玻色—爱因斯坦凝聚体中的量子涨落
摘要
激光制冷与蒸发制冷技术的结合促成了宏观量子现象——玻色-爱因斯坦凝聚体(BEC)—在稀薄原子气体中的实现。随着研究工作的进一步深入,人们的研究兴趣逐步深入到玻色-爱因斯坦凝聚与相关方向的交叉方面,如BCS-BEC结、BECs中的量子纠缠与量子信息、BECs的激发与宇宙膨胀的相似性研究等。本论文主要研究了约束势场下玻色-爱因斯坦凝聚体中的量子涨落及其效应,得出了一些相关的结果,并与宇宙学的有关方法结果作了简单的对比。
全文共四章,第一章前言中简单介绍玻色-爱因斯坦凝聚的形成、描述及其研究进展,以及本论文的研究目的和具体内容。第二章,主要采用粒子数守恒平均场近似方法研究了BECs中的量子涨落及其反作用力,并与有效相互作用方法做一简单比较。最后,就一维情况详细讨论了涨落对凝聚体半径的影响。第三章中采用了同样的近似讨论了随时间变化谐约束势下二维BECs中的量子涨落。通过引入了一对自共轭算符将涨落项满足的方程线性化,进而得出密度和相涨落所满足的方程。并就静态凝聚态情形,得出本征模方程和激发模频率的依赖关系。最后,在Thomas-Fermi近似下讨论了外势或相互作用随时间变化对量子涨落的影响,发现,可以控制外势或相互作用变化来打破绝热环境,从而达到抑制甚至冻结量子涨落的目的。第四章,结论与展望。
The combination of laser cooling and evaporative cooling led to the formation of a kind of macroscopic quantum phenomenon—Bose-Einstein condensation—in dilute atomic vapors in experiment. Along with research work's thorough, people's research interests have extended step-by-step to intersection of BEC with relevant fields, such as BCS-BEC Crossover, Quantum Entanglement and Quantum Information in BECs, similarity between elementary excitation in BECs and cosmic inflation and so on. This thesis studies the quantum fluctuations and related effects of BECsin trap potential, and gets some corresponding results. In addition, we make some simple comparison with effective-action method.
This thesis comprises four parts. we first present the formation, discription and research progress of BEC.,then introduce the background of this thesis and the major content of this thesis in the first chapter. In the following, adopting particle-number-conserving mean-field ansatz, we put emphasis on the study of the Quantum Fluctuations and Quantum Back-reaction of quantum fluid in BECs in chapter 2, in addition, we make a simple comparison with the effective-action method. At last, we take into account the quasi—1D case.,and calculate carefully the influence on the condensates radius from quantum fluctuations.
In chapter 3, we discuss the quantum fluctuations in time-dependent harmonic potential, adopting the same ansatz. By introducing a pair of self-conjugate operators, we may linearize the equation of fluctuation term., and get the equations of density and phase fluctuations. In addition, we get the eigenmode equation and frequency relation of excitation modes on initial static BECs. At last, in Thomas-Fermi approximation we discuss the influence on quantum fluctuation brought by the variation of external potential or interaction intensity. We find that one can regulate the external potential or interaction intensity to control adiabaticity so as to restrain or freeze quantum fluctuation.
In the last chapter of the thesis, we give a simple summary of the above-mentioned works and the outlook.
全文共四章,第一章前言中简单介绍玻色-爱因斯坦凝聚的形成、描述及其研究进展,以及本论文的研究目的和具体内容。第二章,主要采用粒子数守恒平均场近似方法研究了BECs中的量子涨落及其反作用力,并与有效相互作用方法做一简单比较。最后,就一维情况详细讨论了涨落对凝聚体半径的影响。第三章中采用了同样的近似讨论了随时间变化谐约束势下二维BECs中的量子涨落。通过引入了一对自共轭算符将涨落项满足的方程线性化,进而得出密度和相涨落所满足的方程。并就静态凝聚态情形,得出本征模方程和激发模频率的依赖关系。最后,在Thomas-Fermi近似下讨论了外势或相互作用随时间变化对量子涨落的影响,发现,可以控制外势或相互作用变化来打破绝热环境,从而达到抑制甚至冻结量子涨落的目的。第四章,结论与展望。
The combination of laser cooling and evaporative cooling led to the formation of a kind of macroscopic quantum phenomenon—Bose-Einstein condensation—in dilute atomic vapors in experiment. Along with research work's thorough, people's research interests have extended step-by-step to intersection of BEC with relevant fields, such as BCS-BEC Crossover, Quantum Entanglement and Quantum Information in BECs, similarity between elementary excitation in BECs and cosmic inflation and so on. This thesis studies the quantum fluctuations and related effects of BECsin trap potential, and gets some corresponding results. In addition, we make some simple comparison with effective-action method.
This thesis comprises four parts. we first present the formation, discription and research progress of BEC.,then introduce the background of this thesis and the major content of this thesis in the first chapter. In the following, adopting particle-number-conserving mean-field ansatz, we put emphasis on the study of the Quantum Fluctuations and Quantum Back-reaction of quantum fluid in BECs in chapter 2, in addition, we make a simple comparison with the effective-action method. At last, we take into account the quasi—1D case.,and calculate carefully the influence on the condensates radius from quantum fluctuations.
In chapter 3, we discuss the quantum fluctuations in time-dependent harmonic potential, adopting the same ansatz. By introducing a pair of self-conjugate operators, we may linearize the equation of fluctuation term., and get the equations of density and phase fluctuations. In addition, we get the eigenmode equation and frequency relation of excitation modes on initial static BECs. At last, in Thomas-Fermi approximation we discuss the influence on quantum fluctuation brought by the variation of external potential or interaction intensity. We find that one can regulate the external potential or interaction intensity to control adiabaticity so as to restrain or freeze quantum fluctuation.
In the last chapter of the thesis, we give a simple summary of the above-mentioned works and the outlook.
引文
[1]P.Kapitza, "Viscosity of liquid Helium below the λ-point", Nature 141,74(1938); J.F.Allen, and A.D.Misener, "Flow of liquid helium Ⅱ", Nature 141,75(1938).
[2]F.London, "The λ-phenomenon of liquid helium and the Bose-Einstein degeneracy", Nature 141,643(1938)
[3]M.H.Anderson, J.R.Ensher, M.R.Matthews, et al. "observation of Bose-Einstein condensation in a Dilute Atomic Vapor", Science 269,198-201(1995)
[4]汪志诚热力学统计物理[M],北京：高等教育出版社,2003
[5]C.J.Pethick and H. Smith, Bose-Einstein condensation in Dilute gases[M]. UK: Cambridge University Press,2002
[6]E.A.Cornell, " very cold indeed:the nanokelvin physics of Bose-Einstein condensation." J.Res.Natl.Inst.Stand.Technol.101,419(1996)
[7]E.A.Cornell and C.E.Wieman, "Nobel Lecture:Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. " Rev.Mod.Phys.74,875(2001)
[8]W.Ketterle, D.S.Durfee and D.M.Stamper-kurn, "Making, probing and understanding BEC", proc. Intern. school of physics-Enrico Fermi, M.Inguscio, S.Stringari and C.Wieman eds.(1999)
[9]C.C.Bradley, C.A.Sackett, and R.GHulet, "Bose-Einstein condensation of Lithium: observation of limited condensate number",Phys.Rev.Lett.78,985(1997)
[10]S.L.Cornish, N.R.Claussen, J.L.Roberts, et al. "Stable 85Rb Bose-Einstein condensates with widely tunable interactions", Phys.Rev.Lett.85,1795(2000)
[11]GModugno, GFerrari, G.Roati,et al. "Bose-Einstein condensation of Potassium atoms by sympathetic cooling", Science 294,1320(2001)
[12]T.Weber, J.Herbig, M.Mark, et al. "Bose-Einstein condensation of cesium", Science 299,232(2003)
[13]D.G.Fried, T.C.Killian, L.Willmann, et al. "Bose-Einstein condensation of atomic hydrogen", Phys.Rev.Lett 81,3811(1998)
[14]A.Robert, O.Sirjean,A.Browaeys,et al. "A Bose-Einstein condensate of metastable atoms", Science 292,461(2001)
[15]F.Pereira Dos Santos, J.Leonard, Junmin Wang, et al. "Bose-Einstein condensation of metastable helium", Phys.Rev.Lett 86,3459(2001)
[16]Y.Takasu,K.Honda, K.Komori, et al. "High-density trapping of cold ytterbium atoms by an opitical dipole force", Phys.Rev.Lett.90,023003(2003)
[17]C.A.Reagal, M.Greiner and D.S.Jin, "observation of resonace condensation of fermionic atom pairs", Phys.Rev.Lett.92,040403(2004)
[18]M.W.Zwinenin, C.A.Stan, C.H.Schunck, et al. "condensation of pairs of fermionic atoms near a feshbach resonace", Phys.Rev.Lett.92,12043(2004)
[19]M.W.Zwienein, C.A.Stan, C.H.Schunck, et al."observation of Bose-Einstein condensation of molecules" Phys.Rev.Lett.91,250401(2003)
[20]S.Jochim, M.Bartenstein, A.Altmeyer,et al."Bose-Einstein condensation of molecules" Science 13,1706(2003)
[21]M.Greiner, C.A.Regal, D.S.Jin, "Emergence of a molecular Bose-Einstein condensation from a Fermi gas"Nature 426,537(2003)
[22]O.Penrose and L.Onsager, "Bose-Einstein condensation and liquid Helium" Phys.Rev.104,576(1956)
[23]W.Krauth, "Quantum Monte Carlo calculations for a large number of bosons in a harmonic trap", Phys.Rev.Lett.77,3695(1996)
[24]A.J.Leggett, "Bose-Einstein condensation in the alkali gases:some fundamental concepts" Rev.Mod.Phys.73,307(2001)
[25]N.Bogoliubov, J.Phys(moscow) 11,423(1947)
[26]J.E.Williams, "the preparation of topological modes in a strong-coupled two-component BEC", Ph.D.thesis, University of Colorado,1999
[27]A.Arnold, "preparation and manipulation of an 87Rb BEC", Ph.D.thesis, University of Sussex,1999
[28]张礼,葛墨林,量子力学的前沿问题。北京：清华大学出版社
[29]M.Inguscio, S.Stringari, and C.E.Wieman, "Bose-Einstein condensation in atomic gases",Amsterdam, lops Press,(1999)
[30]A.Griffin, D.W.Snoke, and S.Stringari, "Bose-Einstein condensation",Cam, Cambridge University Press,1995
[31]L.Pitaevskii and S.Stringari, "Bose-Einstein Condensation",Oxford,Oxford University Press,2003
[32]H.T.C.Stoof, Phys.Rev.A,49,3824(1994)
[33]C.Orzel, A.K.Tuchman, M.L.Fenselau et al. "Squeezed states in a Bose-Einstein condensate",Science,291,2386(2001)
[34]M.J.Holland, D.S.Jin, M.L.Chiofalo et al. "Emergence of interaction effection in Bose-Einstein condensation." Phys.Rev.Lett.78,3801(1997)
[35]王义遒《量子力学新进展》第三辑2003
[36]M.Greiner, O.Mandel, T.Esslinger, T.W.Hansch, and I.Bloch, Nature 415,39(2002)
[37]J.R.Abo-Shneer, C.Raman, J.M.Vogels,et al."Obervation of vortex lattices in BECs" Science 292,476(2001)
[38]arXiv:0712.1429vl[gr-qc] 10.Dec.2007
[39]Aseem Paranjape,"Backreaction of Cosmological perturbations in covariant macroscopic gravity" Phys.Rev.D 78063522 (2008)
[40]Guillem Perez-Nadal, Albert Roura, Enric verdagner Class. Quant. Grav.25 154013
[41]Michavel Uhlmann, Yan Xu and Ralf Schutzhold,"Aspects of cosmic inflation in expanding Bose-Einstein condensates", New Journal of Physics 7248
[42]Ralf Schutzhold, Michael Uhlmann and Yan Xu "Quantum backreaction in dilute Bose-Einstein condensates"Phys. Rev. D 72105005(2005)
[43]T D Lee and C N Yang Phys. Rev.1051119(1957); T D Lee, K Huang and C N Yang ibid.1061135; E Timmermans, P Tommasini and K Huang Phys. Rev. A 55 3645(1997)
[44]W.GUnruh, "Experimental Black-Hole Evaporation?" Phys.Rev.Lett 46,1351(1981)
[45]N.D.Birrell and P.C.Davies, Quantum Fields in Curved Space[M], UK: Cambridge University Press,1982
[46]M.Stone,"Acoustic energy and momentum in a moving medium",Phys.Rev.E 62,1341(2000)
[47]J.Math.Phys.(N.Y.)4,195
[48]F Dalfovo, S.Giorgini, "Theory of Bose-Einstein condensation in trapped gases" Rev.Mod.Phys 71463(1999)
[49]M Olshanii, "Atomic Scattering in the presence of an External confinement and a Gas of Impenetrable Bosons"Phys. Rev. Lett.81,938(1998)
[50]A Gorlitz,J.M.Vogels,A.E.Leanhardt et al."Realization of Bose-Einstein condensates in Lower Dimentions" Phys. Rev. Lett.87130402(2001)
[51]P G Gennes Superconductivity of Metals and Alloys (W A Benjamin, New York,1966)
[52]Klaus von Klitzing,"The quantized Hall effect" Rev.Mod.Phys.58,519(1986)
[53]E.M.Lifshitz and L.P.Pitaevskii, Statistical Physics,part 2(Pergamon Press,Oxford,1980)
[54]V.N.Popov,Functional Integrals in Uuantum Field Theory and Statistical Physics,(D.Reidel Pub,Dordrecht,1983)
[55]V.L.Berezmskii,Sov.Phys.JETP 32,493(1971)
[56]H.Gauck,M.Hartl,D.Schneble et al,"Quasi-2D Gas of Laser Cooled Atoms in a Planar Matter Waveguide",Phys.Rev.Lett.,81,5298(1998)
[57]D.S.Petrov, M.Holzmann, and G.V.Shlyapnikov, "Bose-Einstein condensation in quasi2D trapped gases." Phys.Rev.Lett.84,2551 (2000)
[58]Y.Castin and R.Dum " Bose-Einstein Condensates in Time Dependent Traps",Phys.Rev.Lett.77,5315(1996)
[59]Yu.Kagan,E.L.Surkov,and GV.Shlyapnikov, "Evolution of a Bose-condensed gas under variations of the confining potential" Phys.Rev.A 54,r1753(1996)
[60]Y.Castin, "Simple theoretical tools for low dimension Bose gases",J.Phys.IV France 116,89-132(2004)
[61]L.Fetter, Ann.Phys.(N.Y.)70,67(1972)
[62]M.Lewenstein and L.You, "Quantum phase Diffusion of a Bose-Einstein Condensate" Phys.Rev.Lett.77,3489(1996)
[63]M.Naraschewski and R.J.Glauber, "Spatial coherence and density correlations of trapped Bose gases" Phys.Rev.A 59,4595(1999)
[64]S.Weinfurtner, A.White, and M.Visser, "Trans-Planckian physcs and signature change events in Bose gas hydrodynamics." Phys.Rev.D 76,124008(2007)
[2]F.London, "The λ-phenomenon of liquid helium and the Bose-Einstein degeneracy", Nature 141,643(1938)
[3]M.H.Anderson, J.R.Ensher, M.R.Matthews, et al. "observation of Bose-Einstein condensation in a Dilute Atomic Vapor", Science 269,198-201(1995)
[4]汪志诚热力学统计物理[M],北京：高等教育出版社,2003
[5]C.J.Pethick and H. Smith, Bose-Einstein condensation in Dilute gases[M]. UK: Cambridge University Press,2002
[6]E.A.Cornell, " very cold indeed:the nanokelvin physics of Bose-Einstein condensation." J.Res.Natl.Inst.Stand.Technol.101,419(1996)
[7]E.A.Cornell and C.E.Wieman, "Nobel Lecture:Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments. " Rev.Mod.Phys.74,875(2001)
[8]W.Ketterle, D.S.Durfee and D.M.Stamper-kurn, "Making, probing and understanding BEC", proc. Intern. school of physics-Enrico Fermi, M.Inguscio, S.Stringari and C.Wieman eds.(1999)
[9]C.C.Bradley, C.A.Sackett, and R.GHulet, "Bose-Einstein condensation of Lithium: observation of limited condensate number",Phys.Rev.Lett.78,985(1997)
[10]S.L.Cornish, N.R.Claussen, J.L.Roberts, et al. "Stable 85Rb Bose-Einstein condensates with widely tunable interactions", Phys.Rev.Lett.85,1795(2000)
[11]GModugno, GFerrari, G.Roati,et al. "Bose-Einstein condensation of Potassium atoms by sympathetic cooling", Science 294,1320(2001)
[12]T.Weber, J.Herbig, M.Mark, et al. "Bose-Einstein condensation of cesium", Science 299,232(2003)
[13]D.G.Fried, T.C.Killian, L.Willmann, et al. "Bose-Einstein condensation of atomic hydrogen", Phys.Rev.Lett 81,3811(1998)
[14]A.Robert, O.Sirjean,A.Browaeys,et al. "A Bose-Einstein condensate of metastable atoms", Science 292,461(2001)
[15]F.Pereira Dos Santos, J.Leonard, Junmin Wang, et al. "Bose-Einstein condensation of metastable helium", Phys.Rev.Lett 86,3459(2001)
[16]Y.Takasu,K.Honda, K.Komori, et al. "High-density trapping of cold ytterbium atoms by an opitical dipole force", Phys.Rev.Lett.90,023003(2003)
[17]C.A.Reagal, M.Greiner and D.S.Jin, "observation of resonace condensation of fermionic atom pairs", Phys.Rev.Lett.92,040403(2004)
[18]M.W.Zwinenin, C.A.Stan, C.H.Schunck, et al. "condensation of pairs of fermionic atoms near a feshbach resonace", Phys.Rev.Lett.92,12043(2004)
[19]M.W.Zwienein, C.A.Stan, C.H.Schunck, et al."observation of Bose-Einstein condensation of molecules" Phys.Rev.Lett.91,250401(2003)
[20]S.Jochim, M.Bartenstein, A.Altmeyer,et al."Bose-Einstein condensation of molecules" Science 13,1706(2003)
[21]M.Greiner, C.A.Regal, D.S.Jin, "Emergence of a molecular Bose-Einstein condensation from a Fermi gas"Nature 426,537(2003)
[22]O.Penrose and L.Onsager, "Bose-Einstein condensation and liquid Helium" Phys.Rev.104,576(1956)
[23]W.Krauth, "Quantum Monte Carlo calculations for a large number of bosons in a harmonic trap", Phys.Rev.Lett.77,3695(1996)
[24]A.J.Leggett, "Bose-Einstein condensation in the alkali gases:some fundamental concepts" Rev.Mod.Phys.73,307(2001)
[25]N.Bogoliubov, J.Phys(moscow) 11,423(1947)
[26]J.E.Williams, "the preparation of topological modes in a strong-coupled two-component BEC", Ph.D.thesis, University of Colorado,1999
[27]A.Arnold, "preparation and manipulation of an 87Rb BEC", Ph.D.thesis, University of Sussex,1999
[28]张礼,葛墨林,量子力学的前沿问题。北京：清华大学出版社
[29]M.Inguscio, S.Stringari, and C.E.Wieman, "Bose-Einstein condensation in atomic gases",Amsterdam, lops Press,(1999)
[30]A.Griffin, D.W.Snoke, and S.Stringari, "Bose-Einstein condensation",Cam, Cambridge University Press,1995
[31]L.Pitaevskii and S.Stringari, "Bose-Einstein Condensation",Oxford,Oxford University Press,2003
[32]H.T.C.Stoof, Phys.Rev.A,49,3824(1994)
[33]C.Orzel, A.K.Tuchman, M.L.Fenselau et al. "Squeezed states in a Bose-Einstein condensate",Science,291,2386(2001)
[34]M.J.Holland, D.S.Jin, M.L.Chiofalo et al. "Emergence of interaction effection in Bose-Einstein condensation." Phys.Rev.Lett.78,3801(1997)
[35]王义遒《量子力学新进展》第三辑2003
[36]M.Greiner, O.Mandel, T.Esslinger, T.W.Hansch, and I.Bloch, Nature 415,39(2002)
[37]J.R.Abo-Shneer, C.Raman, J.M.Vogels,et al."Obervation of vortex lattices in BECs" Science 292,476(2001)
[38]arXiv:0712.1429vl[gr-qc] 10.Dec.2007
[39]Aseem Paranjape,"Backreaction of Cosmological perturbations in covariant macroscopic gravity" Phys.Rev.D 78063522 (2008)
[40]Guillem Perez-Nadal, Albert Roura, Enric verdagner Class. Quant. Grav.25 154013
[41]Michavel Uhlmann, Yan Xu and Ralf Schutzhold,"Aspects of cosmic inflation in expanding Bose-Einstein condensates", New Journal of Physics 7248
[42]Ralf Schutzhold, Michael Uhlmann and Yan Xu "Quantum backreaction in dilute Bose-Einstein condensates"Phys. Rev. D 72105005(2005)
[43]T D Lee and C N Yang Phys. Rev.1051119(1957); T D Lee, K Huang and C N Yang ibid.1061135; E Timmermans, P Tommasini and K Huang Phys. Rev. A 55 3645(1997)
[44]W.GUnruh, "Experimental Black-Hole Evaporation?" Phys.Rev.Lett 46,1351(1981)
[45]N.D.Birrell and P.C.Davies, Quantum Fields in Curved Space[M], UK: Cambridge University Press,1982
[46]M.Stone,"Acoustic energy and momentum in a moving medium",Phys.Rev.E 62,1341(2000)
[47]J.Math.Phys.(N.Y.)4,195
[48]F Dalfovo, S.Giorgini, "Theory of Bose-Einstein condensation in trapped gases" Rev.Mod.Phys 71463(1999)
[49]M Olshanii, "Atomic Scattering in the presence of an External confinement and a Gas of Impenetrable Bosons"Phys. Rev. Lett.81,938(1998)
[50]A Gorlitz,J.M.Vogels,A.E.Leanhardt et al."Realization of Bose-Einstein condensates in Lower Dimentions" Phys. Rev. Lett.87130402(2001)
[51]P G Gennes Superconductivity of Metals and Alloys (W A Benjamin, New York,1966)
[52]Klaus von Klitzing,"The quantized Hall effect" Rev.Mod.Phys.58,519(1986)
[53]E.M.Lifshitz and L.P.Pitaevskii, Statistical Physics,part 2(Pergamon Press,Oxford,1980)
[54]V.N.Popov,Functional Integrals in Uuantum Field Theory and Statistical Physics,(D.Reidel Pub,Dordrecht,1983)
[55]V.L.Berezmskii,Sov.Phys.JETP 32,493(1971)
[56]H.Gauck,M.Hartl,D.Schneble et al,"Quasi-2D Gas of Laser Cooled Atoms in a Planar Matter Waveguide",Phys.Rev.Lett.,81,5298(1998)
[57]D.S.Petrov, M.Holzmann, and G.V.Shlyapnikov, "Bose-Einstein condensation in quasi2D trapped gases." Phys.Rev.Lett.84,2551 (2000)
[58]Y.Castin and R.Dum " Bose-Einstein Condensates in Time Dependent Traps",Phys.Rev.Lett.77,5315(1996)
[59]Yu.Kagan,E.L.Surkov,and GV.Shlyapnikov, "Evolution of a Bose-condensed gas under variations of the confining potential" Phys.Rev.A 54,r1753(1996)
[60]Y.Castin, "Simple theoretical tools for low dimension Bose gases",J.Phys.IV France 116,89-132(2004)
[61]L.Fetter, Ann.Phys.(N.Y.)70,67(1972)
[62]M.Lewenstein and L.You, "Quantum phase Diffusion of a Bose-Einstein Condensate" Phys.Rev.Lett.77,3489(1996)
[63]M.Naraschewski and R.J.Glauber, "Spatial coherence and density correlations of trapped Bose gases" Phys.Rev.A 59,4595(1999)
[64]S.Weinfurtner, A.White, and M.Visser, "Trans-Planckian physcs and signature change events in Bose gas hydrodynamics." Phys.Rev.D 76,124008(2007)