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				Floquet engineering based on the strong light-matter interaction is expected to drive quantum materials into nonequilibrium states on an ultrafast timescale, thereby engineering their electronic structure and physical properties, and achieving novel physical effects which have no counterpart in equilibrium states. In recent years, Floquet engineering has attracted a lot of research interest, and there have been numerous rich theoretical predictions. In addition, important experimental research progress has also been made in several representative materials such as topological insulators, graphene, and black phosphorus. Herein, we briefly introduce the important theoretical and experimental progress in this field, and prospect the research future, experimental challenges, and development directions.- 
													Keywords:
													
- Floquet engineering /
- light-matter interaction /
- nonequilibrium states /
- topological materials /
- two-dimensional materials
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Phys. 91 015006  Google Scholar Google Scholar[20] Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T 2014 Nature 515 237  Google Scholar Google Scholar[21] Görg F, Messer M, Sandholzer K, Jotzu G, Desbuquois R, Esslinger T 2018 Nature 553 481  Google Scholar Google Scholar[22] Rechtsman M C, Zeuner J M, Plotnik Y, et al. 2013 Nature 496 196  Google Scholar Google Scholar[23] Maczewsky L J, Zeuner J M, Nolte S, Szameit A 2017 Nat. Commun. 8 13756  Google Scholar Google Scholar[24] Mukherjee S, Spracklen A, Valiente M, Andersson E, Öhberg P, Goldman N, Thomson R R 2017 Nat. Commun. 8 13918  Google Scholar Google Scholar[25] Ezawa M 2013 Phys. Rev. Lett. 110 026603  Google Scholar Google Scholar[26] Claassen M, Jia C, Moritz B, Devereaux T P 2016 Nat. Commun. 7 13074  Google Scholar Google Scholar[27] Wang R, Wang B G, Shen R, Sheng L, Xing D Y 2014 EPL 105 17004  Google Scholar Google Scholar[28] Chan C K, Lee P A, Burch K S, Han J H, Ran Y 2016 Phys. Rev. Lett. 116 026805  Google Scholar Google Scholar[29] Hübener H, Sentef M A, De Giovannini U, Kemper A F, Rubio A 2017 Nat. Commun. 8 13940  Google Scholar Google Scholar[30] Yan Z B, Wang Z 2016 Phys. Rev. Lett. 117 087402  Google Scholar Google Scholar[31] Ezawa M 2017 Phys. Rev. B 96 041205(R  Google Scholar Google Scholar[32] Deng T W, Zheng B B, Zhan F Y, Fan J, Wu X Z, Wang R 2020 Phys. Rev. B 102 201105  Google Scholar Google Scholar[33] Liu H, Sun J T, Cheng C, Liu F, Meng S 2018 Phys. Rev. Lett. 120 237403  Google Scholar Google Scholar[34] Chan C K, Oh Y T, Han J H, Lee P A 2016 Phys. Rev. B 94 121106(R  Google Scholar Google Scholar[35] Zhou L W, Gong J B 2018 Phys. Rev. B 98 205417  Google Scholar Google Scholar[36] Wu H, An J H 2022 Phys. Rev. B 105 L121113  Google Scholar Google Scholar[37] Topp G E, Jotzu G, McIver J W, Xian L, Rubio A, Sentef M A 2019 Phys. Rev. Res. 1 023031  Google Scholar Google Scholar[38] Rodriguez-Vega M, Vogl M, Fiete G A 2021 Ann. Phys. 435 168434  Google Scholar Google Scholar[39] Li Y, Fertig H A, Seradjeh B 2020 Phys. Rev. Res. 2 043275  Google Scholar Google Scholar[40] Vogl M, Rodriguez-Vega M, Fiete G A 2020 Phys. Rev. B 101 235411  Google Scholar Google Scholar[41] Vogl M, Rodriguez-Vega M, Fiete G A 2020 Phys. Rev. B 101 241408(R  Google Scholar Google Scholar[42] Kim H, Dehghani H, Aoki H, Martin I, Hafezi M 2020 Phys. Rev. Res. 2 043004  Google Scholar Google Scholar[43] Shin D, Hubener H, De Giovannini U, Jin H, Rubio A, Park N 2018 Nat. Commun. 9 638  Google Scholar Google Scholar[44] Mentink J H, Balzer K, Eckstein M 2015 Nat. Commun. 6 6708  Google Scholar Google Scholar[45] Kitamura S, Oka T, Aoki H 2017 Phys. Rev. B 96 014406  Google Scholar Google Scholar[46] Claassen M, Jiang H C, Moritz B, Devereaux T P 2017 Nat. Commun. 8 1192  Google Scholar Google Scholar[47] Katz O, Refael G, Lindner N H 2020 Phys. Rev. B 102 155123  Google Scholar Google Scholar[48] Morimoto T, Kitamura S, Nagaosa N 2023 J. Phys. Soc. Jpn. 92 072001  Google Scholar Google Scholar[49] Sie E J, McIver J W, Lee Y H, Fu L, Kong J, Gedik N 2015 Nat. Mater. 14 290  Google Scholar Google Scholar[50] Kim J, Hong X, Jin C, Shi S F, Chang C Y, Chiu M H, Li L J, Wang F 2014 Science 346 1205  Google Scholar Google Scholar[51] Sie E J, Lui C H, Lee Y H, Fu L, Kong J, Gedik N 2017 Science 355 1066  Google Scholar Google Scholar[52] Shan J Y, Ye M, Chu H, Lee S, Park J G, Balents L, Hsieh D 2021 Nature 600 235  Google Scholar Google Scholar[53] McIver J W, Schulte B, Stein F U, Matsuyama T, Jotzu G, Meier G, Cavalleri A 2020 Nat. Phys. 16 38  Google Scholar Google Scholar[54] Park S, Lee W, Jang S, Choi Y B, Park J, Jung W, Watanabe K, Taniguchi T, Cho G Y, Lee G H 2022 Nature 603 421  Google Scholar Google Scholar[55] Smallwood C L, Kaindl R A, Lanzara A 2016 EPL 115 27001  Google Scholar Google Scholar[56] Sobota J A, He Y, Shen Z X 2021 Rev. Mod. Phys. 93 025006  Google Scholar Google Scholar[57] Zhang H, Pincelli T, Jozwiak C, Kondo T, Ernstorfer R, Sato T, Zhou S 2022 Nat. Rev. Methods Primers 2 1  Google Scholar Google Scholar[58] Wang Y H, Steinberg H, Jarillo-Herrero P, Gedik N 2013 Science 342 453  Google Scholar Google Scholar[59] Mahmood F, Chan C-K, Alpichshev Z, Gardner D, Lee Y, Lee P A, Gedik N 2016 Nat. 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图 2 (a)弗洛凯调控诱导的拓扑相变[7]; (b)在周期光场驱动前后的转角石墨烯平带电子结构[47]; (c)交换作用强度变化随时间的演化曲线[44]; (d)弗洛凯调控调节材料磁性的示意图[48] Fig. 2. (a) Floquet engineering induced topological phase transition[7]; (b) flat band of twisted graphene before and after light driving[47]; (c) the evolution of exchange strength with time[44]; (d) a schematic for manipulating magnetic properties of materials by Floquet engineering[48]. 图 3 (a)单层WS2中观测到的能谷选择的光学斯塔克效应[49]; (b) MnPS3中观测到的弗洛凯调控对于光学非线性系数的调控[52]; (c)石墨烯中观测到光诱导的反常霍尔效应[53]; (d)石墨烯-铝约瑟夫森结中在微波激发下的复制隧穿谱[54] Fig. 3. (a) Observation of valley selective optical stark effect in monolayer WS2[49]; (b) manipulation of optical nonlinear coefficients in MnPS3 by Floquet engineering[52]; (c) observation of light-induced anomalous Hall effect in graphene[53]; (d) replica tunneling spectrum under the excitation of microwaves in graphene-aluminum Josephson junction[54]. 图 4 (a)拓扑绝缘体Bi2Se3的超快电子能谱, 实现弗洛凯能带调控[58,59]; (b)拓扑绝缘体Bi2Te3的亚周期分辨的超快电子能谱和弗洛凯边带的形成过程[60]; (c)半导体黑磷的超快电子能谱, 实现弗洛凯能带调控[61] Fig. 4. (a) TrARPES spectra of Floquet engineering in topological insulator Bi2Se3[58,59]; (b) sub-cycle resolved TrARPES spectra of topological insulator Bi2Te3 to show the formation of Floquet sidebands[60]; (c) TrARPES spectra of Floquet engineering in a semiconductor black phosphorus[61]. 
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[1] Warren B E 1990 X-Ray Diffraction(Courier Corporation [2] Long D A 1977 Raman Spectroscopy (New York and London: McGraw-Hill [3] Henderson B, Imbusch G F 2006 Optical Spectroscopy of Inorganic Solids (Cambridge: Oxford University Press [4] Damascelli A, Hussain Z, Shen Z X 2003 Rev. Mod. Phys. 75 473  Google Scholar Google Scholar[5] Hüfner S 2013 Photoelectron Spectroscopy: Principles and Applications (Springer Science & Business Media [6] Basov D N, Averitt R D, Hsieh D 2017 Nat. Mater. 16 1077  Google Scholar Google Scholar[7] Bao C H, Tang P Z, Sun D, Zhou S Y 2021 Nat. Rev. Phys. 4 33  Google Scholar Google Scholar[8] Oka T, Aoki H 2009 Phys. Rev. B 79 081406(R  Google Scholar Google Scholar[9] Kitagawa T, Oka T, Brataas A, Fu L, Demler E 2011 Phys. Rev. B 84 235108  Google Scholar Google Scholar[10] Lindner N H, Refael G, Galitski V 2011 Nat. Phys. 7 490  Google Scholar Google Scholar[11] Ashcroft N W, Mermin N D 1976 Solid State Physics (Philadelphia: Saunders College [12] Sambe H 1973 Phys. Rev. A 7 2203  Google Scholar Google Scholar[13] Haldane F D M 1988 Phys. Rev. Lett. 61 2015  Google Scholar Google Scholar[14] Oka T, Kitamura S 2019 Annu. Rev. Condens. Matter Phys. 10 387  Google Scholar Google Scholar[15] Rudner M S, Lindner N H 2020 Nat. Rev. Phys. 2 229  Google Scholar Google Scholar[16] de la Torre A, Kennes D M, Claassen M, Gerber S, McIver J W, Sentef M A 2021 Rev. Mod. Phys. 93 041002  Google Scholar Google Scholar[17] Eckardt A 2017 Rev. Mod. Phys. 89 011004  Google Scholar Google Scholar[18] Cooper N R, Dalibard J, Spielman I B 2019 Rev. Mod. Phys. 91 015005  Google Scholar Google Scholar[19] Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M C, Schuster D, Simon J, Zilberberg O, Carusotto I 2019 Rev. Mod. Phys. 91 015006  Google Scholar Google Scholar[20] Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T 2014 Nature 515 237  Google Scholar Google Scholar[21] Görg F, Messer M, Sandholzer K, Jotzu G, Desbuquois R, Esslinger T 2018 Nature 553 481  Google Scholar Google Scholar[22] Rechtsman M C, Zeuner J M, Plotnik Y, et al. 2013 Nature 496 196  Google Scholar Google Scholar[23] Maczewsky L J, Zeuner J M, Nolte S, Szameit A 2017 Nat. Commun. 8 13756  Google Scholar Google Scholar[24] Mukherjee S, Spracklen A, Valiente M, Andersson E, Öhberg P, Goldman N, Thomson R R 2017 Nat. Commun. 8 13918  Google Scholar Google Scholar[25] Ezawa M 2013 Phys. Rev. Lett. 110 026603  Google Scholar Google Scholar[26] Claassen M, Jia C, Moritz B, Devereaux T P 2016 Nat. Commun. 7 13074  Google Scholar Google Scholar[27] Wang R, Wang B G, Shen R, Sheng L, Xing D Y 2014 EPL 105 17004  Google Scholar Google Scholar[28] Chan C K, Lee P A, Burch K S, Han J H, Ran Y 2016 Phys. Rev. Lett. 116 026805  Google Scholar Google Scholar[29] Hübener H, Sentef M A, De Giovannini U, Kemper A F, Rubio A 2017 Nat. Commun. 8 13940  Google Scholar Google Scholar[30] Yan Z B, Wang Z 2016 Phys. Rev. Lett. 117 087402  Google Scholar Google Scholar[31] Ezawa M 2017 Phys. Rev. B 96 041205(R  Google Scholar Google Scholar[32] Deng T W, Zheng B B, Zhan F Y, Fan J, Wu X Z, Wang R 2020 Phys. Rev. B 102 201105  Google Scholar Google Scholar[33] Liu H, Sun J T, Cheng C, Liu F, Meng S 2018 Phys. Rev. Lett. 120 237403  Google Scholar Google Scholar[34] Chan C K, Oh Y T, Han J H, Lee P A 2016 Phys. Rev. B 94 121106(R  Google Scholar Google Scholar[35] Zhou L W, Gong J B 2018 Phys. Rev. B 98 205417  Google Scholar Google Scholar[36] Wu H, An J H 2022 Phys. Rev. B 105 L121113  Google Scholar Google Scholar[37] Topp G E, Jotzu G, McIver J W, Xian L, Rubio A, Sentef M A 2019 Phys. Rev. Res. 1 023031  Google Scholar Google Scholar[38] Rodriguez-Vega M, Vogl M, Fiete G A 2021 Ann. Phys. 435 168434  Google Scholar Google Scholar[39] Li Y, Fertig H A, Seradjeh B 2020 Phys. Rev. Res. 2 043275  Google Scholar Google Scholar[40] Vogl M, Rodriguez-Vega M, Fiete G A 2020 Phys. Rev. B 101 235411  Google Scholar Google Scholar[41] Vogl M, Rodriguez-Vega M, Fiete G A 2020 Phys. Rev. B 101 241408(R  Google Scholar Google Scholar[42] Kim H, Dehghani H, Aoki H, Martin I, Hafezi M 2020 Phys. Rev. Res. 2 043004  Google Scholar Google Scholar[43] Shin D, Hubener H, De Giovannini U, Jin H, Rubio A, Park N 2018 Nat. Commun. 9 638  Google Scholar Google Scholar[44] Mentink J H, Balzer K, Eckstein M 2015 Nat. Commun. 6 6708  Google Scholar Google Scholar[45] Kitamura S, Oka T, Aoki H 2017 Phys. Rev. B 96 014406  Google Scholar Google Scholar[46] Claassen M, Jiang H C, Moritz B, Devereaux T P 2017 Nat. Commun. 8 1192  Google Scholar Google Scholar[47] Katz O, Refael G, Lindner N H 2020 Phys. Rev. B 102 155123  Google Scholar Google Scholar[48] Morimoto T, Kitamura S, Nagaosa N 2023 J. Phys. Soc. Jpn. 92 072001  Google Scholar Google Scholar[49] Sie E J, McIver J W, Lee Y H, Fu L, Kong J, Gedik N 2015 Nat. Mater. 14 290  Google Scholar Google Scholar[50] Kim J, Hong X, Jin C, Shi S F, Chang C Y, Chiu M H, Li L J, Wang F 2014 Science 346 1205  Google Scholar Google Scholar[51] Sie E J, Lui C H, Lee Y H, Fu L, Kong J, Gedik N 2017 Science 355 1066  Google Scholar Google Scholar[52] Shan J Y, Ye M, Chu H, Lee S, Park J G, Balents L, Hsieh D 2021 Nature 600 235  Google Scholar Google Scholar[53] McIver J W, Schulte B, Stein F U, Matsuyama T, Jotzu G, Meier G, Cavalleri A 2020 Nat. Phys. 16 38  Google Scholar Google Scholar[54] Park S, Lee W, Jang S, Choi Y B, Park J, Jung W, Watanabe K, Taniguchi T, Cho G Y, Lee G H 2022 Nature 603 421  Google Scholar Google Scholar[55] Smallwood C L, Kaindl R A, Lanzara A 2016 EPL 115 27001  Google Scholar Google Scholar[56] Sobota J A, He Y, Shen Z X 2021 Rev. Mod. Phys. 93 025006  Google Scholar Google Scholar[57] Zhang H, Pincelli T, Jozwiak C, Kondo T, Ernstorfer R, Sato T, Zhou S 2022 Nat. Rev. Methods Primers 2 1  Google Scholar Google Scholar[58] Wang Y H, Steinberg H, Jarillo-Herrero P, Gedik N 2013 Science 342 453  Google Scholar Google Scholar[59] Mahmood F, Chan C-K, Alpichshev Z, Gardner D, Lee Y, Lee P A, Gedik N 2016 Nat. Phys. 12 306  Google Scholar Google Scholar[60] Ito S, Schüler M, Meierhofer M, et al. 2023 Nature 616 696  Google Scholar Google Scholar[61] Zhou S H, Bao C H, Fan B S, et al. 2023 Nature 614 75  Google Scholar Google Scholar[62] Aeschlimann S, Sato S A, Krause R, Chávez-Cervantes M, De Giovannini U, Hübener H, Forti S, Coletti C, Hanff K, Rossnagel K, Rubio A, Gierz I 2021 Nano Lett. 21 5028  Google Scholar Google Scholar[63] Jung S W, Ryu S H, Shin W J, Sohn Y, Huh M, Koch R J, Jozwiak C, Rotenberg E, Bostwick A, Kim K S 2020 Nat. Mater. 19 277  Google Scholar Google Scholar[64] Zhou S H, Bao C H, Fan B S, Wang F, Zhong H Y, Zhang H Y, Tang P Z, Duan W H, Zhou S Y 2023 Phys. Rev. Lett. 131 116401  Google Scholar Google Scholar[65] Bao C H, Zhong H Y, Zhou S H, Feng R, Wang Y H, Zhou S Y 2022 Rev. Sci. Instrum. 93 013902  Google Scholar Google Scholar[66] Bao C H, Li Q, Xu S, et al. 2022 Nano Lett. 22 1138  Google Scholar Google Scholar[67] Sato S A, McIver J W, Nuske M, Tang P Z, Jotzu G, Schulte B, Hübener H, De Giovannini U, Mathey L, Sentef M A, Cavalleri A, Rubio A 2019 Phys. Rev. B 99 214302  Google Scholar Google Scholar[68] Sato S A, Tang P Z, Sentef M A, Giovannini U D, Hübener H, Rubio A 2019 New J. Phys. 21 093005  Google Scholar Google Scholar[69] Nuske M, Broers L, Schulte B, Jotzu G, Sato S A, Cavalleri A, Rubio A, McIver J W, Mathey L 2020 Phys. Rev. Res. 2 043408  Google Scholar Google Scholar
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