使用約瑟夫森分支放大器的量子測量之研究

Abstract

最近，一種新型的放大器，稱為約瑟夫森分支放大 （JBA），用以測量超導量子位元（qubit），已經被提議和建造出來。JBA 解決了建構在傳統超導 Josephson junction 量子位元測量裝置的散熱問題，此惱人的散熱問題是由此裝置的電壓切換到 normal state 所引起 。本論文旨在模擬使用 JBA 測量量子位元的過程，並提供對理解量子測量問題所必需的相關知識。我們一開始回顧一些基本的超導量子電路元件，並介紹兩種不同類型的量子位元：flux qubit 和 charge qubit。由於 Josephson junction 的非線性電感，JBA 的數學模型可由驅動非線性振盪器所描述，此數學模型被稱為 Duffing 振子。因此，我們著重於量子 Duffing 振子的性質和介紹 JBA 的運作原理。測量量子位元的過程本身是一個開放量子系統的問題。為了來描述它的行為，我們推導了驅動 Duffing 振子和量子位元系統的縮減密度矩陣的 quantum master equation。我們區分了熱環境和測量裝置對系統的影響，並使用 Floquet formalism 處理時間上的週期性問題。並在最後提出一些 Duffing 振子和量子位元測量的模擬結果。

Recently, a new type of amplifier, called the Josephson bifurcation amplifier (JBA), to read out the state of a superconducting quantum bit (qubit), has been proposed and constructed. This JBA has solved the annoying dissipation problem of voltage switching to the normal state in traditional superconducting Josephson junction based qubit measurement devices. This thesis aims to model the qubit readout process by the JBA, and to provide the essential input toward the understanding of the quantum measurement problem. We first review some basic elements of superconducting quantum circuit, and introduce two different types of qubits: flux qubits and charge qubits. Due to the nonlinear inductance of a Josephson junction, the mathematical model of the JBA can be linked to a driven non-linear oscillator, known as the Duffing oscillator. So we focus on the properties of the quantum Duffing oscillator and present the operation principles of the JBA. The qubit readout process is itself an open quantum system problem. To describe its dynamics, we derive the quantum master equation for the reduced density matrix of the combined driven quantum Duffing oscillator and qubit system. We distinguish the influence of the thermal environment on the combined system from that of the measurement device, and use the Floquet formalism to tackle the time-periodical driven problem. Simulation results of the Duffing oscillator and qubit measurement will be presented.