克尔非线性黑体的统计属性
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摘要
在量子光学领域中，光场量子特性的研究一直是人们关注的课题。本文主要研究了克尔非线性黑体中辐射场的统计属性，得出了一些有意义的结论，全文共分为以下五部分：
第一部分，研究了克尔非线性黑体光场的振幅平方压缩效应，发现克尔非线性黑体光场在转变温度T_c下可以同时处于振幅压缩和振幅平方压缩态，并且光场的振幅平方压缩效应要比振幅压缩效应大；克尔非线性黑体光场的振幅平方压缩效应随着温度T和克尔非线性系数γ的增加而减小；随着频率ω_(?)的增大，振幅平方压缩效应先随增大后减小，在频率ω_m=6.7×10~(13)s~(-1)处，振幅平方压缩效应最强；在相同参数下，克尔非线性黑体的某个振幅平方变量比普通黑体的振幅平方变量要小。
第二部分，研究了克尔非线性黑体光场的相位对称性。当温度T处于不同范围时，克尔非线性黑体光场处于不同的态。利用准几率分布函数即Q函数分布详细讨论了克尔非线性黑体光场处于不同态的空间相位。发现温度T和克尔非线性系数γ对Q函数分布有很大的影响。随着温度T的降低，克尔非线性黑体热辐射场由常态变为压缩态，光场经历了一个一级相变，其相位对称自发破缺。
第三部分，研究了克尔非线性黑体的统计性质。讨论了在温度低于转变温度T_c时克尔非线性黑体光场的二阶关联函数，光子数分布函数以及相位分布函数。研究表明，二阶关联函数g~((2))≥2，即光子表现出经典的群聚效应，光场为超泊松分布，而且温度T，克尔非线性系数γ和频率ω_k对泊松分布统计有很大的影响；光子数分布函数P(n)呈现振动形态，振动行为只发生在光子数少量的情况下，是光场的压缩属性的体现；相位分布函数P(θ)呈现称结构，在±π／2处出现双峰，随着温度T和克尔非线性系数γ的减小，相位分布函数P(θ)的振幅增大，波宽变窄，即光场的压缩效应越来越强。同时，给出了这些量的测量方法。
第四部分，研究了克尔非线性黑体辐射场的非经典程度。分别采用两种方法，即Lee的相位式法和Malbouisson的距离式衡量法讨论了在温度低于转变温度T_c时，辐射场的非经典程度。相比较而言，Lee的相位式衡量法比较复杂，不易于处理。同时发现随着温度T和克尔非线性系数γ以及频率ω_k的增大，克尔非线性黑体辐射场的非经度降低；并且克尔非线性黑体辐射场的非经典度比普通黑体的非经典度要高。
第五部分，研究了带电谐振子与克尔非线性黑体相互作用的耦合系统。带电谐振子处于谐势井中，在偶极耦合下与光场发生相互作用。通过海森堡运动方程，建立了谐振子的朗之万方程。研究表明，当温度低于转变温度T_c时，量子朗之万方程的记忆函数是温度T和克尔非线性系数γ的递减函数；在大的截止频率极限下，在克尔非线性黑体中所观测到的谐振子质量和自由能位移要比普通黑体中的谐振子质量和自由能位移大。自由能位移不再简单的与T~2成正比。在普通黑体中，谐振子的能级位移是负值，但是在克尔非线性黑体中，只是在零温度附近，能级位移是负值，而在接近转变温度的情况下，能级位移是一个很大的正值。当温度高于转变温度T_c时，其结论跟普通黑体中的情况一样。
The characteristics of quantum field have attracted much attention in the area ofquantum optics. The statistical properties of radiation thermal field in a Kerr nonlinearblackbody have been investigated, and some significative and new results have obtained.There are five parts in this thesis.
First, the photon field in a Kerr-nonlinear blackbody is in an amplitude-squaredsqueezed state. The amplitude-squared squeezing effect enhances with increasingtemperature T and parameterγ. As frequencyω_k increases, the amplitude-squaredsqueezing effect firstly enhances as frequencyω_k increases. Atω_m=6.7×10~(13)s~(-1), theamplitude-squared squeezing effect is strongest, then the squeezing effect weakens as asfrequencyω_k increases. The amount of the amplitude-squared squeezing in aKerr-nonlinear blackbody is much larger than the corresponding squeezing in normalblackbody, and the degree of amplitude-squared squeezing is much larger than theamplitude squeezing for the same range of parameters in a Kerr-nonlinear blackbody.
Second, the phase symmetry breaking is investigated in a Kerr-nonlinear blackbodyin terms of the quasiprobability distribution. The quasiprobability (Q function)distribution is discussed in detail in the different ranges of temperature T. It is foundthat the distribution of Q function strongly depends on the temperature T and couplingparameterγ. Non-classical effects of quadrature squeezing have been observed. In thetransition from the normal to the squeezed thermal radiation state, the phase symmetry ofthe photon system in a Kerr-nonlinear blackbody is spontaneously broken.
Third, we study the statistical properties of thermal radiation in a Kerr nonlinearblackbody. The second-order correlation function, the phase space distribution function and the photon number distribution are considered. It is found that blow a transitiontemperature T-c, the second-order correlation function g~(2) (0)≥2, which shows that thephoton system is classical bunching, and the statistical behavior always super-Poissonianin the Kerr-nonlinear blackbody, and the super-Poissonian statistics are stronglydependent on the temperature T and the coupling parameterγ; the photon numberdistribution P(n) presents nonclassical oscillations, while the oscillatory behaviorsonly occur at very low number of photons, which shows the photon filed in a Kerrnonlinear blackbody is in a squeezed state; the phase distribution P(θ) has a doubletstructure and the peaks are at±π/2, but with broader width and more significantamplitude as the temperature T and the coupling parameterγincrease.
Fourth, the degree of nonclassicality of a photon field is considered in a Kerrnonlinear blackbody. The space-phase and distance-type measures are employed,respectively. It is shown that blow a transition temperature T_c, the photon field becomesless nonclassical in the Kerr-nonlinear blackbody as temperature T, Kerr nonlinearcoefficientγand frequencyω_k increase. Furthermore, the degree of nonclassicality of aKerr nonlinear blackbody is always higher than that of a normal blackbody.
Fifth, we derive a quantum Langevin equation for the macroscopic description of acharged oscillator in a harmonic potential well interacting with a Kerr nonlinearblackbody radiation field via dipole coupling. It is shown that below a transitiontemperature T_c, the memory function in our quantum Langevin equation is a decreasingfunction of temperature T and Kerr nonlinear coefficientγ. In the large-cutoff limit, theobserved mass and free energy shift of a charged oscillator in a Kerr nonlinear blackbodyare larger than those in a normal blackbody. The free energy shift is a complex functionof temperature which is no longer a simply T~2-dependent function. The energy level shift of an oscillator in a normal blackbody is a negative, while in our Kerr nonlinearblackbody, only near zero temperature the corresponding energy level shift is a negative.As the temperature increases to transition temperature T, the corresponding energy levelshift becomes a large positive value. Above T_c, our results are the same as those in anormal blackbody.
第一部分，研究了克尔非线性黑体光场的振幅平方压缩效应，发现克尔非线性黑体光场在转变温度T_c下可以同时处于振幅压缩和振幅平方压缩态，并且光场的振幅平方压缩效应要比振幅压缩效应大；克尔非线性黑体光场的振幅平方压缩效应随着温度T和克尔非线性系数γ的增加而减小；随着频率ω_(?)的增大，振幅平方压缩效应先随增大后减小，在频率ω_m=6.7×10~(13)s~(-1)处，振幅平方压缩效应最强；在相同参数下，克尔非线性黑体的某个振幅平方变量比普通黑体的振幅平方变量要小。
第二部分，研究了克尔非线性黑体光场的相位对称性。当温度T处于不同范围时，克尔非线性黑体光场处于不同的态。利用准几率分布函数即Q函数分布详细讨论了克尔非线性黑体光场处于不同态的空间相位。发现温度T和克尔非线性系数γ对Q函数分布有很大的影响。随着温度T的降低，克尔非线性黑体热辐射场由常态变为压缩态，光场经历了一个一级相变，其相位对称自发破缺。
第三部分，研究了克尔非线性黑体的统计性质。讨论了在温度低于转变温度T_c时克尔非线性黑体光场的二阶关联函数，光子数分布函数以及相位分布函数。研究表明，二阶关联函数g~((2))≥2，即光子表现出经典的群聚效应，光场为超泊松分布，而且温度T，克尔非线性系数γ和频率ω_k对泊松分布统计有很大的影响；光子数分布函数P(n)呈现振动形态，振动行为只发生在光子数少量的情况下，是光场的压缩属性的体现；相位分布函数P(θ)呈现称结构，在±π／2处出现双峰，随着温度T和克尔非线性系数γ的减小，相位分布函数P(θ)的振幅增大，波宽变窄，即光场的压缩效应越来越强。同时，给出了这些量的测量方法。
第四部分，研究了克尔非线性黑体辐射场的非经典程度。分别采用两种方法，即Lee的相位式法和Malbouisson的距离式衡量法讨论了在温度低于转变温度T_c时，辐射场的非经典程度。相比较而言，Lee的相位式衡量法比较复杂，不易于处理。同时发现随着温度T和克尔非线性系数γ以及频率ω_k的增大，克尔非线性黑体辐射场的非经度降低；并且克尔非线性黑体辐射场的非经典度比普通黑体的非经典度要高。
第五部分，研究了带电谐振子与克尔非线性黑体相互作用的耦合系统。带电谐振子处于谐势井中，在偶极耦合下与光场发生相互作用。通过海森堡运动方程，建立了谐振子的朗之万方程。研究表明，当温度低于转变温度T_c时，量子朗之万方程的记忆函数是温度T和克尔非线性系数γ的递减函数；在大的截止频率极限下，在克尔非线性黑体中所观测到的谐振子质量和自由能位移要比普通黑体中的谐振子质量和自由能位移大。自由能位移不再简单的与T~2成正比。在普通黑体中，谐振子的能级位移是负值，但是在克尔非线性黑体中，只是在零温度附近，能级位移是负值，而在接近转变温度的情况下，能级位移是一个很大的正值。当温度高于转变温度T_c时，其结论跟普通黑体中的情况一样。
The characteristics of quantum field have attracted much attention in the area ofquantum optics. The statistical properties of radiation thermal field in a Kerr nonlinearblackbody have been investigated, and some significative and new results have obtained.There are five parts in this thesis.
First, the photon field in a Kerr-nonlinear blackbody is in an amplitude-squaredsqueezed state. The amplitude-squared squeezing effect enhances with increasingtemperature T and parameterγ. As frequencyω_k increases, the amplitude-squaredsqueezing effect firstly enhances as frequencyω_k increases. Atω_m=6.7×10~(13)s~(-1), theamplitude-squared squeezing effect is strongest, then the squeezing effect weakens as asfrequencyω_k increases. The amount of the amplitude-squared squeezing in aKerr-nonlinear blackbody is much larger than the corresponding squeezing in normalblackbody, and the degree of amplitude-squared squeezing is much larger than theamplitude squeezing for the same range of parameters in a Kerr-nonlinear blackbody.
Second, the phase symmetry breaking is investigated in a Kerr-nonlinear blackbodyin terms of the quasiprobability distribution. The quasiprobability (Q function)distribution is discussed in detail in the different ranges of temperature T. It is foundthat the distribution of Q function strongly depends on the temperature T and couplingparameterγ. Non-classical effects of quadrature squeezing have been observed. In thetransition from the normal to the squeezed thermal radiation state, the phase symmetry ofthe photon system in a Kerr-nonlinear blackbody is spontaneously broken.
Third, we study the statistical properties of thermal radiation in a Kerr nonlinearblackbody. The second-order correlation function, the phase space distribution function and the photon number distribution are considered. It is found that blow a transitiontemperature T-c, the second-order correlation function g~(2) (0)≥2, which shows that thephoton system is classical bunching, and the statistical behavior always super-Poissonianin the Kerr-nonlinear blackbody, and the super-Poissonian statistics are stronglydependent on the temperature T and the coupling parameterγ; the photon numberdistribution P(n) presents nonclassical oscillations, while the oscillatory behaviorsonly occur at very low number of photons, which shows the photon filed in a Kerrnonlinear blackbody is in a squeezed state; the phase distribution P(θ) has a doubletstructure and the peaks are at±π/2, but with broader width and more significantamplitude as the temperature T and the coupling parameterγincrease.
Fourth, the degree of nonclassicality of a photon field is considered in a Kerrnonlinear blackbody. The space-phase and distance-type measures are employed,respectively. It is shown that blow a transition temperature T_c, the photon field becomesless nonclassical in the Kerr-nonlinear blackbody as temperature T, Kerr nonlinearcoefficientγand frequencyω_k increase. Furthermore, the degree of nonclassicality of aKerr nonlinear blackbody is always higher than that of a normal blackbody.
Fifth, we derive a quantum Langevin equation for the macroscopic description of acharged oscillator in a harmonic potential well interacting with a Kerr nonlinearblackbody radiation field via dipole coupling. It is shown that below a transitiontemperature T_c, the memory function in our quantum Langevin equation is a decreasingfunction of temperature T and Kerr nonlinear coefficientγ. In the large-cutoff limit, theobserved mass and free energy shift of a charged oscillator in a Kerr nonlinear blackbodyare larger than those in a normal blackbody. The free energy shift is a complex functionof temperature which is no longer a simply T~2-dependent function. The energy level shift of an oscillator in a normal blackbody is a negative, while in our Kerr nonlinearblackbody, only near zero temperature the corresponding energy level shift is a negative.As the temperature increases to transition temperature T, the corresponding energy levelshift becomes a large positive value. Above T_c, our results are the same as those in anormal blackbody.
引文
[1] Planck M. On the Law of Distribution of Energy in the Normal Spectrum, Annalen der Physik, 1901,4 (3): 553-563
[2] Planck M. Elementary quanta of matter and electricity, Annalen der Physik, 1901,4 (3):564-566
[3] Landau L D, Lifshitz E M. Statistical physics. Third Edition. Oxford: Reed Educational, 1980,183-186
[4] Scully M O, Zubairy M S. Quantum optics, First Edition. The United Kingdom:Cambridge University, 1997
[5] Cheng Z. Quantum effects of thermal radiation in a Kerr nonlinear blackbody. J. Opt.Soc. Am. B, 2002,19(7): 1692-1705
[6] Cheng Z. Radiation properties of a Kerr nonlinear blackbody. Phys. Rev. A, 2005,71(3),033808:1-9
[7] Cheng Z. Squeezed thermal radiation state in a Kerr-nonlinear black body. Phys. Lett.A, 2001,291(1): 4-10
[8] Cheng Z. Spectral energy density and radiation pressure in a Kerr nonlinear blackbody. Phys. Lett. A, 2004, 331(3): 170-174
[9] Cheng Z. Thermodynamics of phase transitions of a Kerr nonlinear blackbody. Chin.Phys.Lett.2008,25(9): 3264-3267
[10]Cheng Z. Photonic superfluidity in a Kerr nonlinear blackbody. Chin. Phys. Lett.,2005, 22(4): 880-883
[11]Heitler M. The quantum theory of radiation. New York: Oxford Clarendon Press Educational, 1954
[12]Loudon L. The quantum theory of light. New York: Oxford University Press, 1973
[13] Walls D F, Milburn G J. Quantum optics, First Edition. Berlin: Springer-Verlag, 1994, 91-98
[14]彭金生，李高翔．近现代量子光学导论(第一版)．北京：科学出版社，1996．
[15]Barnett S M, Radmore P M. Method in theoretical quantum optics, First Edition. Oxford:Clarendon Press. 1997
[16]Mandel L, Wolf E. Optical coherence and quantum optics. London: Cambridge University Press. 1995
[17] Shen Y R. Quantum statistics of nonlinear optics. Phys.Rev. 1967,155(3):921-931
[18]Walls D F. Squeezed states of light. Nature,1983,306(11):141-146
[19]Mandel L. Sub-Poissonian photon statistics in resonance fluorescence. Opt.Lett. 1979, 4(7):205-207
[20]Paul H. Photon antibunching. Rev.Mod. Phys.1982,54(4):1061-1102
[21] Einstein A. Zur Elektrodynamik bewegter Krrper. Annalen der Physik, 1905, 17(10): 891-921
[22]Compton A H. A quantum theory of the scattering of X-rays by light elements. Phys. Rev. 1923, 21(5): 483-502
[23] Brown R H,Twiss R Q. Correlation between photons in two coherent beams of light. Nature. 1956,177(4497):27-29
[24]Brown R H, Twiss R Q. A test of a new type of stellar interferometer on sirius. Nature. 178(10): 1046-1048
[25]Haken H. Laser light emission from interacting chaotic fields Z.Phys.1975, 265(2): 105-118
[26]Haken H. Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Rev.Mod.Phys. 1975,47(1):67-121
[27] Risken H, Vollmer H D. On the application of truncated generalized Fokker-Planck equations Z. Phys.B. 1979,35(3):313-315
[28]Benkert C, Scully M O, Bergou Jet al, Role of pumping statistics in laser dynamics: Quantum Langevin approach. Phys. Rev.A,1990,41(5):2756-2765
[29] Harris S E, Field J E, Imamoglu. Nonlinear optical processes using electro-magnetically induced transparency. Phys.Rev.Lett. 1990,64(10): 1107-1110
[30] Yee T K, Gustafson T K. Diagrammatic analysis of the density operator for nonlinear optical calculations: Pulsed and cw responses. Phys. Rev. A. 1978,18(4): 1597-1617
[31]Boyd G F, Kleiman D A. Parametric interaction of focused. Gaussian light beams.J.Appl.Phys.l968,39(8):3597-3670
[32] Fork R L, Brito C H, Becker P C et al. Compression of optical pulses to six femtoseconds by using cubic phase compensation. Optics Letters. 1987, 12(7): 483-485
[33]Stenholem S, Gas laser with saturable absorber. I .Single mode characteristics. Phys.Rev. A.1973,8(5):2695-2711
[34] Knight P L, Milonni P W. The Rabi frequency in optical spectra. Phys.Rep. 1980,66(2):21-107
[35]Haroche S, Kleppner D, Cavity quantum electrodynamics. Physics Today. 1989,42(1): 24-30
[36] Shore B W, Knight P L. The Jaynes-Cummings model. J. Mod. Opt. 1993, 40(7):1195-1238
[37]Frank O G Application of a one-dimensional classical model of atom-oscillator scattering to atom-surface accommodation coefficient theory. Surface Science. 1979,60(1):45-64
[38] Lamb W E, Retherford R C. Fine structure of the hydrogen atom by microwave Method. Phys. Rev. 1947,72(3):241-243
[39]Schawlow A L, Townes C H. Infrared and optical masers. Phys. Rev.1958,112(6):1940-1949
[40]Fleischhauer M, Schleich W. Revivals made simple: Poisson summation. Phys. Rev.A.1992,47(5):4258-4269
[41] Sherman B, Kurizki G Preparation and detection of macroscopic quantum superpositions by two-photon field-atom interactions. Phys.Rev. A. 1992,45(11): R7674-R7677
[42] Eberly J H, Wodkiewicz K. The time-dependent physical spectrum of light. J. Opt. Soc. Am. 1977, 67(9): 1252-1261
[43]Jaynes E T, Cummings F W. Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc.IEEE, 1963, 51 (1): 89-109
[44]Alsing P, Cardimona D A, Suppression of fluorescence in a lossless cavity. Phys. Rev.A, 1992,45(7): 1793-1796
[45]Brune M, Raimond J M, Haroehe S; Theory of the Rydberg-atom two-Phonon micromaser. Phys.Rev.A, 1987,35(1): 154-157
[46] Stolen H R, AshkiA.Optical Kerr effect in glass waveguide. Applied Physics Letters. 1973, 22(6):294-296
[47]石顺祥，陈果夫，赵卫．非线性光学．西安：西安电子科技大学出版社．2003
[48]方俊鑫，陈栋．固体物理学．第六版，上海：上海科技出版社．1987：196-200
[49]Korolev A V, Liberman M A. Polaritons and a built-in condensate of excitons in a high magnetic field slid state communications. 1996, 100(2):89-94
[50]Ichimiya M, Watanabe M, Hayashi T. Polarization dependence of biexciton formation in CuC1 studied by pump-probe spectroscopy. Journal of Luminescence. 2000, 87-89:272-274
[51]Baumberg J J, Savvidis P G, Lagoudakis P G; et al. Polariton traps in semiconductor microcavities, Physica E. 2002,13(2-4):385-389
[52]Lu Y Q, Zhu Y Y, Chen Y F et al. Optical properties of an ionic-type phononic crustal. Science. 1999,284(5421): 1822-1824
[53]Cheng Z. Pairing effect of photons in nonlinear polar crystals. Phys. Rev. Lett. 1991, 67(20): 2788-2791
[54] Cheng Z. Photonic superguiding state in nonlinear polar crystals. Phys. Rev. A, 1995, 51(1): 675-691
[55]Cheng Z, Xu Hong-Tao. Spectral energy density in a Kerr nonlinear blackbody.Commun. Theor. Phys., 2004,42(6): 884-886
[56] Yin M, Cheng Z. Inhibited spontaneous emission in a Kerr nonlinear blackbody.Phys. Rev. A. 2008,78(6), 063829:1-7
[57]Wan J Y, Cheng Z. Study of nonpolaritons in a Kerr nonlinear optical resonator.Commun. Theor. Phys.2006, 45(6):1107-1111
[58] Cheng Z. Unified quantum field theory of Raman scattering of light in piezoelectric crystals. Acta Physica Sinica. 2005,54(11): 5435-5444
[59]Loudon R, Knight P L. Squeezed light. J.Mod.Opt.,1987,34(6/7):709-759
[60]Hillery M. Sum and difference squeezing of the electromagnetic field. Phys. Rev. A,1989,40(6):3147-3155
[61]Yuen H P. Two-Photon coherent states of the radiation field. Phys.Rev.A,1976, 13(6):222-2213
[62]Glaube R J. coherent and incoherent states of the radiation field. Phys. Rev. 1963,131(6):2766-2788
[63]Kimble H J, Dgenais M, Mandel. Photon antibunching in resonance flurescence.Phys. Rev.Lett.l977,39(11):691-695
[64]Stoler D. Equivalence classes of minimum uncertainty packets. Phys.Rev. D, 1970,1(12):3217-3219
[65]Hong C K, Mandel L. Generation of higher-order squeezing of quantum electro-magnetic fields. Phys.Rev.A,1985,32(2):971-982
[66]Hillery M. Amplitude-squared squeezing of the electromagnetic field. Phys. Rev. A,1987,36(8):379-3802
[67] Zhang Z M, Xu L, Chai J L et al. A new kind of higher-order squeezing of radiation field. Phys.Lett. A,1990,150(1):27-30
[68]Bergou J A, Hillery M, Yu Daoqi. Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states. Phys.Rev.A,1991,43(1):515-520
[69]董传华．高阶压缩的另一种定义．光学学报．1996，16(11)：1543-1548
[70]Kozhekin A, Kurizki G, Sherma B. Quantum-state control by a single conditional measurement: The periodically switched Jaynes-Cummings model. Phys.Rev.A,1996, 54(4): 3535-3538
[71]Greenberger D M, Horne M, Zeilinger A. In Bell's theorem, Quanium theory, and conceptions of the universe. Edited by M.Kafatos. Kluwer: Dordreeht. 1989
[72] Greenberger D M, Horne M, Zeilinger A. Bell's theorem without inequalities. Am. J.Phys. 1990, 58(12):1131-1143
[73]Ficek Z, Tonas R, Kielich S.Amplitude-squared squeezing in two-atom resonance fluorescence. Opt.Coum.,1988,69(1):20-24
[74]Hu X, Nori F. Phonon squeezed states:quantum noise reduction in solids. Physica B: Condensed Matter, 1999,263-264:16-29
[75] Hu X, Nori F. Phonon squeezed states generated by second-order Raman scattering. Phys. Rev. Lett. 1997, 79(23):4605-4608
[76]Nguyen Ba An. Higher-order amplitude squeezing of photons propa-gating through a semiconductor. Phys. Rev. A, 1997, 234(1):45-52
[77] Drummond P D, Shelby R. M, Friberg S R, et al. Quantum Solitons in Optical Fibers. Nature. 1993,365(23): 307-313
[78]Deutsch I H, Chiao R Y, Garrison J C. Two-photon bound states: The photon bullet in dispersive self-focusing media. Phys. Rev. A, 1993, 47(4): 3330-3336
[79] Hasegawa A. Optical Solitons in Fibers, 2nd ed. Berlin: Springer-Verlag. 1990
[80]Lai Y, Haus H A. Quantum theory of solitons in optical fibers. I. Time- dependent Hartree approximation. Phys. Rev. A, 1989, 40(2): 844-853
[81] Cheng Z, Kurizki G.Theory of one-dimensional quantum gap solitons. Phys. Rev. A, 1996, 54(4):3576-3591
[82]Vitali D, Fortunato M, Tombesi P. Complete Quantum Teleportation with a Kerr Nonlinearity. Phys. Rev. Lett., 2000, 85(2): 445-448
[83]Olsen M K, Kruglov V I, Collett M.J. Effects of χ(3) nonlinearities in second-harmonic generation. Phys. Rev. A, 2001, 63(3): 033801-033809
[84]Kenichi Hirosawa, Hiroto Furumochi, Atsushi Tada et al. Photon Number Squeezing of Ultrabroadband Laser Pulses Generated by Microstructure Fibers. Phys. Rev.Lett.,2005, 94(20): 203601/1-4
[85]Buzek V, Jex I. Dynamics of a two-level atom in a Kerr-like medium. Opt. Commun.1990, 78(6):425-435
[86]Buzek V, Jex I. Emission spectra of a two-level atom in a Kerr-like medium.J.Mod.Opt. 1991, 38(5):987-996
[87] Brown R. A brief account of microscopical observations made in the months of June.1928,4(2):161-173
[88] Einstein A. On the motion-required by the molecular kinetic theory of heat of small particles suspended in a stationary liquid. Ann.der Phys. 1905,17(8): 549-560
[89] Einstein A. Investigations on the theory of the Brownian movement. New York:Dover. 1956.
[90]Perrin J B. Molecular agitation and the brownian movement. Comptes Rendus. 1908,146:967-970
[91]Perrin J B. Brownian Movement and Molecular Reality. London : Taylor and Francis.1910,1-93
[92]Langevin M P. Sur la theorie du mouvement brownien. Comptes Rendus Acad. Sci.,Paris. 1908,146:530-533
[93]Chandrasekhar S. Stochastic Problems in Physics and Astronomy. Rev. Mod.Phys.1943, 15(1):1-89
[94]Nyquist R. Thermal agitation of electric charge in conductors. Phys. Rev. 1928,
[95]Callen H B, Welton T A. Irreversibility and generalized noise. Phys. Rev. 1951,83(1):34-40
[96]Zwanzig R. Nonlinear Generalized Langevin Equations. J. Stat. Phys. 1973, 9(3):215-220
[97]Senitzky I R. Dissipation in Quantum Mechanics. The Harmonic Oscillator. Phys.Rev. 1960,119(2):670-679
[98] Lax M. Quantum Noise. ?. Quantum Theory of Noise Sources. Phys. Rev. 1966,145(1):110-129
[99] Ford G W, Kac M, Mazur P. Statistical Mechanics of Assemblies of Coupled Oscillators. J. Math. Phys. 1965, 6(4): 504-515
[100] Mori H. Transport, Collective Motion, and Brownian Motion. Prog. Theor. Phys.1965, 33(3): 423-455
[101] Benguria R, Kac M. Quantum Langevin Equation. Phys.Rev. Lett.1981, 46(1): 1-4
[102] Ford G W, Kac M. On the Quantum Langevin Equation. J. Stat. Phys. 1987,46(5-6): 803-810
[103] Kubo R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 1966, 29(1):255-284
[104] Kubo R. Brownian Motion and Nonequilibrium Statistical Mechanics. Science 1986,233(4761):330-334
[105] Caldeira A O, Leggett A J. Quantum Tunneling in a Dissipative System. Ann. Phys.1983, 149(1):374-456
[106] Grabert H, Schramm H, Ingold G L. Quantum Brownian motion: The functional integral approach. Phys. Rep. 1988, 168(3): 115-207
[107] Weiss U. Quantum Dissipative Systems. Singapore :World Scientific. 1993
[108] Hayes W, Loudon R. Scattering of Light by Crystals. New York: Wiley. 1978 [109] Shen Y R, The Principles of Nonlinear Optics. New York: Wiley. 1984 [110] Sal eh B E, Teach M C. Can the channel capacity of a light-wave communication
system be increased by the use of photon-number-squeezed light. Phys. Rev. lett. 1987, 58 (25): 2656-2659
[111] Pe(?)ina J, Perinova V, Kodiysek J, On the relations of antibunching, sub-Poissonian statistics and squeezing. Opt. Commum. 1984,49 (3): 210-214
[112] Kielich S, Tanas R, Zawodny R. Squeezing in the third-harmonic field generated by self-squeezed light. J. Opt. Soc. Am. B.1987,4 (10):1627-1632
[113] Reid M D, Walls D F. Generation of squeezed states via degenerate four-wave mixing. Phys. Rev. A 1985,31(3): 1622-1635
[114] Pe(?)ina J, Perinova V, Sibilia C, et al. Quantum statistics of four-wave mixing. Opt.Commum. 1984, 49 (4):285-289.
[115] Perinov(?) V. Pe(?)ina J. Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping. Opt. Acta. 1981,28 (6): 769-793
[116] Hillery M. Amplitude squeezing of the electromagnetic field. Opt. Commum. 1987,62 (2): 135-138
[117] Gerry C. Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium. Phys.Rev. A. 1999,59 (5):4095-4098
[118] Mahran M H. Squeezing and higher-order squeezing in the three-level, two-mode system .Phys. Rev. A .1990,42 (7):4199-4209
[119] Xie R H, Smith Jr V H. Generation of higher-order atomic dipole squeezing in a high-Q micromaser cavity. (?) Discussion of the nondegenerate two-photon Jaynes-Cummings model. Physica A .2002, 307 (1-2):207-220
[120] Lynch R, Mavromatis H A, Puri'S criterion and higher-order squeezing. Phys. Rev.A. 1995, 52(1):55-57
[121] Giri D K, Gupta P S, The Squeezing of radiation in four-wave mixing processes. J.Opt. B: Quantum Semiclass. Opt. 2004, 6 (1): 91-96
[122] Hu X, Nori F. Phonon squeezed states generated by second-order Raman scattering. Phys. Rev. Lett. 1997,79 (23):4605-4608
[123] Chizhov A V, Haus J W, Yeong K C. Higher-order squeezing in a boson-coupled two-mode system. Phys. Rev. A. 1995,52 (2): 1698-1703
[124] Onnes H K. The resistance of pure mercury at helium temperatures. Commum. Phys. Lab. Univ. Leiden.1911, 120b:17-19
[125] Kapitza P L. Superfluidity observed. Nature. 1932, 141(3558):74-75
[126] Anderson M H, Ensher J R, Matthews M R, et al. Observation of. Bose-Einstein Condensation in a Dilute Atomic Vapor. Scinence. 1995, 269(5221):198-201
[127] Greiner M, MandelO, Essling T, et al. Quantum phase transition from a superfluid to a mott insulator in a gas ofultracold atoms. Nature. 2002, 415: 39-44
[128] Yoichiro Nambu. Quasi-Particles and gauge invariance in the theory of superconductivity. Phys. Rev. 1960, 117(3), 648-663
[129] 李政道，粒子物理和场论简引，北京：科学出版社，1984
[130] Cooper L N. Bound Electron Pairs in a Degenerate Fermi Gas. Phys. Rev. 1956, 104 (4):1189-1190
[131] Bardeen J, Cooper L N, Schrieffer J R. Microscopic Theory of Super-conductivity. Phys. Rev. 1957,108 (1) 1175-1204
[132] Sudarshan E C G. Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys.Rev. Lett. 1963,10(7):277-279
[133] Wigner E. On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev. 1932, 40(5):749-759
[134] Avelar A T, Baseia B, de Almeida N G. Complementary coherent state for measuring the Q-function: generation and properties. J. Opt. B: Quantum Semiclass. Opt. 2004, 6(4): 41-45
[135] Marian P. Higher-order squeezing and photon statistics for squeezed thermal states. Phys.Rev. A. 1992,45 (3):2044-2051
[136] Chizhov A V, Gantsog Ts, Murzakhmetov B K. Phase distributions of squeezed number states and squeezed thermal states. Quantum Opt. 1993, 5 (2):85-95
[137] Avelar A T, de Souza L A, da Rocha Filho T M, et al. Generation of superposed phase states via Raman interaction . J. Opt. B: Quantum Semiclass. Opt. 2004,6(10): 383-386
[138] Albano L, Mundarain D F, Stephany J. et al .On the squeezed number states and their phase space representations. L Albano. J. Opt. B: Quantum Semiclass. Opt.2002, 4(5):352-357
[139] Vourdas A. Optical signals with thermal noise. Phys.Rev. A 1989, 39 (1):206- 213
[140] Masayoshi Nakano, Kizashi Yamaguchi. Electron-correlation dynamics of a one-dimensional H2 model in a quantized photon. Chem. Phys. Lett. 1998, 317(1-2):103-108
[141] Moyal J. Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc.1949, 45-99
[142] Kano Yutaka. A New Phase-Space Distribution Function in the Statistical Theory of the Electromagnetic Field. J.Math.Phys. 1965,6(12):1913-1915
[143] Klauder J, McKenna J, Currie D. On "Diagonal" coherent-state representations for quantum-mechanical density matrices. J. Math. Phys. 1965,6(5):734-739
[144] Mehta C L, Sudarshan E C G Relation between Quantum and Semiclassical Description of Optical Coherence.Phys. Rev. 1965,138(1B):B274-B280
[145] Cohen L. Generalized Phase-Space Distribution Functions. J.Math.Phys. 1966,7(5): 781-786
[146] Agarwal G S, Wolf E. Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. ?. A generalized wick theorem and multitime mapping. Phys. Rev. D,1970,2(10):2206-2225
[147] Carruthers P, Zachariasen F. Relativistic quantum transport theory approach to multiparticle production. Phys. Rev.D,1976,13(4):950-960
[148] Hillery M, O'Connell R F, Scully M O, et al. Distribution functions in physics:Fundamentals, Phys. Rep. 1984,106(3): 121-167
[149] Cahill K E, Glauber R J. Ordered Expansions in Boson Amplitude Operators. Phys.Rev. 1969, 177 (5): 1857-1881.
[150] Kim M S, De Oliveira A M F, Knight.P.L. Properties of squeezed number states and squeezed thermal states. Phys.Rev. A, 1989,40 (5):2494-2503.
[151] Kim M S, Oliveriva de F A M, Knight P H. Photon number distribution for squeezed numbers states and squeezed thermal states. Opt.Coum. 1989,72 (1/2): 99-103
[152] Das P K, Ghosh A. Phase Changes in Nonlinear Processes in. Interacting Fock Space, Int. J. Mod. Phys. B. 2006, 20 (4):433-443
[153] Burnham D C, Weinberg D L. Observation of simultaneity in parametric production of photon pairs. Phys. Rev. Lett. 1970,25 (2):84-87
[154] Kwiat P, Mattle K, Weinfurter H, et al. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 1995,75 (24):4337-4341
[155] Schleich W, Walls D F, Wheeler J A. Area of overlap and interference in phase space versus Wigner pseudo-probabilities. Phys.Rev. A.1988,38 (3): 1177-1186
[156] Peina J, Bajer J. Origin of oscillations in photon distributions of squeezed states.Phys. Rev. A.1990,41(1):516-518
[157] Erdelyi A, Magnus A, Oberhettinger F, et al. Higher Transcendental Functions.New York: McGraw-Hill Book Company. 1953
[158] Schleich W, Horowicz R J, Varro S. Bifurcation in the phase probability distribution of a highly squeezed state. Phys. Rev. A.1989,40 (12): 7405-7408
[159] Musslimani Z H, Ben-Aryeh Y. Quantum Phase Distribution of Thermal Phase-Squeezed State. Phys. Rev. A.1998, 57(2):1451-1453
[160] Pregnell K L,Pegg D T. Binomial states and the phase distribution measurement of weak optical fields. Phys. Rev. A. 2003, 67(6):063814-063822
[161] Pregnell K L. Single-shot measurement of quantum optical phasePhys. Rev. Lett.2002, 89(17):173601-173605
[162] Pregnell K L. Measuring the elements of the optical density matrix. Phys. Rev. A.2002,66(1):013810-013815
[163] Das P K. Quasiprobability Distribution and Phase Distribution in Interacting Fock Space. Int. J. Theoret. Phys. 2002,41 (10):2013-2024
[164] Valverde C, Baseia B, Malbouisson J M C. On the Measurement of the Phase Distribution of Field States . Mod. Phys. Lett. B. 2002,16(19):701-709
[165] Smithey D T, Beck M, Cooper J, et al. Measurement of number-phase uncertainty relations of optical fields.Phys. Rev. A. 1993,48 (4):3159-3167
[166] Pegg D T, Barnett S M. Phase properties of the quantized single-mode electro-magnetic field.Phys. Rev. A.1989,39 (4): 1665-1675
[167] Tanas R, Miranowicz A, Gantsog Ts. Progress in Optics, Ed by Wolf E.Amsterdam: Elsevier. 1996
[168] Cahill K E, Glauber R J. Density operators and quasiprobability distributions.Phys. Rev. 1969,177(5):1882-1902
[169] Lee C T. Measure of the nonclassicality of nonclassical states. Phys. Rev. A. 1991,44 (5): R2775-R2778
[170] Lee C T. Theorem on nonclassical states. Phys. Rev. A 1995, 52(4):3374-3376
[171] L(?)tkenhaus N, Barnett M. Nonclassical effects in phase space. Phys. Rev. A. 1995,51 (4): 3340-3342
[172] Hillery M. Nonclassical distance in quantum optics. Phys. Rev. A. 1987, 35(2):725-732
[173] Hillery M. Total noise and nonclassical states. Phys. Rev. A. 1989, 39(6):2994-3002
[174] Dodonov V V, Man'ko O V, Man'ko V I et al. Energy-sensitive and "Classical-like" Distances between Quantum States. Physica Scripta.1999, 59: 81-89
[175] Dodonov V V, Man'ko O V, Man'ko V I, et al. Hilbert-Schmidt distance and non-classicality of states in quantum optics. Journal of Modern Optics, 2000,47(4): 633-654
[176] Marian P, Marian T A, Scutaru H. Quantifying nonclassicality of one-mode Gaussian states of the radiation field. Phys. Rev. Lett. 2002,. 88(15):153601-153604
[177] Klimov A B, S(?)nchez-Soto L L, Yustas E C,et al. Distance-based degrees of polarization for a quantum field. Phys. Rev. A. 2005,72(3): 033813-033819
[178] Malbouisson J M C , Baseia B. On the measure of nonclassciality of field state.Phys. Scripta 2003, 67(2): 93-98
[179] Malbouisson J M C, Baseia B. Higher-generation Schrodinger cat states in cavity QED. J. Mod. opt. 1999,46(14): 2015-2041
[180] Malbouisson J M C. Comment on nonclassical depth of the phase state. Physics Lett. A. 2001,286(6):405-406
[181] Francis NC Paraan, Mikhail P Solon, Esguerra J P. Brownian motion of a charged particle driven internally by correlated noise. Phys. Rev. E.2008, 77(2):022101-022105
[182] Bao J D, Zhou Y Z. Ballistic diffusion induced by a thermal broadband noise. Phys.Rev. Lett. 2003, 91(13):138104-138107
[183] Cao L, Wu D J, Lin L. First-order-like transition for colored saturation models of dye lasers: Effects of quantum noise. Phys. Rev. A. 1994,49(1):506-516
[184] Ford G W, Lewis J T, O'Connell R F. The Quantum Langevin Equation. Phys. Rev.A. 1988, 37(11):4419-4428
[185] Ford G W, Lewis J T, O'Connell R F. Quantum Oscillator in a Blackbody Radiation Field. Phys. Rev. Lett. 1985, 55(21):2273-2276
[186] Ford G W, Lewis J T, O'Connell R F. Stark shifts due to black-body radiation. J.Phys. B: At. Mol. Phys. 1986,19(2): LA1-L46
[187] Gallagher T F, Cooke W E. Interactions of Blackbody Radiation with Atoms. Phys.Rev .Lett. 1979, 42(13): 835-839
[188] Ford G W, O'Connell R F. Quantum thermodynamic functions for an oscillator coupled to a heat bath. Phys. Rev. B, 2007, 75(13): 134301-134307
[189] Hollberg L, Hall J L. Measurement of the Shift of Rydberg Energy Levels by Blackbody Radiation. Phys. Rev. Lett. 1984, 53(3):230-233
[190] Caldeira A O, Leggett A J. Quantum tunneling in a dissipative system. Ann. Phys.1983,149(1): 374-456
[191] Ford G W, Lewis J T, O'Connell R F. Quantum Oscillator in a Blackbody Radiation Field ?. Direct Calculation of the Energy using the Fluctuation-Dissipation Theorem. Ann. Phys. 1988,185(2):270-283
[192] Li X L, Ford G W, O'Connell R F. Charged oscillator in a heat bath in the presence of a magnetic field. Phys. Rev. A 1990,42(8):4519-4527
[193] Ford G W, Lewis J T, O'Connell R F. Dissipative Quantum Tunneling: Quantum Langevin Equation Approach. Phys. Lett. A. 1988,128(1-2): 29-34
[194] Ford G W, Lewis J T, O'Connell R F. Stark Shifts Due to Blackbody Radiation. J.Phys.B. 1986,19(2):L41-L46
[195] Ford G W, Lewis J T, O'Connell R F. On the Thermodynamics of Quantum -Electrody-namic Frequency Shifts. J. Phys. B. 1987,20(5):899-906
[2] Planck M. Elementary quanta of matter and electricity, Annalen der Physik, 1901,4 (3):564-566
[3] Landau L D, Lifshitz E M. Statistical physics. Third Edition. Oxford: Reed Educational, 1980,183-186
[4] Scully M O, Zubairy M S. Quantum optics, First Edition. The United Kingdom:Cambridge University, 1997
[5] Cheng Z. Quantum effects of thermal radiation in a Kerr nonlinear blackbody. J. Opt.Soc. Am. B, 2002,19(7): 1692-1705
[6] Cheng Z. Radiation properties of a Kerr nonlinear blackbody. Phys. Rev. A, 2005,71(3),033808:1-9
[7] Cheng Z. Squeezed thermal radiation state in a Kerr-nonlinear black body. Phys. Lett.A, 2001,291(1): 4-10
[8] Cheng Z. Spectral energy density and radiation pressure in a Kerr nonlinear blackbody. Phys. Lett. A, 2004, 331(3): 170-174
[9] Cheng Z. Thermodynamics of phase transitions of a Kerr nonlinear blackbody. Chin.Phys.Lett.2008,25(9): 3264-3267
[10]Cheng Z. Photonic superfluidity in a Kerr nonlinear blackbody. Chin. Phys. Lett.,2005, 22(4): 880-883
[11]Heitler M. The quantum theory of radiation. New York: Oxford Clarendon Press Educational, 1954
[12]Loudon L. The quantum theory of light. New York: Oxford University Press, 1973
[13] Walls D F, Milburn G J. Quantum optics, First Edition. Berlin: Springer-Verlag, 1994, 91-98
[14]彭金生，李高翔．近现代量子光学导论(第一版)．北京：科学出版社，1996．
[15]Barnett S M, Radmore P M. Method in theoretical quantum optics, First Edition. Oxford:Clarendon Press. 1997
[16]Mandel L, Wolf E. Optical coherence and quantum optics. London: Cambridge University Press. 1995
[17] Shen Y R. Quantum statistics of nonlinear optics. Phys.Rev. 1967,155(3):921-931
[18]Walls D F. Squeezed states of light. Nature,1983,306(11):141-146
[19]Mandel L. Sub-Poissonian photon statistics in resonance fluorescence. Opt.Lett. 1979, 4(7):205-207
[20]Paul H. Photon antibunching. Rev.Mod. Phys.1982,54(4):1061-1102
[21] Einstein A. Zur Elektrodynamik bewegter Krrper. Annalen der Physik, 1905, 17(10): 891-921
[22]Compton A H. A quantum theory of the scattering of X-rays by light elements. Phys. Rev. 1923, 21(5): 483-502
[23] Brown R H,Twiss R Q. Correlation between photons in two coherent beams of light. Nature. 1956,177(4497):27-29
[24]Brown R H, Twiss R Q. A test of a new type of stellar interferometer on sirius. Nature. 178(10): 1046-1048
[25]Haken H. Laser light emission from interacting chaotic fields Z.Phys.1975, 265(2): 105-118
[26]Haken H. Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems. Rev.Mod.Phys. 1975,47(1):67-121
[27] Risken H, Vollmer H D. On the application of truncated generalized Fokker-Planck equations Z. Phys.B. 1979,35(3):313-315
[28]Benkert C, Scully M O, Bergou Jet al, Role of pumping statistics in laser dynamics: Quantum Langevin approach. Phys. Rev.A,1990,41(5):2756-2765
[29] Harris S E, Field J E, Imamoglu. Nonlinear optical processes using electro-magnetically induced transparency. Phys.Rev.Lett. 1990,64(10): 1107-1110
[30] Yee T K, Gustafson T K. Diagrammatic analysis of the density operator for nonlinear optical calculations: Pulsed and cw responses. Phys. Rev. A. 1978,18(4): 1597-1617
[31]Boyd G F, Kleiman D A. Parametric interaction of focused. Gaussian light beams.J.Appl.Phys.l968,39(8):3597-3670
[32] Fork R L, Brito C H, Becker P C et al. Compression of optical pulses to six femtoseconds by using cubic phase compensation. Optics Letters. 1987, 12(7): 483-485
[33]Stenholem S, Gas laser with saturable absorber. I .Single mode characteristics. Phys.Rev. A.1973,8(5):2695-2711
[34] Knight P L, Milonni P W. The Rabi frequency in optical spectra. Phys.Rep. 1980,66(2):21-107
[35]Haroche S, Kleppner D, Cavity quantum electrodynamics. Physics Today. 1989,42(1): 24-30
[36] Shore B W, Knight P L. The Jaynes-Cummings model. J. Mod. Opt. 1993, 40(7):1195-1238
[37]Frank O G Application of a one-dimensional classical model of atom-oscillator scattering to atom-surface accommodation coefficient theory. Surface Science. 1979,60(1):45-64
[38] Lamb W E, Retherford R C. Fine structure of the hydrogen atom by microwave Method. Phys. Rev. 1947,72(3):241-243
[39]Schawlow A L, Townes C H. Infrared and optical masers. Phys. Rev.1958,112(6):1940-1949
[40]Fleischhauer M, Schleich W. Revivals made simple: Poisson summation. Phys. Rev.A.1992,47(5):4258-4269
[41] Sherman B, Kurizki G Preparation and detection of macroscopic quantum superpositions by two-photon field-atom interactions. Phys.Rev. A. 1992,45(11): R7674-R7677
[42] Eberly J H, Wodkiewicz K. The time-dependent physical spectrum of light. J. Opt. Soc. Am. 1977, 67(9): 1252-1261
[43]Jaynes E T, Cummings F W. Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proc.IEEE, 1963, 51 (1): 89-109
[44]Alsing P, Cardimona D A, Suppression of fluorescence in a lossless cavity. Phys. Rev.A, 1992,45(7): 1793-1796
[45]Brune M, Raimond J M, Haroehe S; Theory of the Rydberg-atom two-Phonon micromaser. Phys.Rev.A, 1987,35(1): 154-157
[46] Stolen H R, AshkiA.Optical Kerr effect in glass waveguide. Applied Physics Letters. 1973, 22(6):294-296
[47]石顺祥，陈果夫，赵卫．非线性光学．西安：西安电子科技大学出版社．2003
[48]方俊鑫，陈栋．固体物理学．第六版，上海：上海科技出版社．1987：196-200
[49]Korolev A V, Liberman M A. Polaritons and a built-in condensate of excitons in a high magnetic field slid state communications. 1996, 100(2):89-94
[50]Ichimiya M, Watanabe M, Hayashi T. Polarization dependence of biexciton formation in CuC1 studied by pump-probe spectroscopy. Journal of Luminescence. 2000, 87-89:272-274
[51]Baumberg J J, Savvidis P G, Lagoudakis P G; et al. Polariton traps in semiconductor microcavities, Physica E. 2002,13(2-4):385-389
[52]Lu Y Q, Zhu Y Y, Chen Y F et al. Optical properties of an ionic-type phononic crustal. Science. 1999,284(5421): 1822-1824
[53]Cheng Z. Pairing effect of photons in nonlinear polar crystals. Phys. Rev. Lett. 1991, 67(20): 2788-2791
[54] Cheng Z. Photonic superguiding state in nonlinear polar crystals. Phys. Rev. A, 1995, 51(1): 675-691
[55]Cheng Z, Xu Hong-Tao. Spectral energy density in a Kerr nonlinear blackbody.Commun. Theor. Phys., 2004,42(6): 884-886
[56] Yin M, Cheng Z. Inhibited spontaneous emission in a Kerr nonlinear blackbody.Phys. Rev. A. 2008,78(6), 063829:1-7
[57]Wan J Y, Cheng Z. Study of nonpolaritons in a Kerr nonlinear optical resonator.Commun. Theor. Phys.2006, 45(6):1107-1111
[58] Cheng Z. Unified quantum field theory of Raman scattering of light in piezoelectric crystals. Acta Physica Sinica. 2005,54(11): 5435-5444
[59]Loudon R, Knight P L. Squeezed light. J.Mod.Opt.,1987,34(6/7):709-759
[60]Hillery M. Sum and difference squeezing of the electromagnetic field. Phys. Rev. A,1989,40(6):3147-3155
[61]Yuen H P. Two-Photon coherent states of the radiation field. Phys.Rev.A,1976, 13(6):222-2213
[62]Glaube R J. coherent and incoherent states of the radiation field. Phys. Rev. 1963,131(6):2766-2788
[63]Kimble H J, Dgenais M, Mandel. Photon antibunching in resonance flurescence.Phys. Rev.Lett.l977,39(11):691-695
[64]Stoler D. Equivalence classes of minimum uncertainty packets. Phys.Rev. D, 1970,1(12):3217-3219
[65]Hong C K, Mandel L. Generation of higher-order squeezing of quantum electro-magnetic fields. Phys.Rev.A,1985,32(2):971-982
[66]Hillery M. Amplitude-squared squeezing of the electromagnetic field. Phys. Rev. A,1987,36(8):379-3802
[67] Zhang Z M, Xu L, Chai J L et al. A new kind of higher-order squeezing of radiation field. Phys.Lett. A,1990,150(1):27-30
[68]Bergou J A, Hillery M, Yu Daoqi. Minimum uncertainty states for amplitude-squared squeezing: Hermite polynomial states. Phys.Rev.A,1991,43(1):515-520
[69]董传华．高阶压缩的另一种定义．光学学报．1996，16(11)：1543-1548
[70]Kozhekin A, Kurizki G, Sherma B. Quantum-state control by a single conditional measurement: The periodically switched Jaynes-Cummings model. Phys.Rev.A,1996, 54(4): 3535-3538
[71]Greenberger D M, Horne M, Zeilinger A. In Bell's theorem, Quanium theory, and conceptions of the universe. Edited by M.Kafatos. Kluwer: Dordreeht. 1989
[72] Greenberger D M, Horne M, Zeilinger A. Bell's theorem without inequalities. Am. J.Phys. 1990, 58(12):1131-1143
[73]Ficek Z, Tonas R, Kielich S.Amplitude-squared squeezing in two-atom resonance fluorescence. Opt.Coum.,1988,69(1):20-24
[74]Hu X, Nori F. Phonon squeezed states:quantum noise reduction in solids. Physica B: Condensed Matter, 1999,263-264:16-29
[75] Hu X, Nori F. Phonon squeezed states generated by second-order Raman scattering. Phys. Rev. Lett. 1997, 79(23):4605-4608
[76]Nguyen Ba An. Higher-order amplitude squeezing of photons propa-gating through a semiconductor. Phys. Rev. A, 1997, 234(1):45-52
[77] Drummond P D, Shelby R. M, Friberg S R, et al. Quantum Solitons in Optical Fibers. Nature. 1993,365(23): 307-313
[78]Deutsch I H, Chiao R Y, Garrison J C. Two-photon bound states: The photon bullet in dispersive self-focusing media. Phys. Rev. A, 1993, 47(4): 3330-3336
[79] Hasegawa A. Optical Solitons in Fibers, 2nd ed. Berlin: Springer-Verlag. 1990
[80]Lai Y, Haus H A. Quantum theory of solitons in optical fibers. I. Time- dependent Hartree approximation. Phys. Rev. A, 1989, 40(2): 844-853
[81] Cheng Z, Kurizki G.Theory of one-dimensional quantum gap solitons. Phys. Rev. A, 1996, 54(4):3576-3591
[82]Vitali D, Fortunato M, Tombesi P. Complete Quantum Teleportation with a Kerr Nonlinearity. Phys. Rev. Lett., 2000, 85(2): 445-448
[83]Olsen M K, Kruglov V I, Collett M.J. Effects of χ(3) nonlinearities in second-harmonic generation. Phys. Rev. A, 2001, 63(3): 033801-033809
[84]Kenichi Hirosawa, Hiroto Furumochi, Atsushi Tada et al. Photon Number Squeezing of Ultrabroadband Laser Pulses Generated by Microstructure Fibers. Phys. Rev.Lett.,2005, 94(20): 203601/1-4
[85]Buzek V, Jex I. Dynamics of a two-level atom in a Kerr-like medium. Opt. Commun.1990, 78(6):425-435
[86]Buzek V, Jex I. Emission spectra of a two-level atom in a Kerr-like medium.J.Mod.Opt. 1991, 38(5):987-996
[87] Brown R. A brief account of microscopical observations made in the months of June.1928,4(2):161-173
[88] Einstein A. On the motion-required by the molecular kinetic theory of heat of small particles suspended in a stationary liquid. Ann.der Phys. 1905,17(8): 549-560
[89] Einstein A. Investigations on the theory of the Brownian movement. New York:Dover. 1956.
[90]Perrin J B. Molecular agitation and the brownian movement. Comptes Rendus. 1908,146:967-970
[91]Perrin J B. Brownian Movement and Molecular Reality. London : Taylor and Francis.1910,1-93
[92]Langevin M P. Sur la theorie du mouvement brownien. Comptes Rendus Acad. Sci.,Paris. 1908,146:530-533
[93]Chandrasekhar S. Stochastic Problems in Physics and Astronomy. Rev. Mod.Phys.1943, 15(1):1-89
[94]Nyquist R. Thermal agitation of electric charge in conductors. Phys. Rev. 1928,
[95]Callen H B, Welton T A. Irreversibility and generalized noise. Phys. Rev. 1951,83(1):34-40
[96]Zwanzig R. Nonlinear Generalized Langevin Equations. J. Stat. Phys. 1973, 9(3):215-220
[97]Senitzky I R. Dissipation in Quantum Mechanics. The Harmonic Oscillator. Phys.Rev. 1960,119(2):670-679
[98] Lax M. Quantum Noise. ?. Quantum Theory of Noise Sources. Phys. Rev. 1966,145(1):110-129
[99] Ford G W, Kac M, Mazur P. Statistical Mechanics of Assemblies of Coupled Oscillators. J. Math. Phys. 1965, 6(4): 504-515
[100] Mori H. Transport, Collective Motion, and Brownian Motion. Prog. Theor. Phys.1965, 33(3): 423-455
[101] Benguria R, Kac M. Quantum Langevin Equation. Phys.Rev. Lett.1981, 46(1): 1-4
[102] Ford G W, Kac M. On the Quantum Langevin Equation. J. Stat. Phys. 1987,46(5-6): 803-810
[103] Kubo R. The fluctuation-dissipation theorem. Rep. Prog. Phys. 1966, 29(1):255-284
[104] Kubo R. Brownian Motion and Nonequilibrium Statistical Mechanics. Science 1986,233(4761):330-334
[105] Caldeira A O, Leggett A J. Quantum Tunneling in a Dissipative System. Ann. Phys.1983, 149(1):374-456
[106] Grabert H, Schramm H, Ingold G L. Quantum Brownian motion: The functional integral approach. Phys. Rep. 1988, 168(3): 115-207
[107] Weiss U. Quantum Dissipative Systems. Singapore :World Scientific. 1993
[108] Hayes W, Loudon R. Scattering of Light by Crystals. New York: Wiley. 1978 [109] Shen Y R, The Principles of Nonlinear Optics. New York: Wiley. 1984 [110] Sal eh B E, Teach M C. Can the channel capacity of a light-wave communication
system be increased by the use of photon-number-squeezed light. Phys. Rev. lett. 1987, 58 (25): 2656-2659
[111] Pe(?)ina J, Perinova V, Kodiysek J, On the relations of antibunching, sub-Poissonian statistics and squeezing. Opt. Commum. 1984,49 (3): 210-214
[112] Kielich S, Tanas R, Zawodny R. Squeezing in the third-harmonic field generated by self-squeezed light. J. Opt. Soc. Am. B.1987,4 (10):1627-1632
[113] Reid M D, Walls D F. Generation of squeezed states via degenerate four-wave mixing. Phys. Rev. A 1985,31(3): 1622-1635
[114] Pe(?)ina J, Perinova V, Sibilia C, et al. Quantum statistics of four-wave mixing. Opt.Commum. 1984, 49 (4):285-289.
[115] Perinov(?) V. Pe(?)ina J. Quantum statistics of quadratic optical parametric processes with intense coherent or stochastic pumping. Opt. Acta. 1981,28 (6): 769-793
[116] Hillery M. Amplitude squeezing of the electromagnetic field. Opt. Commum. 1987,62 (2): 135-138
[117] Gerry C. Generation of optical macroscopic quantum superposition states via state reduction with a Mach-Zehnder interferometer containing a Kerr medium. Phys.Rev. A. 1999,59 (5):4095-4098
[118] Mahran M H. Squeezing and higher-order squeezing in the three-level, two-mode system .Phys. Rev. A .1990,42 (7):4199-4209
[119] Xie R H, Smith Jr V H. Generation of higher-order atomic dipole squeezing in a high-Q micromaser cavity. (?) Discussion of the nondegenerate two-photon Jaynes-Cummings model. Physica A .2002, 307 (1-2):207-220
[120] Lynch R, Mavromatis H A, Puri'S criterion and higher-order squeezing. Phys. Rev.A. 1995, 52(1):55-57
[121] Giri D K, Gupta P S, The Squeezing of radiation in four-wave mixing processes. J.Opt. B: Quantum Semiclass. Opt. 2004, 6 (1): 91-96
[122] Hu X, Nori F. Phonon squeezed states generated by second-order Raman scattering. Phys. Rev. Lett. 1997,79 (23):4605-4608
[123] Chizhov A V, Haus J W, Yeong K C. Higher-order squeezing in a boson-coupled two-mode system. Phys. Rev. A. 1995,52 (2): 1698-1703
[124] Onnes H K. The resistance of pure mercury at helium temperatures. Commum. Phys. Lab. Univ. Leiden.1911, 120b:17-19
[125] Kapitza P L. Superfluidity observed. Nature. 1932, 141(3558):74-75
[126] Anderson M H, Ensher J R, Matthews M R, et al. Observation of. Bose-Einstein Condensation in a Dilute Atomic Vapor. Scinence. 1995, 269(5221):198-201
[127] Greiner M, MandelO, Essling T, et al. Quantum phase transition from a superfluid to a mott insulator in a gas ofultracold atoms. Nature. 2002, 415: 39-44
[128] Yoichiro Nambu. Quasi-Particles and gauge invariance in the theory of superconductivity. Phys. Rev. 1960, 117(3), 648-663
[129] 李政道，粒子物理和场论简引，北京：科学出版社，1984
[130] Cooper L N. Bound Electron Pairs in a Degenerate Fermi Gas. Phys. Rev. 1956, 104 (4):1189-1190
[131] Bardeen J, Cooper L N, Schrieffer J R. Microscopic Theory of Super-conductivity. Phys. Rev. 1957,108 (1) 1175-1204
[132] Sudarshan E C G. Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys.Rev. Lett. 1963,10(7):277-279
[133] Wigner E. On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev. 1932, 40(5):749-759
[134] Avelar A T, Baseia B, de Almeida N G. Complementary coherent state for measuring the Q-function: generation and properties. J. Opt. B: Quantum Semiclass. Opt. 2004, 6(4): 41-45
[135] Marian P. Higher-order squeezing and photon statistics for squeezed thermal states. Phys.Rev. A. 1992,45 (3):2044-2051
[136] Chizhov A V, Gantsog Ts, Murzakhmetov B K. Phase distributions of squeezed number states and squeezed thermal states. Quantum Opt. 1993, 5 (2):85-95
[137] Avelar A T, de Souza L A, da Rocha Filho T M, et al. Generation of superposed phase states via Raman interaction . J. Opt. B: Quantum Semiclass. Opt. 2004,6(10): 383-386
[138] Albano L, Mundarain D F, Stephany J. et al .On the squeezed number states and their phase space representations. L Albano. J. Opt. B: Quantum Semiclass. Opt.2002, 4(5):352-357
[139] Vourdas A. Optical signals with thermal noise. Phys.Rev. A 1989, 39 (1):206- 213
[140] Masayoshi Nakano, Kizashi Yamaguchi. Electron-correlation dynamics of a one-dimensional H2 model in a quantized photon. Chem. Phys. Lett. 1998, 317(1-2):103-108
[141] Moyal J. Quantum mechanics as a statistical theory. Proc. Cambridge Phil. Soc.1949, 45-99
[142] Kano Yutaka. A New Phase-Space Distribution Function in the Statistical Theory of the Electromagnetic Field. J.Math.Phys. 1965,6(12):1913-1915
[143] Klauder J, McKenna J, Currie D. On "Diagonal" coherent-state representations for quantum-mechanical density matrices. J. Math. Phys. 1965,6(5):734-739
[144] Mehta C L, Sudarshan E C G Relation between Quantum and Semiclassical Description of Optical Coherence.Phys. Rev. 1965,138(1B):B274-B280
[145] Cohen L. Generalized Phase-Space Distribution Functions. J.Math.Phys. 1966,7(5): 781-786
[146] Agarwal G S, Wolf E. Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. ?. A generalized wick theorem and multitime mapping. Phys. Rev. D,1970,2(10):2206-2225
[147] Carruthers P, Zachariasen F. Relativistic quantum transport theory approach to multiparticle production. Phys. Rev.D,1976,13(4):950-960
[148] Hillery M, O'Connell R F, Scully M O, et al. Distribution functions in physics:Fundamentals, Phys. Rep. 1984,106(3): 121-167
[149] Cahill K E, Glauber R J. Ordered Expansions in Boson Amplitude Operators. Phys.Rev. 1969, 177 (5): 1857-1881.
[150] Kim M S, De Oliveira A M F, Knight.P.L. Properties of squeezed number states and squeezed thermal states. Phys.Rev. A, 1989,40 (5):2494-2503.
[151] Kim M S, Oliveriva de F A M, Knight P H. Photon number distribution for squeezed numbers states and squeezed thermal states. Opt.Coum. 1989,72 (1/2): 99-103
[152] Das P K, Ghosh A. Phase Changes in Nonlinear Processes in. Interacting Fock Space, Int. J. Mod. Phys. B. 2006, 20 (4):433-443
[153] Burnham D C, Weinberg D L. Observation of simultaneity in parametric production of photon pairs. Phys. Rev. Lett. 1970,25 (2):84-87
[154] Kwiat P, Mattle K, Weinfurter H, et al. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 1995,75 (24):4337-4341
[155] Schleich W, Walls D F, Wheeler J A. Area of overlap and interference in phase space versus Wigner pseudo-probabilities. Phys.Rev. A.1988,38 (3): 1177-1186
[156] Peina J, Bajer J. Origin of oscillations in photon distributions of squeezed states.Phys. Rev. A.1990,41(1):516-518
[157] Erdelyi A, Magnus A, Oberhettinger F, et al. Higher Transcendental Functions.New York: McGraw-Hill Book Company. 1953
[158] Schleich W, Horowicz R J, Varro S. Bifurcation in the phase probability distribution of a highly squeezed state. Phys. Rev. A.1989,40 (12): 7405-7408
[159] Musslimani Z H, Ben-Aryeh Y. Quantum Phase Distribution of Thermal Phase-Squeezed State. Phys. Rev. A.1998, 57(2):1451-1453
[160] Pregnell K L,Pegg D T. Binomial states and the phase distribution measurement of weak optical fields. Phys. Rev. A. 2003, 67(6):063814-063822
[161] Pregnell K L. Single-shot measurement of quantum optical phasePhys. Rev. Lett.2002, 89(17):173601-173605
[162] Pregnell K L. Measuring the elements of the optical density matrix. Phys. Rev. A.2002,66(1):013810-013815
[163] Das P K. Quasiprobability Distribution and Phase Distribution in Interacting Fock Space. Int. J. Theoret. Phys. 2002,41 (10):2013-2024
[164] Valverde C, Baseia B, Malbouisson J M C. On the Measurement of the Phase Distribution of Field States . Mod. Phys. Lett. B. 2002,16(19):701-709
[165] Smithey D T, Beck M, Cooper J, et al. Measurement of number-phase uncertainty relations of optical fields.Phys. Rev. A. 1993,48 (4):3159-3167
[166] Pegg D T, Barnett S M. Phase properties of the quantized single-mode electro-magnetic field.Phys. Rev. A.1989,39 (4): 1665-1675
[167] Tanas R, Miranowicz A, Gantsog Ts. Progress in Optics, Ed by Wolf E.Amsterdam: Elsevier. 1996
[168] Cahill K E, Glauber R J. Density operators and quasiprobability distributions.Phys. Rev. 1969,177(5):1882-1902
[169] Lee C T. Measure of the nonclassicality of nonclassical states. Phys. Rev. A. 1991,44 (5): R2775-R2778
[170] Lee C T. Theorem on nonclassical states. Phys. Rev. A 1995, 52(4):3374-3376
[171] L(?)tkenhaus N, Barnett M. Nonclassical effects in phase space. Phys. Rev. A. 1995,51 (4): 3340-3342
[172] Hillery M. Nonclassical distance in quantum optics. Phys. Rev. A. 1987, 35(2):725-732
[173] Hillery M. Total noise and nonclassical states. Phys. Rev. A. 1989, 39(6):2994-3002
[174] Dodonov V V, Man'ko O V, Man'ko V I et al. Energy-sensitive and "Classical-like" Distances between Quantum States. Physica Scripta.1999, 59: 81-89
[175] Dodonov V V, Man'ko O V, Man'ko V I, et al. Hilbert-Schmidt distance and non-classicality of states in quantum optics. Journal of Modern Optics, 2000,47(4): 633-654
[176] Marian P, Marian T A, Scutaru H. Quantifying nonclassicality of one-mode Gaussian states of the radiation field. Phys. Rev. Lett. 2002,. 88(15):153601-153604
[177] Klimov A B, S(?)nchez-Soto L L, Yustas E C,et al. Distance-based degrees of polarization for a quantum field. Phys. Rev. A. 2005,72(3): 033813-033819
[178] Malbouisson J M C , Baseia B. On the measure of nonclassciality of field state.Phys. Scripta 2003, 67(2): 93-98
[179] Malbouisson J M C, Baseia B. Higher-generation Schrodinger cat states in cavity QED. J. Mod. opt. 1999,46(14): 2015-2041
[180] Malbouisson J M C. Comment on nonclassical depth of the phase state. Physics Lett. A. 2001,286(6):405-406
[181] Francis NC Paraan, Mikhail P Solon, Esguerra J P. Brownian motion of a charged particle driven internally by correlated noise. Phys. Rev. E.2008, 77(2):022101-022105
[182] Bao J D, Zhou Y Z. Ballistic diffusion induced by a thermal broadband noise. Phys.Rev. Lett. 2003, 91(13):138104-138107
[183] Cao L, Wu D J, Lin L. First-order-like transition for colored saturation models of dye lasers: Effects of quantum noise. Phys. Rev. A. 1994,49(1):506-516
[184] Ford G W, Lewis J T, O'Connell R F. The Quantum Langevin Equation. Phys. Rev.A. 1988, 37(11):4419-4428
[185] Ford G W, Lewis J T, O'Connell R F. Quantum Oscillator in a Blackbody Radiation Field. Phys. Rev. Lett. 1985, 55(21):2273-2276
[186] Ford G W, Lewis J T, O'Connell R F. Stark shifts due to black-body radiation. J.Phys. B: At. Mol. Phys. 1986,19(2): LA1-L46
[187] Gallagher T F, Cooke W E. Interactions of Blackbody Radiation with Atoms. Phys.Rev .Lett. 1979, 42(13): 835-839
[188] Ford G W, O'Connell R F. Quantum thermodynamic functions for an oscillator coupled to a heat bath. Phys. Rev. B, 2007, 75(13): 134301-134307
[189] Hollberg L, Hall J L. Measurement of the Shift of Rydberg Energy Levels by Blackbody Radiation. Phys. Rev. Lett. 1984, 53(3):230-233
[190] Caldeira A O, Leggett A J. Quantum tunneling in a dissipative system. Ann. Phys.1983,149(1): 374-456
[191] Ford G W, Lewis J T, O'Connell R F. Quantum Oscillator in a Blackbody Radiation Field ?. Direct Calculation of the Energy using the Fluctuation-Dissipation Theorem. Ann. Phys. 1988,185(2):270-283
[192] Li X L, Ford G W, O'Connell R F. Charged oscillator in a heat bath in the presence of a magnetic field. Phys. Rev. A 1990,42(8):4519-4527
[193] Ford G W, Lewis J T, O'Connell R F. Dissipative Quantum Tunneling: Quantum Langevin Equation Approach. Phys. Lett. A. 1988,128(1-2): 29-34
[194] Ford G W, Lewis J T, O'Connell R F. Stark Shifts Due to Blackbody Radiation. J.Phys.B. 1986,19(2):L41-L46
[195] Ford G W, Lewis J T, O'Connell R F. On the Thermodynamics of Quantum -Electrody-namic Frequency Shifts. J. Phys. B. 1987,20(5):899-906