ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID DYNAMICS |
Prev
Next
|
|
|
Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model |
Yue Chen(陈越)1,2, Maoxin Liu(刘卯鑫)3, and Xiaosong Chen(陈晓松)3,† |
1 CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China; 2 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China; 3 School of Systems Science, Beijing Normal University, Beijing 100875, China |
|
|
Abstract We investigate the two-mode quantum Rabi model (QRM) describing the interaction between a two-level atom and a two-mode cavity field. The quantum phase transitions are found when the ratio $ \eta $ of transition frequency of atom to frequency of cavity field approaches infinity. We apply the Schrieffer-Wolff (SW) transformation to derive the low-energy effective Hamiltonian of the two-mode QRM, thus yielding the critical point and rich phase diagram of quantum phase transitions. The phase diagram consists of four regions: a normal phase, an electric superradiant phase, a magnetic superradiant phase and an electromagnetic superradiant phase. The quantum phase transition between the normal phase and the electric (magnetic) superradiant phase is of second order and associates with the breaking of the discrete $ Z_2 $ symmetry. On the other hand, the phase transition between the electric superradiant phase and the magnetic superradiant phase is of first order and relates to the breaking of the continuous $U(1)$ symmetry. Several important physical quantities, for example the excitation energy and average photon number in the four phases, are derived. We find that the excitation spectra exhibit the Nambu-Goldstone mode. We calculate analytically the higher-order correction and finite-frequency exponents of relevant quantities. To confirm the validity of the low-energy effective Hamiltonians analytically derived by us, the finite-frequency scaling relation of the averaged photon numbers is calculated by numerically diagonalizing the two-mode quantum Rabi Hamiltonian.
|
Received: 06 May 2023
Revised: 22 June 2023
Accepted manuscript online: 26 July 2023
|
PACS:
|
42.50.Ct
|
(Quantum description of interaction of light and matter; related experiments)
|
|
05.30.Rt
|
(Quantum phase transitions)
|
|
64.60.an
|
(Finite-size systems)
|
|
12.38.Bx
|
(Perturbative calculations)
|
|
Fund: We thank Gaoke Hu for his helpful discussion on the finite-frequency scaling. This work was supported by the National Natural Science Foundation of China (Grant No. 12135003). The authors acknowledge HPC Cluster of ITP-CAS for supplying computation resources. |
Corresponding Authors:
Xiaosong Chen
E-mail: chenxs@bnu.edu.cn
|
Cite this article:
Yue Chen(陈越), Maoxin Liu(刘卯鑫), and Xiaosong Chen(陈晓松) Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model 2023 Chin. Phys. B 32 104213
|
[1] Fink J M, Dombi A, Vukics A, Wallraff A and Domokos P 2017 Phys. Rev. X 7 011012 [2] Sun Z H, Cai J Q, Tang Q C, Hu Y and Fan H 2020 Ann. Phys. 532 1900270 [3] Casteels W, Fazio R and Ciuti C 2017 Phys. Rev. A 95 012128 [4] Zhu H J, Xu K, Zhang G F and Liu W M 2020 Phys. Rev. Lett. 125 050402 [5] Garbe L, Bina M, Keller A, Paris M G A and Felicetti S 2020 Phys. Rev. Lett. 124 120504 [6] Puebla R, Smirne A, Huelga S F and Plenio M B 2020 Phys. Rev. Lett. 124 230602 [7] Chen X Y, Zhang Y Y, Fu L and Zheng H 2020 Phys. Rev. A 101 033827 [8] Zhang Y, Mao B B, Xu D, Zhang Y Y, You W L, Liu M and Luo H G 2020 J. Phys. A: Math. Theor. 53 315302 [9] Forn-Díaz P, Lamata L, Rico E, Kono J and Solano E 2019 Rev. Mod. Phys. 91 025005 [10] Zhu C J, Ping L L, Yang Y P and Agarwal G S 2020 Phys. Rev. Lett. 124 073602 [11] Wang Y Z, He S, Duan L and Chen Q H 2021 Phys. Rev. B 103 205106 [12] Leppäkangas J, Braumüller J, Hauck M, Reiner J M, Schwenk I, Zanker S, Fritz L, Ustinov A V, Weides M and Marthaler M 2018 Phys. Rev. A 97 052321 [13] Abdi M 2019 Phys. Rev. B 100 184310 [14] Garbe L, Egusquiza I L, Solano E, Ciuti C, Coudreau T, Milman P and Felicetti S 2017 Phys. Rev. A 95 053854 [15] Hwang M J, Puebla R and Plenio M B 2015 Phys. Rev. Lett. 115 180404 [16] Xie Q, Zhong H, Batchelor M T and Lee C 2017 J. Phys. A: Math. Theor. 50 113001 [17] Rabi I I 1936 Phys. Rev. 49 324 [18] Rabi I I 1937 Phys. Rev. 51 652 [19] Jaynes E and Cummings F 1963 Proc. IEEE 51 89 [20] Raimond J M, Brune M and Haroche S 2001 Rev. Mod. Phys. 73 565 [21] Holstein T 1959 Ann. Phys. 8 325 [22] Crespi A, Longhi S and Osellame R 2012 Phys. Rev. Lett. 108 163601 [23] Cai M L, Liu Z D, Zhao W D, Wu Y K, Mei Q X, Jiang Y, He L, Zhang X, Zhou Z C and Duan L M 2021 Nat. Commun. 12 1126 [24] Chen X, Wu Z, Jiang M, Lü X Y, Peng X and Du J 2021 Nat. Commun. 12 6281 [25] Blais A, Huang R S, Wallraff A, Girvin S M and Schoelkopf R J 2004 Phys. Rev. A 69 062320 [26] Wallraff A, Schuster D I, Blais A, Frunzio L, Huang R S, Majer J, Kumar S, Girvin S M and Schoelkopf R J 2004 Nature 431 162 [27] 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 [28] 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 [29] Chen Z, Wang Y, Li T, Tian L, Qiu Y, Inomata K, Yoshihara F, Han S, Nori F, Tsai J S and You J Q 2017 Phys. Rev. A 96 012325 [30] Forn-Díaz P, García-Ripoll J J, Peropadre B, Orgiazzi J L, Yurtalan M A, Belyansky R, Wilson C M and Lupascu A 2017 Nat. Phys. 13 39 [31] Yoshihara F, Fuse T, Ashhab S, Kakuyanagi K, Saito S and Semba K 2017 Nat. Phys. 13 44 [32] Irish E K 2007 Phys. Rev. Lett. 99 173601 [33] Irish E K 2007 Phys. Rev. Lett. 99 259901 [34] Zhong H, Xie Q, Batchelor M T and Lee C 2013 J. Phys. A: Math. Theor. 46 415302 [35] Xie Q T, Cui S, Cao J P, Amico L and Fan H 2014 Phys. Rev. X 4 021046 [36] Ying Z J, Liu M, Luo H G, Lin H Q and You J Q 2015 Phys. Rev. A 92 053823 [37] Casanova J, Romero G, Lizuain I, García-Ripoll J J and Solano E 2010 Phys. Rev. Lett. 105 263603 [38] Gan C J and Zheng H 2010 Eur. Phys. J. D 59 473 [39] Braak D 2011 Phys. Rev. Lett. 107 100401 [40] Larson J 2012 Phys. Rev. Lett. 108 033601 [41] Yu L, Zhu S, Liang Q, Chen G and Jia S 2012 Phys. Rev. A 86 015803 [42] Chen Q H, Wang C, He S, Liu T and Wang K L 2012 Phys. Rev. A 86 023822 [43] Ashhab S 2013 Phys. Rev. A 87 013826 [44] De Liberato S 2014 Phys. Rev. Lett. 112 016401 [45] Liu M, Ying Z J, An J H and Luo H G 2015 New J. Phys. 17 043001 [46] Cong L, Sun X M, Liu M, Ying Z J and Luo H G 2017 Phys. Rev. A 95 063803 [47] Wang Y, You W L, Liu M, Dong Y L, Luo H G, Romero G and You J Q 2018 New J. Phys. 20 053061 [48] Mao B B, Li L, Wang Y, You W L, Wu W, Liu M and Luo H G 2019 Phys. Rev. A 99 033834 [49] Mahmoodian S 2019 Phys. Rev. Lett. 123 133603 [50] Kockum A F, Miranowicz A, De Liberato S, Savasta S and Nori F 2019 Nat. Rev. Phys. 1 295 [51] Forn-Díaz P, Lamata L, Rico E, Kono J and Solano E 2019 Rev. Mod. Phys. 91 025005 [52] Xie W, Mao B B, Li G, Wang W, Sun C, Wang Y, You W L and Liu M 2020 J. Phys. A: Math. Theor. 53 095302 [53] Le Boité A 2020 Adv. Quantum Technol. 3 1900140 [54] Frisk K A, Miranowicz A, De Liberato S, Savasta S and Nori F 2019 Nat. Rev. Phys. 1 19 [55] Liu M, Chesi S, Ying Z J, Chen X, Luo H G and Lin H Q 2017 Phys. Rev. Lett. 119 220601 [56] Boller K J, Imamoǧlu A and Harris S E 1991 Phys. Rev. Lett. 66 2593 [57] Fleischhauer M and Lukin M D 2000 Phys. Rev. Lett. 84 5094 [58] Bergmann K, Theuer H and Shore B W 1998 Rev. Mod. Phys. 70 1003 [59] Bruß D and Macchiavello C 2002 Phys. Rev. Lett. 88 127901 [60] Cerf N J, Bourennane M, Karlsson A and Gisin N 2002 Phys. Rev. Lett. 88 127902 [61] Zhou Z, Chu S I and Han S 2002 Phys. Rev. B 66 054527 [62] Sjöqvist E, Tong D M, Andersson L M, Hessmo B, Johansson M and Singh K 2012 New J. Phys. 14 103035 [63] Hayn M, Emary C and Brandes T 2011 Phys. Rev. A 84 053856 [64] Cordero S, Nahmad-Achar E, López-Peña R and Castaños O 2015 Phys. Rev. A 92 053843 [65] Baksic A and Ciuti C 2014 Phys. Rev. Lett. 112 173601 [66] Fan J, Yang Z, Zhang Y, Ma J, Chen G and Jia S 2014 Phys. Rev. A 89 023812 [67] Léonard J, Morales A, Zupancic P, Esslinger T and Donner T 2017 Nature 543 87 [68] Nambu Y and Jona-Lasinio G 1961 Phys. Rev. 122 345 [69] Goldstone J 1961 Il Nuovo Cimento 19 154 [70] Goldstone J, Salam A and Weinberg S 1962 Phys. Rev. 127 965 [71] Popov V N and Fedotov S A 1982 Theor. Math. Phys. 51 363 [72] Ye J and Zhang C 2011 Phys. Rev. A 84 023840 [73] Yu Y X, Ye J and Liu W M 2013 Sci. Rep. 3 3476 [74] Vidal J and Dusuel S 2006 Europhys. Lett. 74 817 [75] Dusuel S and Vidal J 2004 Phys. Rev. Lett. 93 237204 [76] Dusuel S and Vidal J 2005 Phys. Rev. B 71 224420 |
No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
Altmetric
|
blogs
Facebook pages
Wikipedia page
Google+ users
|
Online attention
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.
View more on Altmetrics
|
|
|