玻色爱因斯坦凝聚中的量子相干性和量子度量学的研究
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摘要
玻色爱因斯坦凝聚是一种新的物质形态,它在大尺度上展现了量子特性。在玻色爱因斯坦凝聚提出后的几十年中,人们一直都没有发现类似于玻色爱因斯坦凝聚的现象。终于在20世纪90年代,人们在实验上实现了碱金属气体的玻色爱因斯坦凝聚。之后随着实验条件的不断提高和完善,玻色爱因斯坦凝聚越来越受到关注。本文主要讨论了双模玻色爱因斯坦凝聚和旋量玻色爱因斯坦凝聚,介绍了我们在玻色爱因斯坦凝聚中的量子度量和量子相干性等方面的工作。本文的主要内容如下：
在第一章中首先回顾了玻色爱因斯坦凝聚的研究历史与背景,介绍了玻色爱因斯坦凝聚在实验中的实现、应用前景以及其对各个学科领域的贡献。
在第二章中介绍了玻色爱因斯坦凝聚的一些基本理论,主要内容包括：稀薄气体中玻色爱因斯坦凝聚产生的条件,GP方程的具体推导,旋量玻色爱因斯坦凝聚的具体实现方法以及自旋为1的旋量玻色爱因斯坦凝聚的波函数。
在第三章中,我们利用费舍信息来探测经典分叉和约瑟夫森振荡到自囚禁动力学转变。我们利用具有经典分叉和约瑟夫森振荡到自囚禁的动力学转变的双模玻色爱因斯坦凝聚模型,演示了费舍信息的动力学行为。结果表明,费舍信息精确的刻画了系统的经典分叉和约瑟夫森振荡到自囚禁动力学转变。根据量子费舍信息在度量学中的物理意义,在给定一个初态后,我们发现对于束缚在双势阱中的玻色爱因斯坦凝聚模型的演化态对SU(2)转动的敏感度在不稳定区域接近于海森堡极限,而在稳定区域仅达到散粒噪声的水平。同样在约瑟夫森振荡区域,态的敏感度接近于海森堡极限,而在自囚禁区域仅达到标准量子极限(散粒噪声)。
在第四章中,我们分别介绍了束缚在双阱中的玻色爱因斯坦凝聚系统在具有噪声和耗散情况下的量子相干性。我们首先研究了在没有退相干情况下的时间平均量子相干,发现其相干性在约瑟夫森振荡区域和自囚禁区域的边界达到最大值。后面讨论了在有环境耦合情况下的量子相干性,在不同的耦合方式下,系统的相干性展现出不同的动力学行为。并且发现在某些特殊情况下,系统和环境之间的耦合甚至可以提高系统的量子相干性。
在第五章中,我们介绍了旋量玻色爱因斯坦凝聚中宏观的量子相干性。在单模近似的框架下,考虑原子束缚在一个不对称的势阱中,我们将一个自旋为1的Rb原子系统约化到一个双轴的量子磁体系统。根据束缚阱的几何形状,我们给出了该系统的基态结构。我们还研究了基态的量子相干性,发现当系统中有一个外磁场沿着hard轴时,能级劈裂随着外场的变化而出现周期振荡。我们给出了瞬子解的解析结果并且和数值结果非常吻合。最后我们提出了一种利用Landau-Zener隧穿的实验方案来探测能级劈裂的周期振荡行为。
最后在第六章中,我们对本文的工作进行了总结和展望。
Bose Einstein condensate is a new form of matter, it shows quantum prop-erties on a large scale. After70years of the efforts, in the1990s, Bose Einstein condensate was realized in the alkali gases. With the rapid development of the experimental conditions, Bose Einstein condensates exhibit much interesting phe-nomenons. In this thesis, I focus on two aspect, these are two-mode Bose Einstein condensate and spinor condensate, respectively. I introduce Quantum metrology and quantum coherence in the Bose Einstein condensate. The contents and re-sults of the thesis are organized as follows
(1) In chapter1, we review the history and the background of the Bose Ein-stein condensate, we introduce how the Bose Einstein condensate realized in the experiment, the prospects for research and the contribution to other disciplines.
(2) In chapter2, we introduce some basic theory of the Bose Einstein conden-sate, the main contents include:rarefied gas in ideal Bose-Einstein condensates generated conditions, the detail derivation of the GP equation, the spinor Bose Einstein condensate specific implementation method and the spinor condensate wave function.
(3) In chapter3, we employ the model of Bose Einstein condensate in a double well to demonstrate the behaviors of quantum fisher information in the Josephson oscillation regime and self-trapping regime. The results show that Fisher information characterize the fixed-point bifurcation and the transition from the Josephson oscillation to the self-trapping. We find that the precision of the parameter estimation on the evolved state in the unstable regime is much higher than that in the stable regime, which lies at the shot-noise level. We also show that the parameter sensitivity of the evolved state approaches the Heisenberg limit in the Josephson oscillation regime, while it is just scaled as the shot-noise limit in the self-trapping regime.
(4) In chapter4, We discuss the coherence dynamics of a two mode Bose-Einstein condensate coupled with the environment in the mean field approxima- tion. We give an analytical result of the time average coherence in the absence of system-environment coupling, and find that the time average coherence attains its maximum value at the critical point which corresponds to the boundary be-tween the self-trapping regime and the Josephson oscillation regime. The effects of noise and dissipation on dynamical coherence are also considered by analyzing the couplings of the condensate to the environment. With the first kind of cou-pling as dissipation, the coherence finally stabilizes at a fixed value. Meanwhile, we show that the presence of the dissipation can even enhance the coherence for some particular interaction. With the second kind of coupling as collisional de-phasing, the noise leads a sudden transition of the coherence which then subjects to an exponential decay.
(5) In chapter5, We discuss the macroscopic quantum spin coherence of a spinor condensate confined in an anisotropic potential. Under the single-mode approximation, we show that a spin-1Rb condensate can be modeled as a biaxial quantum magnet. A direct consequence of the biaxial anisotropy is that the tunneling splitting oscillates as a function of the external magnetic field applied along the hard axis. We also propose an experimental scheme to detect the oscillatory behavior of the tunneling splitting by employing the Landau-Zener tunneling.
The summary is given in chapter6.
在第一章中首先回顾了玻色爱因斯坦凝聚的研究历史与背景,介绍了玻色爱因斯坦凝聚在实验中的实现、应用前景以及其对各个学科领域的贡献。
在第二章中介绍了玻色爱因斯坦凝聚的一些基本理论,主要内容包括：稀薄气体中玻色爱因斯坦凝聚产生的条件,GP方程的具体推导,旋量玻色爱因斯坦凝聚的具体实现方法以及自旋为1的旋量玻色爱因斯坦凝聚的波函数。
在第三章中,我们利用费舍信息来探测经典分叉和约瑟夫森振荡到自囚禁动力学转变。我们利用具有经典分叉和约瑟夫森振荡到自囚禁的动力学转变的双模玻色爱因斯坦凝聚模型,演示了费舍信息的动力学行为。结果表明,费舍信息精确的刻画了系统的经典分叉和约瑟夫森振荡到自囚禁动力学转变。根据量子费舍信息在度量学中的物理意义,在给定一个初态后,我们发现对于束缚在双势阱中的玻色爱因斯坦凝聚模型的演化态对SU(2)转动的敏感度在不稳定区域接近于海森堡极限,而在稳定区域仅达到散粒噪声的水平。同样在约瑟夫森振荡区域,态的敏感度接近于海森堡极限,而在自囚禁区域仅达到标准量子极限(散粒噪声)。
在第四章中,我们分别介绍了束缚在双阱中的玻色爱因斯坦凝聚系统在具有噪声和耗散情况下的量子相干性。我们首先研究了在没有退相干情况下的时间平均量子相干,发现其相干性在约瑟夫森振荡区域和自囚禁区域的边界达到最大值。后面讨论了在有环境耦合情况下的量子相干性,在不同的耦合方式下,系统的相干性展现出不同的动力学行为。并且发现在某些特殊情况下,系统和环境之间的耦合甚至可以提高系统的量子相干性。
在第五章中,我们介绍了旋量玻色爱因斯坦凝聚中宏观的量子相干性。在单模近似的框架下,考虑原子束缚在一个不对称的势阱中,我们将一个自旋为1的Rb原子系统约化到一个双轴的量子磁体系统。根据束缚阱的几何形状,我们给出了该系统的基态结构。我们还研究了基态的量子相干性,发现当系统中有一个外磁场沿着hard轴时,能级劈裂随着外场的变化而出现周期振荡。我们给出了瞬子解的解析结果并且和数值结果非常吻合。最后我们提出了一种利用Landau-Zener隧穿的实验方案来探测能级劈裂的周期振荡行为。
最后在第六章中,我们对本文的工作进行了总结和展望。
Bose Einstein condensate is a new form of matter, it shows quantum prop-erties on a large scale. After70years of the efforts, in the1990s, Bose Einstein condensate was realized in the alkali gases. With the rapid development of the experimental conditions, Bose Einstein condensates exhibit much interesting phe-nomenons. In this thesis, I focus on two aspect, these are two-mode Bose Einstein condensate and spinor condensate, respectively. I introduce Quantum metrology and quantum coherence in the Bose Einstein condensate. The contents and re-sults of the thesis are organized as follows
(1) In chapter1, we review the history and the background of the Bose Ein-stein condensate, we introduce how the Bose Einstein condensate realized in the experiment, the prospects for research and the contribution to other disciplines.
(2) In chapter2, we introduce some basic theory of the Bose Einstein conden-sate, the main contents include:rarefied gas in ideal Bose-Einstein condensates generated conditions, the detail derivation of the GP equation, the spinor Bose Einstein condensate specific implementation method and the spinor condensate wave function.
(3) In chapter3, we employ the model of Bose Einstein condensate in a double well to demonstrate the behaviors of quantum fisher information in the Josephson oscillation regime and self-trapping regime. The results show that Fisher information characterize the fixed-point bifurcation and the transition from the Josephson oscillation to the self-trapping. We find that the precision of the parameter estimation on the evolved state in the unstable regime is much higher than that in the stable regime, which lies at the shot-noise level. We also show that the parameter sensitivity of the evolved state approaches the Heisenberg limit in the Josephson oscillation regime, while it is just scaled as the shot-noise limit in the self-trapping regime.
(4) In chapter4, We discuss the coherence dynamics of a two mode Bose-Einstein condensate coupled with the environment in the mean field approxima- tion. We give an analytical result of the time average coherence in the absence of system-environment coupling, and find that the time average coherence attains its maximum value at the critical point which corresponds to the boundary be-tween the self-trapping regime and the Josephson oscillation regime. The effects of noise and dissipation on dynamical coherence are also considered by analyzing the couplings of the condensate to the environment. With the first kind of cou-pling as dissipation, the coherence finally stabilizes at a fixed value. Meanwhile, we show that the presence of the dissipation can even enhance the coherence for some particular interaction. With the second kind of coupling as collisional de-phasing, the noise leads a sudden transition of the coherence which then subjects to an exponential decay.
(5) In chapter5, We discuss the macroscopic quantum spin coherence of a spinor condensate confined in an anisotropic potential. Under the single-mode approximation, we show that a spin-1Rb condensate can be modeled as a biaxial quantum magnet. A direct consequence of the biaxial anisotropy is that the tunneling splitting oscillates as a function of the external magnetic field applied along the hard axis. We also propose an experimental scheme to detect the oscillatory behavior of the tunneling splitting by employing the Landau-Zener tunneling.
The summary is given in chapter6.
引文
[1]S. N. Bose, Z.Phys,26,178 (1924).
[2]A. Einstein, Sitzungsber, Kgl.Preuss, Akad.Wiss 261,3 (1924).
[3]M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cor-nell, Science 269,198 (1995).
[4]K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. Van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett.75,1969 (1995).
[5]C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75,1687(1995).
[6]Steven Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57,314(1986).
[7]A. Migdall, J. prodan, W. Phillips, T. Bergeman, and H. Metealf, Phys. Rev. Lett.54,2596 (1985).
[8]T. Esslinger, I. Bloch, and T.W. Hanseh, Phys. Rev. A 54, R2664 (1998).
[9]J. Reichel, W. Hansel, and T.W. Hanseh, Phys. Rev. Lett.83,3398 (1999).
[10]W. Petrieh, M.H. Anderson, J.R. Ensher, and E.A. Comell,Phys. Rev. Lett. 74,3352 (1995).
[11]J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.77,4984 (1996).
[12]D.G. Fried, T.C. Killian, L Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, Phys. Rev. Lett.81,3811 (1996).
[13]G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, M. Inguscio, Science 294,1320 (2001).
[14]A. Robert,O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, A. Aspect, Science 292,461 (2001).
[15]T. Weber, J. Herbig, M. Mark, Hanns-Christoph Nagerl, R. Grimm, Science 299,232 (2002).
[16]Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett.91,040404 (2003).
[17]A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett 94,160401 (2003).
[18]D.M. Stamper-Kurn, M.R. Andrews, A.p. Chikkaiur, S. Inouye, H.J. Mies-ner, J. Stenger, and W. Ketterie, Phys. Rev. Lett.80,2027 (1997).
[19]T.L. Ho,Phys. Rev. Lett.81,742 (1998).
[20]T. Ohmi and K. Machida, J. Phys. Soe. JPn.67,1822 (1998).
[21]C.K. Law, H. Pu, and N.P. Bigelow, Phys. Rev. Lett.81,5257 (1998).
[22]M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterie, Phys. Rev. Lett.78,582 (1996).
[23]M.R. Mathews, B.P. Anderson, P.C. Haljna, D.S.Hall, C.E.Wieman,and E.A. Cornell, Phys. Rev. Lett.83,2498 (1999).
[24]K.W. Madison, F, Chevy, W. Wohlleben, and J. Dalibadr, Phys. Rev. Lett.84,806 (2000).
[25]J.E. Williams and M.J. Holland, Nature 401,568 (1999).
[26]J. R. Abo-Shaeer, C. Raman, J. M. Vogels, W. Ketterie, Science 292,476 (2001).
[27]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.83,3358 (1999).
[28]K. W. Madison, F. Chevy, V. Bretin, and J. Dalibard, Phys. Rev. Lett.86, 4443 (2001).
[29]M. Greiner, O. Mandell, T. Esslinger, T.W. Hansch, and I. Bloch, Nature 415 39 (2002).
[30]A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam,1982).
[31]C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York,1976).
[32]S. L. Braunstein and C. M. Caves, Phys. Rev. Lett.72,3439 (1994).
[33]S. Braunstein C. M. Caves, and G. J. Milburn, Ann. Phys. (NY) 247,135 (1996).
[34]J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature (London) 455,1216 (2008).
[35]G. R. Jin and S. W. Kim, Phys. Rev. Lett.99,170405 (2007).
[36]L. Pezze and A. Smerzi, Phys. Rev. Lett.102,100401 (2009).
[37]C. Brukner, V. Vedral, and A. Zeilinger, Phys. Rev. A 73,012110 (2006).
[38]S. Y. Cho and R. H. McKenzie, Phys. Rev. A 73,012109 (2006).
[39]J. Ma and X. Wang, Phys. Rev. A 80,012318 (2009).
[40]C. Invernizzi, M. Korbman, L. C. Venuti, and M. G. A. Paris, Phys. Rev. A 78,042106 (2008).
[41]Z. Sun, J. Ma, X. Lu, and X. Wang, Phys. Rev. A 82,022306 (2010).
[42]J. Ma, Y. Huang, X. Wang, and C. P. Sun Phys. Rev. A 84,022302 (2011).
[43]J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996).
[44]J. P. Dowling, Phys. Rev. A 57,4736 (1998).
[45]V. Giovannetti, S. Lloyd, and L. Maccone, Science 306,1330 (2004).
[46]A. Sorensen, L.-M Duan, J. I. Cirac and P. Zoller, Nature (London) 409,63 (2001).
[47]G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[48]A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett.79, 4950 (1997).
[49]S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59,620 (1999).
[50]G. F. Wang, L. Fu, and Jie Liu, Phys. Rev. A 73,013619 (2006).
[51]M. Albiez, R. Gati, J. Foiling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett.95,010402 (2005).
[52]R. Wu, and Y. Shi, Phys. Rev. A 84,063610 (2011).
[53]S. Schneider, and G. J. Milburn, Phys. Rev. A 65,042107 (2002).
[54]A. P. Hines, C. M. Dawson, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 70,022303 (2004).
[55]X. W. Hou, J. H. Chen, and B. Hu, Phys. Rev. A 71,034302 (2005).
[56]K. W. Mahmud, H. Perry, and W. P. Reinhardt, Phys. Rev. A 71,023615 (2005).
[57]A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 67,013609 (2003).
[58]L. B. Fu and J. Liu, Phys. Rev. A 74,063614 (2006).
[59]J. Vidal, G. Palacios, and C. Aslangul, Phys. Rev. A 70,062304 (2004).
[60]C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York,1976.
[61]M. Grifoni and P. Hanggi, Phys. Rep.304,229 (1998).
[62]O. Zobay and B. M. Garraway, Phys. Rev. A 61,033603 (2000).
[63]F. Kh. Abdullaev and R. A. Kraenkel, Phys. Rev. A 62,023613 (2000).
[64]F. Meier and W. Zwerger, Phys. Rev. A 64,033610 (2001).
[65]J. I. Cirac, M. Lewenstein, K. M(?)lmer, and P. Zoller, Phys. Rev. A 57,1208 (1998).
[66]M. J. Steel and M. J. Collett, Phys. Rev. A 57,2920 (1998).
[67]L. M. Kuang and Z. W. Ouyang, Phys. Rev. A 61,023604 (2000).
[68]J. M. Radcliffe, J. Phys. A 4,313 (1971).
[69]F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[70]T. F. Viscondi, K. Furuya, and M. C. de Oliveira, Ann. Phys. (N.Y.) 324, 1837 (2009).
[71]H. Kuratsuji and T. Suzuki, J. Math. Phys.21,472 (1980).
[72]Y. Zhou, H. Zhai, R. Lu, Z. Xu, and L. Chang, Phys. Rev. A 67,043606 (2003).
[73]Biao Wu, Jie Liu, and Qian Niu Phys. Rev. Lett.88,034101 (2002).
[74]U. V. Poulsen and K. M(?)lmer, Phys. Rev. A 64,013616 (2001).
[75]Y. Li, Y. Castin, and A. Sinatra, Phys. Rev. Lett.100,210401 (2008).
[76]A. Widera, S. Trotzky, P. Cheinet, S. Folling, F. Gerbier, I. Bloch, V. Grit-sev, M. D. Lukin, and E. Demler, Phys. Rev. Lett.100,140401 (2008).
[77]D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman E. A. Cornell, Phys. Rev. Lett.81,1539 (1998).
[78]G. R. Jin and C. K. Law, Phys. Rev. A 78,063620 (2008).
[79]C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schuma-cher, Phys. Rev. A 53,2046 (1996).
[80]H. Pu and N. P. Bigelow, Phys. Rev. Lett.80,1130 (1998).
[81]H. Pu and N. P. Bigelow, Phys. Rev. Lett.80,1134 (1998).
[82]L. Yang and Y. Zhang, Phys. Rev. A 74,043604 (2006).
[83]R. M. Wilson, S. Ronen, and J. L. Bohn, Phys. Rev. A 80,023614 (2009).
[84]B. Xiong, J. Gong, H. Pu, W. Bao, and B. Li, Phys. Rev. A 79,013626 (2009).
[85]L. Santos and T. Pfau, Phys. Rev. Lett.96,190404 (2006).
[86]A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett.94,160401 (2005).
[87]Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett.96,080405 (2006).
[88]S. Yi, O. E. Mustecaplioglu, C. P. Sun, and L. You, Phys. Rev. A 66,011601 (2002).
[89]Ying. Wu, Phys. Rev. A 54,4534 (1996).
[90]E. V. Goldstein and P. Meystre, Phys. Rev. A 59,3896 (1999).
[91]D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H. J. Mesner, J. Stenger, and W. Ketterle, Phys. Rev. Lett.80,2027 (1998).
[92]J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature (London) 396,345 (1998).
[93]M. Barret, J. Sauer, and M. S. Chapman, Phys. Rev. Lett.87,010404 (2001).
[94]B. D. Esry, C. H. Greene, J. P. Burke, Jr., and J. L. Bohn, Phys. Rev. Lett. 78,3594 (1997).
[95]C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett.78,586 (1997).
[96]S. Raghavan, A. Smerzi, and V. M. Kenkre, Phys. Rev. A 60, R1787 (1999).
[97]M. Albiez, R. Gati, Jonas Folling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett.95,010402 (2005).
[98]L. Salasnich, Phys. Rev. A 61,015601 (1999).
[99]Franco Dalfovo, Stefano Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys.71,463 (1999).
[100]W. D. Li and J. Liu, Phys. Rev. A 74,063613 (2006).
[101]S. Kohler and F. Sols, Phys. Rev. Lett.89,060403 (2002).
[102]B. B. Wang, P. M. Fu, J. Liu, and B. Wu, Phys. Rev. A 74,063610 (2006).
[103]Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, Phys. Rev. Lett.92,050405 (2004).
[104]J. M. Choi, G. N. Kim, and D. Cho, Phys. Rev. A 77,010501(R) (2008).
[105]Jie Liu, Biao Wu, and Qian Niu, Phys. Rev. Lett.90,170404 (2003).
[106]Jie Liu, Libin Fu, Bi-Yiao Ou, Shi-Gang Chen, Dae-Il Choi, Biao Wu, and Qian Niu, Phys. Rev. A 66,023404 (2002).
[107]S. Giovanazzi, A. Smerzi, and S. Fantoni, Phys. Rev. Lett.84,4521 (2000).
[108]S. Choi and N.P. Bigelow, Phys. Rev. A 72,033612 (2005).
[109]Li-Hua Lu and You-Quan Li, Phys. Rev. A 80,033619 (2009).
[110]J. Ruostekoski and D. F. Walls, Phys. Rev. A 58, R50 (1998).
[111]L. Pitaevskii and S. Stringari, Phys. Rev. Lett.87,180402 (2001).
[112]Yajiang Hao and Qiang Gu, Phys. Rev. A 83,043620 (2011).
[113]Q. Gu and H. Qiu, Phys. Rev. Lett.98,200401 (2007).
[114]Y. Khodorkovsky, G. Kurizki, and A. Vardi, Phys. Rev. Lett.100,220403 (2008).
[115]E. Boukobza, M. Chuchem, D. Cohen, and A. Vardi, Phys. Rev. Lett.102, 180403 (2009).
[116]Y.C. Liu, G.R. Jin, and Y. Lou, Phys. Rev. A 82,045601 (2010).
[117]R. Gati, B. Hemmerling, J. Foiling, M. Albiez, and M. K. Oberthaler, Phys. Rev. Lett.96,130404 (2006).
[118]R. Gati, J. Estve, B. Hemmerling, T. B. Ottenstein, J. Appmeier, A. Weller, and M. K. Oberthaler, New J. Phys.8,189 (2006).
[119]D. Witthaut, F. Trimborn, and S. Wimberger, Phys. Rev. Lett.101,200402 (2008).
[120]D. Witthaut, F. Trimborn, and S. Wimberger, Phys. Rev. A 79,033621 (2009).
[121]D. A. R. Dalvit, J. Dziarmaga, and W. H. Zurek, Phys. Rev. A 62,013607 (2000).
[122]P. J. Y. Louis, P. M. R. Brydon, and C. M. Savage, Phys. Rev. A 64, 053613 (2001).
[123]A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. A 67,013607 (2003).
[124]W. Wang, L.B.Fu, and X.X.Yi, Phys. Rev. A 75,045601 (2007).
[125]A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys.59,1 (1987).
[126]L. Gunther and B. Barbara, eds., Quantum Tunneling of Magnetization-QTMO94, (Kluwer, Dordrecht, Netherlands,1995).
[127]E.M. Chudnovsky, Macroscopic Quantum Tunneling of the Magnetic Mo-ment (Cambridge University Press, Cambridge, U.K.,1998).
[128]Magnetism:Molecules to Materials Ⅲ, edited by J.S. Miller and M. Drillon (Wiley-VCH Verlag GmbH, Weinheim, Germany,2002).
[129]J.R. Friedman and M.P. Sarachik, Annu. Rev. Condens. Matter Phys.1, 109 (2010).
[130]J.R. Friedman, M.P. Sarachik, J. Tejada, and R. Ziolo, Phys. Rev. Lett. 76,3830 (1996).
[131]J.M. Hernandez, X.X. Zhang, F. Luis, J. Bartolome, J. Tejada, and R. Ziolo, Europhys. Lett.35,301 (1996).
[132]L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, Nature 383,145 (1996).
[133]C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D. Gatteschi, Phys. Rev. Lett.78,4645 (1997).
[134]A. Garg, Europhys. Lett.22,205 (1993).
[135]W. Wernsdorfer and R. Sessoli, Science 284,133 (1999).
[136]H. Pu, W. P. Zhang, and P. Meystre, Phys. Rev. Lett.89,090401 (2002)
[137]S. Yi, L. You, and H. Pu, Phys. Rev. Lett.93,040403 (2006).
[138]S. Yi and H. Pu, Phys. Rev. A 73,023602 (2006).
[139]S. Yi and Han Pu, arXiv:0804.0191V1 (2008).
[140]L. Yang and Y. Zhang, Phys. Rev. A 74,043604 (2006).
[141]G. Chen, J.-Q. Liang, and S. Jia, Optics Express 17,19682 (2009).
[142]T. Ohmi and K. Machida, J. Phys. Soc. Jpn.67,1822 (1998).
[143]S. Yi and H. Pu, Phys. Rev. Lett 97,020401 (2006).
[144]C.K. Law, H. Pu, and N.P. Bigelow, Phys. Rev. Lett.81,5257 (1998).
[145]S.P. Kou, J.Q. Liang, Y.B. Zhang, and F.C. Pu, Phys. Rev. B 59 111792 (1999).
[2]A. Einstein, Sitzungsber, Kgl.Preuss, Akad.Wiss 261,3 (1924).
[3]M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cor-nell, Science 269,198 (1995).
[4]K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. Van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett.75,1969 (1995).
[5]C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75,1687(1995).
[6]Steven Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57,314(1986).
[7]A. Migdall, J. prodan, W. Phillips, T. Bergeman, and H. Metealf, Phys. Rev. Lett.54,2596 (1985).
[8]T. Esslinger, I. Bloch, and T.W. Hanseh, Phys. Rev. A 54, R2664 (1998).
[9]J. Reichel, W. Hansel, and T.W. Hanseh, Phys. Rev. Lett.83,3398 (1999).
[10]W. Petrieh, M.H. Anderson, J.R. Ensher, and E.A. Comell,Phys. Rev. Lett. 74,3352 (1995).
[11]J. R. Ensher, D. S. Jin, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.77,4984 (1996).
[12]D.G. Fried, T.C. Killian, L Willmann, D. Landhuis, S.C. Moss, D. Kleppner, and T.J. Greytak, Phys. Rev. Lett.81,3811 (1996).
[13]G. Modugno, G. Ferrari, G. Roati, R. J. Brecha, A. Simoni, M. Inguscio, Science 294,1320 (2001).
[14]A. Robert,O. Sirjean, A. Browaeys, J. Poupard, S. Nowak, D. Boiron, C. I. Westbrook, A. Aspect, Science 292,461 (2001).
[15]T. Weber, J. Herbig, M. Mark, Hanns-Christoph Nagerl, R. Grimm, Science 299,232 (2002).
[16]Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett.91,040404 (2003).
[17]A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett 94,160401 (2003).
[18]D.M. Stamper-Kurn, M.R. Andrews, A.p. Chikkaiur, S. Inouye, H.J. Mies-ner, J. Stenger, and W. Ketterie, Phys. Rev. Lett.80,2027 (1997).
[19]T.L. Ho,Phys. Rev. Lett.81,742 (1998).
[20]T. Ohmi and K. Machida, J. Phys. Soe. JPn.67,1822 (1998).
[21]C.K. Law, H. Pu, and N.P. Bigelow, Phys. Rev. Lett.81,5257 (1998).
[22]M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterie, Phys. Rev. Lett.78,582 (1996).
[23]M.R. Mathews, B.P. Anderson, P.C. Haljna, D.S.Hall, C.E.Wieman,and E.A. Cornell, Phys. Rev. Lett.83,2498 (1999).
[24]K.W. Madison, F, Chevy, W. Wohlleben, and J. Dalibadr, Phys. Rev. Lett.84,806 (2000).
[25]J.E. Williams and M.J. Holland, Nature 401,568 (1999).
[26]J. R. Abo-Shaeer, C. Raman, J. M. Vogels, W. Ketterie, Science 292,476 (2001).
[27]M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell, Phys. Rev. Lett.83,3358 (1999).
[28]K. W. Madison, F. Chevy, V. Bretin, and J. Dalibard, Phys. Rev. Lett.86, 4443 (2001).
[29]M. Greiner, O. Mandell, T. Esslinger, T.W. Hansch, and I. Bloch, Nature 415 39 (2002).
[30]A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam,1982).
[31]C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York,1976).
[32]S. L. Braunstein and C. M. Caves, Phys. Rev. Lett.72,3439 (1994).
[33]S. Braunstein C. M. Caves, and G. J. Milburn, Ann. Phys. (NY) 247,135 (1996).
[34]J. Esteve, C. Gross, A. Weller, S. Giovanazzi, and M. K. Oberthaler, Nature (London) 455,1216 (2008).
[35]G. R. Jin and S. W. Kim, Phys. Rev. Lett.99,170405 (2007).
[36]L. Pezze and A. Smerzi, Phys. Rev. Lett.102,100401 (2009).
[37]C. Brukner, V. Vedral, and A. Zeilinger, Phys. Rev. A 73,012110 (2006).
[38]S. Y. Cho and R. H. McKenzie, Phys. Rev. A 73,012109 (2006).
[39]J. Ma and X. Wang, Phys. Rev. A 80,012318 (2009).
[40]C. Invernizzi, M. Korbman, L. C. Venuti, and M. G. A. Paris, Phys. Rev. A 78,042106 (2008).
[41]Z. Sun, J. Ma, X. Lu, and X. Wang, Phys. Rev. A 82,022306 (2010).
[42]J. Ma, Y. Huang, X. Wang, and C. P. Sun Phys. Rev. A 84,022302 (2011).
[43]J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996).
[44]J. P. Dowling, Phys. Rev. A 57,4736 (1998).
[45]V. Giovannetti, S. Lloyd, and L. Maccone, Science 306,1330 (2004).
[46]A. Sorensen, L.-M Duan, J. I. Cirac and P. Zoller, Nature (London) 409,63 (2001).
[47]G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys. Rev. A 55, 4318 (1997).
[48]A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys. Rev. Lett.79, 4950 (1997).
[49]S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, Phys. Rev. A 59,620 (1999).
[50]G. F. Wang, L. Fu, and Jie Liu, Phys. Rev. A 73,013619 (2006).
[51]M. Albiez, R. Gati, J. Foiling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett.95,010402 (2005).
[52]R. Wu, and Y. Shi, Phys. Rev. A 84,063610 (2011).
[53]S. Schneider, and G. J. Milburn, Phys. Rev. A 65,042107 (2002).
[54]A. P. Hines, C. M. Dawson, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 70,022303 (2004).
[55]X. W. Hou, J. H. Chen, and B. Hu, Phys. Rev. A 71,034302 (2005).
[56]K. W. Mahmud, H. Perry, and W. P. Reinhardt, Phys. Rev. A 71,023615 (2005).
[57]A. P. Hines, R. H. McKenzie, and G. J. Milburn, Phys. Rev. A 67,013609 (2003).
[58]L. B. Fu and J. Liu, Phys. Rev. A 74,063614 (2006).
[59]J. Vidal, G. Palacios, and C. Aslangul, Phys. Rev. A 70,062304 (2004).
[60]C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York,1976.
[61]M. Grifoni and P. Hanggi, Phys. Rep.304,229 (1998).
[62]O. Zobay and B. M. Garraway, Phys. Rev. A 61,033603 (2000).
[63]F. Kh. Abdullaev and R. A. Kraenkel, Phys. Rev. A 62,023613 (2000).
[64]F. Meier and W. Zwerger, Phys. Rev. A 64,033610 (2001).
[65]J. I. Cirac, M. Lewenstein, K. M(?)lmer, and P. Zoller, Phys. Rev. A 57,1208 (1998).
[66]M. J. Steel and M. J. Collett, Phys. Rev. A 57,2920 (1998).
[67]L. M. Kuang and Z. W. Ouyang, Phys. Rev. A 61,023604 (2000).
[68]J. M. Radcliffe, J. Phys. A 4,313 (1971).
[69]F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[70]T. F. Viscondi, K. Furuya, and M. C. de Oliveira, Ann. Phys. (N.Y.) 324, 1837 (2009).
[71]H. Kuratsuji and T. Suzuki, J. Math. Phys.21,472 (1980).
[72]Y. Zhou, H. Zhai, R. Lu, Z. Xu, and L. Chang, Phys. Rev. A 67,043606 (2003).
[73]Biao Wu, Jie Liu, and Qian Niu Phys. Rev. Lett.88,034101 (2002).
[74]U. V. Poulsen and K. M(?)lmer, Phys. Rev. A 64,013616 (2001).
[75]Y. Li, Y. Castin, and A. Sinatra, Phys. Rev. Lett.100,210401 (2008).
[76]A. Widera, S. Trotzky, P. Cheinet, S. Folling, F. Gerbier, I. Bloch, V. Grit-sev, M. D. Lukin, and E. Demler, Phys. Rev. Lett.100,140401 (2008).
[77]D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman E. A. Cornell, Phys. Rev. Lett.81,1539 (1998).
[78]G. R. Jin and C. K. Law, Phys. Rev. A 78,063620 (2008).
[79]C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schuma-cher, Phys. Rev. A 53,2046 (1996).
[80]H. Pu and N. P. Bigelow, Phys. Rev. Lett.80,1130 (1998).
[81]H. Pu and N. P. Bigelow, Phys. Rev. Lett.80,1134 (1998).
[82]L. Yang and Y. Zhang, Phys. Rev. A 74,043604 (2006).
[83]R. M. Wilson, S. Ronen, and J. L. Bohn, Phys. Rev. A 80,023614 (2009).
[84]B. Xiong, J. Gong, H. Pu, W. Bao, and B. Li, Phys. Rev. A 79,013626 (2009).
[85]L. Santos and T. Pfau, Phys. Rev. Lett.96,190404 (2006).
[86]A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett.94,160401 (2005).
[87]Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett.96,080405 (2006).
[88]S. Yi, O. E. Mustecaplioglu, C. P. Sun, and L. You, Phys. Rev. A 66,011601 (2002).
[89]Ying. Wu, Phys. Rev. A 54,4534 (1996).
[90]E. V. Goldstein and P. Meystre, Phys. Rev. A 59,3896 (1999).
[91]D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H. J. Mesner, J. Stenger, and W. Ketterle, Phys. Rev. Lett.80,2027 (1998).
[92]J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature (London) 396,345 (1998).
[93]M. Barret, J. Sauer, and M. S. Chapman, Phys. Rev. Lett.87,010404 (2001).
[94]B. D. Esry, C. H. Greene, J. P. Burke, Jr., and J. L. Bohn, Phys. Rev. Lett. 78,3594 (1997).
[95]C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett.78,586 (1997).
[96]S. Raghavan, A. Smerzi, and V. M. Kenkre, Phys. Rev. A 60, R1787 (1999).
[97]M. Albiez, R. Gati, Jonas Folling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, Phys. Rev. Lett.95,010402 (2005).
[98]L. Salasnich, Phys. Rev. A 61,015601 (1999).
[99]Franco Dalfovo, Stefano Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys.71,463 (1999).
[100]W. D. Li and J. Liu, Phys. Rev. A 74,063613 (2006).
[101]S. Kohler and F. Sols, Phys. Rev. Lett.89,060403 (2002).
[102]B. B. Wang, P. M. Fu, J. Liu, and B. Wu, Phys. Rev. A 74,063610 (2006).
[103]Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E. Pritchard, and A. E. Leanhardt, Phys. Rev. Lett.92,050405 (2004).
[104]J. M. Choi, G. N. Kim, and D. Cho, Phys. Rev. A 77,010501(R) (2008).
[105]Jie Liu, Biao Wu, and Qian Niu, Phys. Rev. Lett.90,170404 (2003).
[106]Jie Liu, Libin Fu, Bi-Yiao Ou, Shi-Gang Chen, Dae-Il Choi, Biao Wu, and Qian Niu, Phys. Rev. A 66,023404 (2002).
[107]S. Giovanazzi, A. Smerzi, and S. Fantoni, Phys. Rev. Lett.84,4521 (2000).
[108]S. Choi and N.P. Bigelow, Phys. Rev. A 72,033612 (2005).
[109]Li-Hua Lu and You-Quan Li, Phys. Rev. A 80,033619 (2009).
[110]J. Ruostekoski and D. F. Walls, Phys. Rev. A 58, R50 (1998).
[111]L. Pitaevskii and S. Stringari, Phys. Rev. Lett.87,180402 (2001).
[112]Yajiang Hao and Qiang Gu, Phys. Rev. A 83,043620 (2011).
[113]Q. Gu and H. Qiu, Phys. Rev. Lett.98,200401 (2007).
[114]Y. Khodorkovsky, G. Kurizki, and A. Vardi, Phys. Rev. Lett.100,220403 (2008).
[115]E. Boukobza, M. Chuchem, D. Cohen, and A. Vardi, Phys. Rev. Lett.102, 180403 (2009).
[116]Y.C. Liu, G.R. Jin, and Y. Lou, Phys. Rev. A 82,045601 (2010).
[117]R. Gati, B. Hemmerling, J. Foiling, M. Albiez, and M. K. Oberthaler, Phys. Rev. Lett.96,130404 (2006).
[118]R. Gati, J. Estve, B. Hemmerling, T. B. Ottenstein, J. Appmeier, A. Weller, and M. K. Oberthaler, New J. Phys.8,189 (2006).
[119]D. Witthaut, F. Trimborn, and S. Wimberger, Phys. Rev. Lett.101,200402 (2008).
[120]D. Witthaut, F. Trimborn, and S. Wimberger, Phys. Rev. A 79,033621 (2009).
[121]D. A. R. Dalvit, J. Dziarmaga, and W. H. Zurek, Phys. Rev. A 62,013607 (2000).
[122]P. J. Y. Louis, P. M. R. Brydon, and C. M. Savage, Phys. Rev. A 64, 053613 (2001).
[123]A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. A 67,013607 (2003).
[124]W. Wang, L.B.Fu, and X.X.Yi, Phys. Rev. A 75,045601 (2007).
[125]A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys.59,1 (1987).
[126]L. Gunther and B. Barbara, eds., Quantum Tunneling of Magnetization-QTMO94, (Kluwer, Dordrecht, Netherlands,1995).
[127]E.M. Chudnovsky, Macroscopic Quantum Tunneling of the Magnetic Mo-ment (Cambridge University Press, Cambridge, U.K.,1998).
[128]Magnetism:Molecules to Materials Ⅲ, edited by J.S. Miller and M. Drillon (Wiley-VCH Verlag GmbH, Weinheim, Germany,2002).
[129]J.R. Friedman and M.P. Sarachik, Annu. Rev. Condens. Matter Phys.1, 109 (2010).
[130]J.R. Friedman, M.P. Sarachik, J. Tejada, and R. Ziolo, Phys. Rev. Lett. 76,3830 (1996).
[131]J.M. Hernandez, X.X. Zhang, F. Luis, J. Bartolome, J. Tejada, and R. Ziolo, Europhys. Lett.35,301 (1996).
[132]L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, and B. Barbara, Nature 383,145 (1996).
[133]C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D. Gatteschi, Phys. Rev. Lett.78,4645 (1997).
[134]A. Garg, Europhys. Lett.22,205 (1993).
[135]W. Wernsdorfer and R. Sessoli, Science 284,133 (1999).
[136]H. Pu, W. P. Zhang, and P. Meystre, Phys. Rev. Lett.89,090401 (2002)
[137]S. Yi, L. You, and H. Pu, Phys. Rev. Lett.93,040403 (2006).
[138]S. Yi and H. Pu, Phys. Rev. A 73,023602 (2006).
[139]S. Yi and Han Pu, arXiv:0804.0191V1 (2008).
[140]L. Yang and Y. Zhang, Phys. Rev. A 74,043604 (2006).
[141]G. Chen, J.-Q. Liang, and S. Jia, Optics Express 17,19682 (2009).
[142]T. Ohmi and K. Machida, J. Phys. Soc. Jpn.67,1822 (1998).
[143]S. Yi and H. Pu, Phys. Rev. Lett 97,020401 (2006).
[144]C.K. Law, H. Pu, and N.P. Bigelow, Phys. Rev. Lett.81,5257 (1998).
[145]S.P. Kou, J.Q. Liang, Y.B. Zhang, and F.C. Pu, Phys. Rev. B 59 111792 (1999).