中性原子量子處理器中受相位波動場驅動之量子閘保真度分析

Abstract

對於物質的量子態操縱，普遍使用外加場與量子系統耦合，促使量子態隨著交互作用而隨時間演化。以中性原子作為量子系統時，通常使用雷射耦合電子能態，達成電子能態的居量、相位等等的同調控制，雷射各個參數的穩定性則成為至關重要的控制變因，在諸多場合皆有相對高的要求。本作以中性原子作為量子處理器所涉及之量子態控制為背景，利用線性響應函數探討雷射的相位雜訊對單位元量子邏輯閘的保真度影響。

本作以二能階系統之單光子過程切入，找出描述量子系統演化的哈密頓量作為後續分析的基礎，並更進一步考慮同為實驗上常見的，牽涉三能階的雙光子過程，以絕熱消去法近似為等效的二能階系統，使得雜訊的分析能適用得更廣泛。

隨機過程是替雜訊建立模型的良好理論基礎，本作透過維納-辛欽定理(Wiener-Khintchine theorem)推導隨機訊號之功率頻譜密度，以傅立葉頻率分析時變的雜訊為我們帶來對雷射穩定性直觀、方便的描述方式。

經過前述的理論基礎，我們將雜訊從雷射與量子系統交互作用之哈密頓量中分離出，並視雜訊為微小量，透過微擾理論建構保真度的形式，並且得出線性響應函數，其描述了與雜訊頻譜密度配合的權重關係，進而得出單次單位元量子邏輯閘施作後的保真度。

For the manipulation of quantum states of matter, external fields are commonly used to couple with quantum systems, driving quantum states to evolve over time through interactions. When using neutral atoms as quantum systems, lasers are typically employed to couple electronic energy states, achieving coherent control of electronic state populations, phases, and other properties. The stability of various laser parameters becomes a crucial control variable with relatively high requirements in many scenarios. This work, using quantum state control in neutral atom-based quantum processors as the background, employs linear response functions to investigate the effects of laser phase noise on the fidelity of single-qubit quantum logic gates.

This work begins with a single-photon process in a two-level system, identifying the Hamiltonian that describes quantum system evolution as the foundation for subsequent analysis. We further consider the two-photon process involving three energy levels, which is also common in experiments, and approximate it as an equivalent two-level system using the adiabatic elimination method, making the noise analysis more broadly applicable.

Stochastic process theory provides a solid theoretical foundation for modelling noise. This work derives the power spectral density of stochastic signals through the Wiener-Khintchine theorem. Analysing time-varying noise in the Fourier frequency domain provides us with an intuitive and convenient way to describe laser stability.

Based on the aforementioned theoretical foundations, we separate the noise from the Hamiltonian of laser-quantum system interaction and treat the noise as a small perturbation. Through perturbation theory, we construct the form of the fidelity and obtain the linear response function, which describes the weighting relationship associated with the noise spectral density, thereby determining the fidelity after a single application of a single-qubit quantum logic gate.