Abstract The Jaynes-Cummings model with or without rotating-wave approximation plays a major role to study the interaction between atom and light. We investigate the Jaynes-Cummings model beyond the rotating-wave approximation. Treating the counter-rotating terms as periodic drivings, we solve the model in the extended Floquet space. It is found that the full energy spectrum folded in the quasi-energy bands can be described by an effective Hamiltonian derived in the high-frequency regime. In contrast to the Z2 symmetry of the original model, the effective Hamiltonian bears an enlarged U(1) symmetry with a unique photon-dependent atom-light detuning and coupling strength. We further analyze the energy spectrum, eigenstate fidelity and mean photon number of the resultant polaritons, which are shown to be in accordance with the numerical simulations in the extended Floquet space up to an ultra-strong coupling regime and are not altered significantly for a finite atom-light detuning. Our results suggest that the effective model provides a good starting point to investigate the rich physics brought by counter-rotating terms in the frame of Floquet theory.
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11875195) and the Foundation of Beijing Education Committees, China (Grant Nos. CIT&TCD201804074 and KZ201810028043).
Corresponding Authors:
Qing Sun
E-mail: sunqing@cnu.edu.cn
Cite this article:
Yi-Fan Wang(王伊凡), Hong-Hao Yin(尹洪浩), Ming-Yue Yang(杨明月), An-Chun Ji(纪安春), and Qing Sun(孙青) Effective Hamiltonian of the Jaynes-Cummings model beyond rotating-wave approximation 2021 Chin. Phys. B 30 064204
[1] Jaynes E T and Cummings F W 1963 Proc. IEEE51 89 [2] Shore B W and Knight P L 1993 J. Mod. Opt.40 1195 [3] Cummings F W 1965 Phys. Rev.140 A1051 [4] Eberly J H, Narozhny N B and Sanchez-Mondragon J J 1980 Phys. Rev. Lett.44 1323 [5] Narozhny N B, Sanchez-Mondragon J J and Eberly J H 1981 Phys. Rev. A23 236 [6] Knight P L and Radmore P M 1982 Phys. Rev. A26 676 [7] Rempe G, Walther H and Klein N 1987 Phys. Rev. Lett.58 353 [8] Phoenix S J D and Knight P L 1991 Phys. Rev. A44 6023 [9] Brune M, Haroche S, Raimond J M, Davidovich L and Zagury N 1992 Phys. Rev. A45 5193 [10] Bužek V, Moya-Cessa H, Knight P L and Phoenix S J D 1992 Phys. Rev. A45 8190 [11] Brune M, Hagley E, Dreyer J, Maître X, Maali A, Wunderlich C, Raimond J M and Haroche S 1996 Phys. Rev. Lett.77 4887 [12] Furuya K, Nemes M C and Pellegrino G Q 1998 Phys. Rev. Lett.80 5524 [13] Sun Q, Hu J, Wen L, Pu H and Ji A C 2018 Phys. Rev. A98 033801 [14] Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R-S, Majer J, Kumar S, Girvin S M and Schoelkopf R J 2004 Nature431 162 [15] Hofheinz M, Wang H, Ansmann M, Bialczak R C, Lucero E, Neeley M, O'Connell A D, Sank D, Wenner J, Martinis J M and Cleland A N 2009 Nature459 546 [16] LaHaye M D, Suh J, Echternach P M, Schwab K C and Roukes M L 2009 Nature459 960 [17] Niemczyk T, Deppe F, Huebl H, Menzel E P, Hocke F, Schwarz M J, Garcia-Ripoll J J, Zueco D, Hümmer T, Solano E, Marx A and Gross R 2010 Nat. Phys.6 772 [18] Casanova J, Romero G, Lizuain I, García-Ripoll J J and Solano E 2010 Phys. Rev. Lett.105 263603 [19] Forn-Díaz P, Lisenfeld J, Marcos D, García-Ripoll J J, Solano E, Harmans C J P M and Mooij J E 2010 Phys. Rev. Lett.105 237001 [20] Chen Q H, Li L, Liu T and Wang K L 2012 Chin. Phys. Lett.29 014208 [21] Baust A, Hoffmann E, Haeberlein M, Schwarz M J, Eder P, Goetz J, Wulschner F, Xie E, Zhong L, Quijandría F, Zueco D, García Ripoll J J, García-Álvarez L, Romero G, Solano E, Fedorov K G, Menzel E P, Deppe F, Marx A and Gross R 2016 Phys. Rev. B93 214501 [22] Forn-Díaz P, Lamata L, Rico E, Kono J and Solano E 2019 Rev. Mod. Phys.91 025005 [23] Milonni P W, Ackerhalt J R and Galbraith H W 1983 Phys. Rev. Lett.50 966 [24] Emary C and Brandes T 2003 Phys. Rev. E67 066203 [25] Chen Q H, Yang Y, Liu T and Wang K L 2010 Phys. Rev. A82 052306 [26] Liu T, Feng M and Wang K L 2011 Phys. Rev. A84 062109 [27] Naderi M H 2011 J. Phys. A: Math. Theor.44 055304 [28] Wang Y M and Du G and Liang J Q 2012 Chin. Phys. B21 044207 [29] Tang N, Xu T T and Zeng H S 2013 Chin. Phys. B22 030304 [30] Mirzaee M and Batavani M 2015 Chin. Phys. B24 040306 [31] Rabi I I 1936 Phys. Rev.49 324 [32] Rabi I I 1937 Phys. Rev.51 652 [33] Braak D 2011 Phys. Rev. Lett.107 100401 [34] Braak D 2019 Symmetry11 1259 [35] Dong Y H, Zhang W J, Liu J and Xie X T 2019 Chin. Phys. B28 114202 [36] Feranchuk I D, Komarov L I and Ulyanenkov A P 1996 J. Phys. A: Math. Gen.29 4035 [37] Tur É A 2000 Opt. Spectrosc.89 574 [38] Pan F, Guan X, Wang Y and Draayer J P 2010 J. Phys. B: At. Mol. Opt. Phys.43 175501 [39] Chen Q H, Liu T, Zhang Y Y and Wang K L 2011 Europhys. Lett.96 14003 [40] He S, Wang C, Chen Q H, Ren X Z, Liu T and Wang K L 2012 Phys. Rev. A86 033837 [41] Irish E K, Gea-Banacloche J, Martin I and Schwab K C 2005 Phys. Rev. B72 195410 [42] Irish E K 2007 Phys. Rev. Lett.99 173601 [43] Zhang Y Y, Chen Q H and Zhao Y 2013 Phys. Rev. A87 033827 [44] Zhang Y W, Chen G, Yu L X, Liang Q F, Liang J Q and Jia S T 2011 Phys. Rev. A83 065802 [45] Yu L X, Zhu S Q, Liang Q F, Chen G and Jia S T 2012 Phys. Rev. A86 015803 [46] Liu M X, Ying Z J, An J H and Luo H G 2015 New J. Phys.17 043001 [47] Ying Z J, Liu M X, Luo H G, Lin H Q and You J Q 2015 Phys. Rev. A92 053823 [48] Mao B B, Liu M X, Wu W, Li L S, Ying Z J and Luo H G 2018 Chin. Phys. B27 054219 [49] Gan C J and Zheng H 2010 Eur. Phys. J. D59 473 [50] Mirzaee M and Kamani N 2013 Chin. Phys. B22 094203 [51] Wang Z H and Zhou D L 2013 Chin. Phys. B22 114205 [52] Wang Y M and Haw J Y 2015 Phys. Lett. A379 779 [53] Cong L, Sun X M, Liu M X, Ying Z J and Luo H G 2017 Phys. Rev. A95 063803 [54] Rahav S, Gilary I and Fishman S 2003 Phys. Rev. A68 013820 [55] Goldman N and Dalibard J 2014 Phys. Rev. X4 031027 [56] Eckardt A and Anisimovas E 2015 New J. Phys.17 093039 [57] Bukov M, D'Alessio L and Polkovnikov A 2015 Adv. Phys.64 139 [58] Eckardt A 2017 Rev. Mod. Phys.89 011004 [59] Rodriguez-Vega M, Lentz M and Seradjeh B 2018 New J. Phys.20 093022 [60] Oka T and Aoki H 2009 Phys. Rev. B79 081406(R) [61] Hemmerich A 2010 Phys. Rev. A81 063626 [62] Bermudez A, Schaetz T and Porras D 2012 New J. Phys.14 053049 [63] Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D and Esslinger T 2014 Nature515 237 [64] Gulácsi B and Dóra B 2015 Phys. Rev. Lett.115 160402 [65] Meinert F, Mark M J, Lauber K, Daley A J and Nägerl H C 2016 Phys. Rev. Lett.116 205301 [66] Mikami T, Kitamura S, Yasuda K, Tsuji N, Oka T and Aoki H 2016 Phys. Rev. B93 144307 [67] Rodriguez-Vega M and Seradjeh B 2018 Phys. Rev. Lett.121 036402
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
