A method of determining microwave dissipation of Josephson junctions with non-linear frequency response

Chen Heng-Jie; Xue Hang; Li Shao-Xiong; Wang Zhen

doi:10.7498/aps.68.20190167

Abstract

Based on Josephson junction (JJ), superconducting quantum bit (qubit) is operated at frequencies of several GHz. Dissipation of JJs in this frequency range can cause energy relaxation in qubits, and limit coherence time, therefore it is highly concerned and needs to be determined quantitatively. The dissipation of JJs can be quantified by microwave quality factor. It is usually done at very low temperature (~mK) to determine whether a JJ is suitable for qubit devices by measuring the quality factor. In this paper, a method based on nonlinear frequency response of JJs is proposed to determine the quality factor. This method can be used in thermal activation regime, which may bring great conveniences to experiments. To analyze high frequency properties of JJs, the dynamic equation of a current-biased JJ, which describes high frequency oscillation of the JJ, is introduced. A fourth-order potential approximation is used to obtain the analytical equation of non-linear response. The dependence on quality factor, as well as on amplitude, of difference between JJ’s plasma frequency and resonant frequency, is derived from the equation. The approximate treatment is quantitatively validated by our numerical simulations with practical JJ parameters including different environment influences. Thus, based on nonlinear frequency response of JJs, a reliable and simple method to determine quality factor of JJ is proposed, which is desirable for exploring JJ based microwave devices such as parametric amplifier, superconducting qubit. Being driven well into the nonlinear microwave response regime, due to frequency-amplitude interaction, the resonant frequency of a current bias JJ deviates from the JJ’s plasma frequency. The deviation is directly related to the microwave quality factor. Hence, the quality factor can be deducted from the experimental measurement of the resonant frequency deviation, with different microwave power values applied. In comparison with linear resonance experiment, the nonlinear resonance used by the proposed method produces stronger signal. Therefore it is more robust against external noise. When being conducted at high temperature, the proposed method is more reliable. The accuracy of the measured quality factor primarily depends on those of the JJ’s parameters such as critical current and capacitance, while those parameters can be experimentally determined with high precision.

Keywords:

PACS:

85.25.Cp	(Josephson devices)
05.45.–a

Authors and contacts

1.
State Key Laboratory of Functional Materials for Informatics, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China
2.
CAS Center for Excellence in Superconducting Electronics (CENSE), Shanghai 200050, China
3.
University of Chinese Academy of Sciences, Beijing 100049, China

Corresponding author: Li Shao-Xiong, sxli@mail.sim.ac.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61771459).

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1. 引　言

超导约瑟夫森结是超导电子学应用的关键元件之一. 尤其是在最近二十年里, 它在超导量子信息处理器件中有着至关重要的应用^[1]. 随着结制备水平的进步、材料损耗研究的深入和电路设计的优化, 以约瑟夫森结为基础的超导量子比特有了巨大的进步. 为了不断提升约瑟夫森结器件水平, 建立一种适用于约瑟夫森结微波器件, 特别是量子比特器件, 而且简单、可靠的结质量评估方法是十分有意义的.

通常测量约瑟夫森结的电流-电压(I-V)关系曲线是一个简单评估结质量的方法. 例如对于在超导电子学中有广泛应用的Nb/AlO_x/Nb约瑟夫森结, 可定义一个量V_m来衡量结的品质. V_m为测得的I-V曲线中2 mV处的能隙内电阻与结临界电流的乘积^[2]. V_m值越大, 表明得到的结质量越好. 但是, V_m不直接反映结的高频性质. 在结的质量比较好、没有额外漏电流的情况下, 能隙内电阻值的大小取决于能隙内低能激发的准粒子数, 因此V_m基本反映了结中准粒子的情况. 在高频应用中, 准粒子会带来能量耗散^[3]. 而超导量子比特器件通常工作在数吉赫兹的频率, 它在这一频域上的能量损耗是最受关注的性质之一. 高频损耗会导致量子比特的能量弛豫, 能量弛豫率1/T₁的大小直接正比于能量耗散的大小^[4,5]. 能量弛豫时间T₁, 相位退相干时间T_ø 是限制量子比特量子相干时间T₂的两个主要因素(1/T₂ = 1/(2T₁) + 1/T_ø). 因此, 测量结的高频耗散, 能有效地评估制备的约瑟夫森结在量子比特器件中的适用性. 约瑟夫森结中高频耗散主要来自三个方面: 准粒子损耗、辐射损耗、介电损耗^[6]. 其中准粒子损耗会随着温度降低而指数减少^[7], 而且从I-V曲线中得到的V_m反映的是热平衡下的准粒子平均数, 不反映准粒子的涨落情况^[8,9], 因此V_m不能直接全面表征结的高频性质. 高频耗散的准确确定通常需要复杂的动力学建模, 不合理的系统建模会给出不可靠的结果^[10,11]. 另外, 直接测量能量弛豫时间, 会有严格的实验温度要求(通常在mK范围), 为此需要极其谨慎地实验排除所有强度可比于该温度涨落的噪声干扰^[12,13].

基于结的非线性频率响应, 本文提出一种温度限制相对宽松的测量结微波损耗的简单可靠方法. 若外加微波频率在约瑟夫森结的线性等离子振荡频率附近, 随着微波驱动能量的增强, 结因非线性效应产生的共振频率偏离结的线性等离子振荡频率. 这一偏离的幅度与振荡经历的耗散直接相关. 这种非线性频率响应效应可以被用来确定微波耗散.

本文首先介绍电流偏置约瑟夫森结的动力学描述. 采用结的等效势阱的四阶近似, 得到约瑟夫森结的非线性响应解析结果. 从得到的非线性响应方程中建立结的共振频率-振幅-微波品质因子关系. 用数值方法模拟未做势阱四阶近似的电流偏置约瑟夫森结微波动力学行为. 数值结果证实了非线性响应方程的定量关系可在约瑟夫森结中应用. 最后讨论提出的方法在约瑟夫森结中的实验应用情况.

2. 约瑟夫森结中的非线性响应

约瑟夫森结是超导量子比特器件中至关重要的非线性元件. 电流偏置的约瑟夫森结在较强微波驱动下会表现出比较强的非线性响应效应. 为了能够定量描述这一非线性响应, 需要考虑约瑟夫森结动力学具体行为. 电流偏置的约瑟夫森结的动力学行为, 可以由电阻电容并联结(RCSJ)模型描述^[14,15]:

$\frac{{\hbar C}}{{2e}}\frac{{{{\rm{d}}^2}\varphi }}{{{\rm{d}}{t^2}}} + \frac{\hbar }{{2eR}}\frac{{{\rm{d}}\varphi }}{{{\rm{d}}t}} + {I_{{\rm{c}}0}}\sin \varphi - {I_{{\rm{dc}}}} = {I_{{\rm{rf}}}}\cos \left({{\omega _{{\rm{rf}}}}t} \right),$

(1)

其中φ是通过约瑟夫森结的相位差, $\hbar$ 是约化普朗克常数, e是电荷电量, C是结电容, R是损耗带来的有效阻抗, I_c0是结临界电流, I_dc是偏置电流, ${I_{{\rm{rf}}}}\cos\left({{\omega _{{\rm{rf}}}}t} \right)$ 为外加微波驱动. 在固定偏置电流(< I_c0)下, (1)式描述的是一个虚拟相位粒子在洗衣板势某个势阱中的高频振荡. 为方便分析及数值计算, 用约瑟夫森频率 ${\omega _{{\rm{p}}0}}= \big[\displaystyle{2e{I_{{\rm{c}}0}}}/({{\hbar C}})\big]^{1/2}$ 为时间尺度归一化时间 $\tau = {\omega _{{\rm{p0}}}}t$ , 可得

$\frac{{{{\rm{d}}^2}\varphi }}{{{\rm{d}}{\tau ^2}}} + \frac{1}{Q}\frac{{{\rm{d}}\varphi }}{{{\rm{d}}\tau }} + \sin \varphi - {i_{{\rm{dc}}}} = {i_{{\rm{rf}}}}\cos \left({\gamma \tau } \right),$

(2)

其中 ${i_{{\rm{dc}}}} = {I_{{\rm{dc}}}}/{I_{{\rm{c0}}}}$ 是归一化的偏置电流, $Q = {\omega _{{\rm{p}}0}}RC$ 是品质因子. 运动方程(2)式中对应的非线性势能(洗衣板势)为

$U = \cos \varphi + {i_{{\rm{dc}}}}\varphi .$

(3)

考虑虚拟粒子是在势阱的极小值附近振动, 可将势函数U在极小值 ${\varphi _0} = \arcsin{i_{{\rm{dc}}}}$ 附近展开, 取四阶近似, 运动方程(2)式可简化为

$\frac{{{{\rm{d}}^2}\varphi }}{{{\rm{d}}{\tau ^2}}} + \frac{1}{Q}\frac{{{\rm{d}}\varphi }}{{{\rm{d}}\tau }} + \omega _{\rm{p}}^2\varphi + a_2^{}{\varphi ^2} + a_3^{}{\varphi ^3} = {i_{{\rm{rf}}}}\cos \left({\gamma \tau } \right),$

(4)

其中 ${\omega _{\rm{p}}} = {\left({1 - i_{{\rm{dc}}}^2} \right)^{{1 / 4}}}$ , ${a_2} = - i_{{\rm{dc}}}^{}/2$ , ${a_3} = - (1 -$ $i_{{\rm{dc}}^2} )^{{1 / 2}}/6$ . 当外加微波 ${i_{{\rm{rf}}}}\cos\left({\gamma \tau } \right)$ 强度较弱, $\gamma = {\omega _{\rm{p}}}$ 时发生线性共振. 当外加微波强度较强时, 势能展开中的非线性项会引起频率-振幅相互作用. 发生共振时共振频率 ${\gamma _{{\rm{res}}}} = {\omega _{\rm{p}}} + \kappa {b^2}$ 偏离结的线性等离子频率 ${\omega _{\rm{p}}}$ , 其中 $\kappa = \displaystyle\frac{{3{a_3}}}{{8{\omega _{\rm{p}}}}} - \frac{{5a_2^2}}{{12\omega _{\rm{p}}^3}}$ , b为振幅. 因此采用了上述四阶近似势之后, 由非线性振动理论可得结的非线性响应方程^[16]:

${b^2}\left[ {{{\left(\gamma - {\omega _{\rm{p}}} - \kappa {b^2}\right)}^2} + \frac{1}{{4{Q^2}}}} \right] = \frac{{i_{{\rm{rf}}}^{\rm{2}}}}{{4{\omega _{\rm{p}}}^2}}.$

(5)

发生共振时, 振幅b达到最大值,

${b_{\max }} = \frac{{{i_{{\rm{rf}}}}}}{{{\omega _{\rm{p}}}}}Q.$

(6)

因此结线性等离子频率与非线性共振频率的差 $\Delta \omega = {\gamma _{{\rm{res}}}} - {\omega _{\rm{p}}}$ 与微波品质因子Q有一个直接简单的关系:

$\sqrt {\frac{{\Delta \omega }}{\kappa }} {\omega _{\rm{p}}} = i_{{\rm{rf}}}^{}{Q^{}}.$

(7)

利用这个关系, 可以通过测量不同强度微波驱动下的频率差值来确定结的微波品质因子.

3. 数值模拟验证

由于(7)式是采用了势阱近似得到的, 为了验证这一关系可以被定量地应用于约瑟夫森结中, 我们对微波驱动的约瑟夫森结动力学行为进行了数值模拟. 采用四阶Runge-Kutta方法数值求解方程(1), 式中势能U未做前述四阶近似处理. 数值计算步长为0.05. 为获得虚拟相位粒子在相空间中完整的运动图像, 总计算步长数为1800. 选取常见的结面积为4 µm²的结, 其他所有用到的结参数值见表1. 表2列出了所有数值模拟用到的实验条件参数值. 其中结的直流偏置选取在0.5左右是考虑到实验在热激发逃离势阱为主导的高温区进行, 相较于量子隧穿逃离为主的低温区, 高温区的势阱需要更深才能容许较大的微波驱动^[17]. 由于 ${\omega _{\rm{p}}} = (1 -$ $\left. i_{\rm{dc}}^2 \right)^{{1 / 4}}$ , 不同的直流偏置电流选择会导致工作频率的不同. 但是我们的数值模拟计算中采用了归一化因子 ${\omega _{{\rm{p}}0}}$ , 实验上总是可以通过磁场调节结的临界电流值达到将工作频率调整到一个感兴趣范围的目的.

表 1 数值模拟采用的结参数

Table 1. Parameters of Josephson junctions used in numerical simulations.

结参数	结1	结2	结3
临界电流密度j_c/A·cm^–2	100	150	200
比电容C_s/fF·μm^–2	42.9	46.4	50.5
品质因子Q	163.0	508.6	1459.1

下载: 导出CSV

| 显示表格

表 2 数值模拟采用的实验参数

Table 2. Experiment settings used in numerical simulations.

实验参数	表示符号	取值
归一直流偏置	i_dc	0.46—0.52
归一微波电流	i_rf	1.6 × 10^–4—5.5 × 10^–4
归一微波频率	γ	0.9306

下载: 导出CSV

| 显示表格

图1为数值计算得到的一个典型的虚拟相位粒子在相空间的运动轨迹. 由于模拟计算时采用的初始条件不一定处于相位粒子的稳定态, 采用的总计算步长数必须保证粒子运动轨迹能够达到一个不变稳定态. 稳定态的运动轨迹(例如图1中内圈轨迹被重复了很多遍因而更密集)可被认定是方程(1)的稳定解. 运动轨迹在位置坐标上的变化幅度对应粒子的振动幅度.

$图 1 计算得到的相位粒子在相空间中的运动轨迹　纵坐标$ v = {{{\rm{d}}\phi } / {{\mathop{\scriptsize\rm d}\nolimits} \tau }}$, 计算采用的实验参数为${i_{{\rm{dc}}}} = {\rm 0.473},{i_{{\rm{rf}}}} =$$ {\rm 5.5} \times {\rm 10^{ - 4}}$, γ = 0.9306\r\nFig. 1. Calculated trajectory of phase particle with experiment parameters for junction 3: ${i_{{\rm{dc}}}} = 0.473$, ${i_{{\rm{rf}}}} = 5.5 \times $10–4, γ = 0.9306. Vertical axis $v = {{{\rm{d}}\phi } / {{\rm{d}}\tau }}$.$

图 1 计算得到的相位粒子在相空间中的运动轨迹　纵坐标

$v = {{{\rm{d}}\phi } / {{\mathop{\scriptsize\rm d}\nolimits} \tau }}$

, 计算采用的实验参数为

${i_{{\rm{dc}}}} = {\rm 0.473},{i_{{\rm{rf}}}} =$

${\rm 5.5} \times {\rm 10^{ - 4}}$

, γ = 0.9306

Fig. 1. Calculated trajectory of phase particle with experiment parameters for junction 3:

${i_{{\rm{dc}}}} = 0.473$

${i_{{\rm{rf}}}} = 5.5 \times$

10^–4, γ = 0.9306. Vertical axis

$v = {{{\rm{d}}\phi } / {{\rm{d}}\tau }}$

下载: 全尺寸图片幻灯片

微波频率响应通常需要变化外加微波频率. 但是实验上测量约瑟夫森结的微波响应时, 选取固定外加微波频率, 改变偏置电流 ${i_{{\rm{dc}}}}$ 的方法(即改变 ${\omega _{\rm{p}}}$ )可以保证耦合到结的实际微波功率的可定标性. 为方便与实验结果比较, 在不同的微波强度驱动条件下, 在计算非线性响应时只改变 ${i_{{\rm{dc}}}}$ , 这对应于(5)式中固定γ, 改变 ${\omega _{\rm{p}}}$ . 图2给出了数值模拟得到的结非线性响应结果与(5)式计算结果的比较, 可以看到两者定量符合. 图2中, 响应幅度b最大处对应约瑟夫森结发生共振吸收. 发生共振时共振频率 ${\gamma _{{\rm{res}}}} = \gamma$ , 对应的 ${\omega _{\rm{p}}}$ 由最大振幅处的 ${i_{{\rm{dc}}}}$ 决定. 从计算的结果可见当外加微波强度增大, 约瑟夫森结非线性共振时对应的 ${i_{{\rm{dc}}}}$ 变小, 从而相应的 ${\omega _{\rm{p}}}$ 变大, 这意味着共振频率相对更小, 这说明较强微波驱动下约瑟夫森结振子会软化. 为进一步验证由非线性频率响应确定微波品质因子的准确性及在约瑟夫森结中的适用性, 图3给出了表1中三个样品结模拟得到的结果与(7)式结果的比较. 考虑到同样工艺制备的结可能被运用到不同的器件, 遇到不同的电路环境, 图4为结2在不同环境带来不同损耗情况下的模拟结果比较. 同样, (7)式与模拟结果定量符合. 从图3和图4的结果可知, 只要实验确定了约瑟夫森结的共振峰对应的 ${i_{{\rm{dc}}}}$ 、结临界电流、结电容参数, 结的微波品质便可定量确定(图3和图4中直线斜率大小对应微波品质因子). 由于模拟选取了常见的约瑟夫森结参数, 而且在模拟的品质因子变化范围内的约瑟夫森结中已有实验观测到了量子相干态^[18], 我们认为由非线性频率响应来确定微波品质因子的方法完全适用于约瑟夫森结系统, 且可用来评估制备的约瑟夫森结在相关器件应用中的微波性能.

$图 2 结(${I_{\rm{C}}} = 8$ μA, Q = 515.7)在不同的微波驱动下, (5)式(曲线)和模拟(误差棒)得到的响应振幅b随归一直流偏置${i_{\rm{dc}}}$的函数变化关系, 响应曲线按其最大振幅从小到大顺序对应的外加微波强度分别为${i_{{\rm{rf}}}}={\rm{ 1}}.{\rm{63}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 2}}.{\rm{44}} \times $${\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$和$ {\rm{5}}.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$, 点划线显示了最大振幅对应的${i_{{\rm{dc}}}}$随外加微波强度的变化\r\nFig. 2. Microwave response curves obtained by Eq. (5) (curves) and numerical simulation (error bars), for junction (${I_{\rm{C}}} = 8$ μA, Q = 515.7) with applied microwave ${i_{{\rm{rf}}}}=$${\rm{ 1}}.{\rm{63}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 2}}.{\rm{44}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}\;{\rm{ and }}\;5.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$ for the curves with the maximum amplitude from small to large respectively. Dot-dash line shows the dependence of ${i_{{\rm{dc}}}}$ where corresponding to the maximum oscillation amplitude on the power of the applied microwave.$

图 2 结(

${I_{\rm{C}}} = 8$

μA, Q = 515.7)在不同的微波驱动下, (5)式(曲线)和模拟(误差棒)得到的响应振幅b随归一直流偏置

${i_{\rm{dc}}}$

的函数变化关系, 响应曲线按其最大振幅从小到大顺序对应的外加微波强度分别为

${i_{{\rm{rf}}}}={\rm{ 1}}.{\rm{63}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 2}}.{\rm{44}} \times$

${\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$

和

${\rm{5}}.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$

, 点划线显示了最大振幅对应的

${i_{{\rm{dc}}}}$

随外加微波强度的变化

Fig. 2. Microwave response curves obtained by Eq. (5) (curves) and numerical simulation (error bars), for junction (

${I_{\rm{C}}} = 8$

μA, Q = 515.7) with applied microwave

${i_{{\rm{rf}}}}=$

${\rm{ 1}}.{\rm{63}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 2}}.{\rm{44}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}\;{\rm{ and }}\;5.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$

for the curves with the maximum amplitude from small to large respectively. Dot-dash line shows the dependence of

${i_{{\rm{dc}}}}$

where corresponding to the maximum oscillation amplitude on the power of the applied microwave.

下载: 全尺寸图片幻灯片

$图 3 表1中三个样品结的$ \tilde v = \sqrt {{{\Delta \omega }}/{k}} {\omega _{\rm{p}}}$随$ {i_{{\rm{rf}}}}$的变化关系图, 线是(7)式的结果, 误差棒是$\tilde v $的数值模拟结果, 图中直线斜率从小到大分别对应Q值163.0(结1), 508.6(结2), 1549.1(结3)\r\nFig. 3. $\tilde v = \sqrt {{{\Delta \omega }}/{k}} {\omega _{\rm{p}}}$ as a function of ${i_{{\rm{rf}}}}$ for different parameters of sample Josephson junctions in Table 1. Lines are results of Eq. (7). Error bars are numerical simulation results of $\tilde v$. The lines of slops from small to large corresponding to Q values: 163.0 (junction 1), 508.6 (junction 2), 1549.1 (junction 3).$

图 3 表1中三个样品结的

$\tilde v = \sqrt {{{\Delta \omega }}/{k}} {\omega _{\rm{p}}}$

随

${i_{{\rm{rf}}}}$

的变化关系图, 线是(7)式的结果, 误差棒是

$\tilde v$

的数值模拟结果, 图中直线斜率从小到大分别对应Q值163.0(结1), 508.6(结2), 1549.1(结3)

Fig. 3.

$\tilde v = \sqrt {{{\Delta \omega }}/{k}} {\omega _{\rm{p}}}$

as a function of

${i_{{\rm{rf}}}}$

for different parameters of sample Josephson junctions in Table 1. Lines are results of Eq. (7). Error bars are numerical simulation results of

$\tilde v$

. The lines of slops from small to large corresponding to Q values: 163.0 (junction 1), 508.6 (junction 2), 1549.1 (junction 3).

下载: 全尺寸图片幻灯片

$图 4 结2在不同的电路环境下, 导致不同的品质因子Q (51.6, 257.8, 515.7, 1533.0, 对应直线斜率从小到大)时, $\tilde v$随${i_{{\rm{rf}}}}$的变化关系, 直线是(7)式的结果, 误差棒是$\tilde v$的数值模拟结果\r\nFig. 4. $\tilde v$ as a function of ${i_{{\rm{rf}}}}$ for different quality factors. Lines are Eq. (7)’s results. Error bars are numerical simulation results of $\tilde v$. The lines of slops from small to large corresponding to Q values: 51.6, 257.8, 515.7, 1533.0 accounting for junction 2 with different environment influences.$

图 4 结2在不同的电路环境下, 导致不同的品质因子Q (51.6, 257.8, 515.7, 1533.0, 对应直线斜率从小到大)时,

$\tilde v$

随

${i_{{\rm{rf}}}}$

的变化关系, 直线是(7)式的结果, 误差棒是

$\tilde v$

的数值模拟结果

Fig. 4.

$\tilde v$

as a function of

${i_{{\rm{rf}}}}$

for different quality factors. Lines are Eq. (7)’s results. Error bars are numerical simulation results of

$\tilde v$

. The lines of slops from small to large corresponding to Q values: 51.6, 257.8, 515.7, 1533.0 accounting for junction 2 with different environment influences.

下载: 全尺寸图片幻灯片

4. 讨　论

对于过阻尼的约瑟夫森结, 其线性等离子振动频率( ${\omega _{\rm{p}}}$ )也受微波损耗影响很大. 但是大多数约瑟夫森微波器件, 尤其是量子比特器件, 要求比较高的微波品质(Q $\gg$ 1). 欠阻尼约瑟夫森结的等离子振动频率几乎不随微波品质变化. 对于欠阻尼约瑟夫森结, 将其驱动到较强的非线性区域, 可由非线性频率响应来确定微波品质. 虽然约瑟夫森结的线性微波响应也与微波损耗相关^[19], 但是与线性响应相比, 非线性响应信号更强, 不容易受额外的噪声干扰, 可以在相对较高的温区准确测量. 图3和图4的结果显示, 结的微波非线性响应(图中直线斜率)随微波品质变化很大, 已有的实验结果也已经显示微波品质因子严重影响非线性振动^[20]. Mao等^[21]用数值模拟比对由较强微波驱动的、测量温度在4.2 K的共振激发实验数据得到的约瑟夫森结高频等效阻抗. 因为约瑟夫森结的共振频率、临界电流、结电容等参数都是实验可测量的, 我们提出的非线性频率响应模型方法不需要数值模拟便可直接实验确定微波品质因子. 而且由(7)式可知微波品质因子的定量准确性也是主要由结参数—临界电流、结电容参数的准确性保证的^[13,22]. 考虑到共振激发的实验测量可以适用于单个电流偏置的约瑟夫森结中^[19,21], 不需要额外的电路设计, 数值模拟也证实了非线性响应模型在约瑟夫森结中的定量准确性, 因此利用非线性频率响应来测量微波损耗的方法是方便可靠的.

References

[1]	Devoret M H Schoelkopf R J 2013 Science 339 1169 Google Scholar
[2]	van Theodore D, Charles W T 1998 Principles of Superconductive Devices and Circuits Second Edition (Upper Saddle River: Prentice Hall) p194
[3]	Mattis D C, Bardeen J 1958 Phys. Rev. 111 412 Google Scholar
[4]	Leggett A J, Chakravarty S, Dorsey A T, Fisher M P A, Garg A, Zwerger W 1987 Rev. Mod. Phys. 59 1 Google Scholar
[5]	Makhlin Y, Schön G, Shnirman A 2001 Rev. Mod. Phys. 73 357 Google Scholar
[6]	Martinis J M, Cooper K B, McDermott R, Steffen M, Ansmann M, Osborn K D, Cicak K, Oh S, Pappas D P, Simmonds R W, Yu C C 2005 Phys. Rev. Lett. 95 210503 Google Scholar
[7]	Tinkham M 2004 Introduction to Superconductivity (2nd Ed.) (Dover) p76
[8]	Pop I M, Geerlings K, Catelani G, Schoelkopf R J, Glazman L I, Devoret M H 2014 Nature 508 369 Google Scholar
[9]	Yan F, Gustavsson S, Kamal A, Birenbaum J, Sears A P, Hover D, Gudmundsen T J, RosenBerg D, Samach G, Weber S, Yoder J L, Orlando T P, Clarke J, Kerman A J, Oliver W D 2016 Nat. Commun. 7 12964 Google Scholar
[10]	Cosmelli C, Carelli P, Castellano M G, Chiarello F, Diambrini Palazzi G, Leoni R, Torrioli G 1999 Phys. Rev. Lett. 82 5357 Google Scholar
[11]	Han S, Rouse R 2001 Phys. Rev. Lett. 86 4191 Google Scholar
[12]	Dutta S K, Xu H, Berkley A J, Ramos R C, Gubrud M A, Anderson J R, Lobb C J, Wellstood F C 2004 Phys. Rev. B 70 140502 Google Scholar
[13]	Han S, Yu Y, Chu X, Chu S, Wang Z 2001 Science 293 1457 Google Scholar
[14]	McCumber D E 1968 J. Appl. Phys. 39 3113 Google Scholar
[15]	Stewart W C 1968 Appl. Phys. Lett. 12 277 Google Scholar
[16]	Landau L D, Lifshitz E M 2007 Mechanics Third Edition (Beijing: World Publishing Corporation) p88
[17]	Li S X, Yu Y, Zhang Y, Qiu W, Han S, Wang Z 2002 Phys. Rev. Lett. 89 098301 Google Scholar
[18]	Martinis J M, Nam S, Aumentado J 2002 Phys. Rev. Lett. 89 117901 Google Scholar
[19]	Devoret M H, Esteve D, Martinis J M, Cleland A, Clarke J 1987 Phys. Rev. B 36 58 Google Scholar
[20]	Manucharyan V E, Boaknin E, Metcalfe M, Vijay R, Siddiqi I, Devoret M 2007 Phys. Rev. B 76 014524 Google Scholar
[21]	Mao B, Han S 2007 IEEE Trans. Appl. Supercond. 17 94 Google Scholar
[22]	Sun G, Chen J, Ji Z, Xu W, Kang L, Wu P, Dong N, Mao G, Yu Y, Xing D 2006 App. Phys. Lett. 89 082516 Google Scholar

Cited By

期刊类型引用(1)

董全睿，陈涛，高世杰，刘永凯，张建强，吴昊. 光电精跟踪系统的改进差分进化算法研究. 中国光学. 2020(06): 1314-1323 .

百度学术

其他类型引用(1)

图 1 计算得到的相位粒子在相空间中的运动轨迹　纵坐标 $v = {{{\rm{d}}\phi } / {{\mathop{\scriptsize\rm d}\nolimits} \tau }}$ , 计算采用的实验参数为 ${i_{{\rm{dc}}}} = {\rm 0.473},{i_{{\rm{rf}}}} =$ ${\rm 5.5} \times {\rm 10^{ - 4}}$ , γ = 0.9306

Figure 1. Calculated trajectory of phase particle with experiment parameters for junction 3: ${i_{{\rm{dc}}}} = 0.473$ , ${i_{{\rm{rf}}}} = 5.5 \times$ 10^–4, γ = 0.9306. Vertical axis $v = {{{\rm{d}}\phi } / {{\rm{d}}\tau }}$ .

DownLoad: Full-Size Img PowerPoint

图 2 结( ${I_{\rm{C}}} = 8$ μA, Q = 515.7)在不同的微波驱动下, (5)式(曲线)和模拟(误差棒)得到的响应振幅b随归一直流偏置 ${i_{\rm{dc}}}$ 的函数变化关系, 响应曲线按其最大振幅从小到大顺序对应的外加微波强度分别为 ${i_{{\rm{rf}}}}={\rm{ 1}}.{\rm{63}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 2}}.{\rm{44}} \times$ ${\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$ 和 ${\rm{5}}.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$ , 点划线显示了最大振幅对应的 ${i_{{\rm{dc}}}}$ 随外加微波强度的变化

Figure 2. Microwave response curves obtained by Eq. (5) (curves) and numerical simulation (error bars), for junction ( ${I_{\rm{C}}} = 8$ μA, Q = 515.7) with applied microwave ${i_{{\rm{rf}}}}=$ ${\rm{ 1}}.{\rm{63}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 2}}.{\rm{44}} \times {\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}\;{\rm{ and }}\;5.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$ for the curves with the maximum amplitude from small to large respectively. Dot-dash line shows the dependence of ${i_{{\rm{dc}}}}$ where corresponding to the maximum oscillation amplitude on the power of the applied microwave.

DownLoad: Full-Size Img PowerPoint

图 3 表1中三个样品结的 $\tilde v = \sqrt {{{\Delta \omega }}/{k}} {\omega _{\rm{p}}}$ 随 ${i_{{\rm{rf}}}}$ 的变化关系图, 线是(7)式的结果, 误差棒是 $\tilde v$ 的数值模拟结果, 图中直线斜率从小到大分别对应Q值163.0(结1), 508.6(结2), 1549.1(结3)

Figure 3. $\tilde v = \sqrt {{{\Delta \omega }}/{k}} {\omega _{\rm{p}}}$ as a function of ${i_{{\rm{rf}}}}$ for different parameters of sample Josephson junctions in Table 1. Lines are results of Eq. (7). Error bars are numerical simulation results of $\tilde v$ . The lines of slops from small to large corresponding to Q values: 163.0 (junction 1), 508.6 (junction 2), 1549.1 (junction 3).

DownLoad: Full-Size Img PowerPoint

图 4 结2在不同的电路环境下, 导致不同的品质因子Q (51.6, 257.8, 515.7, 1533.0, 对应直线斜率从小到大)时, $\tilde v$ 随 ${i_{{\rm{rf}}}}$ 的变化关系, 直线是(7)式的结果, 误差棒是 $\tilde v$ 的数值模拟结果

Figure 4. $\tilde v$ as a function of ${i_{{\rm{rf}}}}$ for different quality factors. Lines are Eq. (7)’s results. Error bars are numerical simulation results of $\tilde v$ . The lines of slops from small to large corresponding to Q values: 51.6, 257.8, 515.7, 1533.0 accounting for junction 2 with different environment influences.

DownLoad: Full-Size Img PowerPoint

表 1 数值模拟采用的结参数

Table 1. Parameters of Josephson junctions used in numerical simulations.

结参数结1 结2 结3

临界电流密度j_c/A·cm^–2 100 150 200
比电容C_s/fF·μm^–2 42.9 46.4 50.5
品质因子Q 163.0 508.6 1459.1

DownLoad: CSV

表 2 数值模拟采用的实验参数

Table 2. Experiment settings used in numerical simulations.

实验参数表示符号取值

归一直流偏置 i_dc 0.46—0.52
归一微波电流 i_rf 1.6 × 10^–4—5.5 × 10^–4
归一微波频率 γ 0.9306

DownLoad: CSV

[1]	Devoret M H Schoelkopf R J 2013 Science 339 1169 Google Scholar
[2]	van Theodore D, Charles W T 1998 Principles of Superconductive Devices and Circuits Second Edition (Upper Saddle River: Prentice Hall) p194
[3]	Mattis D C, Bardeen J 1958 Phys. Rev. 111 412 Google Scholar
[4]	Leggett A J, Chakravarty S, Dorsey A T, Fisher M P A, Garg A, Zwerger W 1987 Rev. Mod. Phys. 59 1 Google Scholar
[5]	Makhlin Y, Schön G, Shnirman A 2001 Rev. Mod. Phys. 73 357 Google Scholar
[6]	Martinis J M, Cooper K B, McDermott R, Steffen M, Ansmann M, Osborn K D, Cicak K, Oh S, Pappas D P, Simmonds R W, Yu C C 2005 Phys. Rev. Lett. 95 210503 Google Scholar
[7]	Tinkham M 2004 Introduction to Superconductivity (2nd Ed.) (Dover) p76
[8]	Pop I M, Geerlings K, Catelani G, Schoelkopf R J, Glazman L I, Devoret M H 2014 Nature 508 369 Google Scholar
[9]	Yan F, Gustavsson S, Kamal A, Birenbaum J, Sears A P, Hover D, Gudmundsen T J, RosenBerg D, Samach G, Weber S, Yoder J L, Orlando T P, Clarke J, Kerman A J, Oliver W D 2016 Nat. Commun. 7 12964 Google Scholar
[10]	Cosmelli C, Carelli P, Castellano M G, Chiarello F, Diambrini Palazzi G, Leoni R, Torrioli G 1999 Phys. Rev. Lett. 82 5357 Google Scholar
[11]	Han S, Rouse R 2001 Phys. Rev. Lett. 86 4191 Google Scholar
[12]	Dutta S K, Xu H, Berkley A J, Ramos R C, Gubrud M A, Anderson J R, Lobb C J, Wellstood F C 2004 Phys. Rev. B 70 140502 Google Scholar
[13]	Han S, Yu Y, Chu X, Chu S, Wang Z 2001 Science 293 1457 Google Scholar
[14]	McCumber D E 1968 J. Appl. Phys. 39 3113 Google Scholar
[15]	Stewart W C 1968 Appl. Phys. Lett. 12 277 Google Scholar
[16]	Landau L D, Lifshitz E M 2007 Mechanics Third Edition (Beijing: World Publishing Corporation) p88
[17]	Li S X, Yu Y, Zhang Y, Qiu W, Han S, Wang Z 2002 Phys. Rev. Lett. 89 098301 Google Scholar
[18]	Martinis J M, Nam S, Aumentado J 2002 Phys. Rev. Lett. 89 117901 Google Scholar
[19]	Devoret M H, Esteve D, Martinis J M, Cleland A, Clarke J 1987 Phys. Rev. B 36 58 Google Scholar
[20]	Manucharyan V E, Boaknin E, Metcalfe M, Vijay R, Siddiqi I, Devoret M 2007 Phys. Rev. B 76 014524 Google Scholar
[21]	Mao B, Han S 2007 IEEE Trans. Appl. Supercond. 17 94 Google Scholar
[22]	Sun G, Chen J, Ji Z, Xu W, Kang L, Wu P, Dong N, Mao G, Yu Y, Xing D 2006 App. Phys. Lett. 89 082516 Google Scholar

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期刊类型引用(1)
1. 董全睿，陈涛，高世杰，刘永凯，张建强，吴昊. 光电精跟踪系统的改进差分进化算法研究. 中国光学. 2020(06): 1314-1323 . 百度学术
其他类型引用(1)

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