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
Fund Project:Project supported by the National Natural Science Foundation of China (Grant No. 61771459).
Received Date:28 January 2019
Accepted Date:21 March 2019
Available Online:01 June 2019
Published Online:05 June 2019
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:Josephson junction/ nonlinear resonance/ quality factor/ qubit
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 $${\rm{1}}{{\rm{0}}^{ - 4}},{\rm{ 3}}.{\rm{67}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$和$ {\rm{5}}.{\rm{50}} \times {\rm{1}}{{\rm{0}}^{ - 4}}$, 点划线显示了最大振幅对应的${i_{{\rm{dc}}}}$随外加微波强度的变化 Figure2. Microwave response curves obtained by Eq. (5) (curves) and numerical simulation (error bars), for junction (${I_{\rm{C}}} = 8$ μA, Q = 515.7) with applied microwave ${i_{{\rm{rf}}}}=$${\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 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 $的数值模拟结果, 图中直线斜率从小到大分别对应Q值163.0(结1), 508.6(结2), 1549.1(结3) Figure3.$\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 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 Figure4.$\tilde v$ 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 1 计算得到的相位粒子在相空间中的运动轨迹 纵坐标
图 2 结(
图 3 表1中三个样品结的
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, 
