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In this paper, we investigate the finite-temperature properties and phase transition of the Dicke model. Converting the atomic pseudo-spin operator to the two-mode Fermi operators, we obtain the partition function in terms of the imaginary-time path integral. The atomic population and average photon number as analytic functions of the atom-photon coupling strength are found from the thermodynamic equilibrium equation, which leads to the stationary state at a finite temperature and is determined by the variation in an extremum-condition of the Euclidean action with respect to the bosonic field. In particular we study the phase transition from normal to superradiation phase at a fixed low-temperature, in which the phase transition is dominated by quantum fluctuations. The phase transition induced by the variation of the atom-photon coupling strength indeed obeys the Landau continuous phase-transition theory, in which the average photon-number can serve as an order parameter with non-zero value that characterizes the superradiation phase. In the zero temperature limit our results recover exactly all those obtained from the quantum phase transition theory at zero temperature. In addition, we discuss the thermodynamic properties and compare the difference between finite-temperature phase transition and zero-temperature quantum phase transition. It is discovered that the average photon-number and mean energy in the low-temperature stationary state coincide with the corresponding values of zero-temperature in the strong coupling region. The entropy of the superradiation phase decays rapidly to zero with the increase of coupling strength.
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Keywords:
- Dicke model /
- imaginary-time path integral /
- phase transition
[1] Dicke R H 1954 Phys. Rev. 93 99
[2] Scully M O, Zubairy M S 1997 Quantum Optics (Cambridge University Press) p196
[3] Jurčo B 1989 J. Math. Phys. 30 1289
[4] Bogoliubov N M, Bullough R K, Timonen J 1996 J. Phys. A: Math. Gen. 29 6305
[5] Amico L, Hikami K 2005 Eur. Phys. J. B 43 387
[6] Klein A, Marshalek E R 1991 Rev. Mod. Phys. 63 375
[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] Weiss U 2008 Quantum Dissipative Systems (Singapore:World Scientific) p31
[9] Carollo A C M, Pachos J K 2005 Phys. Rev. Lett. 95 157203
[10] Osterloh A, Amico L, Falci G, Fazio R 2002 Nature 416 608
[11] Zhu S L 2006 Physics 35 11 (in Chinese) [朱诗亮 2006 物理 35 11]
[12] Vidal G, Lorre J I, Rico E, atKitaev A 2003 Phys. Rev. Lett. 90 227902
[13] 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]
[14] Hioes F T 1973 Phys. Rev. A 8 1440
[15] Sachdev S 1999 Quantum Phase Transitions(UK:Cambridge University Press)
[16] Hepp K, Lieb E H 1973 Ann. Phys. 76 360
[17] Wang Y K, Hioe F T 1973 Phys. Rev. A 7 831
[18] Emary Clive, Brandes Tobias 2003 Phys. Rev. E67 066203
[19] Chen G, Li J Q, Liang J Q 2006 Phys. Rev. A 74 054101
[20] Yang X Y, Xue H B, Liang J Q 2013 Acta Phys. Sin. 62 114205 (in Chinese) [杨晓勇, 薛海斌, 梁九卿 2013 62 114205]
[21] Lian J L, Zhang Y W, Liang J Q 2012 Chin. Phys. Lett. 29 060302
[22] Zhao X Q, Liu N, Liang J Q 2014 Phys. Rev. A 90 023622
[23] Yu L X, Liang Q F, Wang L R, Zhu S Q 2014 Acta Phys. Sin. 63 134204 (in Chinese) [俞立先, 梁奇锋, 汪丽蓉, 朱士群 2014 63 134204]
[24] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
[25] Bastidas V M, Emary C, Regler B, Brandes T 2012 Phys. Rev. Lett. 108 043003
[26] Nagy D, Kónya G, Szirmai G, Domokos P 2010 Phys. Rev. Lett. 104 130401
[27] Zhang Y W, Lian J L, Liang J Q, Chen G, Jia S T 2013 Phys. Rev. A 87 013616
[28] Liu N, Li J D, Liang J Q 2013 Phys. Rev. A 87 053623
[29] Liu N, Lian J L, Ma J, Xiao L T, Chen G, Liang J Q, Jia S T 2011 Phys. Rev. A 83 033601
[30] Popov V N, Fedotov S A 1982 Theor. Math. Phys 51 73
[31] Aparicio Alcalde M, de Lemos A L L, Svaiter N F 2007 J. Phys. A :Math. Theor 40 11961
[32] Popov V N, Fedotov S A 1988 Sov. Phys. JETP 67 535
[33] Aparicio Alcalde M, Pimentel B M 2011 Physic A 390 3385
[34] Kir'yanov V B, Yarunin V S 1982 Teoret. Mat. Fiz 51 456
[35] Liang J Q, Wei L F 2011 Advances In Quantum Physics (Beijing: Science Press) p95 (in Chinese) [梁九卿, 韦联福 2011 量子物理新进展(北京: 科学出版社)第95页]
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[1] Dicke R H 1954 Phys. Rev. 93 99
[2] Scully M O, Zubairy M S 1997 Quantum Optics (Cambridge University Press) p196
[3] Jurčo B 1989 J. Math. Phys. 30 1289
[4] Bogoliubov N M, Bullough R K, Timonen J 1996 J. Phys. A: Math. Gen. 29 6305
[5] Amico L, Hikami K 2005 Eur. Phys. J. B 43 387
[6] Klein A, Marshalek E R 1991 Rev. Mod. Phys. 63 375
[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] Weiss U 2008 Quantum Dissipative Systems (Singapore:World Scientific) p31
[9] Carollo A C M, Pachos J K 2005 Phys. Rev. Lett. 95 157203
[10] Osterloh A, Amico L, Falci G, Fazio R 2002 Nature 416 608
[11] Zhu S L 2006 Physics 35 11 (in Chinese) [朱诗亮 2006 物理 35 11]
[12] Vidal G, Lorre J I, Rico E, atKitaev A 2003 Phys. Rev. Lett. 90 227902
[13] 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]
[14] Hioes F T 1973 Phys. Rev. A 8 1440
[15] Sachdev S 1999 Quantum Phase Transitions(UK:Cambridge University Press)
[16] Hepp K, Lieb E H 1973 Ann. Phys. 76 360
[17] Wang Y K, Hioe F T 1973 Phys. Rev. A 7 831
[18] Emary Clive, Brandes Tobias 2003 Phys. Rev. E67 066203
[19] Chen G, Li J Q, Liang J Q 2006 Phys. Rev. A 74 054101
[20] Yang X Y, Xue H B, Liang J Q 2013 Acta Phys. Sin. 62 114205 (in Chinese) [杨晓勇, 薛海斌, 梁九卿 2013 62 114205]
[21] Lian J L, Zhang Y W, Liang J Q 2012 Chin. Phys. Lett. 29 060302
[22] Zhao X Q, Liu N, Liang J Q 2014 Phys. Rev. A 90 023622
[23] Yu L X, Liang Q F, Wang L R, Zhu S Q 2014 Acta Phys. Sin. 63 134204 (in Chinese) [俞立先, 梁奇锋, 汪丽蓉, 朱士群 2014 63 134204]
[24] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
[25] Bastidas V M, Emary C, Regler B, Brandes T 2012 Phys. Rev. Lett. 108 043003
[26] Nagy D, Kónya G, Szirmai G, Domokos P 2010 Phys. Rev. Lett. 104 130401
[27] Zhang Y W, Lian J L, Liang J Q, Chen G, Jia S T 2013 Phys. Rev. A 87 013616
[28] Liu N, Li J D, Liang J Q 2013 Phys. Rev. A 87 053623
[29] Liu N, Lian J L, Ma J, Xiao L T, Chen G, Liang J Q, Jia S T 2011 Phys. Rev. A 83 033601
[30] Popov V N, Fedotov S A 1982 Theor. Math. Phys 51 73
[31] Aparicio Alcalde M, de Lemos A L L, Svaiter N F 2007 J. Phys. A :Math. Theor 40 11961
[32] Popov V N, Fedotov S A 1988 Sov. Phys. JETP 67 535
[33] Aparicio Alcalde M, Pimentel B M 2011 Physic A 390 3385
[34] Kir'yanov V B, Yarunin V S 1982 Teoret. Mat. Fiz 51 456
[35] Liang J Q, Wei L F 2011 Advances In Quantum Physics (Beijing: Science Press) p95 (in Chinese) [梁九卿, 韦联福 2011 量子物理新进展(北京: 科学出版社)第95页]
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