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By presenting the quantum evolution with the trajectories of points on the Bloch sphere, the Majorana representation provides an intuitive way to study a high dimensional quantum evolution. In this work, we study the dynamical evolution of the nonlinear two-mode boson system both in the mean-field model by one point on the Bloch sphere and the second-quantized model by the Majorana points, respectively. It is shown that the evolution of the state in the mean-field model and the self-trapping effect can be perfectly characterized by the motion of the point, while the quantum evolution in the second-quantized model can be expressed by an elegant formula of the Majorana points. We find that the motions of states in the two models are the same in linear case. In the nonlinear case, the contribution of the boson interactions to the formula of Majorana points in the second quantized model can be decomposed into two parts:one is the single point part which equals to the nonlinear part of the equation in mean-field model under lager boson number limit; the other one is related to the correlations between the Majorana points which cannot be found in the equation of the point in mean-field model. This means that, the quantum fluctuation which is neglected in the mean-field model can be represented by these correlations. To illustrate our results and shed more light on these two different models, we discussed the quantum state evolution and corresponding self-trapping phenomenon with different boson numbers and boson interacting strength by using the fidelity between the states of the two models and the correlation between the Majoranapoints and the single points in the mean-field model. The result show that the dynamics evolution of the two models are quite different with small boson numbers, since the correlation between the Majorana stars cannot be neglected. However, the second-quantized evolution and the mean-field evolution still vary in both the fidelity population difference between the two boson modes and the fidelity of the states in the two models. The difference between the continuous changes of the second quantized evolution with the boson interacting strength and the critical behavior of the mean-field evolution which related to the self-trapping effect is also discussed. These results can help us to investigate how to include the quantum fluctuation into the mean-field model and find a method beyond the mean field approach.
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Keywords:
- Majorana representation /
- self trapping /
- mean-field approach
[1] Bloch F, Rabi I I 1945 Rev. Mod. Phys. 17 237
[2] Majorana E 1932 Nuovo Cim. 9 43
[3] Stamper-Kurn D M, Ueda M 2013 Rev. Mod. Phys. 85 1191
[4] Zhu Q, Wu B 2015 Chin. Phys. B 24 050507
[5] Lian B, Ho T L, Zhai H 2012 Phys. Rev. A 85 051606
[6] Cui X, Lian B, Ho T L, Lev B L, Zhai H 2013 Phys. Rev. A 88 011601
[7] Devi A R U, Sudha, Rajagopal A K 2012 Quantum Inf. Process. 11 685
[8] Bruno P 2012 Phys. Rev. Lett. 108 240402
[9] Liu H D, Fu L B 2014 Phys. Rev. Lett. 113 240403
[10] Liu H D, Fu L B 2016 Phys. Rev. A 94 022123
[11] Tamate S, Ogawa K, Kitano M 2011 Phys. Rev. A 84 052114
[12] Aulbach M, Markham D, Murao M 2010 New J. Phys. 12 073025
[13] Martin J, Giraud O, Braun P A, Braun D, Bastin T 2010 Phys. Rev. A 81 062347
[14] Bastin T, Krins S, Mathonet P, Godefroid M, Lamata L, Solano E 2009 Phys. Rev. Lett. 103 070503
[15] Ribeiro P, Mosseri R 2011 Phys. Rev. Lett. 106 180502
[16] Ganczarek W, Kuś M,Życzkowski K 2012 Phys. Rev. A 85 032314
[17] Wang Z, Markham D 2012 Phys. Rev. Lett. 108 210407
[18] Wang Z, Markham D 2013 Phys. Rev. A 87 12104
[19] Cao H 2013 Acta Phys. Sin. 62 030303 (in Chinese)[曹辉2013 62 030303]
[20] Barnett R, Podolsky D, Refael G 2009 Phys. Rev. B 80 024420
[21] Kawaguchi Y, Ueda M 2012 Phys. Rep. 520 253
[22] Yang C, Guo H, Fu L B, Chen S 2015 Phys. Rev. B 91 125132
[23] Milburn G J, Corney J, Wright E M, Walls D F 1997 Phys. Rev. A 55 4318
[24] Micheli A, Jaksch D, Cirac J I, Zoller P 2003 Phys. Rev. A 67 013607
[25] Wu B, Niu Q 2000 Phys. Rev. A 61 023402
[26] Liu J, Wu B, Niu Q 2003 Phys. Rev. Lett. 90 170404
[27] Wu B, Niu Q, New J 2012 Physics 5 104
[28] Chen Y A, Huber S D, Trotzky S, Bloch I, Altman E 2011 Nat. Phys. 7 61
[29] Chen Z D, Liang J Q, Shen S Q, Xie W F 2004 Phys. Rev. A 69 23611
[30] Tonel A P, Links J, Foerster A 2005 J. Phys. A 38 1235
[31] Fu L, Liu J 2006 Phys. Rev. A 74 063614
[32] Ma Y, Fu L B, Yang Z A, Liu J 2006 Acta Phys. Sin. 55 5623 (in Chinese)[马云, 傅立斌, 杨志安, 刘杰2006 55 5623]
[33] Gong J B, Morales-Molina L, Hänggi P 2009 Phys. Rev. Lett. 103 133002
[34] Pang M M, Hao Y 2016 Chin. Phys. B 25 40501
[35] Wang G F, Fu L B, Liu L 2006 Phys. Rev. A 73 13619
[36] Cirac J I, Lewenstein M, Mo K, Zoller P 1998 Phys. Rev. A 57 1208
[37] Leggett A J 2001 Rev. Mod. Phys. 73 307
[38] Li S C, Duan W S 2009 Acta Phys. Sin. 58 4396 (in Chinese)[栗生长, 段文山2009 58 4396]
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[1] Bloch F, Rabi I I 1945 Rev. Mod. Phys. 17 237
[2] Majorana E 1932 Nuovo Cim. 9 43
[3] Stamper-Kurn D M, Ueda M 2013 Rev. Mod. Phys. 85 1191
[4] Zhu Q, Wu B 2015 Chin. Phys. B 24 050507
[5] Lian B, Ho T L, Zhai H 2012 Phys. Rev. A 85 051606
[6] Cui X, Lian B, Ho T L, Lev B L, Zhai H 2013 Phys. Rev. A 88 011601
[7] Devi A R U, Sudha, Rajagopal A K 2012 Quantum Inf. Process. 11 685
[8] Bruno P 2012 Phys. Rev. Lett. 108 240402
[9] Liu H D, Fu L B 2014 Phys. Rev. Lett. 113 240403
[10] Liu H D, Fu L B 2016 Phys. Rev. A 94 022123
[11] Tamate S, Ogawa K, Kitano M 2011 Phys. Rev. A 84 052114
[12] Aulbach M, Markham D, Murao M 2010 New J. Phys. 12 073025
[13] Martin J, Giraud O, Braun P A, Braun D, Bastin T 2010 Phys. Rev. A 81 062347
[14] Bastin T, Krins S, Mathonet P, Godefroid M, Lamata L, Solano E 2009 Phys. Rev. Lett. 103 070503
[15] Ribeiro P, Mosseri R 2011 Phys. Rev. Lett. 106 180502
[16] Ganczarek W, Kuś M,Życzkowski K 2012 Phys. Rev. A 85 032314
[17] Wang Z, Markham D 2012 Phys. Rev. Lett. 108 210407
[18] Wang Z, Markham D 2013 Phys. Rev. A 87 12104
[19] Cao H 2013 Acta Phys. Sin. 62 030303 (in Chinese)[曹辉2013 62 030303]
[20] Barnett R, Podolsky D, Refael G 2009 Phys. Rev. B 80 024420
[21] Kawaguchi Y, Ueda M 2012 Phys. Rep. 520 253
[22] Yang C, Guo H, Fu L B, Chen S 2015 Phys. Rev. B 91 125132
[23] Milburn G J, Corney J, Wright E M, Walls D F 1997 Phys. Rev. A 55 4318
[24] Micheli A, Jaksch D, Cirac J I, Zoller P 2003 Phys. Rev. A 67 013607
[25] Wu B, Niu Q 2000 Phys. Rev. A 61 023402
[26] Liu J, Wu B, Niu Q 2003 Phys. Rev. Lett. 90 170404
[27] Wu B, Niu Q, New J 2012 Physics 5 104
[28] Chen Y A, Huber S D, Trotzky S, Bloch I, Altman E 2011 Nat. Phys. 7 61
[29] Chen Z D, Liang J Q, Shen S Q, Xie W F 2004 Phys. Rev. A 69 23611
[30] Tonel A P, Links J, Foerster A 2005 J. Phys. A 38 1235
[31] Fu L, Liu J 2006 Phys. Rev. A 74 063614
[32] Ma Y, Fu L B, Yang Z A, Liu J 2006 Acta Phys. Sin. 55 5623 (in Chinese)[马云, 傅立斌, 杨志安, 刘杰2006 55 5623]
[33] Gong J B, Morales-Molina L, Hänggi P 2009 Phys. Rev. Lett. 103 133002
[34] Pang M M, Hao Y 2016 Chin. Phys. B 25 40501
[35] Wang G F, Fu L B, Liu L 2006 Phys. Rev. A 73 13619
[36] Cirac J I, Lewenstein M, Mo K, Zoller P 1998 Phys. Rev. A 57 1208
[37] Leggett A J 2001 Rev. Mod. Phys. 73 307
[38] Li S C, Duan W S 2009 Acta Phys. Sin. 58 4396 (in Chinese)[栗生长, 段文山2009 58 4396]
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