玻色—爱因斯坦凝聚体中孤子和干涉的理论研究
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摘要
自从玻色‐爱因斯坦凝聚体在稀薄碱金属气体中实现后，人们在玻色‐爱因斯坦凝聚体方向上做了大量的理论和实验研究，例如在凝聚体中关于孤立子、涡旋、约瑟夫森震荡、混沌、对称性破缺的研究。另一方面玻色‐爱因斯坦凝聚体具有非常好的相干性可以作为理想相干光源来用于研究物质波的干涉。现在大家都采用平均场近似理论研究玻色‐爱因斯坦凝聚体，这种理论为研究玻色‐爱因斯坦凝聚体性质提供了方向。我们采用平均场理论研究玻色‐爱因斯坦凝聚体中孤子和干涉行为，具体工作分成三部分。
第一，旋量玻色‐爱因斯坦凝聚体具有内部自由度，这种自由度来源于原子自旋。当旋量玻色‐爱因斯坦凝聚体被囚禁在磁势阱中时，其自旋自由被冻结，即所有原子只有一个自由度，此时可形成单组分玻色‐爱因斯坦凝聚体。然而当旋量玻色‐爱因斯坦凝聚体囚禁在光势阱中，其自旋自由度被释放。这里我们只考虑spin-1的玻色凝聚体系。数值求解spin-1的玻色‐爱因斯坦凝聚在谐振子势阱中的一维含时GP方程。结果表明，当GP方程的耦合参数为负时，我们在spin-1的玻色‐爱因斯坦凝聚体的单组分、双组分和三组分解中发现了孤子呼吸子解。当散射长度随时间变化时，在凝聚体的单组分态上可形成孤子呼吸子。在双组分态中具有种间排斥相互作用和种内吸引相互作用的两团凝聚体发生弹性碰撞，此时凝聚体中出现了孤子呼吸子；没有种间相互作用的两团凝聚体形成了稳定的亮孤子；具有种间排斥相互作用和种内排斥相互作用的两团凝聚体发生非弹性碰撞，此时凝聚体中没有孤子产生。三组分态在吸引相互作用参数条件下，三团凝聚体中都出现了孤子呼吸子，并且出现的自旋交换反应。此时之外，在本论文中我们还给出了这些孤子呼吸子产生的原因。
第二，柱状势阱中双组分玻色‐爱因斯坦凝聚体的动力学可通过数值求解耦合GP方程来实现。我们展示了由于种间相互作用和初始状态中不同比例的双组分原子混合可导致不同数目的环状暗孤子同时在双组分中产生。而这些孤子在密度最低处或者密度为零处会有相位突变，这些相突变点处的粒子具有非常大的运动速度，如果伴随非常大的几率流密度则此处为灰孤子，如果其几率流密度为零则此处为暗孤子。这些孤子都是不稳定的，会经过非常短的时间就会演化成其他的孤子态。
最后，两团玻色-爱因斯坦凝聚体的干涉可以通过数值求解一维含时GP方程来实现。两团玻色爱因斯坦凝聚体初始时被置放在由两个在不同位置截断的谐振子势阱组成的双势阱中。在本文中我们展示了两团玻色爱因斯坦凝聚体可以形成周期性的干涉，并且当两团凝聚体重叠时，则会出现暗孤立子。两团凝聚体的周期性干涉可以用一维谐振子波函数的来解释，我们发现由于原子间的相互作用部分原子会在基态与激发态中进行跃迁，这种跃迁与凝聚体的能量变化一致。
Since the realization of Bose-Einstein condensates (BECs) in dilute alkali-metalgases, lots of theoretical and experimental studies of BECs have been reported such assolitons, vortices, josephson oscillation, chaos and symmetry breaking. In anotherregard, the Bose-Einstein condensates have long coherence time and highcontrollability, they as ideal coherent sources have been widely used to study matterwave interference. In the mean-field approximation, the condensates may bedescribed by a macroscopic wavefunction. This theory paves a way toward studyingthe properties of condensates theoretically. In this thesis, we theoretically investigatethe soliton and interference in Bose-Einstein condensates. Our detailed research workcan be divided into three sections.
In the first section, spinor Bose–Einstein condensates has internal degree offreedom, this freedom is due to the hyperfine spin of atom. When spinor condensatesare trapped in a magnetic potential, the spin degree of freedom is frozen. While whenspinor condensates are trapped in an optical potential, the spin degree of freedomcomes into play. Here we consider spin-1Bose–Einstein condensates which aretrapped in a harmonic potential with different nonlinearity coefficients. We illustratethe dynamics of soliton breathers in one-component, two-component andthree-component states by numerically solving one-dimensional time-dependentcoupled Gross–Pitaecskii (GP) equations. The condensate in one-component stateform soliton breather when scattering length are changed with time. In thetwo-component state, two condensates with repulsive interspecies interactions and attractive interaction make elastic collision and novel soliton breathers are created; thetwo condensates without interspecies interaction form stable bright soliton; the twocondensates with repulsive interspecies interaction and repulsive intraspeciesinteraction make inelastic collision and no soliton created in two condensates. Inthree-component state soliton breathers are generated in three codensates, spinorexchange collision is found. Besides, possible reasons for creating those solitonbreathers are discussed in thesis.
In the second section, we demonstrate the the dynamical evolution oftwo-component Bose–Einstein condensates which are trapped in a cylindrical well bysolving the coupled GP equations numerically. We present that, due to intercomponentinteraction and different initial component populations, different numbers of ring darksolitons are generated in two components at the same time. These solitons all havedensity zeros (minima) accompanied with phase jumps, the phase jumps can causelarge superfluid velocities. At phase jump points those ring dark solitons have zerosuperfluid currents, while those ring gray solitons have large superfluid currents. Allsolitons are unstable and will evolve into other soliton states after a brief time.
In the end, we study the interference between two condensates with repulsiveinteraction, which is investigated by numerical solution of the one-dimensionaltime-dependent GP equation. The two condensates are initially prepared in adouble-well potential. This potential consists two truncated harmonic wells centeredat different positions. The two condensates can form periodic interference pattern,while dark solitons are generated when two condensates overlap. The evolution oftwo condensates is described by the ground state and excited states ofone-dimensional harmonic oscillator wavefunctions. Due to the existence ofatom-atom interactions, many atoms are transferred among ground state and excitedstates, atom transfer coincides with the variation of the energy of system.
第一，旋量玻色‐爱因斯坦凝聚体具有内部自由度，这种自由度来源于原子自旋。当旋量玻色‐爱因斯坦凝聚体被囚禁在磁势阱中时，其自旋自由被冻结，即所有原子只有一个自由度，此时可形成单组分玻色‐爱因斯坦凝聚体。然而当旋量玻色‐爱因斯坦凝聚体囚禁在光势阱中，其自旋自由度被释放。这里我们只考虑spin-1的玻色凝聚体系。数值求解spin-1的玻色‐爱因斯坦凝聚在谐振子势阱中的一维含时GP方程。结果表明，当GP方程的耦合参数为负时，我们在spin-1的玻色‐爱因斯坦凝聚体的单组分、双组分和三组分解中发现了孤子呼吸子解。当散射长度随时间变化时，在凝聚体的单组分态上可形成孤子呼吸子。在双组分态中具有种间排斥相互作用和种内吸引相互作用的两团凝聚体发生弹性碰撞，此时凝聚体中出现了孤子呼吸子；没有种间相互作用的两团凝聚体形成了稳定的亮孤子；具有种间排斥相互作用和种内排斥相互作用的两团凝聚体发生非弹性碰撞，此时凝聚体中没有孤子产生。三组分态在吸引相互作用参数条件下，三团凝聚体中都出现了孤子呼吸子，并且出现的自旋交换反应。此时之外，在本论文中我们还给出了这些孤子呼吸子产生的原因。
第二，柱状势阱中双组分玻色‐爱因斯坦凝聚体的动力学可通过数值求解耦合GP方程来实现。我们展示了由于种间相互作用和初始状态中不同比例的双组分原子混合可导致不同数目的环状暗孤子同时在双组分中产生。而这些孤子在密度最低处或者密度为零处会有相位突变，这些相突变点处的粒子具有非常大的运动速度，如果伴随非常大的几率流密度则此处为灰孤子，如果其几率流密度为零则此处为暗孤子。这些孤子都是不稳定的，会经过非常短的时间就会演化成其他的孤子态。
最后，两团玻色-爱因斯坦凝聚体的干涉可以通过数值求解一维含时GP方程来实现。两团玻色爱因斯坦凝聚体初始时被置放在由两个在不同位置截断的谐振子势阱组成的双势阱中。在本文中我们展示了两团玻色爱因斯坦凝聚体可以形成周期性的干涉，并且当两团凝聚体重叠时，则会出现暗孤立子。两团凝聚体的周期性干涉可以用一维谐振子波函数的来解释，我们发现由于原子间的相互作用部分原子会在基态与激发态中进行跃迁，这种跃迁与凝聚体的能量变化一致。
Since the realization of Bose-Einstein condensates (BECs) in dilute alkali-metalgases, lots of theoretical and experimental studies of BECs have been reported such assolitons, vortices, josephson oscillation, chaos and symmetry breaking. In anotherregard, the Bose-Einstein condensates have long coherence time and highcontrollability, they as ideal coherent sources have been widely used to study matterwave interference. In the mean-field approximation, the condensates may bedescribed by a macroscopic wavefunction. This theory paves a way toward studyingthe properties of condensates theoretically. In this thesis, we theoretically investigatethe soliton and interference in Bose-Einstein condensates. Our detailed research workcan be divided into three sections.
In the first section, spinor Bose–Einstein condensates has internal degree offreedom, this freedom is due to the hyperfine spin of atom. When spinor condensatesare trapped in a magnetic potential, the spin degree of freedom is frozen. While whenspinor condensates are trapped in an optical potential, the spin degree of freedomcomes into play. Here we consider spin-1Bose–Einstein condensates which aretrapped in a harmonic potential with different nonlinearity coefficients. We illustratethe dynamics of soliton breathers in one-component, two-component andthree-component states by numerically solving one-dimensional time-dependentcoupled Gross–Pitaecskii (GP) equations. The condensate in one-component stateform soliton breather when scattering length are changed with time. In thetwo-component state, two condensates with repulsive interspecies interactions and attractive interaction make elastic collision and novel soliton breathers are created; thetwo condensates without interspecies interaction form stable bright soliton; the twocondensates with repulsive interspecies interaction and repulsive intraspeciesinteraction make inelastic collision and no soliton created in two condensates. Inthree-component state soliton breathers are generated in three codensates, spinorexchange collision is found. Besides, possible reasons for creating those solitonbreathers are discussed in thesis.
In the second section, we demonstrate the the dynamical evolution oftwo-component Bose–Einstein condensates which are trapped in a cylindrical well bysolving the coupled GP equations numerically. We present that, due to intercomponentinteraction and different initial component populations, different numbers of ring darksolitons are generated in two components at the same time. These solitons all havedensity zeros (minima) accompanied with phase jumps, the phase jumps can causelarge superfluid velocities. At phase jump points those ring dark solitons have zerosuperfluid currents, while those ring gray solitons have large superfluid currents. Allsolitons are unstable and will evolve into other soliton states after a brief time.
In the end, we study the interference between two condensates with repulsiveinteraction, which is investigated by numerical solution of the one-dimensionaltime-dependent GP equation. The two condensates are initially prepared in adouble-well potential. This potential consists two truncated harmonic wells centeredat different positions. The two condensates can form periodic interference pattern,while dark solitons are generated when two condensates overlap. The evolution oftwo condensates is described by the ground state and excited states ofone-dimensional harmonic oscillator wavefunctions. Due to the existence ofatom-atom interactions, many atoms are transferred among ground state and excitedstates, atom transfer coincides with the variation of the energy of system.
引文
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