BCS-BEC渡越理论在高密度费米物质态的超流性研究方面的应用
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摘要
根据量子统计理论的观点,玻色子和费米子具有完全不同的量子统计特征：对玻色子体系,当温度低于某一临界值时,将有宏观数量的粒子凝聚到一个或几个量子态的现象,称为玻色-爱因斯坦凝聚(BEC)。而对于费米子体系,在低于费米温度时,由于泡利不相容原理,粒子将从最低能级开始一个一个的填充能量低于某一确定值的所有能量态,这个确定的能量值我们称为费米能级。
发现超导现象之后,人们开始了对超导微观机理的研究。1957年Bardeen, Cooper和Schrieffer (BCS)建立了超导的微观理论。在BCS理论中,超导的产生要经过两个基本的过程：(1)电子通过某种吸引相互作用形成配对；(2)配对的电子相位发生相干凝聚,形成超导。1986年,人们又发现了高温超导现象,之后有理论提出,当改变粒子间的散射长度从负到正时,费米子系统将发生从BCS到BEC的连续的转变。通过Feshbach共振调节原子间的散射长度,在实验上实现了从BCS到BEC的连续转变。之后人们对超冷费米气体和BCS-BEC渡越过程进行了大量的研究。
本篇论文组织如下：第一章分为四节,分别介绍玻色气体和费米气体的统计性质及其在绝对零度下的性质及出现的现象,Feshbach共振、分子BEC和本文的研究意义。第二章主要介绍BCS理论的成功及不足,BCS-BEC过渡理论的出现、具体的内容及近几年人们针对BCS-BEC渡越区域的研究进展。第三章运用BCS-BEC渡越理论研究高密度费米物质态。这一章是本篇论文的重点,利用BCS-BEC渡越理论,研究不同的相互作用强度下,在给定的有限范围的相互作用形式(P. Nozieres和S. Schmitt-Rink在研究费米气体的BCS-BEC渡越理论时所使用的势能的形式,简称NSR)时,费米物质态的超流转变温度及超流能隙等随密度的变化,并在此基础上研究系统的热力学行为如能量、比热、熵等随温度的变化。我们着重研究要正极限或大散射长度的情形,因为他们同费米原子气体及核物质密切相关。研究结果表明,虽然超流转变温度开始时随密度增加而增加,但过高的密度对超流转变温度会有抑制作用。
According to the point of view in statistical theory, in a boson system, macroscopic numbers of particles are condensed to the lowest-energy state when the temperature falls below a critical temperature, which is defined as Bose-Einstein condensation (BEC). However, in a Fermi system, as the temperature of the system is below the Fermi temperature, no quantum state can be occupied by more than one particle in light of the Pauli principle. All states having energy below a certain value are inhabited, while the states with higher energy surpassing this critical energy remain unoccupied. The threshold energy is named as Fermi energy.
Since the superconductor was discovered, immense physical qualities on it had been studied. Bardeen, Cooper, and Schrieffer (BCS) gave the theory on conventional superconductor in 1957. According the BCS theory, there are two processes to make ordinary metals evolve from normal state to superconducting state:Firstly, two electrons are attracted into each other forming cooper pairs by an attracting interaction by means of phonons; Second, energy of cooper pairs is collectively condensed into a certain lowest energy. In 1986, cooper-based high-temperature superconductor was discovered. Numerous theories relevant to HTC superconducting mechanism was put forward in the process of research, namely: with the increase of attractive interaction strength, the ground-state wave function of cooper pairs evolves continuously from BCS situation to BEC situation. From the view of experiment, the BCS-BEC crossover can be achieved by using a Feshbach resonance in a magnetic field to tune the interaction. Subsequently, most of work in this field mainly focuses on the research of ultracold fermionic gases and the BCS-BEC crossover.
This thesis is arranged as follows:
In chapter 1, firstly, we discuss the properties and the phase transition at zero temperature in both the Boson and Fermi system, then, we turn to background introduction and fundamental theories of Fechbach resonance. Finally, Fermi superfluidity, molecules BEC and the means to write this paper were discussed.
In chapter 2, we introduce the BEC theory which is the most successful theories in condensed matter physics, the background and development of the BCS-BEC crossover theory and the research results on the BCS-BEC crossover problems.
In chapter 3, we apply the standard mean-field description of BCS-BES crossover theory to high density Fermi matter and study the properties of superfluidity in high density Fermi matter. We assume the interacting form of high density Fermi matter is NSR, and research the influence of density on superfluid critical temperature and superfluid gap of Fermi matter. Based on the above research, we study the thermodynamic properties of high density Fermi matter. Our researches are mainly focusing on the situation with the scattering lengthα→±∞, because it is similar with the nuclear matter and Fermi atom gases. The result shows that the superfluid transition temperature increases with density, but the relevant increase will be suppressed by the extremely high density.
发现超导现象之后,人们开始了对超导微观机理的研究。1957年Bardeen, Cooper和Schrieffer (BCS)建立了超导的微观理论。在BCS理论中,超导的产生要经过两个基本的过程：(1)电子通过某种吸引相互作用形成配对；(2)配对的电子相位发生相干凝聚,形成超导。1986年,人们又发现了高温超导现象,之后有理论提出,当改变粒子间的散射长度从负到正时,费米子系统将发生从BCS到BEC的连续的转变。通过Feshbach共振调节原子间的散射长度,在实验上实现了从BCS到BEC的连续转变。之后人们对超冷费米气体和BCS-BEC渡越过程进行了大量的研究。
本篇论文组织如下：第一章分为四节,分别介绍玻色气体和费米气体的统计性质及其在绝对零度下的性质及出现的现象,Feshbach共振、分子BEC和本文的研究意义。第二章主要介绍BCS理论的成功及不足,BCS-BEC过渡理论的出现、具体的内容及近几年人们针对BCS-BEC渡越区域的研究进展。第三章运用BCS-BEC渡越理论研究高密度费米物质态。这一章是本篇论文的重点,利用BCS-BEC渡越理论,研究不同的相互作用强度下,在给定的有限范围的相互作用形式(P. Nozieres和S. Schmitt-Rink在研究费米气体的BCS-BEC渡越理论时所使用的势能的形式,简称NSR)时,费米物质态的超流转变温度及超流能隙等随密度的变化,并在此基础上研究系统的热力学行为如能量、比热、熵等随温度的变化。我们着重研究要正极限或大散射长度的情形,因为他们同费米原子气体及核物质密切相关。研究结果表明,虽然超流转变温度开始时随密度增加而增加,但过高的密度对超流转变温度会有抑制作用。
According to the point of view in statistical theory, in a boson system, macroscopic numbers of particles are condensed to the lowest-energy state when the temperature falls below a critical temperature, which is defined as Bose-Einstein condensation (BEC). However, in a Fermi system, as the temperature of the system is below the Fermi temperature, no quantum state can be occupied by more than one particle in light of the Pauli principle. All states having energy below a certain value are inhabited, while the states with higher energy surpassing this critical energy remain unoccupied. The threshold energy is named as Fermi energy.
Since the superconductor was discovered, immense physical qualities on it had been studied. Bardeen, Cooper, and Schrieffer (BCS) gave the theory on conventional superconductor in 1957. According the BCS theory, there are two processes to make ordinary metals evolve from normal state to superconducting state:Firstly, two electrons are attracted into each other forming cooper pairs by an attracting interaction by means of phonons; Second, energy of cooper pairs is collectively condensed into a certain lowest energy. In 1986, cooper-based high-temperature superconductor was discovered. Numerous theories relevant to HTC superconducting mechanism was put forward in the process of research, namely: with the increase of attractive interaction strength, the ground-state wave function of cooper pairs evolves continuously from BCS situation to BEC situation. From the view of experiment, the BCS-BEC crossover can be achieved by using a Feshbach resonance in a magnetic field to tune the interaction. Subsequently, most of work in this field mainly focuses on the research of ultracold fermionic gases and the BCS-BEC crossover.
This thesis is arranged as follows:
In chapter 1, firstly, we discuss the properties and the phase transition at zero temperature in both the Boson and Fermi system, then, we turn to background introduction and fundamental theories of Fechbach resonance. Finally, Fermi superfluidity, molecules BEC and the means to write this paper were discussed.
In chapter 2, we introduce the BEC theory which is the most successful theories in condensed matter physics, the background and development of the BCS-BEC crossover theory and the research results on the BCS-BEC crossover problems.
In chapter 3, we apply the standard mean-field description of BCS-BES crossover theory to high density Fermi matter and study the properties of superfluidity in high density Fermi matter. We assume the interacting form of high density Fermi matter is NSR, and research the influence of density on superfluid critical temperature and superfluid gap of Fermi matter. Based on the above research, we study the thermodynamic properties of high density Fermi matter. Our researches are mainly focusing on the situation with the scattering lengthα→±∞, because it is similar with the nuclear matter and Fermi atom gases. The result shows that the superfluid transition temperature increases with density, but the relevant increase will be suppressed by the extremely high density.
引文
[1]Eagles D M, Possible Pairing without Superconductivity at Low Carrier Concentrations in Bulk and Thin-Film Superconducting Semiconductors, Phys. Rev,1969,186(2):456-463.
[2]Leggett A J, Modern Trends in the Theory of Condensed Matter, edited by A. Pekalski and J. Przgstawa., Springer-Verlag:Berlin,1980,13-27.
[3]O'Hara K M, Hemmer S L, Gehm M E, et al. Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms, Science,2002,298(5601):2179-2182.
[4]Jochim S, Bartenstein M, Altmeyer A, et al. Bose-Einstein Condensation of Molecules, Science,2003,302(5653):2101-2103.
[5]Greiner M, Regal C A, J in D S, Emergence of a molecular Bose-Einstein condensate from a Fermi gas, Nature,2003,426(6966):537-540.
[6]Zwierlein M W, Stan C A, Schunck C H, et al. Observation of Bose-Einstein Condensation of Molecules Phys. Rev. Lett,2003,91(25):250401.
[7]Bartenstein M, Altmeyer A, Riedl S, et al. Crossover from a Molecular Bose-Einstein Condensate to a Degenerate Fermi Gas, Phys. Rev. Lett,2004,92(12):120401.
[8]Regal C A, Greiner M, Jin D S, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett,2004,92(4):040403.
[9]Zwierlein M W, Stan C A, Schunck C H, et al. Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance, Phys. Rev. Lett,2004,92(12):120403.
[10]Kinast J, Hemmer S L, Gehm M E, et al. Evidence for Superfluidity in a Resonantly Interacting Fermi Gas, Phys. Rev. Lett,2004,92(15):150402.
[11]Bourdel T, Khaykovich L, Cubizolles J, et al. Experimental Study of the BEC-BCS Crossover Region in Lithium 6, Phys. Rev. Lett,2004,93(5):50401.
[12]Part ridge GB, St recker K E, Kamar R I, et al, Preprint cond-mat/0505353.
[13]Yi Yu, Q.J. Chen, Superfluidity in atomic Fermi gases, Physica. C,2010,470:s900-s903.
[14]汪志诚,热力学统计物理,北京,高等教育出版社,2006：298-317.
[15]熊宏伟,吕宝龙,刘淑娟,詹明生,超冷费米气体研究的新进展,物理学进展,2005,25(3)：296-316.
[16]张礼,费米原子对的玻色-爱因斯坦凝聚,物理与工程,2005,1(1)：6-9.
[17]DeMarco B, Jin D S, Onset of Fermi Degeneracy in a Trapped Atomic Gas, Science,1999, 285(5434):1703-1706.
[18]H.Feshbach, Theoretical Nuclear Physics, NewYork, wiley,1992
[19]Stwalley W C, Stability of Spin-Aligned Hydrogen at Low Temperatures and High Magnetic Fields:New Field-Dependent Scattering Resonances and Predissociations, Phys. Rev. Lett,1976,37(24):1628-1631.
[20]E.Tiesinga, B.J.Verhaar and H.T.C.Stoof, Threshold and resonance Phenomena in ultracold ground-state collisions, Phy. Rev,1992, A47:4114-4122.
[21]A.J. Moerdijk, B.J. Verhaar and A. Avelsson, Resonances in Untracold Collisions of 6Li, 7Li, and 23Na, Phy. Rev,1995, A51:4852.
[22]S. Inouye, Pfau T, Gupta S, et al. Phase-coherent amplification of atomic matter waves, Nature,1998,402 (6762):641-644.
[23]Chu S, Hollberg L, Bjorkholm J E, et al, Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure, Phy. Rev. Lett,1986,55(1):48-51.
[24]Kasebich M, Chu S, Laser Cooling Below a Photon Recoil with Three-level Atoms, Phy. Rev. Lett,1992,69(12):1741-1744.
[25]Asnkin A, Trapping of Atoms by Resonance Radiation Pressure, Phy. Rev. Lett,1978, 40(12):729-723.
[26]D.G. Yakovlev, K.P. Levenfish, and Yu A.Shibanov, Phys. Usp,1999,825:169.
[27]I. Bombaci, EOS for Dense Isospin Asymmetric Nuclear Matter For Astrophysical Applications in the book Isospin Physics in Heavy-Ion Collsions at Intermediate Energies, edited by B. A. Li, and W. U.schroder, Nova. Science Publishers, Inc, New York,2001.
[28]J.R. Schrieffer, Theory of Superconductivity,3rd ed. Perseus Books, Reading, MA,1983.
[29]Cooper L N. Bound Electron Pairs in a Degnerate Fermi Gas, Phys. Rev,1956,104:1189.
[30]Q.J. Chen, Generalization of BCS theory to short coherence length superconductors: A BCS-Bose-Einstein crossover scenario, Ph.D Dissertation, Chicago, University of Chicago,2000.
[31]Renyuan Liao, Fermi pair and BCS-BEC crossover in novel System, Kent, Kent state university,2008,14-19.
[32]Cote R. Griffin A, Cooper pair-condensate fluctuations and plasmons in layered superconductors, Phys. Rev. B,1993,48:10404-10425.
[33]P. Nozieres and S. Schmitt-Rink, J. Low Temp, Bose Condensation in an Attractive Fermion Gas:From Weak to Strong Coupling Superconductibity, J. Low Temp. Phys,1985, 59:195-211.
[34]Q.J. Chen, J. Stajic, S.N. Tan, K. Levin, BCS-BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep,2005,412:1-88.
[35]S Schmitt, C.M. Varma, and A.E. Ruckenstein, Pairing in two dimensions, Phys. Rev. Lett, 1989,63:445-448.
[36]R. Haussmann, Crossover from BCS superconductivity to Bose-Einstein condensation: A self-consistent theory, Z. Phys. B,1993,91:291.
[37]R. Haussmann, Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation, Phys. Rev. B, 1994,49:12975-12983.
[38]B. Janko, J. Maly, and K. Levin, Pseudogap effects induced by resonant pair scattering, Phys. Rev. B,1997,56:R11407-R11410.
[39]J. Maly, B. Janko, and K. Levin. Superconductivity from a pseudogapped normal state:A mode-coupling approach to precursor superconductivity, Phys. Rev. B,1999,59: 1354-1357.
[40]J. Maly, B. Janko, and K. Levin, Numerical studies of the s-wave pseudogap state and related Tc:the "pairing approximation" theory, Physica C,1999,321:113.
[41]Chin C, Bartenstein M, Altmeyer A, et al. Observation of the Pairing Gap in a Strongly Interacting Fermi Gas, Science,2004,305(5687):1128-1130.
[42]Kinnunen J, Rodriguez M, Torma P, Pairing Gap and In-Gap Excitations in Trapped Fermionic Superfluids, Science,2004,305(5687):1131-1133.
[43]Zwierlein M W, A bo-Shaeer J R,Schirotzek A, et al. Vortices and superfluidity in a strongly interacting Fermi gas, Nature,2005,435(7045):1047-1051.
[44]李正中,固体理论,北京,高等教育出版社,2002.
[45]A.L费特,J.D瓦立克,多粒子系统的量子理论,科学出版社,1984.
[2]Leggett A J, Modern Trends in the Theory of Condensed Matter, edited by A. Pekalski and J. Przgstawa., Springer-Verlag:Berlin,1980,13-27.
[3]O'Hara K M, Hemmer S L, Gehm M E, et al. Observation of a Strongly Interacting Degenerate Fermi Gas of Atoms, Science,2002,298(5601):2179-2182.
[4]Jochim S, Bartenstein M, Altmeyer A, et al. Bose-Einstein Condensation of Molecules, Science,2003,302(5653):2101-2103.
[5]Greiner M, Regal C A, J in D S, Emergence of a molecular Bose-Einstein condensate from a Fermi gas, Nature,2003,426(6966):537-540.
[6]Zwierlein M W, Stan C A, Schunck C H, et al. Observation of Bose-Einstein Condensation of Molecules Phys. Rev. Lett,2003,91(25):250401.
[7]Bartenstein M, Altmeyer A, Riedl S, et al. Crossover from a Molecular Bose-Einstein Condensate to a Degenerate Fermi Gas, Phys. Rev. Lett,2004,92(12):120401.
[8]Regal C A, Greiner M, Jin D S, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett,2004,92(4):040403.
[9]Zwierlein M W, Stan C A, Schunck C H, et al. Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance, Phys. Rev. Lett,2004,92(12):120403.
[10]Kinast J, Hemmer S L, Gehm M E, et al. Evidence for Superfluidity in a Resonantly Interacting Fermi Gas, Phys. Rev. Lett,2004,92(15):150402.
[11]Bourdel T, Khaykovich L, Cubizolles J, et al. Experimental Study of the BEC-BCS Crossover Region in Lithium 6, Phys. Rev. Lett,2004,93(5):50401.
[12]Part ridge GB, St recker K E, Kamar R I, et al, Preprint cond-mat/0505353.
[13]Yi Yu, Q.J. Chen, Superfluidity in atomic Fermi gases, Physica. C,2010,470:s900-s903.
[14]汪志诚,热力学统计物理,北京,高等教育出版社,2006：298-317.
[15]熊宏伟,吕宝龙,刘淑娟,詹明生,超冷费米气体研究的新进展,物理学进展,2005,25(3)：296-316.
[16]张礼,费米原子对的玻色-爱因斯坦凝聚,物理与工程,2005,1(1)：6-9.
[17]DeMarco B, Jin D S, Onset of Fermi Degeneracy in a Trapped Atomic Gas, Science,1999, 285(5434):1703-1706.
[18]H.Feshbach, Theoretical Nuclear Physics, NewYork, wiley,1992
[19]Stwalley W C, Stability of Spin-Aligned Hydrogen at Low Temperatures and High Magnetic Fields:New Field-Dependent Scattering Resonances and Predissociations, Phys. Rev. Lett,1976,37(24):1628-1631.
[20]E.Tiesinga, B.J.Verhaar and H.T.C.Stoof, Threshold and resonance Phenomena in ultracold ground-state collisions, Phy. Rev,1992, A47:4114-4122.
[21]A.J. Moerdijk, B.J. Verhaar and A. Avelsson, Resonances in Untracold Collisions of 6Li, 7Li, and 23Na, Phy. Rev,1995, A51:4852.
[22]S. Inouye, Pfau T, Gupta S, et al. Phase-coherent amplification of atomic matter waves, Nature,1998,402 (6762):641-644.
[23]Chu S, Hollberg L, Bjorkholm J E, et al, Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure, Phy. Rev. Lett,1986,55(1):48-51.
[24]Kasebich M, Chu S, Laser Cooling Below a Photon Recoil with Three-level Atoms, Phy. Rev. Lett,1992,69(12):1741-1744.
[25]Asnkin A, Trapping of Atoms by Resonance Radiation Pressure, Phy. Rev. Lett,1978, 40(12):729-723.
[26]D.G. Yakovlev, K.P. Levenfish, and Yu A.Shibanov, Phys. Usp,1999,825:169.
[27]I. Bombaci, EOS for Dense Isospin Asymmetric Nuclear Matter For Astrophysical Applications in the book Isospin Physics in Heavy-Ion Collsions at Intermediate Energies, edited by B. A. Li, and W. U.schroder, Nova. Science Publishers, Inc, New York,2001.
[28]J.R. Schrieffer, Theory of Superconductivity,3rd ed. Perseus Books, Reading, MA,1983.
[29]Cooper L N. Bound Electron Pairs in a Degnerate Fermi Gas, Phys. Rev,1956,104:1189.
[30]Q.J. Chen, Generalization of BCS theory to short coherence length superconductors: A BCS-Bose-Einstein crossover scenario, Ph.D Dissertation, Chicago, University of Chicago,2000.
[31]Renyuan Liao, Fermi pair and BCS-BEC crossover in novel System, Kent, Kent state university,2008,14-19.
[32]Cote R. Griffin A, Cooper pair-condensate fluctuations and plasmons in layered superconductors, Phys. Rev. B,1993,48:10404-10425.
[33]P. Nozieres and S. Schmitt-Rink, J. Low Temp, Bose Condensation in an Attractive Fermion Gas:From Weak to Strong Coupling Superconductibity, J. Low Temp. Phys,1985, 59:195-211.
[34]Q.J. Chen, J. Stajic, S.N. Tan, K. Levin, BCS-BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep,2005,412:1-88.
[35]S Schmitt, C.M. Varma, and A.E. Ruckenstein, Pairing in two dimensions, Phys. Rev. Lett, 1989,63:445-448.
[36]R. Haussmann, Crossover from BCS superconductivity to Bose-Einstein condensation: A self-consistent theory, Z. Phys. B,1993,91:291.
[37]R. Haussmann, Properties of a Fermi liquid at the superfluid transition in the crossover region between BCS superconductivity and Bose-Einstein condensation, Phys. Rev. B, 1994,49:12975-12983.
[38]B. Janko, J. Maly, and K. Levin, Pseudogap effects induced by resonant pair scattering, Phys. Rev. B,1997,56:R11407-R11410.
[39]J. Maly, B. Janko, and K. Levin. Superconductivity from a pseudogapped normal state:A mode-coupling approach to precursor superconductivity, Phys. Rev. B,1999,59: 1354-1357.
[40]J. Maly, B. Janko, and K. Levin, Numerical studies of the s-wave pseudogap state and related Tc:the "pairing approximation" theory, Physica C,1999,321:113.
[41]Chin C, Bartenstein M, Altmeyer A, et al. Observation of the Pairing Gap in a Strongly Interacting Fermi Gas, Science,2004,305(5687):1128-1130.
[42]Kinnunen J, Rodriguez M, Torma P, Pairing Gap and In-Gap Excitations in Trapped Fermionic Superfluids, Science,2004,305(5687):1131-1133.
[43]Zwierlein M W, A bo-Shaeer J R,Schirotzek A, et al. Vortices and superfluidity in a strongly interacting Fermi gas, Nature,2005,435(7045):1047-1051.
[44]李正中,固体理论,北京,高等教育出版社,2002.
[45]A.L费特,J.D瓦立克,多粒子系统的量子理论,科学出版社,1984.