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Josephson junction, as the core nonlinear element underpinning superconducting electronics, is characterized by its current-phase relation (CPR), which fundamentally determines the dynamical properties and functional capabilities of superconducting quantum devices.Traditional Josephson junctions typically exhibit a traditional sinusoidal CPR; however, the junctions characterized by non-sinusoidal CPR have recently attracted considerable attention due to their distinctive physical properties and promising quantum device applications. In this work, a numerical model tailored specifically for junctions exhibiting non-sinusoidal CPR is developed by integrating experimentally measured current–voltage (I-V) characteristics from Nb/Al-AlOx/Nb junctions into a resistively and capacitively shunted junction (RCSJ) framework. By leveraging this refined model, the influence of CPR skewness on Josephson junction dynamics is systematically investigated. Our results indicate that in underdamped junctions, the critical current significantly diminishes with the increase of CPR skewness, a behavior reminiscent of the adjustable critical currents typically observed in DC superconducting quantum interference devices (SQUIDs). Conversely, in overdamped junctions, the influence of CPR skewness on the I-V characteristics is found to be negligible. However, our numerical simulations under microwave irradiation indicate that nonsinusoidal CPRs readily promote the emergence of half-integer Shapiro steps in overdamped junctions, thereby establishing CPR skewness as a plausible microscopic origin for this phenomenon. In addition, the advanced design system (ADS) simulations is employed to model nonlinear resonators and DC SQUID circuits, offering a detailed investigation into how nonsinusoidal CPRs modulate the Josephson inductance and magnetic flux response. Our findings reveal that engineering the CPR of Josephson junctions provides substantial flexibility in the design of superconducting qubits, parametric amplifiers, and non-magnetic nonreciprocal devices. This tunability underscores significant opportunities for developing next-generation superconducting electronic components. The Josephson junctions with engineered CPR offer expanded functionality for superconducting quantum technologies. This study suggests that customized CPR can enhance control over the dynamical behavior of junctions, and promote the optimized designs of superconducting qubits, parametric amplifiers, and nonmagnetic nonreciprocal devices.
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Keywords:
- Josephson junction /
- half-integer Shapiro steps /
- RCSJ model /
- DC-SQUID /
- superconducting electronics
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图 1 约瑟夫森结示意图 (a) Nb/Al-AlOx/Nb约瑟夫森结典型的三明治结构示意图 (b) 约瑟夫森结RCSJ电路模型示意图
Figure 1. Schematic diagram of a Josephson junction: (a) Schematic diagram of a typical sandwich structure of a Nb/Al-AlOx/Nb Josephson junction; (b) Josephson junction RCSJ circuit model diagram.
图 2 欠阻尼结和过阻尼结约瑟夫森相位随时间的变化曲线 (a) 正弦型CPR约瑟夫森结相位随时间的变化曲线; (b) 正弦型CPR约瑟夫森结相位导数随时间的变化曲线; (c) 非正弦型CPR约瑟夫森结相位随时间的变化曲线($ T_{n} $ = 0.999); (d) 非正弦型CPR约瑟夫森结相位导数随时间的变化曲线($ T_{n} $ = 0.999)
Figure 2. The time evolution of the Josephson phase for underdamped and overdamped junctions (a) Time evolution of the phase for a sinusoidal CPR Josephson junction; (b) Time evolution of the phase derivative for a sinusoidal CPR Josephson junction; (c) Time evolution of the phase for a non-sinusoidal CPR Josephson junction ($ T_{n} $ = 0.999); (d) Time evolution of the phase derivative for a non-sinusoidal CPR Josephson junction ($ T_{n} $ = 0.999).
图 3 约瑟夫森结I-V曲线 (a) 约瑟夫森结电流相位关系曲线; (b) 约瑟夫森结中非线性电阻模拟欠阻尼结迟滞曲线, 非线性电阻的电流是准粒子电流不再是超流; (c) 基于RCSJ模型数值计算欠阻尼结在不同CPR时的I-V曲线, 这里只考虑超流的变化, 不同CPR带来不同的临界电流; (d) 基于RCSJ模型数值计算过阻尼结在不同CPR时的I-V曲线, 过阻尼结下不同CPR约瑟夫森结几乎相同的I-V曲线; (e) 实验测量的Nb/Al-AlOx/Nb约瑟夫分森结的I-V曲线, 这是欠阻尼结形式, 可见实际的约瑟夫森结具有超导电流以及迟滞的准粒子电流; (f) 对Nb/Al-AlOx/Nb约瑟夫分森结的临界电流多次扫描, 观察其临界电流稳定性
Figure 3. Josephson-junction I-V curves: (a) current–phase relation; (b) simulated underdamped hysteresis with quasiparticle current; (c) underdamped RCSJ, CPR-dependent $ I_{\mathrm{c}} $; (d) overdamped RCSJ, CPR-independent; (e) measured Nb/Al-AlOx/Nb underdamped hysteresis; (f) repeated $ I_{\mathrm{c}} $ sweeps demonstrating consistent, time-stable critical current
图 4 约瑟夫森结夏皮罗台阶数值计算结果 (a) 正弦CPR过阻尼结的夏皮罗台阶; (b) 正弦CPR欠阻尼结的夏皮罗台阶; (c) 非正弦CPR过阻尼结的夏皮罗台阶; (d) 非正弦CPR欠阻尼结的夏皮罗台阶
Figure 4. Numerically calculated Shapiro-step responses of Josephson junctions: (a) overdamped junction with sinusoidal CPR; (b) underdamped junction with sinusoidal CPR; (c) overdamped junction with non-sinusoidal CPR; (d) underdamped junction with non-sinusoidal CPR
图 5 正弦CPR约瑟夫森结微分电阻数值计算结果 (a) 过阻尼结微分电阻和相应夏皮罗台阶; (b) 欠阻尼结微分电阻和相应的夏皮罗台阶; (c) 过阻尼结微分电阻和偏置电路以及微波幅值的伪彩色三维图; (d) 欠阻尼结微分电阻和偏置电路以及微波幅值的伪彩色三维图
Figure 5. Numerically calculated differential resistance of Josephson junctions with sinusoidal CPR: (a) overdamped junction—differential resistance and corresponding Shapiro steps; (b) underdamped junction—differential resistance and corresponding Shapiro steps; (c) pseudocolor 3-D map of differential resistance versus bias current and microwave amplitude for the overdamped junction; (d) same 3-D map for the underdamped junction
图 6 非正弦CPR约瑟夫森结微分电阻和偏置电流以及微波幅值的伪彩三维图 (a) $ T_{n} $ = 0.01; (b) $ T_{n} $ = 0.5; (c) $ T_{n} $ = 0.9; (d) $ T_{n} $ = 0.999
Figure 6. Pseudocolor three-dimensional maps of differential resistance versus bias current and microwave amplitude for Josephson junctions with non-sinusoidal CPR: (a) $T_n = 0.01$; (b) $T_n = 0.5$; (c) $T_n = 0.9$; (d) $T_n = 0.999$
图 7 ADS中构建的基于约瑟夫森结的非线性谐振器S11参数随频率和结临界电流变化的伪彩色三维图 (a) 正弦型CPR约瑟夫森结; (b) $T_n = 0.01$; (c) $T_n = 0.9$; (d) $T_n = 0.999$
Figure 7. Pseudocolor three-dimensional maps of the $S_{11}$ parameter versus frequency and junction critical current for ADS-simulated Josephson-junction nonlinear resonators: (a) sinusoidal CPR; (b) $T_{n} = 0.01$; (c) $T_{n} = 0.9$; (d) $T_{n} = 0.999$
图 8 ADS中构建的DC-SQUID电压和偏置磁通以及DC-SQUID电流关系的伪彩色三维图 (a) 基于正弦CPR约瑟夫森结的DC-SQUID; (b) $T_n = 0.01$; (c) $T_n = 0.9$; (d) $T_n = 0.999$
Figure 8. Pseudocolor three-dimensional maps of DC-SQUID voltage versus bias flux and SQUID current from ADS simulations: (a) DC-SQUID with sinusoidal-CPR Josephson junctions; (b) $T_{n} = 0.01$; (c) $T_{n} = 0.9$; (d) $T_{n} = 0.999$
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