厄密和非厄密Bose-Hubbard二聚体中多体量子态的操控
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摘要
单粒子系统的物理性质及其量子操控一直是物理学家关注的焦点,并已取得了很大的进展。近年来,原子囚禁和冷却技术的发展为多体相互作用系统的研究提供了实验基础。光格中超冷原子玻色-爱因斯坦凝聚体(BEC)的实验实现和原子芯片的成功制备是两个典型的例子。它们在量子信息处理等高新技术领域有着潜在的应用价值,也为原子干涉,原子激光,原子二极管和原子三级管等方面的研究提供了较为理想的系统。另一方面,人们发现,被精密操控的光格囚禁的原子系统可用于模拟一些复杂的固态现象,例如：Mott绝缘-超流相变,约瑟夫森效应,安德森局域化以及超流-玻璃相变等。特别是,用原子去模拟电子材料和电路已被认为是一个新兴的领域,这种原子替代品可用于生产新的纳米尺度的装置。
在对相互作用玻色子系统的研究中,量子态的操控是一个最基本的问题,它既是量子相干控制的核心内容,也是量子信息处理的基础。由于BEC是一个便于操控的多体系统,通过激光外场有效控制它的量子态可成为量子操控的范例。因此,本文以一个基于BEC的Bose-Hubbard二聚体为研究对象,分析和数值研究多体量子态的激光操控。该二聚体可以是一个双势阱耦合的BEC或一个两分量BEC。本文的结构如下：
第一章,我们介绍了玻色-爱因斯坦凝聚理论及两模Bose-Hubbard模型。
第二章,我们研究了含时驱动Bose-Hubbard二聚体精确的相干控制。利用平均场近似以及外势和内部相互作用平衡的条件,我们得出了系统精确的Floquet态。对于平衡区域中不同的参数,Floquet态分别精确地描述了宏观量子自囚禁,隧穿的非线性相干增益和破灭。因此可通过适当地调节系统参数去精确地控制凝聚体的宏观量子运动。
第三章,我们研究怎样利用单光子共振去操控Bose-Hubbard二聚体的多体量子态.通过将含时驱动的两模Bose Hubbard哈密顿量中的周期驱动作为微扰,我们给出了一阶和二阶近似下的跃迁几率。我们发现多体量子态的一级量子跃迁遵从一个选择定则,并揭示了选择定则与纠缠熵之间的关系,也就是说,跃迁仅仅只能在不同纠缠熵的态之间发生,禁锢跃迁与两态间的熵平衡相关联。因此我们可以通过利用单光子共振同时去操控系统的多体量子态及纠缠熵。
第四章,我们研究在没有PT对称性的非厄密Bose-Hubbard二聚体中量子态的非相干控制.我们发现,对于一个没有P丁对称性的非厄密两格点Bose-Hubburd哈密顿量,相互作用强度和非厄密性间的平衡条件可导致实能谱的产生,相应的量子态为一个不随时间演化衰减的定态。不同平衡条件导致不同的实能量和非简并定态的产生,改变参数值去满足一个新的平衡条件可产生一个新的实能量和一个新的定态。因此,通过利用Feshbach共振技术调节相互作用强度去满足不同的平衡条件,可增加一个开放多体系统的残存几率并可非相干地控制定态间的量子跃迁。
第五章,我们研究了开放的两分量BEC在Bloch分量表示形式下的动力学及其布居转换控制。我们发现不同的耗散形式导致原子偏向不同的稳定态,特别是,可通过利用耗散和非线性的复合效应去控制凝聚体。例如控制两个超精细能级上凝聚体自囚禁态间的转换,或者控制一个自囚禁态向宏观量子隧穿区域转换。
第六章,结论和展望。整个文章中,我们研究具有粒子间相互作用的Bose-Hubbard二聚体系统中的多体量子态的操控,动力学行为和耗散现象。作者的研究工作主要集中在第二,三,四,五章。
The physical properties and quantum control of single-particle systems always are investigative focus of physicist, which has acquired great advances. In recent years, the development of atom coolling and trapping technology has provided ex-perimental condition for the investigation of interacting many-body system. The realization of Bose-Einstein condensate (BEC) of ultra-cold atoms in optical lat-tices and the creation of atom chips are two typical examples of many-body system. They not only have potential applications in quantum information processing and other advanced technology but also provide an ideal system for the investigation of atom interferometers and lasers, atom diodes and transistors. The precise tailor-ing and manipulation of optical lattices, on the other hand, enable us to investigate complex solidstate phenomena, such as the transition from Mott-insulator to su-perfiuid, Josephson effect, Anderson localization, and the Bose-glass transitions. Especially, it is envisioned that the emerging field of atomtronics, i.e. the atom analog of electronics materials and circuits, will be able to provide nanoscale de-vices of unprecedented quality.
Among all the exciting issues raised in the framework of interacting Bose sys-tems, manipulation of quantum state is a basic issue, which not only is the heart of quantum control, but also is one of major tasks in quantum engineering and quantum information processing. Because BEC is a many-body system charac-terized by convenient manipulation, manipulating its many-body quantum state by adjusting external laser field can become typical example of quantum control. Therefore, this thesis focuses on studying manipulation of many-body quantum state via utilizing external field for a Bose-Hubbard dimer that models a BEC in a double well potential or a two-component BEC. The works are structured as follows:
In the first chapter, we introduce Bose-Einstein condensate theory and two mode Bose-Hubbard model.
In Chapter 2, we initiate the study of exact coherent control to time-dependent driving two mode Bose-Hubbard model. By employing mean-field approximation and balance conditions we construct new exact Floquet solutions with the de-generate Floquet energy, and give the balance region on the parameter space. It is revealed that for some different parameters in the balance region the Floquet solutions exactly describe the macroscopic quantum self-trapping, nonlinear coher-ent construction and destruction of quantum tunneling, respectively. Therefore, we can accurately perform the coherent control by suitably adjusting the system parameters.
In Chapter 3, we show manipulation of many-body quantum states via single-photon resonance for a Bose-Hubbard dimer. By treating the periodic driving in time-dependent driving two mode Bose-Hubbard Hamiltonian as a weak pertur-bation, the transition probabilities up to second-order approximation are given as functions of the driving parameters, which are considerable only for the single-photon resonance case. Due to some transition matrix elements vanishing, the first-order quantum transition obeys a selection rule. The non-forbidden transi-tions involve states of different entanglement entropies and the forbidden transi-tions relate to the entropy balances between two states. The results provide a new route for manipulating many-body quantum states and entanglement entropies, and controlling the atomic tunnelings of the Bose-Hubbard dimer.
In Chapter 4, We investigate incoherent control in a non-Hermitian Bose-Hubbard dimer without PT symmetry. It is analytically and numerically found that a real energy appears uniquely when a balance between the interaction strength and non-Hermiticity is established. The corresponding quantum state is a station-ary state which does not decay in time evolution. Any one of the balance conditions leads to a unique real energy and a nondegenerate stationary state. Changing the parameter values to fit a new balance condition can produce a new real energy and a new stationary state. This provides a route for enhancing survival probability of an open many-body system and for incoherently controlling quantum transition between the stationary states by modulating the interaction strength to fit the different balance conditions.
In Chapter 5, we study the dynamics and population switch of an open two-component Bose-Einstein condensate in the Bloch representation. We find that different dissipation forms lead to different steady states. Specially, one can obtain the switching between the self-trapping states in the two hyperfine levels or induce the switching to the macroscopic quantum tunneling regime by making use of the combined effect between nonlinearity and dissipation.
In the last part of this paper we give a simple summary and discussion to the above-mentioned works and present some expectations in this field
Throughout this thesis we will approach the manipulation of many-body quan-tum state, dynamics and dissipation phenomenon in a Bose-Hubbard dimer. Our main works are involved in chapters two, three, four and five.
在对相互作用玻色子系统的研究中,量子态的操控是一个最基本的问题,它既是量子相干控制的核心内容,也是量子信息处理的基础。由于BEC是一个便于操控的多体系统,通过激光外场有效控制它的量子态可成为量子操控的范例。因此,本文以一个基于BEC的Bose-Hubbard二聚体为研究对象,分析和数值研究多体量子态的激光操控。该二聚体可以是一个双势阱耦合的BEC或一个两分量BEC。本文的结构如下：
第一章,我们介绍了玻色-爱因斯坦凝聚理论及两模Bose-Hubbard模型。
第二章,我们研究了含时驱动Bose-Hubbard二聚体精确的相干控制。利用平均场近似以及外势和内部相互作用平衡的条件,我们得出了系统精确的Floquet态。对于平衡区域中不同的参数,Floquet态分别精确地描述了宏观量子自囚禁,隧穿的非线性相干增益和破灭。因此可通过适当地调节系统参数去精确地控制凝聚体的宏观量子运动。
第三章,我们研究怎样利用单光子共振去操控Bose-Hubbard二聚体的多体量子态.通过将含时驱动的两模Bose Hubbard哈密顿量中的周期驱动作为微扰,我们给出了一阶和二阶近似下的跃迁几率。我们发现多体量子态的一级量子跃迁遵从一个选择定则,并揭示了选择定则与纠缠熵之间的关系,也就是说,跃迁仅仅只能在不同纠缠熵的态之间发生,禁锢跃迁与两态间的熵平衡相关联。因此我们可以通过利用单光子共振同时去操控系统的多体量子态及纠缠熵。
第四章,我们研究在没有PT对称性的非厄密Bose-Hubbard二聚体中量子态的非相干控制.我们发现,对于一个没有P丁对称性的非厄密两格点Bose-Hubburd哈密顿量,相互作用强度和非厄密性间的平衡条件可导致实能谱的产生,相应的量子态为一个不随时间演化衰减的定态。不同平衡条件导致不同的实能量和非简并定态的产生,改变参数值去满足一个新的平衡条件可产生一个新的实能量和一个新的定态。因此,通过利用Feshbach共振技术调节相互作用强度去满足不同的平衡条件,可增加一个开放多体系统的残存几率并可非相干地控制定态间的量子跃迁。
第五章,我们研究了开放的两分量BEC在Bloch分量表示形式下的动力学及其布居转换控制。我们发现不同的耗散形式导致原子偏向不同的稳定态,特别是,可通过利用耗散和非线性的复合效应去控制凝聚体。例如控制两个超精细能级上凝聚体自囚禁态间的转换,或者控制一个自囚禁态向宏观量子隧穿区域转换。
第六章,结论和展望。整个文章中,我们研究具有粒子间相互作用的Bose-Hubbard二聚体系统中的多体量子态的操控,动力学行为和耗散现象。作者的研究工作主要集中在第二,三,四,五章。
The physical properties and quantum control of single-particle systems always are investigative focus of physicist, which has acquired great advances. In recent years, the development of atom coolling and trapping technology has provided ex-perimental condition for the investigation of interacting many-body system. The realization of Bose-Einstein condensate (BEC) of ultra-cold atoms in optical lat-tices and the creation of atom chips are two typical examples of many-body system. They not only have potential applications in quantum information processing and other advanced technology but also provide an ideal system for the investigation of atom interferometers and lasers, atom diodes and transistors. The precise tailor-ing and manipulation of optical lattices, on the other hand, enable us to investigate complex solidstate phenomena, such as the transition from Mott-insulator to su-perfiuid, Josephson effect, Anderson localization, and the Bose-glass transitions. Especially, it is envisioned that the emerging field of atomtronics, i.e. the atom analog of electronics materials and circuits, will be able to provide nanoscale de-vices of unprecedented quality.
Among all the exciting issues raised in the framework of interacting Bose sys-tems, manipulation of quantum state is a basic issue, which not only is the heart of quantum control, but also is one of major tasks in quantum engineering and quantum information processing. Because BEC is a many-body system charac-terized by convenient manipulation, manipulating its many-body quantum state by adjusting external laser field can become typical example of quantum control. Therefore, this thesis focuses on studying manipulation of many-body quantum state via utilizing external field for a Bose-Hubbard dimer that models a BEC in a double well potential or a two-component BEC. The works are structured as follows:
In the first chapter, we introduce Bose-Einstein condensate theory and two mode Bose-Hubbard model.
In Chapter 2, we initiate the study of exact coherent control to time-dependent driving two mode Bose-Hubbard model. By employing mean-field approximation and balance conditions we construct new exact Floquet solutions with the de-generate Floquet energy, and give the balance region on the parameter space. It is revealed that for some different parameters in the balance region the Floquet solutions exactly describe the macroscopic quantum self-trapping, nonlinear coher-ent construction and destruction of quantum tunneling, respectively. Therefore, we can accurately perform the coherent control by suitably adjusting the system parameters.
In Chapter 3, we show manipulation of many-body quantum states via single-photon resonance for a Bose-Hubbard dimer. By treating the periodic driving in time-dependent driving two mode Bose-Hubbard Hamiltonian as a weak pertur-bation, the transition probabilities up to second-order approximation are given as functions of the driving parameters, which are considerable only for the single-photon resonance case. Due to some transition matrix elements vanishing, the first-order quantum transition obeys a selection rule. The non-forbidden transi-tions involve states of different entanglement entropies and the forbidden transi-tions relate to the entropy balances between two states. The results provide a new route for manipulating many-body quantum states and entanglement entropies, and controlling the atomic tunnelings of the Bose-Hubbard dimer.
In Chapter 4, We investigate incoherent control in a non-Hermitian Bose-Hubbard dimer without PT symmetry. It is analytically and numerically found that a real energy appears uniquely when a balance between the interaction strength and non-Hermiticity is established. The corresponding quantum state is a station-ary state which does not decay in time evolution. Any one of the balance conditions leads to a unique real energy and a nondegenerate stationary state. Changing the parameter values to fit a new balance condition can produce a new real energy and a new stationary state. This provides a route for enhancing survival probability of an open many-body system and for incoherently controlling quantum transition between the stationary states by modulating the interaction strength to fit the different balance conditions.
In Chapter 5, we study the dynamics and population switch of an open two-component Bose-Einstein condensate in the Bloch representation. We find that different dissipation forms lead to different steady states. Specially, one can obtain the switching between the self-trapping states in the two hyperfine levels or induce the switching to the macroscopic quantum tunneling regime by making use of the combined effect between nonlinearity and dissipation.
In the last part of this paper we give a simple summary and discussion to the above-mentioned works and present some expectations in this field
Throughout this thesis we will approach the manipulation of many-body quan-tum state, dynamics and dissipation phenomenon in a Bose-Hubbard dimer. Our main works are involved in chapters two, three, four and five.
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