BCS-BEC渡越过程中超冷费米原子气体的相干特性与非线性效应研究
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摘要
玻色-爱因斯坦凝聚(Bose-Einstein condensation, BEC)是指当温度低于某一临界值时玻色子体系中大量粒子凝聚到一个或几个量子态的现象。BEC是量子统计物理学的基本结论之一,是超流与超导现象的物理根源。尽管费米子不能直接发生BEC,但通过Cooper配对等机制可形成费米子对。这些费米子对如同(准)复合玻色子,从而可以发生BEC,导致费米体系的超导和超流。近年来理论与实验研究表明,当粒子间的相互作用从吸引变为排斥时,费米子体系可以实现从Bardeen-Cooper-Schrieffer (BCS)超流到BEC的转变。事实上,BCS与BEC只不过是BCS-BEC渡越(BCS-BEC crossover)过程的两个极限而已。
由于激光冷却与囚禁技术的发展与成功应用,科学家们已能使稀薄原子气体的温度降低到纳开的数量级,从而实现弱相互作用玻色原子(如87Rb,23Na等)气体的BEC及费米原子(6Li,40K等)气体的超低温冷却与量子简并,并进而通过Feshbach共振技术实现费米原子气体的超流以及从BCS态到BEC的渡越(BCS-BEC crossover)。这些重大进展使得超冷原子物理成为原子分子物理、非线性与量子光学、统计和凝聚态物理等学科十分活跃的交叉前沿研究领域。目前超冷费米气体已成为国际物理学界的研究热点之一。无论是从基础物理研究的角度(包括物质波光学、量子调控、高温超导体等强关联体系的量子模拟、夸克-胶子等离子体和中子星的研究等),还是从发展精密光谱与精密测量等高新技术应用方面(包括研制原子激光、原子干涉仪、原子芯片、原子钟与光钟等)这些研究都有十分重要的意义。
超流费米原子气体理论研究的一个重要方法是使用量子多体理论。由于问题的复杂性,既使对于简化了的哈密顿量,适用于整个BCS-BEC渡越的微观理论至今还未能建立起来。另外,由于实验中所使用的囚禁势阱要求处理非均匀量子多体问题,而微观理论对于非均匀多体问题没有十分有效的理论处理方法。再者,对于非线性集体激发等非线性非平衡动力学问题,用微观多体理论求解也是十分困难的。有鉴于此,我们拟采取的基本研究方案是基于扩展的Thomas-Fermi密度泛函理论和推广的超流流体动力学方程,以及由此导出的序参量方程。基于以上方案本论文详细研究了在整个BCS-BEC渡越区域超流费米气体的相干特性与非线性动力学行为。研究结果包括以下几个方面：
1.超冷费米原子气体的一个重要研究内容是实现费米原子的凝聚和研究相关的超流特性。研究超流特性的一个重要手段是通过观察释放谐振子势和光晶格势后超流费米气体的干涉图案。我们从超流流体动力学方程出发,通过适当计入量子压力效应导出了一个超流序参量方程。基于此方程得到了光晶格势中所有子凝聚体(subcondensates)从BCS超流到BEC的基态分布。然后利用费曼传播子方法研究了关闭谐振子势和光晶格势后子凝聚体的相干演化,得到了整个BCS-BEC渡越区域中超流费米气体经近弹道碰撞后形成的干涉图案。所得计算结果与MIT小组发表在Nature [J. K. Chin et al., Nature 443,961 (2006)]上的著名实验结果符合得很好。
2.超冷费米气体的超流态类似于非线性量子光学中的宏观相干态,可以预测在超流费米气体中也能观测到类似的宏观非线性集体激发,其中孤子最为引人注目。我们基于自己导出的超流序参量方程,利用解析和数值两种方法研究了准一维情形下超流费米气体在整个BCS-BEC渡越中暗孤子的形成和演化。在长波长近似下导出了描述弱非线性激发所满足的Korteweg-de Vries方程,分别给出了适用于BCS极限、BEC极限、以及BCS-BEC渡越区域的孤子解。发现在不同超流区域暗孤子呈现不同的物理特征。在BCS区,暗孤子传播速度较大,宽度较小；而在BEC区,暗孤子传播速度较小,宽度较大。当运动到凝聚体边界时,孤子逐渐减速并伴随有振幅较小的声辐射。在不同超流区域,孤子的演化和声辐射行为是不同的。我们还研究了两个暗孤子的迎面碰撞。发现沿整个BCS-BEC渡越区域由孤子碰撞导致的相移呈非单调变化。所有解析结果与数值模拟结果符合得很好。
3.准一维近似只适用于体系粒子数较小及横向囚禁势很强的情形。然而,目前实验上实现的所有超流费米原子气体都不满足这些条件,所以有必要考虑粒子数较大及横向囚禁势较弱时三维雪茄型费米原子凝聚体中的线性和非线性激发。基于超流序参量方程,我们首先解析求解了适用于整个BCS-BEC渡越的线性集体激发的本征值问题,给出了所有线性集体激发本征值和本征函数的解析表达式,所得到的适用于整个BCS-BEC渡越区域的线性声速值与Duke大学的实验测量结果值符合得很好。另外,我们详细研究了体系非线性激发的动力学行为。导出了大粒子数和长波长情形下集体激发振幅所满足的非线性演化方程,并给出了该方程的孤子解。结果发现,体系的非线性激发对于大粒子数和小粒子数情形有重大差别。在小粒子数情形,所得孤子为暗孤子,其传播速度小于声速(亚声暗孤子)；在大粒子数情形,所得孤子为亮孤子,其传播速度大于声速(超声亮孤子)。物理原因是小粒子数情形下孤子的形成是由于量子压力导致的色散与粒子间相互作用导致的非线性效应之间的平衡所致,而大粒子数情况下体系的量子压力效应可以忽略。该情形下孤子是由于体系的波导色散与粒子间相互作用之间的平衡所形成的。研究表明,在不同超流区域超声速亮孤子呈现不同的物理性质,从而可以用来刻画BCS-BEC渡越中超流费米原子气体在不同超流区域的不同超流特性。
以上得到的研究结果不仅对于了解超流费米原子气体的非平衡非线性动力学性质,解释与预测有关实验结果,而且对于深入探索超冷费米原子气体的物理特性和对其进行有效的相干操控及其应用均有较为重要的意义。
Bose-Einstein condensation(BEC) is a phenomenon of macroscopic occupa-tion in one or several quantum states by a large number of identical bosons when the system temperature is below to a critical temperature. BEC is one of the most important conclusions in quantum statistical physics, and the origin of supercon-ductivity and superfluidity. The Pauli exclusion principle prevents multiple oc-cupancy of identical fermions to share the same quantum state. However, system of fermions with weak attractive interaction is known to form large overlapping Cooper pairs, which behave like (quasi) composite bosons and can condense into the pair state with zero center of mass momentum giving rise to superconduc-tivity and superfluidity. It is revealed that as the interaction between particles changes from attractive to repulsive in Fermi systems, a continuous transition from a Bardeen-Cooper-Schrieffer(BCS) type superfluid of loosely bound pairs to BEC of molecules occurs, while BCS and BEC are the two-limiting cases of the BCS-BEC crossover.
With the spectacular development of laser cooling and trapping techniques, experimenters chill dilute gases of certain atoms to nanoKelvin temperatures, which allows the realizations of BEC in weakly interacting bosonic atomic(i.e. 87Rb,23Na) gases and quantum degenerate fermionic atomic (i.e.6Li,40K) gases, and using a Feshbach resonance they explore the superfluidity as well as BCS-BCS crossover in quantum degenerate Fermi gases. Ultracold atomic physics has so far been in essence a hot boundary field between atomic and molecular, nonlin-ear and quantum optics and condensed matter physics, which not only serves as a test-bed for theories developed in matter-wave optics, quantum manipulation, high temperature superconductors, Quark-gluon plasma, and neutron stars, but also leads to new achievements in atom laser, atom interferometers, atom clocks, and light clocks.
A complete description of ultracold fermionic atomic gases in the BCS-BEC crossover based on quantum many-body theory is still not clear. Because ul-tracold atoms are trapped in external trapping potentials, the imhomogeneous and mesoscopic features of the system make a microscopic approach of nonlinear collective excitations and non-equilibrium problems even more difficult. There-fore, in this dissertation we use an order-parameter equation, which is based on extended Thomas-Fermi functional theory and generalized superfluid hydrody-namics equations, to investigate coherent properties and nonlinear dynamical be-haviors of superfluid Fermi gases in the BCS-BEC crossover in details, including the following aspects:
An important topic in the study of ultracold Fermi gases is a direct obser-vation of superfluid properties in the BCS-BEC crossover. One expects that in-terference patterns appear after superfliud Fermi gases released from combined harmonic oscillator and optical lattice potentials due to the long-range phase co-herence. We first start from superfluid hydrodynamics equations and include a proper quantum pressure term, then obtain an order-parameter equation. We solve the order-parameter equation valid for the crossover from BCS superfluid to BEC to obtain an initial distribution of subcondensates formed in an optical lattice. Then we investigate the coherent evolution of the subcondensates when both harmonic oscillator and optical lattice potentials are switched off. The inter-ference patterns of the superfluid Fermi gas along the BCS-BEC crossover dur-ing a nearly ballistic expansion are calculated by means of Feynman propagator method combined with numerical simulations. The result obtained agrees with the experimental observation[ J. K. Chin et al., Nature 443,961 (2006)].
The existence of macroscopic coherent behavior in superfluid fermionic atomic gases resembles other macroscopic coherent states found in the field of nonlinear and quantum optics, and hence one would expect to observe macroscopic non-linear collective phenomena, of which a soliton is a striking example. We study, both analytically and numerically, the formation and propagation of dark solitons in a quasi-one dimensional superfluid Fermi gas in the crossover from BCS su-perfluid to BEC. Starting from a superfluid order-parameter equation we derive a Korteweg-de Vries(KdV) equation for weak nonlinear excitations under quasi-one-dimensional and long wavelength approximations. We present dark soliton solutions valid for both BCS and BEC limits and also for the crossover, and show that dark solitons in different superfluid regimes possess different features. Par-ticularly, a dark soliton in the BCS (BEC) regime has larger (smaller) propagating velocity and smaller (larger) spatial width. Upon moving to the boundary of the condensate, it generally decelerates and generates small radiations, which dis-play different behavior in different superfluid regimes. We study also a head-on collision between two dark solitons and demonstrate that the phase shift due to the collision changes non-monotonically along the BCS-BEC crossover. All an-alytical results are checked by numerical simulations and good agreements be-tween them are found.
Quasi-one-dimensional condition requires that atomic numbers of systems must be very small and confinements in transverse directions be very strong, however, which are hard to realize experimentally in superfluid fermionic atomic gases. The investigation of the linear and nonlinear sound propagation of cigar-shaped superfluid Fermi gas with a large particle number is hence of particular interest. We first solve analytically the eigenvalue problem of linear collective ex-citations and provide explicit expressions of all eigenvalues and eigenfunctions, which are valid for all superfluid regimes in the BCS-BEC crossover. The lin-ear sound speed obtained agrees well with that of a recent experimental mea-surement. We then consider a weak nonlinear excitation and show that the time evolution of the excitation obeys a KdV equation. Different from the result ob-tained in quasi-one-dimensional case studied previously, where subsonic dark solitons are obtained via the balance between quantum pressure and nonlinear effect, we demonstrate that bright solitons with supersonic propagating velocity can be generated in the present three-dimensional system through the balance between a waveguide-like dispersion and the inter-particle interaction. The su-personic bright solitons obtained display different physical properties in different superfluid regimes and hence can be used to characterize superfluid features of the BCS-BEC crossover.
The results presented here may be useful for understanding coherent and nonlinear properties of superfluid fermionic atomic gases in the BCS-BEC crossover, and guiding experimental and theoretical investigations of ultracold fermi gases more extensively.
由于激光冷却与囚禁技术的发展与成功应用,科学家们已能使稀薄原子气体的温度降低到纳开的数量级,从而实现弱相互作用玻色原子(如87Rb,23Na等)气体的BEC及费米原子(6Li,40K等)气体的超低温冷却与量子简并,并进而通过Feshbach共振技术实现费米原子气体的超流以及从BCS态到BEC的渡越(BCS-BEC crossover)。这些重大进展使得超冷原子物理成为原子分子物理、非线性与量子光学、统计和凝聚态物理等学科十分活跃的交叉前沿研究领域。目前超冷费米气体已成为国际物理学界的研究热点之一。无论是从基础物理研究的角度(包括物质波光学、量子调控、高温超导体等强关联体系的量子模拟、夸克-胶子等离子体和中子星的研究等),还是从发展精密光谱与精密测量等高新技术应用方面(包括研制原子激光、原子干涉仪、原子芯片、原子钟与光钟等)这些研究都有十分重要的意义。
超流费米原子气体理论研究的一个重要方法是使用量子多体理论。由于问题的复杂性,既使对于简化了的哈密顿量,适用于整个BCS-BEC渡越的微观理论至今还未能建立起来。另外,由于实验中所使用的囚禁势阱要求处理非均匀量子多体问题,而微观理论对于非均匀多体问题没有十分有效的理论处理方法。再者,对于非线性集体激发等非线性非平衡动力学问题,用微观多体理论求解也是十分困难的。有鉴于此,我们拟采取的基本研究方案是基于扩展的Thomas-Fermi密度泛函理论和推广的超流流体动力学方程,以及由此导出的序参量方程。基于以上方案本论文详细研究了在整个BCS-BEC渡越区域超流费米气体的相干特性与非线性动力学行为。研究结果包括以下几个方面：
1.超冷费米原子气体的一个重要研究内容是实现费米原子的凝聚和研究相关的超流特性。研究超流特性的一个重要手段是通过观察释放谐振子势和光晶格势后超流费米气体的干涉图案。我们从超流流体动力学方程出发,通过适当计入量子压力效应导出了一个超流序参量方程。基于此方程得到了光晶格势中所有子凝聚体(subcondensates)从BCS超流到BEC的基态分布。然后利用费曼传播子方法研究了关闭谐振子势和光晶格势后子凝聚体的相干演化,得到了整个BCS-BEC渡越区域中超流费米气体经近弹道碰撞后形成的干涉图案。所得计算结果与MIT小组发表在Nature [J. K. Chin et al., Nature 443,961 (2006)]上的著名实验结果符合得很好。
2.超冷费米气体的超流态类似于非线性量子光学中的宏观相干态,可以预测在超流费米气体中也能观测到类似的宏观非线性集体激发,其中孤子最为引人注目。我们基于自己导出的超流序参量方程,利用解析和数值两种方法研究了准一维情形下超流费米气体在整个BCS-BEC渡越中暗孤子的形成和演化。在长波长近似下导出了描述弱非线性激发所满足的Korteweg-de Vries方程,分别给出了适用于BCS极限、BEC极限、以及BCS-BEC渡越区域的孤子解。发现在不同超流区域暗孤子呈现不同的物理特征。在BCS区,暗孤子传播速度较大,宽度较小；而在BEC区,暗孤子传播速度较小,宽度较大。当运动到凝聚体边界时,孤子逐渐减速并伴随有振幅较小的声辐射。在不同超流区域,孤子的演化和声辐射行为是不同的。我们还研究了两个暗孤子的迎面碰撞。发现沿整个BCS-BEC渡越区域由孤子碰撞导致的相移呈非单调变化。所有解析结果与数值模拟结果符合得很好。
3.准一维近似只适用于体系粒子数较小及横向囚禁势很强的情形。然而,目前实验上实现的所有超流费米原子气体都不满足这些条件,所以有必要考虑粒子数较大及横向囚禁势较弱时三维雪茄型费米原子凝聚体中的线性和非线性激发。基于超流序参量方程,我们首先解析求解了适用于整个BCS-BEC渡越的线性集体激发的本征值问题,给出了所有线性集体激发本征值和本征函数的解析表达式,所得到的适用于整个BCS-BEC渡越区域的线性声速值与Duke大学的实验测量结果值符合得很好。另外,我们详细研究了体系非线性激发的动力学行为。导出了大粒子数和长波长情形下集体激发振幅所满足的非线性演化方程,并给出了该方程的孤子解。结果发现,体系的非线性激发对于大粒子数和小粒子数情形有重大差别。在小粒子数情形,所得孤子为暗孤子,其传播速度小于声速(亚声暗孤子)；在大粒子数情形,所得孤子为亮孤子,其传播速度大于声速(超声亮孤子)。物理原因是小粒子数情形下孤子的形成是由于量子压力导致的色散与粒子间相互作用导致的非线性效应之间的平衡所致,而大粒子数情况下体系的量子压力效应可以忽略。该情形下孤子是由于体系的波导色散与粒子间相互作用之间的平衡所形成的。研究表明,在不同超流区域超声速亮孤子呈现不同的物理性质,从而可以用来刻画BCS-BEC渡越中超流费米原子气体在不同超流区域的不同超流特性。
以上得到的研究结果不仅对于了解超流费米原子气体的非平衡非线性动力学性质,解释与预测有关实验结果,而且对于深入探索超冷费米原子气体的物理特性和对其进行有效的相干操控及其应用均有较为重要的意义。
Bose-Einstein condensation(BEC) is a phenomenon of macroscopic occupa-tion in one or several quantum states by a large number of identical bosons when the system temperature is below to a critical temperature. BEC is one of the most important conclusions in quantum statistical physics, and the origin of supercon-ductivity and superfluidity. The Pauli exclusion principle prevents multiple oc-cupancy of identical fermions to share the same quantum state. However, system of fermions with weak attractive interaction is known to form large overlapping Cooper pairs, which behave like (quasi) composite bosons and can condense into the pair state with zero center of mass momentum giving rise to superconduc-tivity and superfluidity. It is revealed that as the interaction between particles changes from attractive to repulsive in Fermi systems, a continuous transition from a Bardeen-Cooper-Schrieffer(BCS) type superfluid of loosely bound pairs to BEC of molecules occurs, while BCS and BEC are the two-limiting cases of the BCS-BEC crossover.
With the spectacular development of laser cooling and trapping techniques, experimenters chill dilute gases of certain atoms to nanoKelvin temperatures, which allows the realizations of BEC in weakly interacting bosonic atomic(i.e. 87Rb,23Na) gases and quantum degenerate fermionic atomic (i.e.6Li,40K) gases, and using a Feshbach resonance they explore the superfluidity as well as BCS-BCS crossover in quantum degenerate Fermi gases. Ultracold atomic physics has so far been in essence a hot boundary field between atomic and molecular, nonlin-ear and quantum optics and condensed matter physics, which not only serves as a test-bed for theories developed in matter-wave optics, quantum manipulation, high temperature superconductors, Quark-gluon plasma, and neutron stars, but also leads to new achievements in atom laser, atom interferometers, atom clocks, and light clocks.
A complete description of ultracold fermionic atomic gases in the BCS-BEC crossover based on quantum many-body theory is still not clear. Because ul-tracold atoms are trapped in external trapping potentials, the imhomogeneous and mesoscopic features of the system make a microscopic approach of nonlinear collective excitations and non-equilibrium problems even more difficult. There-fore, in this dissertation we use an order-parameter equation, which is based on extended Thomas-Fermi functional theory and generalized superfluid hydrody-namics equations, to investigate coherent properties and nonlinear dynamical be-haviors of superfluid Fermi gases in the BCS-BEC crossover in details, including the following aspects:
An important topic in the study of ultracold Fermi gases is a direct obser-vation of superfluid properties in the BCS-BEC crossover. One expects that in-terference patterns appear after superfliud Fermi gases released from combined harmonic oscillator and optical lattice potentials due to the long-range phase co-herence. We first start from superfluid hydrodynamics equations and include a proper quantum pressure term, then obtain an order-parameter equation. We solve the order-parameter equation valid for the crossover from BCS superfluid to BEC to obtain an initial distribution of subcondensates formed in an optical lattice. Then we investigate the coherent evolution of the subcondensates when both harmonic oscillator and optical lattice potentials are switched off. The inter-ference patterns of the superfluid Fermi gas along the BCS-BEC crossover dur-ing a nearly ballistic expansion are calculated by means of Feynman propagator method combined with numerical simulations. The result obtained agrees with the experimental observation[ J. K. Chin et al., Nature 443,961 (2006)].
The existence of macroscopic coherent behavior in superfluid fermionic atomic gases resembles other macroscopic coherent states found in the field of nonlinear and quantum optics, and hence one would expect to observe macroscopic non-linear collective phenomena, of which a soliton is a striking example. We study, both analytically and numerically, the formation and propagation of dark solitons in a quasi-one dimensional superfluid Fermi gas in the crossover from BCS su-perfluid to BEC. Starting from a superfluid order-parameter equation we derive a Korteweg-de Vries(KdV) equation for weak nonlinear excitations under quasi-one-dimensional and long wavelength approximations. We present dark soliton solutions valid for both BCS and BEC limits and also for the crossover, and show that dark solitons in different superfluid regimes possess different features. Par-ticularly, a dark soliton in the BCS (BEC) regime has larger (smaller) propagating velocity and smaller (larger) spatial width. Upon moving to the boundary of the condensate, it generally decelerates and generates small radiations, which dis-play different behavior in different superfluid regimes. We study also a head-on collision between two dark solitons and demonstrate that the phase shift due to the collision changes non-monotonically along the BCS-BEC crossover. All an-alytical results are checked by numerical simulations and good agreements be-tween them are found.
Quasi-one-dimensional condition requires that atomic numbers of systems must be very small and confinements in transverse directions be very strong, however, which are hard to realize experimentally in superfluid fermionic atomic gases. The investigation of the linear and nonlinear sound propagation of cigar-shaped superfluid Fermi gas with a large particle number is hence of particular interest. We first solve analytically the eigenvalue problem of linear collective ex-citations and provide explicit expressions of all eigenvalues and eigenfunctions, which are valid for all superfluid regimes in the BCS-BEC crossover. The lin-ear sound speed obtained agrees well with that of a recent experimental mea-surement. We then consider a weak nonlinear excitation and show that the time evolution of the excitation obeys a KdV equation. Different from the result ob-tained in quasi-one-dimensional case studied previously, where subsonic dark solitons are obtained via the balance between quantum pressure and nonlinear effect, we demonstrate that bright solitons with supersonic propagating velocity can be generated in the present three-dimensional system through the balance between a waveguide-like dispersion and the inter-particle interaction. The su-personic bright solitons obtained display different physical properties in different superfluid regimes and hence can be used to characterize superfluid features of the BCS-BEC crossover.
The results presented here may be useful for understanding coherent and nonlinear properties of superfluid fermionic atomic gases in the BCS-BEC crossover, and guiding experimental and theoretical investigations of ultracold fermi gases more extensively.
引文
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