双组份玻色—爱因斯坦凝聚中的自旋压缩
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摘要
自旋压缩的研究受到广泛关注源于三方面的原因：海森堡不确定关系的普遍存在是量子测量器件精度提高的最根本障碍,如何抑制它对精度的影响有着巨大的应用前景；自旋压缩与量子纠缠的密切关系使得自旋压缩的研究对量子信息有着重要的意义；另外,量子纠缠是物理有待探索的基本问题之一,所以对自旋压缩的研究既有应用方面的意义,也有着基本理论方面的意义。本论文着重解决如下的问题：到目前为止无论是自旋压缩的定义还是判断标准都存在争议,就几种定义和判断标准在第二章作者提出了自己的看法；δJz是线性相互作用,不应对自旋压缩产生影响,所以这一项常被忽略,但有些情况下这一项确实存在,将这一项纳入哈密顿量进行计算时,由于对称性被破坏需要采用新的方法,而且这一项在某些情况下对自旋压缩的产生了影响,文中给出了合理的解释；时间关联函数表示二能级间的耦合,它是可以在实验中被观测的量,已发现一级关联函数与压缩系数之间有特定的关系,则可以利用这种关系将自旋压缩变成直接观测的现象,二级关联函数与之是否也有特定关系,这是计算二级关联函数的动机,计算发现,在特定的条件下二阶关联函数和压缩系数随时间的变化规律非常相似；有人提出新的纠缠态判断标准,论文的也计算了纠缠态的判断系数；粒子计数是物理实验中测量物理量最常用的手段,所以数压缩态的研究有着广泛的应用意义,文中最后研究了数压缩系数。
There are three reasons for people's extensive interest in spin squeezing. Firstly, the spin squeezing is an effective method to overcome the limit in precision improvement set by Heisenberg Uncertainty Principle. Secondly, because of the relationship between spin squeezing and entanglement, the research in spin squeezing is of importance for quantum information. Lastly but not the least, the entanglement is one of fundamental physics problems, so research in thespin squeezing has both practical and theoretical significance. Dispute on the definition and criteria for judgement still exists till now, author gives her own opinions in Chapter 2. Linear interaction is anticipated to bring no influence on spin squeezing, so the termδ(?)z is often neglected in calculation. Under some circumstance this term exists factually. In this thesis, it will be included in Hamiltonia to calculate. New method is introduce to solve problems because of symmetry breaking in some parameters afterδ(?)z added to Hamiltonian. After calculation, the author found that in some circumstance, this linear term brings some negative influence, reasonable explanation will be given in this thesis for this phenomenon. Particular relationship between squeezing parameter and the first-order correlated function has been found. Temporal correlated functions are measurable directly in experiment, so spin squeezing can be understood well with this relationship. What about the second-order function? Is there some particular relationship between them. That is the motivation of calculating second-order functions? After calculation, the answer is yes. New criteria for entanglement is provided recently, the squeezing parameter and the new entanglement criteria will be compared. It is meaningful to do research the number spin squeezing, for particle counting is such a popular measurement in experiment.
There are three reasons for people's extensive interest in spin squeezing. Firstly, the spin squeezing is an effective method to overcome the limit in precision improvement set by Heisenberg Uncertainty Principle. Secondly, because of the relationship between spin squeezing and entanglement, the research in spin squeezing is of importance for quantum information. Lastly but not the least, the entanglement is one of fundamental physics problems, so research in thespin squeezing has both practical and theoretical significance. Dispute on the definition and criteria for judgement still exists till now, author gives her own opinions in Chapter 2. Linear interaction is anticipated to bring no influence on spin squeezing, so the termδ(?)z is often neglected in calculation. Under some circumstance this term exists factually. In this thesis, it will be included in Hamiltonia to calculate. New method is introduce to solve problems because of symmetry breaking in some parameters afterδ(?)z added to Hamiltonian. After calculation, the author found that in some circumstance, this linear term brings some negative influence, reasonable explanation will be given in this thesis for this phenomenon. Particular relationship between squeezing parameter and the first-order correlated function has been found. Temporal correlated functions are measurable directly in experiment, so spin squeezing can be understood well with this relationship. What about the second-order function? Is there some particular relationship between them. That is the motivation of calculating second-order functions? After calculation, the answer is yes. New criteria for entanglement is provided recently, the squeezing parameter and the new entanglement criteria will be compared. It is meaningful to do research the number spin squeezing, for particle counting is such a popular measurement in experiment.
引文
[1]Glauber R J. The quantum theory of optical coherence. [J]Phys Rev,1963, 130: 2529-2539.
[2]Arecchi F T, Courtens E, Gilmore R, et al.[J]Atomic Coherent states in quantum optics.Phys Rev A,1972,6:2211-2237.
[3]Stoler D. Equivalence classes of minimum uncertainty packets [J].Phys Rev D,1970,1:3217-3219.
[4]Kitagawa M ,Ueda M .Squeezed spin states. [J]Phys Rev A,1993,47: 5138-5143.
[5]Pu H ,Meystre P. Creating macroscopic atomic Einstein-Podolsky-Rosen states from Bose-Einstein condensates. [J]Phys Rev Lett,2000,85:3987-3990.
[6]Duan L M,Sφrensen A ,Cirac J I ,et al.[J]Squeezing and entanglement of atomic beams.Phys Rev Lett,2000,85:3991-3994.
[7]Wineland D J, Bollinger J J, Itano W M, et al. [J]Spin squeezing and reduced quantum noise in spectroscopy.Phys Rev A,1992,46:R6797-6800.
[8]Wineland D J,Bollinger J J,Itano W M, et al.Squeezed atomic states and projection noise in spectroscopy.[J] Phys Rev A,1994,55:67-88.
[9]Sφrensen A, Duan L M, Cirac J I, et al. Many-particle entanglement with Bose-Einstein condensates.[J] Nature(London),2001,409:63-66.
[10]Sorensen A,Mφlmer K. Spin-spin interaction and spin squeezing in an sptical lattice .[J]Phys Rev Lett,1999,83:2274-2277.
[11]Julsgaard B,Sherson J ,Cirac J. I, et al.Experimental demonstration of quantum memory for light.[J] Nature(London),2004,432:482-486.
[12]Wall D F. Squeezed states of light. [J]Nature(London),1983,306:141-146
[13]Loudon R, Knight P L. Squeezed light. [J] J Mod Opt 1987,34:709-759.
[14]Thorne K S .Gravitational-wave research: current status and future prospects. [J]Rev. Mod. Phys.,1980,52:285-297.
[15]Caves C M. Quantum-mechanical noise in an interferometer.[J] Phys Rev D,1981,23:1693-1708.
[16]Shapiro J H, Yuen H P, J. A. Machado Mata. Optical communication with two-photon coherent states--Part Ⅱ:Photoemissive detection and structured receiver performance. [J] IEEE Trans Inform Theory,1979,25:179-192.
[17]Yuen H P ,Shapiro J H. Optical communication with two-photon coherent states--Part Ⅲ:Quantum measurements realizable with photoemissive detectors [J]IEEE Trans Inform Theory,1980,26:78-92
[18]Wodkiewicz K. Reduced quantum fluctuations in the Josephson Junction.[J] Phys Rev B,1985,32:4750-4752.
[19]Oztop Baris Alexander K, Shumovsky Alexander S. Spin squeezing and entanglement. [J]Conference on Coherence and Quantum Optics (CQO) 2007 paper: CSuA35OSA. Technical Digest (CD),
[20]Law C K, Ng H T, Leung P T. Coherent control of spin squeezing. [J]Phys Rev A,2001,63:055601.
[21]Jin G R, Liu Y C, Liu W M. Spin squeezing in a generalized one-axis
twisting mode. [J]New J Phys,2009,11:073049.
[22]Geza Toth, Christian Knapp, Otfried Guhne, et al. Spin squeezing and entanglement.[J] Phys Rev A,2009,79:042334.
[23]Korbicz J K, Cirac J I, Lewenstein M .Spin squeezing inequalities and entanglement of N qubit states. [J]Phys Rev Lett,2005,95:120502.
[24]Jin G R ,Kim S W. Spin squeezing and maximal-squeezing time.[J] Phys Rev A,2007,76:043621.
[25]Jin G R,Law C K. Relationship between spin squeezing and single-particle coherence in two-component Bose-Einstein condensates with Josephson coupling.[J]Phys Rev A,2008,78:063620.
[26]Raghavan S, Pu H, Meystre P, et al. Generation of arbitrary Dicke states in spinor Bose-Einstein condensates.[J] Opt Commun,2001,188:149-154.
[27]Helmerson K,You L.Creating massive entanglement of Bose-Einstein condensed atoms. [J]Phys Rev Lett,2001,87:170402.
[28]Kuzmich A ,Bigelow N P ,Mandel L. Atomic quantum non-demolition measurements and squeezing. [J] Europhys lett,1998,42(5):481-486.
[29]Kuzmich A ,Mandel L ,Bigelow N P. Generation of spin squeezing via continuous quantum nondemolition measurement. [J]Phys Rev Lett,2000,85: 1594-1597.
[30]Meiser D ,Ye Jun, Holland M J .Spin squeezing in optical lattice clocks via lattice-based QND measurements.[J] New J Phys,2008,10:073014.
[31]Saffman M,Oblak D, Appel J, et al. Spin squeezing of atomic ensembles by multicolor quantum nondemolition measurements.[J] Phys Rev A,2009,79:023831
[32]Takano T,Fuyama M ,Namiki R ,et al. Spin squeezing of a cold atomic ensemble with the nuclear spin of one-Half. [J] Phys Rev Lett,2009,102:033601.
[33]Kuzmich A,Mφlmer K,Polzik E S. Spin squeezing in an ensemble of atoms illuminated with squeezed light.[J]Phys Rev Lett,1997,79:4782-4785.
[34]Hald J, Sφrensen J L,Schori C, et al. Spin squeezed atoms:a macroscopic entangled ensemble created by light.[J] Phys Rev Lett,1999,83:1319-1322.
[35]Vernac L ,Pinard M,Giacobino E. Spin squeezing in two-level systems. [J]Phys Rev A,2000,62:063812.
[36]Takeuchi M ,Ichihara S ,Takano T, et al. Spin squeezing via one-axis twisting with coherent light.[J] Phys Rev Lett,2005,94:023003.
[37]Fernholz T,Krauter H Jensen K, et al.Spin squeezing of atomic ensembles via nuclear-electronic spin entanglement.[J]Phys Rev Lett,2008,101:073601.
[38]Jaksch D,Cirac J I, Zoller P. Dynamically turning off interactions in a two-component condensate.[J] Phys Rev A,2002,65,033625.
[39]Jin G R ,Kim S W. Storage of spin squeezing in a two-component Bose-Einstein condensate.[J]Phys Rev Lett,2007,99:170405.
[40]Yun Li, Y. Castin, A. Sinatra .Optimum spin squeezing in Bose-Einstein condensates with particle losses. [J]Phys Rev Lett,2008,100:210401.
[41]Rey A M, Jiang L,Lukin M D. Quantum-limited measurements of atomic scattering properties. [J]Phys Rev A,2007,76:053617.
[42]徐在新,[M]高等量子力学,第二版,上海,华东师范大学出版社,1995.4.1,20-27，36-41.
[43]C. J. PETHICK&H. SMITH, [M]BOSE-EINSTEIN CONDENSATION IN DILUTE GASES,第二版,美国,剑桥大学出版社,119-120.
[44]S.Raghavan, H.Pu,P.Meystre,et al. Generation of arbitrary Dicke states in Spinor Bose-Einstein condensates. [J]Opt. Commun.2003,188:149-154.
[45]G.R.Jin, X.W.Wang, D.Li,et al. Atom number squeezing and bipartite entanglement of Two-component Bose-Einstein condensates, submitted to IOP.
[46]G.R.Jin, Yong-Chun Liu and Wu-Ming Liu. Spin squeezing in a generalized one-axis twisting model.[J] New Journal of Physics,2009,11:073049.
[47]Y.Khodorkovsky,G.Kurizki.and A.Vardi, Bosonic Ampli fi cation of Noise-Induced Suppression of Phase Diffusion[J].Phys Rev. Lett.,2008,100:220403.
[48]Hui Jing Mutual coherence and spin squeezing in double-well atomic condensates.[J] Phys. Lett.A,2002,306:91-96.
[49]Luca Pezze and Augusto Smerzi.Entanglement, Nonlinear Dynamics, and the Heisenberg Limit.[J] Phys. Rev. Lett.,2009,102:100401.
[50]Esteve J ,Gross C, Weller A, Giovanazzi S,et al. Squeezing and entanglement in a Bose-Einstein condensate.[J] Nature,2008:1216-1219.
[2]Arecchi F T, Courtens E, Gilmore R, et al.[J]Atomic Coherent states in quantum optics.Phys Rev A,1972,6:2211-2237.
[3]Stoler D. Equivalence classes of minimum uncertainty packets [J].Phys Rev D,1970,1:3217-3219.
[4]Kitagawa M ,Ueda M .Squeezed spin states. [J]Phys Rev A,1993,47: 5138-5143.
[5]Pu H ,Meystre P. Creating macroscopic atomic Einstein-Podolsky-Rosen states from Bose-Einstein condensates. [J]Phys Rev Lett,2000,85:3987-3990.
[6]Duan L M,Sφrensen A ,Cirac J I ,et al.[J]Squeezing and entanglement of atomic beams.Phys Rev Lett,2000,85:3991-3994.
[7]Wineland D J, Bollinger J J, Itano W M, et al. [J]Spin squeezing and reduced quantum noise in spectroscopy.Phys Rev A,1992,46:R6797-6800.
[8]Wineland D J,Bollinger J J,Itano W M, et al.Squeezed atomic states and projection noise in spectroscopy.[J] Phys Rev A,1994,55:67-88.
[9]Sφrensen A, Duan L M, Cirac J I, et al. Many-particle entanglement with Bose-Einstein condensates.[J] Nature(London),2001,409:63-66.
[10]Sorensen A,Mφlmer K. Spin-spin interaction and spin squeezing in an sptical lattice .[J]Phys Rev Lett,1999,83:2274-2277.
[11]Julsgaard B,Sherson J ,Cirac J. I, et al.Experimental demonstration of quantum memory for light.[J] Nature(London),2004,432:482-486.
[12]Wall D F. Squeezed states of light. [J]Nature(London),1983,306:141-146
[13]Loudon R, Knight P L. Squeezed light. [J] J Mod Opt 1987,34:709-759.
[14]Thorne K S .Gravitational-wave research: current status and future prospects. [J]Rev. Mod. Phys.,1980,52:285-297.
[15]Caves C M. Quantum-mechanical noise in an interferometer.[J] Phys Rev D,1981,23:1693-1708.
[16]Shapiro J H, Yuen H P, J. A. Machado Mata. Optical communication with two-photon coherent states--Part Ⅱ:Photoemissive detection and structured receiver performance. [J] IEEE Trans Inform Theory,1979,25:179-192.
[17]Yuen H P ,Shapiro J H. Optical communication with two-photon coherent states--Part Ⅲ:Quantum measurements realizable with photoemissive detectors [J]IEEE Trans Inform Theory,1980,26:78-92
[18]Wodkiewicz K. Reduced quantum fluctuations in the Josephson Junction.[J] Phys Rev B,1985,32:4750-4752.
[19]Oztop Baris Alexander K, Shumovsky Alexander S. Spin squeezing and entanglement. [J]Conference on Coherence and Quantum Optics (CQO) 2007 paper: CSuA35OSA. Technical Digest (CD),
[20]Law C K, Ng H T, Leung P T. Coherent control of spin squeezing. [J]Phys Rev A,2001,63:055601.
[21]Jin G R, Liu Y C, Liu W M. Spin squeezing in a generalized one-axis
twisting mode. [J]New J Phys,2009,11:073049.
[22]Geza Toth, Christian Knapp, Otfried Guhne, et al. Spin squeezing and entanglement.[J] Phys Rev A,2009,79:042334.
[23]Korbicz J K, Cirac J I, Lewenstein M .Spin squeezing inequalities and entanglement of N qubit states. [J]Phys Rev Lett,2005,95:120502.
[24]Jin G R ,Kim S W. Spin squeezing and maximal-squeezing time.[J] Phys Rev A,2007,76:043621.
[25]Jin G R,Law C K. Relationship between spin squeezing and single-particle coherence in two-component Bose-Einstein condensates with Josephson coupling.[J]Phys Rev A,2008,78:063620.
[26]Raghavan S, Pu H, Meystre P, et al. Generation of arbitrary Dicke states in spinor Bose-Einstein condensates.[J] Opt Commun,2001,188:149-154.
[27]Helmerson K,You L.Creating massive entanglement of Bose-Einstein condensed atoms. [J]Phys Rev Lett,2001,87:170402.
[28]Kuzmich A ,Bigelow N P ,Mandel L. Atomic quantum non-demolition measurements and squeezing. [J] Europhys lett,1998,42(5):481-486.
[29]Kuzmich A ,Mandel L ,Bigelow N P. Generation of spin squeezing via continuous quantum nondemolition measurement. [J]Phys Rev Lett,2000,85: 1594-1597.
[30]Meiser D ,Ye Jun, Holland M J .Spin squeezing in optical lattice clocks via lattice-based QND measurements.[J] New J Phys,2008,10:073014.
[31]Saffman M,Oblak D, Appel J, et al. Spin squeezing of atomic ensembles by multicolor quantum nondemolition measurements.[J] Phys Rev A,2009,79:023831
[32]Takano T,Fuyama M ,Namiki R ,et al. Spin squeezing of a cold atomic ensemble with the nuclear spin of one-Half. [J] Phys Rev Lett,2009,102:033601.
[33]Kuzmich A,Mφlmer K,Polzik E S. Spin squeezing in an ensemble of atoms illuminated with squeezed light.[J]Phys Rev Lett,1997,79:4782-4785.
[34]Hald J, Sφrensen J L,Schori C, et al. Spin squeezed atoms:a macroscopic entangled ensemble created by light.[J] Phys Rev Lett,1999,83:1319-1322.
[35]Vernac L ,Pinard M,Giacobino E. Spin squeezing in two-level systems. [J]Phys Rev A,2000,62:063812.
[36]Takeuchi M ,Ichihara S ,Takano T, et al. Spin squeezing via one-axis twisting with coherent light.[J] Phys Rev Lett,2005,94:023003.
[37]Fernholz T,Krauter H Jensen K, et al.Spin squeezing of atomic ensembles via nuclear-electronic spin entanglement.[J]Phys Rev Lett,2008,101:073601.
[38]Jaksch D,Cirac J I, Zoller P. Dynamically turning off interactions in a two-component condensate.[J] Phys Rev A,2002,65,033625.
[39]Jin G R ,Kim S W. Storage of spin squeezing in a two-component Bose-Einstein condensate.[J]Phys Rev Lett,2007,99:170405.
[40]Yun Li, Y. Castin, A. Sinatra .Optimum spin squeezing in Bose-Einstein condensates with particle losses. [J]Phys Rev Lett,2008,100:210401.
[41]Rey A M, Jiang L,Lukin M D. Quantum-limited measurements of atomic scattering properties. [J]Phys Rev A,2007,76:053617.
[42]徐在新,[M]高等量子力学,第二版,上海,华东师范大学出版社,1995.4.1,20-27，36-41.
[43]C. J. PETHICK&H. SMITH, [M]BOSE-EINSTEIN CONDENSATION IN DILUTE GASES,第二版,美国,剑桥大学出版社,119-120.
[44]S.Raghavan, H.Pu,P.Meystre,et al. Generation of arbitrary Dicke states in Spinor Bose-Einstein condensates. [J]Opt. Commun.2003,188:149-154.
[45]G.R.Jin, X.W.Wang, D.Li,et al. Atom number squeezing and bipartite entanglement of Two-component Bose-Einstein condensates, submitted to IOP.
[46]G.R.Jin, Yong-Chun Liu and Wu-Ming Liu. Spin squeezing in a generalized one-axis twisting model.[J] New Journal of Physics,2009,11:073049.
[47]Y.Khodorkovsky,G.Kurizki.and A.Vardi, Bosonic Ampli fi cation of Noise-Induced Suppression of Phase Diffusion[J].Phys Rev. Lett.,2008,100:220403.
[48]Hui Jing Mutual coherence and spin squeezing in double-well atomic condensates.[J] Phys. Lett.A,2002,306:91-96.
[49]Luca Pezze and Augusto Smerzi.Entanglement, Nonlinear Dynamics, and the Heisenberg Limit.[J] Phys. Rev. Lett.,2009,102:100401.
[50]Esteve J ,Gross C, Weller A, Giovanazzi S,et al. Squeezing and entanglement in a Bose-Einstein condensate.[J] Nature,2008:1216-1219.