arXiv 2304.136562023-04quant-ph

计算驱动非线性振荡器有效哈密顿量的图解方法

A diagrammatic method to compute the effective Hamiltonian of driven nonlinear oscillators

Xu Xiao, Jayameenakshi Venkatraman, Rodrigo G. Cortiñas, Shoumik Chowdhury, Michel H. Devoret

AI总结：

提出一种基于相空间自洽微扰展开的图解方法计算驱动非线性振荡器有效哈密顿量，提供生成任意阶哈密顿量的程序，并通过超导电路实例与数值对比验证了其精确性与多功能性。

摘要：

在本工作中，我们提出了一种新的图解方法，用于计算驱动非线性振荡器的有效哈密顿量。我们方法的核心是在相空间中发展的一种自洽微扰展开，它建立了图与代数之间的直接对应关系。每个图对应一个哈密顿量项，其前置因子与 Feynman 图一样，涉及对拓扑等价图的简单计数。利用图解方法的算法简单性，我们提供了一个现成的计算机程序，可生成任意阶的有效哈密顿量。我们展示了我们的方案与 Schrieffer-Wolff 方法等现有微扰方法的一致性。此外，在 $\hbar\rightarrow0$ 的极限下，我们从结果中恢复了经典谐波平衡方案。我们的方法有助于理解量子系统内的动态控制，并达到了推进未来量子信息处理器所必需的精度。为了证明其价值和多功能性，我们分析了来自超导电路领域的五个例子。这些例子包括三足 Schrödinger cat 哈密顿量稳定的实验提案、超导电路实验中能量重整化现象的建模、驱动 transmon 中多光子共振的全面表征、电感分流 transmon 电路的提案，以及驱动振荡器中经典超次谐波分岔的表征。最后，我们通过将该方法与实验数据和精确 Floquet 数值对角化进行比较，对其性能进行了基准测试。

英文摘要：

In this work, we present a new diagrammatic method for computing the effective Hamiltonian of driven nonlinear oscillators. At the heart of our method is a self-consistent perturbation expansion developed in phase space, which establishes a direct correspondence between the diagram and algebra. Each diagram corresponds to a Hamiltonian term, the prefactor of which, like those in Feynman diagrams, involves a simple counting of topologically equivalent diagrams. Leveraging the algorithmic simplicity of our diagrammatic method, we provide a readily available computer program that generates the effective Hamiltonian to arbitrary order. We show the consistency of our schemes with existing perturbation methods such as the Schrieffer-Wolff method. Furthermore, we recover the classical harmonic balance scheme from our result in the limit of $\hbar\rightarrow0$. Our method contributes to the understanding of dynamic control within quantum systems and achieves precision essential for advancing future quantum information processors. To demonstrate its value and versatility, we analyze five examples from the field of superconducting circuits. These include an experimental proposal for the Hamiltonian stabilization of a three-legged Schrödinger cat, modeling of energy renormalization phenomena in superconducting circuits experiments, a comprehensive characterization of multiphoton resonances in a driven transmon, a proposal for an inductively shunted transmon circuit, and a characterization of classical ultra-subharmonic bifurcation in driven oscillators. Lastly, we benchmark the performance of our method by comparing it with experimental data and exact Floquet numerical diagonalization.

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