量子拉比模型及其相关模型中的极化子图像研究 Alternative Title Polaron Picture for the Quantum Rabi Model and its Related Models 丛磊 Thesis Advisor 罗洪刚 2018-04-01 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 光和物质相互作用 极化子方法 量子拉比模型 两光子量子拉比模型 线性和非线性的量子拉比模型 Abstract 光和物质相互作用一直以来都是人们感兴趣的话题。近年来，人工原子（超导比特，N-V色心，量子点等）与腔场相互作用的实验系统提供了研究这一问题的新平台。随着实验技术的进步，新的现象推动着理论上描述光-物质相互作用的最简单的模型——量子拉比模型的发展。对于这一模型及其推广模型的研究不仅有助于我们更好的理解已有的实验现象，而且有助于我们更好的理解光-物质相互作用的本质。在本文中，我们使用一种新的理论方法来求解基本的描述光和物质相互作用的模型，本文将这一方法称之为极化子方法。这篇文章的主要内容是展示我们是如何应用极化子方法来求解描述不同的光和物质相互作用的理论模型。首先，我们将介绍应用于量子拉比模型的已有的主要研究方法。其次，我们将介绍应用于量子拉比模型的极化子方法。再次，我们将介绍如何应用这一方法研究两光子量子拉比模型。最后，我们将在一个全新的理论模型，即线性和非线性混合的量子拉比模型中看到极化子方法对于这一模型中物理现象的理解和在此基础之上求得的解析结果。 首先，我们回顾了已有的对量子拉比模型进行求解的主要方法和结果。直接的数值方法，可以计算体系的能谱，并进一步研究量子拉比模型能谱的统计和动力学性质。而近似方法的研究，如广义旋波近似，广义变分方法和平均光子数相关的变分方法可以在不同情况下描述体系的物理性质，从而在部分参数区域或整个参数区域内取得较好的结果。值得强调的是最近几年在量子拉比模型的解析精确求解上取得的重要进展，新的理论方法在很大程度上促进了对量子拉比模型及其相关模型的进一步研究。 其次，在对量子拉比模型的求解过程中，我们科研小组最早提出极化子方法这一解析方法。这一方法包括两个部分，第一步是对体系物理图像的分析，体系中各项的作用都逐一得到了体现。我们发现这一模型中的单光子耦合项可以使光场项所决定的与自旋上和下态相关的势阱在位置空间中发生分离。而隧穿项，则会在相互分离的势阱中分别诱导出一个附加的势阱。第二步，基于前面的极化子图像，我们可以对模型做进一步解析求解。如构建一个包含极化子与反极化子的基态试探波函数，并通过变分方法加以求解，以此来研究体系基态性质。而且，通过进一步考虑高阶的隧穿过程，我们的两极化子方法还可以推广为系统的多极化子方法。极化子方法用于量子拉比模型有两点优势，一是相比于已有的一些近似方法，我们的多极化子试探态获得的结果适用于整个参数区域，二是相比于传统的相干态展开方法，我们的方法可以更为高效的得到结果。 再次，我们使用极化子方法来研究两光子量子拉比模型。我们发现极化子方法同样适用于这一描述非线性光和物质耦合的理论模型。在这一模型中，极化子方法揭示出的两光子量子拉比模型的物理图像与量子拉比模型中的相比，有不同之处，也有相通的地方。不同之处在于，双光子耦合项可以使得光场项所决定的与自旋上和下态相关的势阱在频率空间中发生分离；相通之处在于，隧穿过程仍然可以理解为在量子拉比模型中所说的两个势阱之间相互交换组份。基于这样的理解，我们可以准确构建出体系基态，并计算得到体系准确的可观测量。此外，极化子方法揭示出的体系物理图像还可用于对这一模型中存在的独特的谱崩塌问题提供新的理解。与传统的解释相比，新的理解充分考虑了隧穿诱导过程的作用，从而更加准确的解释了体系的能谱的定性特征。极化子方法在这一模型中的推广应用，不仅揭示了隧穿诱导的思想在光和物质耦合模型之中的重要作用，还启发了我们极化子方法可能还适用于更多的相关模型。 最后，我们研究一个新的同时包含单光子过程和双光子过程的量子拉比模型。在这一新模型中，同时包含了线性和非线性的光和物质耦合。此前，虽然在实际体系中线性过程和非线性过程往往同时出现，但是由于后者的效果较弱而被忽略。在这一模型中，我们发现当线性光和物质耦合强度迈入强耦合区域时，一个很弱的非线性耦合会导致体系发生巨大转变，造成诸如自发对称破缺、相变等现象。 我们进一步的研究表明，在弱线性耦合下，一个较强的非线性耦合也会造成体系的对称破缺，形成相变。经过全面的研究，我们在这一模型中发现了三种相变机制，在有限频率之下，他们之间会有明显区分，而在低频极限下，所有的相边界会融合在一起。在这一模型中，极化子方法提供的位移与频移共存的极化子图像帮助我们准确的理解了体系基态的性质。在同时考虑隧穿能量与势能的竞争之后，极化子方法给出了一个低频极限下的相边界，与数值的边界符合得很好。极化子方法对理解这一模型中的丰富的物理结果起到了重要作用，同时也加深了人们对光和物质相互作用模型，特别是通常被认为是可忽略的非线性过程作用的重要性的认识。 本文使用极化子方法对光和物质相互作用模型进行了求解，这不仅体现了极化子方法的广泛适用性， 同时也增进了人们对这些理论模型的理解。 Other Abstract Light-matter interaction is a continuously fascinating research area. The system which contains an artificial atom (superconducting qubit, N-V centre, quantum dot) and a cavity field provides a good platform to research this topic. With the advanced experimental technologies, the new phenomena drive the theoretical study of the quantum Rabi model (QRM) which is the simplest model for the light-matter interaction system. Study of this model and it's generations not only contribute to the understanding of the experimental results, but also help us to understand the physical insights of those systems. In this thesis, we provide a new theoretical method for these basic models which describe the light-matter interaction. Firstly, we review the existing research of the quantum Rabi model. Secondly, we introduce the polaron method for the QRM. Thirdly, we extend this method to the two-photon QRM. Finally, we show how polaron method helps us to understand the rich physical phenomena inside a new theoretical model which contains the linear and nonlinear light-matter interaction. And the polaron method also gives out an analytical solution to this model. Firstly, we review the existing main methods and results of the quantum Rabi model. In the eyes of a direct numerical method, the energy spectrum of the system can be calculated, thus the statistical and dynamic properties of the energy spectrum of the quantum Rabi model can be further studied. In addition, approximation methods, such as the generalized rotational wave approximation, the generalized variational method and the mean photon number dependent variational method, show good results in some parameter regions or in the entire parameter region. Most importantly, significant research progress has been made in the analytical solution of the quantum Rabi model in recent years. The newly developed theoretical analytical method has greatly promoted further research on the quantum Rabi model and its related models. Secondly, our team proposed the polaron method with solving the QRM. The polaron method contains two steps. The first one is to provide an analysis of the physical picture where each term of the system plays their own role. We found the linear light-matter interaction causes the spin-state related potential wells separating in the position space. In the meanwhile, the tunneling term induces an additional potential well for each of those two potential wells. The second step is to give the analytical solution of the model based on the previous polaron picture, such as constructing a trial state with polaron and antipolaron to study the properties of the ground state. The two-polaron method can also be extended to be a multi-polaron method by taking the high order tunneling process into consideration. The polaron method has two advantages when applied to the QRM. Firstly, when compared to the existed approximate methods, the polaron method works for the whole coupling regime. Secondly, when compared to the coherent state expansion method, the polaron method can reach a better result more efficiently. Thirdly, we apply the polaron method to the two-photon QRM. It is found that this method also works for the nonlinear model. The difference and similarity between polaron picture here and the previous one is revealed. The difference is that the light-matter interaction term here causes the potential wells separating in the frequency space instead of the position space. The similarity is that the tunneling term also plays a role that helps the two separated potential exchange their own components. Based on such understanding, we can construct the ground state of the system and calculate observables accurately. Moreover, polaron method provides special understanding of the spectral collapse problem based on the understanding of the tunneling induced potential well. The polaron method takes the tunneling inducing process into consideration and thus provides better qualitative understanding of the spectral collapse behavior compared to the existed understanding. The successful application of the polaron method in this model reflects the importance of the idea about tunneling inducing process. It also inspires us to find more application of the polaron method. Finally, we study a new generalized QRM which contains the single-photon and two-photon interaction at the same time. This model is important because it takes the linear and nonlinear light-matter interaction term into consideration at the same time. Linear and nonlinear light-matter interaction process actually appear together in real systems, however the latter is usually neglected because the nonlinear process is believed much weaker than the former. In this new mixed model, it is found that the nonlinear light-matter interaction plays an important role while the linear light-matter coupling enters the strong coupling regime. A weak nonlinear coupling causes great changes of the system properties and leads to the spontaneous symmetry breaking. Further research shows that a strong nonlinear can also lead to the symmetry breaking (phase transition) while the linear coupling is weak. A full research reveals three different phase transition boundary inside this model. They show obvious differences in a finite $\omega$ and merge when $\omega \rightarrow 0$. The physical picture which contains displaced and frequency shifted potential wells contributes to the understanding of this model. Considering the competition between potential energy and tunneling energy, the polaron method can provide an analytical phase transition boundary when $\omega \rightarrow 0$ which matches the numeric result very well. The polaorn method plays an important role for the understanding of the rich physical phenomena, and deepens the knowledge of the models which describe the light-matter interaction, especially with the nonlinear process inside. In this thesis, we show that using the polaron method to study models which describe light-matter interaction can not only contribute to the understanding of these models, but also indicate that the polaron method has general applicability. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/229599 Collection 物理科学与技术学院 Recommended CitationGB/T 7714 丛磊. 量子拉比模型及其相关模型中的极化子图像研究[D]. 兰州. 兰州大学,2018.
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