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Now, many different approaches have been presented to study the different semi-classical models derived from the Dicke Hamiltonian, which reflect a fact that the quantum-mechanical spin possesses no direct classical analog. The Hartree-Fock-type approximation is one of the widely used approaches, with which we derive the Heisenberg equations of motion for the system and replace the operators in these equations with the corresponding expectation values. In the present paper, we investigate the role of quantum phase transition in determining the chaotic property of the time-dependent driven Dicke model. The semi-classical Hamiltonian is derived by evaluating the expectation value of the Dicke Hamiltonian in a state, which is a product state of photonic and atomic coherent states. Based on the inverse of the relations between the position-momentum representation and the Bosonic creation-annihilation operators, the Hamiltonian is rewritten in the position-momentum representation and it undergoes a spontaneous symmetry-breaking phase transition, which is directly analogous to the quantum phase transition of the quantum system. In order to depict the Poincaré sections, which are used to analyze the trajectories through the four-dimensional phase space, we give the equations of motion of system from the derivatives of the semi-classical Hamiltonian for a variety of different parameters and initial conditions. According to the Dicke quantum phase transition observed from the experimental setup , we study the effect of a monochromatic non-adiabatic modulation of the atom-field coupling in Dicke model (i.e., the driven Dicke model) on the system chaos by adjusting the pump laser intensity. The change from periodic track to chaotic figure reflects the quantum properties of the system, especially the quantum phase transition point, which is a key position for people to analyse the shift from periodic orbit to chaos. In an undriven case, the system reduces to the standard Dicke model. We discover from the Poincaré sections that the system undergoes a change from the classical periodic orbit to a number of chaotic trajectories and in the superradiant phase area, the whole phase space is completely chaotic. When the static and driving coupling both exist, the system shows rich chaotic motion. The ground state properties are mainly determined by the static coupling, while the orbit of the system is adjusted by the driving coupling. If the static coupling is greater than the critical coupling, the system displays completely chaotic images in the Poincaré sections, and the periodic orbits in the chaos can also be adjusted by the strong driving coupling. While the static coupling is less than the critical coupling, the system can also show the chaotic images by adjusting the driving coupling strength and oscillation frequency. Moreover, by tuning the oscillation frequency, the Poincaré sections may change from the classical orbits to the chaos, and back to the classical orbits in the normal phase of the system.
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Keywords:
- a monochromatic non-adiabatic modulation /
- Dicke model /
- quantum chaos /
- the Poincaré
[1] Hepp K, Lieb E H 1973 Ann. Phys. 76 360
[2] Hioes F T 1973 Phys. Rev. A 8 1440
[3] Liu N, Lian J L, Ma J, Xiao L T, Chen G, Liang J Q, Jia S T 2011 Phys. Rev. A 83 033601
[4] Graham R, Höhnerbach M 1984 Phys. Lett. A 101 61
[5] Furuya K, Nemes M C, Pellegrino G Q 1998 Phys. Rev. Lett. 80 5524
[6] Song L J, Yan D, Gai Y J, Wang Y B 2011 Acta Phys. Sin. 60 020302 (in Chinese) [宋立军, 严冬, 盖永杰, 王玉波 2011 60 020302]
[7] Song L J, Yan D, Gai Y J, Wang Y B 2010 Acta Phys. Sin. 59 3695 (in Chinese) [宋立军, 严冬, 盖永杰, 王玉波 2010 59 3695]
[8] Zhao W L, Wang J Z, Dou F Q 2012 Acta Phys. Sin. 61 240302 (in Chinese) [赵文垒, 王建忠, 豆福全 2012 61 240302]
[9] Emary C, Brandes T 2003 Phys. Rev. E 67 066203
[10] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
[11] Yu L X, Fan J T, Zhu S Q, Chen G, Jia S T, Nori F 2014 Phys. Rev. A 89 023838
[12] Fan J T, Yang Z W, Zhang Y W, Ma J, Chen G, Jia S T 2014 Phys. Rev. A 89 023812
[13] Zhao X Q, Liu N, Liang J Q 2014 Phys. Rev. A 90 023622
[14] Bastidas V M, Emary C, Regler B, Brandes T 2012 Phys. Rev. Lett. 108 043003
[15] Liu N, Li J D, Liang J Q 2013 Phys. Rev. A 87 053623
[16] Holstein T, Primakoff H 1949 Phys. Rev. A 58 1098
[17] Hillery M, Mlodinow L D 1984 Phys. Rev. A 31 797
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[1] Hepp K, Lieb E H 1973 Ann. Phys. 76 360
[2] Hioes F T 1973 Phys. Rev. A 8 1440
[3] Liu N, Lian J L, Ma J, Xiao L T, Chen G, Liang J Q, Jia S T 2011 Phys. Rev. A 83 033601
[4] Graham R, Höhnerbach M 1984 Phys. Lett. A 101 61
[5] Furuya K, Nemes M C, Pellegrino G Q 1998 Phys. Rev. Lett. 80 5524
[6] Song L J, Yan D, Gai Y J, Wang Y B 2011 Acta Phys. Sin. 60 020302 (in Chinese) [宋立军, 严冬, 盖永杰, 王玉波 2011 60 020302]
[7] Song L J, Yan D, Gai Y J, Wang Y B 2010 Acta Phys. Sin. 59 3695 (in Chinese) [宋立军, 严冬, 盖永杰, 王玉波 2010 59 3695]
[8] Zhao W L, Wang J Z, Dou F Q 2012 Acta Phys. Sin. 61 240302 (in Chinese) [赵文垒, 王建忠, 豆福全 2012 61 240302]
[9] Emary C, Brandes T 2003 Phys. Rev. E 67 066203
[10] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
[11] Yu L X, Fan J T, Zhu S Q, Chen G, Jia S T, Nori F 2014 Phys. Rev. A 89 023838
[12] Fan J T, Yang Z W, Zhang Y W, Ma J, Chen G, Jia S T 2014 Phys. Rev. A 89 023812
[13] Zhao X Q, Liu N, Liang J Q 2014 Phys. Rev. A 90 023622
[14] Bastidas V M, Emary C, Regler B, Brandes T 2012 Phys. Rev. Lett. 108 043003
[15] Liu N, Li J D, Liang J Q 2013 Phys. Rev. A 87 053623
[16] Holstein T, Primakoff H 1949 Phys. Rev. A 58 1098
[17] Hillery M, Mlodinow L D 1984 Phys. Rev. A 31 797
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