多分量超冷玻色原子气的平衡态动力学研究
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摘要
本论文主要研究了多分量玻色-爱因斯坦凝聚体系中的平衡态以及动力学问题,包括对双阱系统及原子-分子转化过程的讨论。在前两章中我们简要回顾了玻色-爱因斯坦凝聚现象的实验及理论背景知识,并对多分量玻色-爱因斯坦凝聚体的相关进展作了介绍。第三章至第六章则对应于我们的研究成果,具体如下：
在第三章中,我们在双阱中研究了双分量玻色-爱因斯坦凝聚体的不动点跃迁问题。由于两分量之间存在异种相互作用,这使得体系的不动点解很复杂。根据其对称性,我们将这些不动点解分为四类：对称解,反对称解,各向同性解,以及不对称解。我们给出了各类解的存在情况随着体系参数的依赖关系。通过对这些不动点解作小量展开,我们讨论了它们的动力学稳定性问题,给出了各类解在参数空间的稳定性相图。我们感兴趣的是这些不动点解在Rosen-Zener型的遂穿强度的绝热调制下的演化行为。这里我们通过外加一个高频的周期调制来引入一个有效的遂穿强度,它的值可以通过改变调制的幅度来调节。我们的数值模拟局限在每个分量在两阱间的相位差均为π。对于初值为对称解时,当最大的有效遂穿强度足够大时,末态则有一定的几率处于另一个对称解上,这依赖于体系参数的具体取值。而对于初值为反对称解的情况,末态的情况则更为复杂,随着最大有效遂穿强度的不同取值,末态可能是对称解和反对称解中的任意一个,甚至可能是超出这些不动点解的混沌态。我们对此做了详细讨论。
在第四章中,我们讨论了在各向同性的三维球谐势中原子-分子混合物的相分离。通过Feshbach共振,我们可以将原子转化为分子,这样我们即在约束势中实现了不对称的两分量玻色-爱因斯坦凝聚。我们通过寻找能量极小值来确定体系的基态,从而确定相应的相图。体系可能的基态只有三种情况,真空态,纯的分子超流态,以及混合的原子-分子超流态。由于转化过程的不对称性,纯的原子超流态是不存在的。我们引入Hessian矩阵来分析每个态的动力学稳定性。在Thomas-Fermi近似下,我们人为定义了两个局域的化学势。在这两个化学势张开的参数空间里,我们给出了体系的基态相图。当原子-分子之间的相互作用改变时,我们在某个特定的临界值附近发现了一个不连续的突变。由于局域化学势空间和约束势实空间之间的映射关系,我们同样给出了原子和分子在实空间的密度分布,并验证了前面的相图分析。
在第五章中,我们提出了一个实验建议,如何操控Ramsey型的磁场脉冲来提高原子-分子转化率。忽略掉约束势的影响,在平均场近似下,我们用一个经典的类摆模型来描述这个体系。通过求得不动点解,我们可以确定体系相应的本征态。我们假设体系的初态全为原子,当磁场为一个定值(单个脉冲)的时候,随着时间演化,体系会在原子态和分子态之间发生拉比振荡。我们给出了振荡周期随能量失谐之间的依赖关系,发现当能量失谐趋于零时,振荡周期趋于发散。相比于单个脉冲实现的分子转化率,利用双脉冲的相同磁场,我们有可能实现比其更高的转化率。在相空间的能量等高图上,我们给出了合理的解释。拉比振荡的轨迹对应于一条能量等高线,而两脉冲之间的时间间隔会使得体系在不同的能量等高线之间跃迁,从而实现高的分子转化率。
在第六章中,我们将玻色-爱因斯坦凝聚体置于光学腔中,并讨论光子对Feshbach共振下原子-分子转化率的影响。当光子频率远大于原子和分子两者各自的跃迁频率，光子与原子和分子之间的相互作用呈弥散型。由于腔中光子的衰减率远大于系统的其他特征频率,因此光子的演化绝热地依赖于原子和分子凝聚体的演化,从而可以得到原子和分子满足的有效哈密顿量。利用平均场近似,我们发现非线性项的存在使得系统可能出现更多的解,尤其是当泵浦激光的强度足够大时,我们会观察到双稳现象。我们提出光子与Feshbach共振模型之间的耦合可以被用来提高原子-分子转化率。我们同样给出了系统相应的量子模型,以对比平均场理论所得到的结果。
This essay focuses on the static and dynamical properties of multi-component Bose-Einstein condensates, including the discussion about double-well system and atom-to-molecule conversion process. In the first two chapters, we briefly review the experimental and theoretical backgrounds of Bose-Einstein condensa-tion, and also introduce the relevant investigations of multi-component conden-sates. Our research achievements correspond to the content from Chapter 3 to Chapter 6 which are listed as follows:
In Chapter 3, we investigate the fixed-point transitions of two-component Bose-Einstein condensates in double-well trapping potentials. Based on the sym-metry between the two kinds of atoms, we classify the fixed-point solutions into four categories:symmetrical, anti-symmetrical, isotropical and asymmetrical. We discuss the existence of each kind of solution and the corresponding dynam-ical stability, and give the stability diagrams in parameter spaces. We apply a high-frequency periodic modulation to the energy bias between the two wells and vary the value of the amplitude to tune the effective tunneling strength. As a result, during the evolution the final state may belong to a different fixed-point solution from the initial one. This is the so-called fixed-point transitions which reply on the parameter values sensitively. We believe this phenomena could be used to transfer condensates between wells.
In Chapter 4, we discuss the phase separation of atomic and molecular Bose-Einstein condensates trapped in isotropic three-dimensional harmonic potential. Under Feshbach resonance, atoms can be converted into molecules. By find-ing the energy minimum, we determine the ground state phase diagram. The possible ground states are vacuum state, pure molecular superfluid state and mixed atomic-molecular superfluid state. The asymmetry between atoms and molecules in the Feshbach term inhibits the existence of pure atomic superfluid state. Hessian matrix is introduced to determine the stability of each phase. Under Thomas-Fermi approximation, we plot the phase diagram. Due to the correspondence between the parameter space and coordinate space, we plot the density distribution of atomic and molecular condensates. Both prove the ex-istence of phase separation. We also notice the discontinuous transition about some critical interaction strength.
In Chapter 5, we give an experimental proposal to show another approach to increase the atom-molecule conversion efficiency by manipulating the Ramsey type of magnetic field pulses. Ignoring the effect of trapping potential, we de-scribe the system with a pendulum-like model under mean-field approximation. We find the fixed-point solutions, and determine the periodicity of Rabi oscilla-tion for one constant magnetic field. The periodicity diverges when the molecular energy approaches the atomic energy. By applying the Ramsey type of magnetic field pulses, we find that if the separation between the two pulses is appropriately tuned, the atom-molecule conversion efficiency could reach the maximum. The basic idea is about the transition of the system between Rabi oscillation orbitals given by different energy contours. We give a simple and easy-understanding picture to explain how our proposal would work.
In Chapter 6, we put Bose-Einstein condensates in a high-finesse optical cavity, and investigate the influence of photons on the atom-molecule conver-sion efficiency under Feshbach resonance. We consider the dispersive interaction between photons and atoms or molecules. Since photons decay very fast, the corresponding evolution adiabatically follows that of atomic and molecular con-densates. Thus we can get the effective Hamiltonian describing pure atoms and molecules. Based on mean-field approximation, we find the existence of nonlin-ear terms would yield more possible solutions. When the pumping strength is above a critical value, we may observe the bistability phenomena which could be proposed as a way to tune the molecular production rate with photons in cavity. We also give the quantum model and the comparison with the mean-field model is expected.
在第三章中,我们在双阱中研究了双分量玻色-爱因斯坦凝聚体的不动点跃迁问题。由于两分量之间存在异种相互作用,这使得体系的不动点解很复杂。根据其对称性,我们将这些不动点解分为四类：对称解,反对称解,各向同性解,以及不对称解。我们给出了各类解的存在情况随着体系参数的依赖关系。通过对这些不动点解作小量展开,我们讨论了它们的动力学稳定性问题,给出了各类解在参数空间的稳定性相图。我们感兴趣的是这些不动点解在Rosen-Zener型的遂穿强度的绝热调制下的演化行为。这里我们通过外加一个高频的周期调制来引入一个有效的遂穿强度,它的值可以通过改变调制的幅度来调节。我们的数值模拟局限在每个分量在两阱间的相位差均为π。对于初值为对称解时,当最大的有效遂穿强度足够大时,末态则有一定的几率处于另一个对称解上,这依赖于体系参数的具体取值。而对于初值为反对称解的情况,末态的情况则更为复杂,随着最大有效遂穿强度的不同取值,末态可能是对称解和反对称解中的任意一个,甚至可能是超出这些不动点解的混沌态。我们对此做了详细讨论。
在第四章中,我们讨论了在各向同性的三维球谐势中原子-分子混合物的相分离。通过Feshbach共振,我们可以将原子转化为分子,这样我们即在约束势中实现了不对称的两分量玻色-爱因斯坦凝聚。我们通过寻找能量极小值来确定体系的基态,从而确定相应的相图。体系可能的基态只有三种情况,真空态,纯的分子超流态,以及混合的原子-分子超流态。由于转化过程的不对称性,纯的原子超流态是不存在的。我们引入Hessian矩阵来分析每个态的动力学稳定性。在Thomas-Fermi近似下,我们人为定义了两个局域的化学势。在这两个化学势张开的参数空间里,我们给出了体系的基态相图。当原子-分子之间的相互作用改变时,我们在某个特定的临界值附近发现了一个不连续的突变。由于局域化学势空间和约束势实空间之间的映射关系,我们同样给出了原子和分子在实空间的密度分布,并验证了前面的相图分析。
在第五章中,我们提出了一个实验建议,如何操控Ramsey型的磁场脉冲来提高原子-分子转化率。忽略掉约束势的影响,在平均场近似下,我们用一个经典的类摆模型来描述这个体系。通过求得不动点解,我们可以确定体系相应的本征态。我们假设体系的初态全为原子,当磁场为一个定值(单个脉冲)的时候,随着时间演化,体系会在原子态和分子态之间发生拉比振荡。我们给出了振荡周期随能量失谐之间的依赖关系,发现当能量失谐趋于零时,振荡周期趋于发散。相比于单个脉冲实现的分子转化率,利用双脉冲的相同磁场,我们有可能实现比其更高的转化率。在相空间的能量等高图上,我们给出了合理的解释。拉比振荡的轨迹对应于一条能量等高线,而两脉冲之间的时间间隔会使得体系在不同的能量等高线之间跃迁,从而实现高的分子转化率。
在第六章中,我们将玻色-爱因斯坦凝聚体置于光学腔中,并讨论光子对Feshbach共振下原子-分子转化率的影响。当光子频率远大于原子和分子两者各自的跃迁频率，光子与原子和分子之间的相互作用呈弥散型。由于腔中光子的衰减率远大于系统的其他特征频率,因此光子的演化绝热地依赖于原子和分子凝聚体的演化,从而可以得到原子和分子满足的有效哈密顿量。利用平均场近似,我们发现非线性项的存在使得系统可能出现更多的解,尤其是当泵浦激光的强度足够大时,我们会观察到双稳现象。我们提出光子与Feshbach共振模型之间的耦合可以被用来提高原子-分子转化率。我们同样给出了系统相应的量子模型,以对比平均场理论所得到的结果。
This essay focuses on the static and dynamical properties of multi-component Bose-Einstein condensates, including the discussion about double-well system and atom-to-molecule conversion process. In the first two chapters, we briefly review the experimental and theoretical backgrounds of Bose-Einstein condensa-tion, and also introduce the relevant investigations of multi-component conden-sates. Our research achievements correspond to the content from Chapter 3 to Chapter 6 which are listed as follows:
In Chapter 3, we investigate the fixed-point transitions of two-component Bose-Einstein condensates in double-well trapping potentials. Based on the sym-metry between the two kinds of atoms, we classify the fixed-point solutions into four categories:symmetrical, anti-symmetrical, isotropical and asymmetrical. We discuss the existence of each kind of solution and the corresponding dynam-ical stability, and give the stability diagrams in parameter spaces. We apply a high-frequency periodic modulation to the energy bias between the two wells and vary the value of the amplitude to tune the effective tunneling strength. As a result, during the evolution the final state may belong to a different fixed-point solution from the initial one. This is the so-called fixed-point transitions which reply on the parameter values sensitively. We believe this phenomena could be used to transfer condensates between wells.
In Chapter 4, we discuss the phase separation of atomic and molecular Bose-Einstein condensates trapped in isotropic three-dimensional harmonic potential. Under Feshbach resonance, atoms can be converted into molecules. By find-ing the energy minimum, we determine the ground state phase diagram. The possible ground states are vacuum state, pure molecular superfluid state and mixed atomic-molecular superfluid state. The asymmetry between atoms and molecules in the Feshbach term inhibits the existence of pure atomic superfluid state. Hessian matrix is introduced to determine the stability of each phase. Under Thomas-Fermi approximation, we plot the phase diagram. Due to the correspondence between the parameter space and coordinate space, we plot the density distribution of atomic and molecular condensates. Both prove the ex-istence of phase separation. We also notice the discontinuous transition about some critical interaction strength.
In Chapter 5, we give an experimental proposal to show another approach to increase the atom-molecule conversion efficiency by manipulating the Ramsey type of magnetic field pulses. Ignoring the effect of trapping potential, we de-scribe the system with a pendulum-like model under mean-field approximation. We find the fixed-point solutions, and determine the periodicity of Rabi oscilla-tion for one constant magnetic field. The periodicity diverges when the molecular energy approaches the atomic energy. By applying the Ramsey type of magnetic field pulses, we find that if the separation between the two pulses is appropriately tuned, the atom-molecule conversion efficiency could reach the maximum. The basic idea is about the transition of the system between Rabi oscillation orbitals given by different energy contours. We give a simple and easy-understanding picture to explain how our proposal would work.
In Chapter 6, we put Bose-Einstein condensates in a high-finesse optical cavity, and investigate the influence of photons on the atom-molecule conver-sion efficiency under Feshbach resonance. We consider the dispersive interaction between photons and atoms or molecules. Since photons decay very fast, the corresponding evolution adiabatically follows that of atomic and molecular con-densates. Thus we can get the effective Hamiltonian describing pure atoms and molecules. Based on mean-field approximation, we find the existence of nonlin-ear terms would yield more possible solutions. When the pumping strength is above a critical value, we may observe the bistability phenomena which could be proposed as a way to tune the molecular production rate with photons in cavity. We also give the quantum model and the comparison with the mean-field model is expected.
引文
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