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We use the mean field approximation method to study the quantum phase transitions of the Jaynes-Cummings lattice model and the Rabi lattice model. The effective Hamiltonians are obtained for the JC and Rabi model including the Kerr nonlinear term. Numerically we diagonalized the Hamiltonian matrix and calculated the superfluidity order parameter and the two-photon correlation function by solving the iteration equations. We have explored the Mott insulating-superfluid quantum phase transition, the bunching-antibunching behavior of light, and the effect of Kerr nonlinear term on the quantum phase transition and photon statistical characteristics. Our results show that in the JC lattice model, by increasing J, a quantum phase transition takes place and the system is driven to a superfluid phase. The phase boundaries of the Mott lobes are N-dependent. However the photon will always be in a bunching statistical behavior irrelevant of the coupling strength between the two-level atom and the phonton and the nonlinear Kerr effect. In the Rabi lattice model, the anti-rotating wave term breaks Mott-lobe structure of the phase diagram and the increase of the two-level atom and photon interaction strength g and the photon transition strength J between the lattices drive the system from the Mott insulating phase to the superfluid phase. The photon statistical behavior changes from the bunching to the antibunching one when considering the anti-rotating wave term, which is important in the strongly coupled systems. Most interestingly, the increase of the Kerr nonlinear coefficient will inhibit the Mott insulating phase-superfluid phase transition, but favor the superfluid phase and the transition from the bunching to anti-bunching statistics. -
Keywords:
- Jaynes-Cummings lattice model /
- Rabi lattice model /
- quantum phase transition /
- photon statistics
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[16] Rebic S, Tan S M, Parkins A S, Walls D F 1999 J. Opt. B 1 490
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[17] Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885
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[19] Greiner M, Mandel O, Esslinger T, Hänsch T W, Bloch I 2002 Nature 415 39
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[34] He Y, Zhu X, Mihalache D, Liu J, Chen Z 2012 Phys. Rev. A 85 013831
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图 1 (a), (b)平均场近似下, 不同晶格模型关于超流序参量
$ A = \braket{a} $ 的$ J \text {-} g $ 相图 (a) JC晶格模型; (b) Rabi 晶格模型. 横坐标为格点之间的光子跃迁强度$ J $ , 纵坐标为二能级原子和光子相互作用强度$ g $ , 横纵坐标的单位为$ \omega_0 $ , 颜色条表示超流序参量$ A = \braket{a} $ 的大小. 深蓝色表示Mott绝缘相, 浅黄色表示超流体相. 其他参量取值为:$ \mathop{\omega_{0} = \omega_{1}} = 1 $ , 光子截断数$ N = 20 $ . (c), (d)对于不同的$ J $ , 不同晶格模型的超流序参量$ A $ 随$ g $ 变化的图像 (c) JC晶格模型; (d) Rabi晶格模型Figure 1. (a), (b) Under the mean field approximation, the
$ J \text {-} g $ phase diagram of different lattice models with respect to the superfluid order parameter$ A = \braket{a} $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the photon transition intensity$ J $ between the lattice, the ordinate is the two-level atom and photon interaction strength$ g $ , the unit of the abscissa and the ordinate is$ \omega_0 $ , and the color bar represents the value of the superfluid order parameter$ A = \braket{a} $ . Dark blue indicates Mott insulating phase, and light yellow indicates superfluid phase. Other parameters are taken as$ \mathop{\omega_{0} = \omega_{1}} = 1 $ , and the number of the photon truncation$ N = 20 $ . (c), (d) For different$ J $ , the superfluid order parameter$ A $ of different lattice models varies with$ g $ : (c) JC lattice model; (d) Rabi lattice model.
图 2 平均场近似下, 不同晶格模型关于二阶关联函数
$ g^{2}(0) $ 的$J\text {-} g$ 相图 (a) JC晶格模型; (b) Rabi晶格模型. 横坐标为格点之间的光子跃迁强度$ J $ , 纵坐标为二能级原子和光子相互作用强度$ g $ , 横纵坐标的单位为$ \omega_0 $ , 颜色条表示二阶关联函数$ g^{2}(0) $ 的值. 其他参量取值为:$ \mathop{\omega_{0} = \omega_{1}} = 1 $ , 光子截断数$ N = 20 $ Figure 2. Under the mean field approximation, the
$ J \text {-} g $ phase diagram of different lattice models with respect to the second-order correlation function$ g^{2}(0) $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the photon transition intensity$ J $ between the lattice, the ordinate is the two-level atom and photon interaction strength$ g $ , the unit of the abscissa and the ordinate is$ \omega_0 $ , the color bar is represented by the value of the second-order correlation function$ g^{2}(0) $ .$ \mathop{\omega_{0} = \omega_{1}} = 1 $ , and the number of photon truncation$ N = 20 $ .
图 3 Kerr效应下不同晶格模型关于超流序参量
$ A = \braket{a} $ 的$ \kappa \text {-} g $ 相图 (a) JC晶格模型; (b) Rabi 晶格模型. 横坐标为Kerr非线性强度$ \kappa $ , 纵坐标为二能级原子和光子相互作用强度$ g $ , 横纵坐标的单位为$ \omega_0 $ , 颜色条表示超流序参量$ A $ 的大小. 其他参量取值为:$ \mathop{\omega_{0} = \omega_{1}} = 1 $ ,$ J = 0.05 $ , 光子截断数$ N = 20 $ Figure 3. The
$ \kappa \text {-} g $ phase diagram of different lattice models under the Kerr effect with respect to the superfluid order parameter$ A = \braket{a} $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the Kerr nonlinear intensity$ \kappa $ , the ordinate is the two-level atom and photon interaction strength$ g $ , the unit of the abscissa and the ordinate is$ \omega_0 $ , and the color bar represents the value of the superfluid order parameter$ A $ . Other parameters are taken as$ \mathop{\omega_{0} = \omega_{1}} = 1 $ ,$ J = 0.05 $ , and the number of photon truncation$ N = 20 $ .
图 4 Kerr效应下不同晶格模型关于二阶关联函数
$ g^{2}(0) $ 的$ \kappa \text {-} g $ 相图 (a) JC晶格模型; (b) Rabi晶格模型. 横坐标为Kerr非线性强度$ \kappa $ , 纵坐标为二能级原子和光子相互作用强度$ g $ , 横纵坐标的单位为$ \omega_0 $ , 颜色条表示二阶关联函数$ g^{2}(0) $ . 其他参量取值为:$ \mathop{\omega_{0} = \omega_{1}} = 1 $ ,$ J = 0.05 $ , 光子截断数$ N = 20 $ Figure 4. The
$ \kappa \text {-} g $ phase diagram of different lattice models under the Kerr effect with respect to the second-order correlation function$ g^{2}(0) $ : (a) JC lattice model; (b) Rabi lattice model. The abscissa is the Kerr nonlinear intensity$ \kappa $ , the ordinate is the two-level atom and photon interaction strength$ g $ , the unit of the abscissa and the ordinate is$ \omega_0 $ , and the color bar represents the value of second-order correlation function$ g^{2}(0) $ . Other parameters are taken as$ \mathop{\omega_{0} = \omega_{1}} = 1 $ ,$ J = 0.05 $ , and the number of photon truncation$ N = 20 $ . -
[1] Hartmann M J, Brandao F G S L, Plenio M B 2006 Nat. Phys. 2 849
Google Scholar
[2] Greentree A D, Tahan C, Cole J H, Hollenberg L C 2006 Nat. Phys. 2 856
Google Scholar
[3] Angelakis D G, Santos M F, Bose S 2007 Phys. Rev. A 76 031805
Google Scholar
[4] Rossini D, Fazio R 2007 Phys. Rev. Lett. 99 186401
Google Scholar
[5] Aichhorn M, Hohenadler M, Tahan C, Littlewood P B 2008 Phys. Rev. Lett. 100 216401
Google Scholar
[6] Na N, Utsunomiya S, Tian L, Yamamoto Y 2008 Phys. Rev. A 77 031803
Google Scholar
[7] Carusotto I, Gerace D, Türeci H E, De Liberato S, Ciuti C, and Imamoğlu A 2009 Phys. Rev. Lett. 103 033601
Google Scholar
[8] Schmidt S, Blatter G 2009 Phys. Rev. Lett. 103 086403
Google Scholar
[9] Koch J, Le Hur K 2009 Phys. Rev. A 80 023811
Google Scholar
[10] Pippan P, Evertz H G, Hohenadler M 2009 Phys. Rev. A 80 033612
Google Scholar
[11] Ferretti S, Andreani L C, Türeci H E, Gerace D 2010 Phys. Rev. A 82 013841
Google Scholar
[12] Umucalilar R O, Carusotto I 2011 Phys. Rev. A 84 043804
Google Scholar
[13] Birnbaum K M, Boca A, Miller R, Boozer A D, Northup T E, Kimble H J 2005 Nature 436 87
[14] Tian L, Carmichael H J 1992 Phys. Rev. A 46 R6801
Google Scholar
[15] Imamoḡlu A, Schmidt H, Woods G, Deutsch M 1997 Phys. Rev. Lett. 79 1467
Google Scholar
[16] Rebic S, Tan S M, Parkins A S, Walls D F 1999 J. Opt. B 1 490
Google Scholar
[17] Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885
Google Scholar
[18] Schmidt S, Koch J 2013 Ann. Phys. 525 395
Google Scholar
[19] Greiner M, Mandel O, Esslinger T, Hänsch T W, Bloch I 2002 Nature 415 39
[20] Lundqvist S, Nilsson N B 1989 Physics of Low-dimensional Systems (Sweden: World Scientific) pp89−95
[21] Fisher M P, Weichman P B, Grinstein G, Fisher D S 1989 Phys. Rev. B 40 546
Google Scholar
[22] van Oosten D, van Der Straten P, Stoof H T C 2001 Phys. Rev. A 63 053601
Google Scholar
[23] van Oosten D, van Der Straten P, Stoof H T C 2003 Phys. Rev. A 67 033606
Google Scholar
[24] Sheshadri K, Krishnamurthy H R, Pandit R, Ramakrishnan T V 1993 Europhys. Lett. 22 257
Google Scholar
[25] Krauth W, Trivedi N 1991 Europhys. Lett. 14 627
Google Scholar
[26] Krauth W, Trivedi N, Ceperley D 1991 Phys. Rev. Lett. 67 2307
Google Scholar
[27] Xie Z W, Liu W M 2004 Phys. Rev. A 70 045602
Google Scholar
[28] Albus A, Illuminati F, Eisert J 2003 Phys. Rev. A 68 023606
Google Scholar
[29] Lewenstein M, Santos L, Baranov M A, Fehrmann H 2004 Phys. Rev. Lett. 92 050401
Google Scholar
[30] Illuminati F, Albus A 2004 Phys. Rev. Lett. 93 090406
Google Scholar
[31] Cramer M, Eisert J, Illuminati F 2004 Phys. Rev. Lett. 93 190405
Google Scholar
[32] Fehrmann H, Baranov M A, Damski B, Lewenstein M, Santos L 2004 Opt. Commun. 243 23
[33] Littlewood P B, Eastham P R, Keeling J M J, Marchetti F M, Simons B D, Szymanska M H 2004 J. Phys. Condens. Matter 16 S3597
Google Scholar
[34] He Y, Zhu X, Mihalache D, Liu J, Chen Z 2012 Phys. Rev. A 85 013831
Google Scholar
[35] Eguchi K, Takagi Y, Nakagawa T, Yokoyama T 2012 Phys. Rev. B 85 174415
Google Scholar
[36] Vitali D, Fortunato M, Tombesi P 2000 Phys. Rev. Lett. 85 445
Google Scholar
[37] Angelakis D G, Dai L, Kwek L C 2010 Europhys. Lett. 91 10003
Google Scholar
[38] Patargias N, Bartzis V, Jannussis A 1995 Phys. Scr. 52 554
Google Scholar
[39] Bu S P, Zhang G F, Liu J, Chen Z Y 2008 Phys. Scr. 78 065008
Google Scholar
[40] Cordero S, Récamier J 2011 J. Phys. B: At. Mol. Opt. Phys. 44 135502
Google Scholar
[41] Schmidt H, Imamoğlu A 1996 Opt. Lett. 21 1936
Google Scholar
[42] Harris S E, Hau L V 1999 Phys. Rev. Lett. 82 4611
Google Scholar
[43] Niu Y, Gong S 2006 Phys. Rev. A 73 053811
Google Scholar
[44] Glauber R J 1963 Phys. Rev. 130 2529
Google Scholar
[45] Gomes C B C, Almeida F A G, Souza A M C 2016 Phys. Lett. A 38 1799
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