 112 050503 [14] Schmitt S, Gefen T, Stürner F M, Unden T, Wolff G, Müller C, Scheuer J, Naydenov B, Markham M, Pezzagna S, Meijer J, Schwarz I, Plenio M, Retzker A, McGuinness L P and Jelezko F 2017 Science 356 832 [15] Zopes J, Sasaki K, Cujia K S, Boss J M, Chang K, Segawa T F, Itoh K M and Degen C L 2017 Phys. Rev. Lett. 119 260501 [16] Nakamura Y, Pashkin Y A, Yamamoto T and Tsai J S 2002 Phys. Rev. Lett. 88 047901 [17] Bylander J, Gustavsson S, Yan F, Yoshihara F, Harrabi K, Fitch G, Cory D G, Nakamura Y, Tsai J S and Oliver W D 2011 Nat. Phys. 7 565 [18] Pokharel B, Anand N, Fortman B and Lidar D A 2018 Phys. Rev. Lett. 121 220502 [19] Eto Y, Ikeda H, Suzuki H, Hasegawa S, Tomiyama Y, Sekine S, Sadgrove M and Hirano T 2013 Phys. Rev. A 88 031602 [20] Muessel W, Strobel H, Linnemann D, Hume D B and Oberthaler M K 2014 Phys. Rev. Lett. 113 103004 [21] Trypogeorgos D, Valdés-Curiel A, Lundblad N and Spielman I B 2018 Phys. Rev. A 97 013407 [22] Anderson R P, Kewming M J and Turner L D 2018 Phys. Rev. A 97 013408 [23] In contrast to coupling all the spin levels during a pulse, remarkably, Kaewuam et al. demonstrated an exquisite strategy for dynamical decoupling implemented in multi-ion clock system, where a pulse only couples upper hyperfine levels of a clock transition to cancel out inhomogeneous broadening associated with the electric quadrupole shift induced by neighboring ions.[6] [24] Hirose M, Aiello C D and Cappellaro P 2012 Phys. Rev. A 86 062320 [25] We note that the low frequency AC signal fields are specified as those with a characteristic time much longer than the measurement time. [26] Giovannetti V, Lloyd S and Maccone L 2004 Science 306 1330 [27] Esteve J, Gross C, Weller A, Giovanazzi S and Oberthaler M K 2008 Nature 455 1216 [28] Hamley C D, Gerving C S, Hoang T M, Bookjans E M and Chapman M S 2012 Nat. Phys. 8 305 [29] Pezzé L, Smerzi A, Oberthaler M K, Schmied R and Treutlein P 2018 Rev. Mod. Phys. 90 035005 [30] Kaubruegger R, Silvi P, Kokail C, van Bijnen R, Rey A M, Ye J, Kaufman A M and Zoller P 2019 Phys. Rev. Lett. 123 260505 [31] Bao H, Duan J, Jin S, Lu X, Li P, Qu W, Wang M, Novikova I, Mikhailov E E, Zhao K F, Molmer K, Shen H and Xiao Y 2020 Nature 581 159 [32] Baamara Y, Sinatra A and Gessner M 2021 Phys. Rev. Lett. 127 160501 [33] Taylor J M, Cappellaro P, Childress L, Jiang L, Budker D, Hemmer P R, Yacoby A, Walsworth R and Lukin M D 2008 Nat. Phys. 4 810 [34] Aiello C D, Hirose M and Cappellaro P 2013 Nat. Commun. 4 1419 [35] Wolf T, Neumann P, Nakamura K, Sumiya H, Ohshima T, Isoya J and Wrachtrup J 2015 Phys. Rev. X 5 041001 [36] Lang J E, Liu R B and Monteiro T S 2015 Phys. Rev. X 5 041016 [37] Boss J M, Cujia K S, Zopes J and Degen C L 2017 Science 356 837 [38] Degen C L, Reinhard F and Cappellaro P 2017 Rev. Mod. Phys. 89 035002 [39] DC signal here refers to a static (or low frequency) magnetic field to be measured, while an observable such as spin average is oscillating with a (high) Larmor frequency. [40] Law C K, Pu H and Bigelow N P 1998 Phys. Rev. Lett. 81 5257 [41] The quadratic Zeeman shift for 87Rb scales as 72 Hz/G2, resulting negligible quadratic energy for our calculation at a typical magnetic field (~ 10 mG). [42] Hahn E L 1950 Phys. Rev. 80 580 [43] Meiboom S and Gill D 1958 Rev. Sci. Inst. 29 688 [44] Solomon I 1959 Phys. Rev. Lett. 2 301 [45] Viola L, Knill E and Lloyd S 1999 Phys. Rev. Lett. 82 2417 [46] Uhrig G S 2007 Phys. Rev. Lett. 98 100504 [47] Cywiński L, Lutchyn R M, Nave C P and Das Sarma S 2008 Phys. Rev. B 77 174509 [48] Uhrig G S 2009 Phys. Rev. Lett. 102 120502 [49] Haeberlen U 1976 High Resolution NMR in Solids Selective Averaging (New York: Academic Press) [50] Zhang J, Han Y, Xu P and Zhang W 2016 Phys. Rev. A 94 053608 [51] Yao Q, Zhang J, Yi X F, You L and Zhang W 2019 Phys. Rev. Lett. 122 010408 [52] Luo X Y, Zou Y Q, Wu L N, Liu Q, Han M F, Tey M K and You L 2017 Science 355 620 [53] Zou Y Q, Wu L N, Liu Q, Luo X Y, Guo S F, Cao J H, Tey M K and You L 2018 Proc. Natl. Acad. Sci. USA 115 6381 [54] Gross C, Zibold T, Nicklas E, Esteve J and Oberthaler M K 2010 Nature 464 1165 [55] Vengalattore M, Higbie J M, Leslie S R, Guzman J, Sadler L E and Stamper-Kurn D M 2007 Phys. Rev. Lett. 98 200801 [56] Kell A, Link M, Breyer M, Hoffmann A, Köhl M and Gao K 2021 Rev. Sci. Instr. 92 093202 [57] Kitagawa M and Ueda M 1993 Phys. Rev. A 47 5138 |