多量子比特Rabi模型的解析解及动力学

发布时间：2018-06-17 02:37

本文选题：Rabi模型 + 超强耦合　；参考：《山西大学》2016年博士论文

【摘要】：量子物质和量子谐振子的相互作用在许多量子光学和凝聚态物理系统中发挥了核心作用,其中量子Rabi模型是最简单和最典型的模型之一。在弱耦合的情况下,可以采用旋波近似将Rabi模型转换为Jaynes-Cummings (J-C)模型。近年来,超强耦合区域的实现在基础量子物理学和量子信息的应用方面取得了令人瞩目的成就。量子比特和单模辐射场的耦合系统是研究量子信息过程中必不可缺的资源。本文主要在当前的实验基础上研究多量子比特Rabi模型,着重研究系统的本征解、动力学演化以及几何相。首先,我们研究非全同的两量子比特Rabi模型的解析解。在平移Fock表象中,类似于Rabi模型我们可以利用截断的方法对系统哈密顿量分块对角化。当量子比特跃迁频率远小于单模光场的频率时,在超强耦合区域内零级近似的结果与数值解符合的很好。通过分析量子态的保真度我们发现了同一宇称内部子空间的能级交叉,对于全同的量子比特系统可以用交换对称性来区分这两个简并本征态,从而可以判断该系统是可积的。另外,我们进一步探索了系统的动力学演化问题,其中包括两量子比特的纠缠演化。结果表明,随着耦合强度的增加量子比特处在初始态的几率随时间演化呈现了量子崩塌与复原现象的出现、保持、最后消失的一个过程。并且非全同的量子比特Rabi模型呈现了新的动力学行为,与全同的量子比特系统完全不同。其次,我们研究了Rabi模型的几何曲率和几何相。在旋波近似下,利用规范不变的Berry曲率通过面积分计算系统本征态的Berry相。另外,我们还对该方法进行了扩展,即定义了一般态的几何曲率,并且重点讨论纯真空态诱导的几何曲率,它对应的几何相与一个周期内的平均光子数有关。更进一步,我们研究了非旋波近似下的几何相,发现了一个异常的突变。它意味着绝热近似在能谱的免交叉点是失效的。最后,我们通过绝热近似方法解析地描述三量子比特Dicke模型的动力学特性,与单量子比特和两量子比特有本质的差异。量子比特系统随时间演化的解析公式展示了三个复原序列,进而产生了三个频率波叠加的拍频。在超强耦合区域内,我们以简单、明晰的方式估量了量子比特与量子化单模光场的纠缠和量子比特内部的两体纠缠行为,发现GHZ态对纠缠死亡的抵抗能力比W态强,即GHZ态具有更强的鲁棒性。
[Abstract]:The interaction between quantum matter and quantum harmonic oscillator plays a central role in many quantum optics and condensed matter physical systems, among which the quantum Rabi model is one of the simplest and most typical models. In the case of weak coupling, the Rabi model can be transformed into Jaynes-Cummings J-C model by using the rotational wave approximation. In recent years, the realization of super coupling regions has made remarkable achievements in the application of basic quantum physics and quantum information. The coupling system of quantum bit and single mode radiation field is an indispensable resource in the study of quantum information process. In this paper, we mainly study the multi-quantum bit Rabi model based on the current experiments, focusing on the intrinsic solution, dynamical evolution and geometric phase of the system. First, we study the analytical solutions of two quantum bits Rabi model which is not completely identical. In the translational Fock representation, similar to the Rabi model, we can use the truncation method to diagonalize the Hamiltonian block of the system. When the quantum bit transition frequency is much smaller than that of the single-mode light field, the results of the zero-order approximation in the ultra-strong coupling region are in good agreement with the numerical solution. By analyzing the fidelity of quantum states, we find the energy level crossover in the same parity inner subspace. For the identical quantum bit system, the exchange symmetry can be used to distinguish the two degenerate eigenstates, and it can be judged that the system is integrable. In addition, we further explore the dynamical evolution of the system, including the entanglement evolution of two quantum bits. The results show that with the increase of coupling strength, the probability of quantum bits in the initial state evolves with time, which presents a process of quantum collapse and recovery, keeping and finally disappearing. Moreover, the non-identical quantum bit Rabi model presents a new dynamic behavior, which is completely different from the identical quantum bit system. Secondly, we study the geometric curvature and phase of Rabi model. In the spin wave approximation, the Berry phase of the intrinsic state of the system is calculated by means of the gauge invariant Berry curvature through the area division. In addition, the method is extended to define the geometric curvature of the general state, and to discuss the geometric curvature induced by the pure vacuum state. The geometric phase is related to the average number of photons in a period. Furthermore, we study the geometric phase in the non-rotational wave approximation and find an abnormal mutation. It means that the adiabatic approximation is invalid at the non-cross point of the energy spectrum. Finally, we analytically describe the dynamic properties of the three-quantum bit Dicke model by adiabatic approximation, which is essentially different from the single quantum bit and the two quantum bits. The analytic formula of quantum bit system evolution with time shows three recovery sequences, and then produces the beat frequency of three frequency wave superposition. In the super coupling region, we estimate the entanglement between quantum bit and quantized single-mode light field and two-body entanglement in quantum bit in a simple and clear way. It is found that GHz state has stronger resistance to entanglement death than W state. That is, GHz states have stronger robustness.
【学位授予单位】：山西大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O413

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