Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

doi:10.1088/1674-1056/acea66

中国物理B ›› 2023, Vol. 32 ›› Issue (10): 104213-104213.doi: 10.1088/1674-1056/acea66

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

收稿日期:2023-05-06 修回日期:2023-06-22 接受日期:2023-07-26 出版日期:2023-09-21 发布日期:2023-10-08
通讯作者: Xiaosong Chen E-mail:chenxs@bnu.edu.cn
基金资助:
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.

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

Received:2023-05-06 Revised:2023-06-22 Accepted:2023-07-26 Online:2023-09-21 Published:2023-10-08
Contact: Xiaosong Chen E-mail:chenxs@bnu.edu.cn
Supported by:
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.

摘要/Abstract

摘要： 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.

关键词: two-mode quantum Rabi model, superradiant phase transition, Nambu-Goldstone mode, finite-frequency scaling, Schrieffer-Wolff (SW) transformation

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.

Key words: two-mode quantum Rabi model, superradiant phase transition, Nambu-Goldstone mode, finite-frequency scaling, Schrieffer-Wolff (SW) transformation

中图分类号: (Quantum description of interaction of light and matter; related experiments)

42.50.Ct

05.30.Rt (Quantum phase transitions) 64.60.an (Finite-size systems) 12.38.Bx (Perturbative calculations)

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[J]. 中国物理B, 2023, 32(10): 104213-104213.

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Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

Ground-state phase diagram, symmetries, excitation spectra and finite-frequency scaling of the two-mode quantum Rabi model

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