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Dicke model describes a collective interaction between the two-level atoms and the light cavity and has been predicted to show a peculiar quantum phase transition, which is a second-order phase transition from a normal phase (in a weak-coupling strength) to a superradiant phase (in a strong-coupling strength). The model plays an important role in illustrating the quantum ground-state properties of many-body macroscopic quantum states. In the experiment, Dicke quantum phase transition in an open system could be formed by a Bose-Einstein condensate coupled to a high-finesse optical cavity. This experiment on the Bose-Einstein condensate trapped in the optical cavity have opened new frontiers, which could combine the cold atoms with quantum optics and makes it possible to enter into the strongly coupled regime of cavity quantum electrodynamics. In strong coupled regime, the atoms exchange the photons many times before spontaneous emission and cavity losses set in. It has become a hot research topic in recent years and plays an important role in many fields of modern physics, such as condensed matter physics, nuclear physics, etc. It can be applied to the manipulation of the geometric phase and entanglement in quantum information and computing. Quantum phase transition has been widely studied for the Dicke model as a typical example. Many different research methods about the mean-field approximation have been used to analyze the ground state properties of the Dicke model. In this paper, we study the ground state properties of two-component Bose-Einstein condensate in a single-mode cavity. Meanwhile, the associated quantum phase transition is described by the spin-coherent-state variational method, whose advantage is that the ground state energy and wave function can be obtained without the thermodynamic limit. By taking the average in the boson coherent state, we obtain an equivalent effective pesudospin Hamiltonian, which will be diagonalized by using the spin coherent state. Finally, we can obtain the energy functional, which is the basics of the variation to obtain the numerical solution of photon number and the expression of the atomic number and the ground state energy. This paper presents a rich phase diagram, which can be manipulated by changing the atom-field coupling imbalance between two components and the atom-field frequency detuning. While in the single-mode Dicke model there exist only the normal phase and the superradiation phase. When the frequency of one component atom is zero or the frequency of the two component atoms are equal in optical cavity, the system returns to the standard Dicke model, in which there occurs the second-order phase transition from the normal phase to the superradiant phase by adjusting the atom-field coupling. In conclusion, we discover that the stimulated radiation comes from the collective state of atomic population inversion, which does not exist in the single-mode Dicke model. Meanwhile, the new stimulated-radiation state S and S, which can only be produced by one component of the atom, are observed in the two-component Bose-Einstein condensates in the single-mode optical cavity. By adjusting the atom-field coupling imbalance and the atom-field frequency detuning (the blue or red detuning), the order of the superradiation state and the stimulated-radiation states can be exchanged between the two components of the atom.
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Keywords:
- quantum phase transition /
- spin-coherent-state transformation /
- energy function /
- stimulated radiation
[1] Dicke R H 1954 Phys. Rev. 93 99
[2] Colombe Y, Steinmetz T, Dubois G, Linke F, Hunger D, Reichel J 2007 Nature 450 272
[3] Baumann K, Mottl R, Brennecke F, Esslinger T 2011 Phys. Rev. Lett. 107 140402
[4] Ritsch H, Domokos P, Brennecke F, Esslinger T 2013 Rev. Mod. Phys. 85 553
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[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 物理学报 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 物理学报 59 3695]
[8] Cao H, Fu L B 2012 Eur. Phys. J. D 66 97
[9] Zhang Y C, Zhou X F, Guo G C, Zhou X, Pu H, Zhou Z W 2014 Phys. Rev. A 89 053624
[10] Zhang Y, Chen G, Zhang C 2013 Sci. Rep. 3 1937
[11] Lian J L, Yu L, Liang J Q, Chen G, Jia S T 2013 Sci. Rep. 3 3166
[12] Huang Y, Hu Z D 2015 Sci. Rep. 5 8006
[13] Hamner C, Qu C, Zhang Y, Chang J, Gong M, Zhang C 2014 Nature Commun. 5 4023
[14] Emary C, Brandes T 2003 Phys. Rev. E 67 066203
[15] Dimer F, Estienne B, Parkins A S, Carmichael H J 2007 Phys. Rev. A 75 013804
[16] Chen G, Wang X G, Liang J Q, Wang Z D 2008 Phys. Rev. A 78 023634
[17] Nagy D, Knya G, Szirmai G, Domokos P 2010 Phys. Rev. Lett. 104 130401
[18] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
[19] Timmermans E 1998 Phys. Rev. Lett. 81 5718
[20] Pu H, Bigelow N P 1998 Phys. Rev. Lett. 80 1130
[21] Dong Y, Ye J W, Pu H 2011 Phys. Rev. A 83 031608
[22] Sasaki K, Suzuki N, Saito H 2011 Phys. Rev. A 83 053606
[23] Sensen A, Duan L M, Cirac J I, Zoller P 2001 Nature 409 63
[24] Gordon D, Savage C M 1999 Phys. Rev. A 59 4623
[25] Andrews M R, Townsend C G, Miesner H J, Durfee D S, Kurn D M, Ketterle W 1997 Science 275 637
[26] Bhattacherjee A B 2014 Phys. Lett. A 378 3244
[27] Liu N, Li J D, Liang J Q 2013 Phys. Rev. A 87 053623
[28] Keeling J, Bhaseen M J, Simons B D 2010 Phys. Rev. Lett. 105 043001
[29] Bhaseen M J, Mayoh J, Simons B D 2012 Phys. Rev. A 85 013817
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[1] Dicke R H 1954 Phys. Rev. 93 99
[2] Colombe Y, Steinmetz T, Dubois G, Linke F, Hunger D, Reichel J 2007 Nature 450 272
[3] Baumann K, Mottl R, Brennecke F, Esslinger T 2011 Phys. Rev. Lett. 107 140402
[4] Ritsch H, Domokos P, Brennecke F, Esslinger T 2013 Rev. Mod. Phys. 85 553
[5] Liang J Q, Liu J L, Li W D, Li Z J 2009 Phys. Rev. A 79 033617
[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 物理学报 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 物理学报 59 3695]
[8] Cao H, Fu L B 2012 Eur. Phys. J. D 66 97
[9] Zhang Y C, Zhou X F, Guo G C, Zhou X, Pu H, Zhou Z W 2014 Phys. Rev. A 89 053624
[10] Zhang Y, Chen G, Zhang C 2013 Sci. Rep. 3 1937
[11] Lian J L, Yu L, Liang J Q, Chen G, Jia S T 2013 Sci. Rep. 3 3166
[12] Huang Y, Hu Z D 2015 Sci. Rep. 5 8006
[13] Hamner C, Qu C, Zhang Y, Chang J, Gong M, Zhang C 2014 Nature Commun. 5 4023
[14] Emary C, Brandes T 2003 Phys. Rev. E 67 066203
[15] Dimer F, Estienne B, Parkins A S, Carmichael H J 2007 Phys. Rev. A 75 013804
[16] Chen G, Wang X G, Liang J Q, Wang Z D 2008 Phys. Rev. A 78 023634
[17] Nagy D, Knya G, Szirmai G, Domokos P 2010 Phys. Rev. Lett. 104 130401
[18] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
[19] Timmermans E 1998 Phys. Rev. Lett. 81 5718
[20] Pu H, Bigelow N P 1998 Phys. Rev. Lett. 80 1130
[21] Dong Y, Ye J W, Pu H 2011 Phys. Rev. A 83 031608
[22] Sasaki K, Suzuki N, Saito H 2011 Phys. Rev. A 83 053606
[23] Sensen A, Duan L M, Cirac J I, Zoller P 2001 Nature 409 63
[24] Gordon D, Savage C M 1999 Phys. Rev. A 59 4623
[25] Andrews M R, Townsend C G, Miesner H J, Durfee D S, Kurn D M, Ketterle W 1997 Science 275 637
[26] Bhattacherjee A B 2014 Phys. Lett. A 378 3244
[27] Liu N, Li J D, Liang J Q 2013 Phys. Rev. A 87 053623
[28] Keeling J, Bhaseen M J, Simons B D 2010 Phys. Rev. Lett. 105 043001
[29] Bhaseen M J, Mayoh J, Simons B D 2012 Phys. Rev. A 85 013817
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