Advances in Condensed Matter Physics
Vol.07 No.01(2018), Article ID:23714,5 pages
10.12677/CMP.2018.71004

Geometric Phase in a Quantum Coherent Tunable Coupling of Superconducting Qubits

Yuanxin Qiao, Zhaoxian Yu

Department of Physics, Beijing Information Science and Technology University, Beijing

Received: Jan. 21st, 2018; accepted: Feb. 2nd, 2018; published: Feb. 9th, 2018

ABSTRACT

By using the Lewis-Riesenfeld invariant theory, we have studied the geometric phase in a quantum coherent tunable coupling of superconducting qubits. The geometric phase has nothing to do with the tunnel splitting, but also the dc energy bias and the HF signal. This result has certain significance to the quantum computation.

Keywords:Geometric Phase, Continuous Monitoring of Rabi Oscillations, Josephson Flux Qubit

超导量子位的量子相干可调谐耦合中的几何相位

乔元新,于肇贤

北京信息科技大学理学院,北京

收稿日期:2018年1月21日;录用日期:2018年2月2日;发布日期:2018年2月9日

摘 要

通过使用Lewis-Riesenfeld不变量理论,我们研究了超导量子位的量子相干可调耦合中的几何相位。发现几何相位与隧道分裂,直流能量偏置和HF信号等无关。这一结果对量子计算有一定的意义。

关键词 :几何相位,连续监测的拉比振荡,约瑟夫流量子位

Copyright © 2018 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

众所周知,Pancharatnam首先引入几何相位的概念来研究古典光在不同极化状态下的干扰。Berry [1] 在循环绝热演化的情况下发现了Pancharatnam相位的量子对应物。Ahyalov和Anandan [2] 对其扩展到了非绝热循环演化。Samuel等人 [3] 通过扩展到非循环演化和顺序投影测量来研究纯态几何相位的一般性。几何相位是量子运动学的结果,因此与状态空间中的路径的动态起源的详细性质无关。Mukunda和Simon [4] 通过将在状态空间中穿过的路径作为几何相位的主要概念给出了运动学方法。通过放宽绝热条件,统一性和演化的循环性,进行了进一步的推广和改进 [5] 。最近,混合状态的几何相位也已经被研究 [6] [7] [8] [9] 。

我们知道Lewis和Riesenfeld [10] 提出的量子不变性理论是用于处理与时间相关的哈密顿量系统的强大工具。通过引入基本不变量的概念的一般性,并用于研究几何相位 [9] [10] [11] [12] [13] ,与时间相关的薛定谔方程的精确解。Berry相位的发现不仅是旧的量子绝热近似理论的突破,也为我们许多物理现象提供了新见解。Berry相位的概念已经在不同的方向得到发展 [10] [14] - [24] 。

自从几何相位发现以来,它对物理理论和应用均产生了重大影响。一方面,其理论层面的研究十分广泛,包含:绝热和循环过程、非绝热和非循环的推广、混态、非对角项、开放系统以及它与微分几何和规范场理论的广泛联系。另一方面,它的应用几乎触及了物理研究的方方面面,例如:分子物理学、凝聚态以及量子信息与计算。其次,量子精确和准精确可解问题无论从理论物理学家还是实验物理学家的角度看都有重要的意义,因为其理论计算的结论可以和实验的结果直接对照。最后,狭义相对论的理论结论超越我们日常的经验而且几乎不可能在平常的低速实验中得到观察,例如:运动的尺缩短和运动的钟变慢等现象。所以,借助于光学实验验证它的结论变得非常有意义。

本文将通过使用Lewis-Riesenfeld不变理论,研究超导量子位的量子相干可调谐耦合中的几何相位。

2. 模板

量子相干可调谐耦合的哈密尔顿算子可以写成 [25] :

H ^ = 1 2 Δ σ x 1 2 ε σ z W ( t ) σ z , (1)

其中 Δ 为隧道分裂, ε 代表直流能量偏置, W ( t ) = W cos ω H F t 为高频信号(HF),对于 W = 0 ,量子位

级别 | 0 | 1 分别具有 1 2 Ω ,其中 Ω = Δ 2 + ε 2 。这些电平之间的拉比振荡在谐振 ω H F Ω / 。在一个

简单周期内,我们设定

A = 1 2 Δ , B = 1 2 ε , β ( t ) = W ( t ) , (2)

所以上述哈密尔顿量可以表示为

H ^ = A σ x + B σ z + β ( t ) σ z , (3)

其中 σ ± = 1 2 [ σ x ± i σ y ] ,且

[ σ + , σ ] = σ z , [ σ z , σ + ] = 2 σ + [ σ z , σ ] = 2 σ . (4)

其中 σ z σ x , y 是泡利矩阵。

3. 动力学与几何相位

为了自洽,我们首先介绍Lewis-Riesenfeld (L-R)不变量理论 [10] 。

对于一维系统的与时间相关的哈密顿量,存在运算符 I ^ ( t ) 的不变量满足方程

i I ^ ( t ) t + [ I ^ ( t ) , H ^ ( t ) ] = 0. (5)

给出了与时间有关 | λ n , t 的特征值方程

I ^ ( t ) | λ n , t = λ n | λ n , t , (6)

λ n t = 0 。此系统的与时间有关的薛定谔方程是

i | ψ ( t ) s t = H ^ ( t ) | ψ ( t ) s (7)

根据L-R不变理论,方程(7)的特定解 | λ n , t s 只有相位因子 exp [ i δ n ( t ) ] I ^ ( t ) 的本征函数 | λ n , t 不同,即

| λ n , t s = exp [ i δ n ( t ) ] | λ n , t (8)

这表明 | λ n , t s ( n = 1 , 2 , ) 形成方程(7)的解的完整集合。那么薛定谔方程(7)的一般解可以写成

| ψ ( t ) s = n C n exp [ i δ n ( t ) ] | λ n , t , (9)

其中

δ n ( t ) = 0 t d t λ n , t | i t H ^ ( t ) | λ n , t , (10)

C n = λ n , 0 | ψ ( 0 ) s .

对方程(3)哈密顿量描述的系统,我们能够定义如下的不变量

I ^ ( t ) = α ( t ) σ + + α * ( t ) σ + β ( t ) σ z (11)

将方程式(3)与方程式(11)代入到方程式(1),我们能够到的一个辅助方程

i α ˙ ( t ) + 2 ( A β α β ) = 0 , i β ˙ + A ( α α * ) = 0. (12)

为了得到一个与时间无关的不变量,我们可以引入酉变换算子 V ^ ( t ) = exp [ ξ ( t ) σ ^ + ξ * ( t ) σ ^ ] 。很容易发现当满足下列关系式

β cos ( 2 | ξ | ) + α ξ 2 | ξ | [ sin 2 | ξ | + 1 2 | ξ | 1 ] = 1 (13)

α + ξ | β | | ξ | [ sin ( 2 | ξ | + 2 | ξ | 1 ) ] = 1 , (14)

I ^ V V ^ ( t ) I ^ ( t ) V ^ ( t ) = σ z . (15)

通过使用 Baker-Campbell-Hausdoff公式 [26]

V ^ ( t ) V ^ ( t ) t = L ^ t + 1 2 ! [ L ^ t , L ^ ] + 1 3 ! [ [ L ^ t , L ^ ] , L ^ ] + 1 4 [ [ [ L ^ t , L ^ ] , L ^ ] , L ^ ] + , (16)

其中 V ^ ( t ) = exp [ L ^ ( t ) ] ,很容易发现,满足以下等式时,

A + B + β ξ | ξ | [ sin ( 2 | ξ | ) + 2 | ξ | 1 ] + 2 ξ ( ξ ξ ˙ * ξ ˙ ξ * ) ( 2 | ξ | ) 3 [ sin ( 2 | ξ | ) 2 | ξ | 3 ] = 0 , (17)

H ^ V ( t ) V ^ ( t ) I ^ ( t ) V ^ ( t ) i V ^ ( t ) V ^ ( t ) t = λ σ z , (18)

其中

λ = i [ 1 cos ( 2 | ξ | ) ] ( ξ ξ ˙ * ξ ˙ ξ * ) + A [ sin 2 | ξ | + 1 2 2 | ξ | 1 ] 2 | ξ | [ α ξ + α * ξ * ] (19)

几何相位具有的关系为

δ ˙ g ( t ) = i [ 1 cos ( 2 | ξ | ) ] ( ξ ξ ˙ ξ ˙ ξ * ) . (20)

该结果具有纯几何的意义,具有实验上的可观察性。

4. 结论

在本文中,通过使用Lewis-Riesenfeld不变量理论,我们研究了超导量子位的量子相干可调谐耦合中的几何相位。发现几何相位与隧道分裂,直流能量偏置和HF信号等无关。这一结果对量子计算有一定的意义。

基金项目

本文得到北京信息科技大学研究生院项目支持。

文章引用

乔元新,于肇贤. 超导量子位的量子相干可调谐耦合中的几何相位
Geometric Phase in a Quantum Coherent Tunable Coupling of Superconducting Qubits[J]. 凝聚态物理学进展, 2018, 07(01): 22-26. http://dx.doi.org/10.12677/CMP.2018.71004

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