自旋轨道耦合冷原子费米气体中的量子效应及其应用
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摘要
由于最近几年实验的飞速进展，自旋轨道耦合冷原子气体引起了人们极大的关注。有了自旋轨道耦合效应之后，冷原子气体会表现出一些全新的性质。人们预言在这个体系中会出现拓扑超流态以及Marjorana费米子。同时，自旋轨道耦合效应的引入，也使得人们可以利用冷原子这样一个非常易于操控的体系，来模拟拓扑绝缘体等其他重要的量子效应。自旋轨道耦合的冷原子气体也成为了实现拓扑量子计算的重要平台。
本文主要研究了自旋轨道耦合冷原子费米气体的量子效应，并且涉及了利用开放量子体系实现量子计算的一些理论问题。
首先，我们研究了单通道模型处理Feshbach共振附近的自旋轨道耦合冷原子费米气体的有效性问题。在不存在自旋轨道耦合时，人们认为当开通道和闭通道的耦合强度很大时，单通道模型和双通道模型等价。我们通过对比存在自旋轨道耦合时两个模型的处理结果，发现此时两个模型等价的条件不仅与开通道和闭通道的耦合强度有关，还与自旋轨道耦合强度相关。我们给出了新的两个模型等价的条件，并且提出了实验上检验的方法。
然后我们研究了二维光晶格中的自旋轨道耦合费米气体，给出了这个体系具有动量分辨能力的射频谱的性质。这种射频谱信号，可以在实验中有效的探测到，并且可以利用它得到体系单粒子能量以及能量本征态的信息。另外，我们通过这种射频谱的信号，发现了体系中一个有趣的自旋动量反转对称性，并在理论中给出了证明。
在开放量子体系的量子信息处理方面，我们首先研究了利用开放量子体系实现量子计算的一种模型：对偶量子计算，对其给出了一个规范统一的描述。然后我们研究了量子操作算子按照幺正矩阵线性展开的性质，给出了相消干涉最小的展开方式以及幺正矩阵个数最小的展开方式。另外我们研究一个一般的量子操作和幺正操作的区别的度量，我们把这种区别叫做这个量子操作的非幺正性。它的大小反映了这个量子操作噪声的强度。我们提出了量子操作非幺正性的度量方式，给出了单比特常见量子信道的计算结果，并且研究了量子操作非幺正性分布及其随环境维度大小的影响。
Following unprecedented experimental development, spin-orbit coupled atomicgases have attracted much attention in recent years. Due to the spin-orbit coupling ef-fect, atomic gases exhibit many new important properties. Topological superfluid andMajorana fermions have been predicted in this system. Also, as spin-orbit atomic gasesare easy to control in experiment, they can be used to simulate other important phys-ical phenomena, such as topological insulators. They are also important resources fortopological quantum computing.
In this thesis, we mainly study the quantum effect of ultracold spin-orbit atomicFermi gases. Some theoretical problems related to quantum computing in open quantumsystems will also be touched upon.
Firstly, we study the validity of the single channel model for a spin-orbit-coupledatomicFermigasnearFeshbachresonances. Itiswidelyaccepted, incaseswithoutspin-orbitcoupling, that when the channel couplingbetweentheclosedandtheopenchannelsis strong, the two-channel model is equivalent to the single-channel model. However,in the presence of spin-orbit coupling, we find that the condition for the equivalencebecomes much more stringent and is related to the strength of the spin-orbit coupling.We give new criteria for the equivalence in the presence of spin-orbit coupling and givea scheme for experiment testing.
Secondly, we study the momentum-resolved radio frequency spectroscopy of spin-orbit atomic Fermi gases in a two-dimension optical lattice. Momentum-resolved ra-dio frequency spectroscopy is a powerful tool to probe single-particle energies andeigenstates. We also find an interesting spin-momentum reversed symmetry from themomentum-resolved radio frequency spectroscopy and prove it theoretically.
In the field of quantum computing in open quantum systems, we first study a circuitmodel, i.e., duality quantum computer model, and give it a unified description. Then westudy the properties of a Kraus operator as a linear combination of unitary matrices. Wefind the expansion with the maximum constructive interference and also the expansionrequiring the minimum number of unitary matrices. Finally, we study the measure of thedistance between a general quantum operation and the unitary operations, which we callthe non-unitarity of the quantum operation. This also qualifies the noise of the quantum operation. We propose a new measure for the non-unitarity of a quantum operation andgivetheresultsforsomeimportantsinglequbitquantumchannels. Wealsostudythedis-tribution of the non-unitarity of quantum operations and the its relation to the dimensionof the environment.
本文主要研究了自旋轨道耦合冷原子费米气体的量子效应，并且涉及了利用开放量子体系实现量子计算的一些理论问题。
首先，我们研究了单通道模型处理Feshbach共振附近的自旋轨道耦合冷原子费米气体的有效性问题。在不存在自旋轨道耦合时，人们认为当开通道和闭通道的耦合强度很大时，单通道模型和双通道模型等价。我们通过对比存在自旋轨道耦合时两个模型的处理结果，发现此时两个模型等价的条件不仅与开通道和闭通道的耦合强度有关，还与自旋轨道耦合强度相关。我们给出了新的两个模型等价的条件，并且提出了实验上检验的方法。
然后我们研究了二维光晶格中的自旋轨道耦合费米气体，给出了这个体系具有动量分辨能力的射频谱的性质。这种射频谱信号，可以在实验中有效的探测到，并且可以利用它得到体系单粒子能量以及能量本征态的信息。另外，我们通过这种射频谱的信号，发现了体系中一个有趣的自旋动量反转对称性，并在理论中给出了证明。
在开放量子体系的量子信息处理方面，我们首先研究了利用开放量子体系实现量子计算的一种模型：对偶量子计算，对其给出了一个规范统一的描述。然后我们研究了量子操作算子按照幺正矩阵线性展开的性质，给出了相消干涉最小的展开方式以及幺正矩阵个数最小的展开方式。另外我们研究一个一般的量子操作和幺正操作的区别的度量，我们把这种区别叫做这个量子操作的非幺正性。它的大小反映了这个量子操作噪声的强度。我们提出了量子操作非幺正性的度量方式，给出了单比特常见量子信道的计算结果，并且研究了量子操作非幺正性分布及其随环境维度大小的影响。
Following unprecedented experimental development, spin-orbit coupled atomicgases have attracted much attention in recent years. Due to the spin-orbit coupling ef-fect, atomic gases exhibit many new important properties. Topological superfluid andMajorana fermions have been predicted in this system. Also, as spin-orbit atomic gasesare easy to control in experiment, they can be used to simulate other important phys-ical phenomena, such as topological insulators. They are also important resources fortopological quantum computing.
In this thesis, we mainly study the quantum effect of ultracold spin-orbit atomicFermi gases. Some theoretical problems related to quantum computing in open quantumsystems will also be touched upon.
Firstly, we study the validity of the single channel model for a spin-orbit-coupledatomicFermigasnearFeshbachresonances. Itiswidelyaccepted, incaseswithoutspin-orbitcoupling, that when the channel couplingbetweentheclosedandtheopenchannelsis strong, the two-channel model is equivalent to the single-channel model. However,in the presence of spin-orbit coupling, we find that the condition for the equivalencebecomes much more stringent and is related to the strength of the spin-orbit coupling.We give new criteria for the equivalence in the presence of spin-orbit coupling and givea scheme for experiment testing.
Secondly, we study the momentum-resolved radio frequency spectroscopy of spin-orbit atomic Fermi gases in a two-dimension optical lattice. Momentum-resolved ra-dio frequency spectroscopy is a powerful tool to probe single-particle energies andeigenstates. We also find an interesting spin-momentum reversed symmetry from themomentum-resolved radio frequency spectroscopy and prove it theoretically.
In the field of quantum computing in open quantum systems, we first study a circuitmodel, i.e., duality quantum computer model, and give it a unified description. Then westudy the properties of a Kraus operator as a linear combination of unitary matrices. Wefind the expansion with the maximum constructive interference and also the expansionrequiring the minimum number of unitary matrices. Finally, we study the measure of thedistance between a general quantum operation and the unitary operations, which we callthe non-unitarity of the quantum operation. This also qualifies the noise of the quantum operation. We propose a new measure for the non-unitarity of a quantum operation andgivetheresultsforsomeimportantsinglequbitquantumchannels. Wealsostudythedis-tribution of the non-unitarity of quantum operations and the its relation to the dimensionof the environment.
引文
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