© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2410.07835 · Superconductor Science and Technology

Numerical analysis of Josephson junction arrays for multi-order quantum voltage steps

Paolo Durandetto

Published May 20, 2025• 1 citation

Abstract

Abstract The dynamics of overdamped Josephson junctions under varying microwave-driving conditions have been studied through numerical simulations using the resistively-shunted junction model, with a focus on primary voltage metrology applications, where a significantly high number of series-connected junctions and stringent uniformity of their electrical parameters are required. The aim is to determine the optimal junction characteristics and external microwave (rf) parameters that maximize the width of quantum voltage levels (Shapiro steps) from order n = 0 to n > 1. Both the rf and dc power requirements, along with the junction parameter spread and power attenuation, are analyzed as key factors that need to be optimized for improved performance of the quantum device. This work aims to advance the development of next-generation programmable Josephson voltage standards with logic architectures that surpass the conventional binary and ternary codifications used in present quantum voltage arrays, while significantly reducing the overall number of junctions as well as the number of sub-arrays and bias lines. Existing technologies exploiting n = 0 and n = ± 1 voltage steps are first discussed and analyzed to verify the validity of the simulation model. They are then further investigated to extend their usability with multi-order quantum steps for n up to 3. From the simulation results, it follows that present junction technologies may be employed with no modifications for the simultaneous operation of quantum steps up to n = 2, although optimal power efficiency would require a retrimming of the junction’s electrical parameters. On the contrary, extending the highest step order to n = 3 strictly requires the junction’s characteristic parameters to be properly adjusted to maintain sustainable power levels as well as acceptable quantum-locking ranges.

DOI: 10.1088/1361-6668/addaee

Related papers

The impact of parameter spreads of Josephson junctions on the performance of quantum-based voltage standards

Guanghong Wen, Yi Zhu, Y. Zheng +4 more

Quantum metrology based on Josephson junction array reproduces the most accurate desired voltage by far, therefore being introduced to provide voltage standards worldwide. In this work, we quantitatively analyzed the dependence of the first Shapiro step height of the junction array at 70 GHz on the parameter spread of 8,000 Josephson junctions with an average critical current (Ic) of 3.5 mA and an average normal state resistance (Rn) of 40 mΩ, using numerical simulations based on the resistively shunted junction (RSJ) model. The results indicate an upper limit spread of the Ic and Rn of the Josephson junctions. Specifically, to keep the maximum first Shapiro step above 0.9 mA, the Ic standard deviation, σ, should not exceed 25%, and for it to stay above 0.5 mA, the Rn deviation should not exceed 9%. This analysis specifically clarifies the required level of manufacturing process sophistication for superconducting devices to realize high-temperature superconducting (HTS) quantum-based voltage standards.

Physica C Superconductivity•Aug 19, 2026•0 citations

The impact of parameter spread of high-temperature superconducting Josephson junctions on the performance of quantum-based voltage standards

Wen, Guanghong, Zhu, Yi, Zheng, Yingxiang +6 more

Quantum metrology based on Josephson junction array reproduces the most accurate desired voltage by far, therefore being introduced to provide voltage standards worldwide. In this work, we quantitatively analyzed the dependence of the first Shapiro step height of the junction array at 50 GHz on the parameter spread of 10,000 Josephson junctions by numerical simulation with resistively shunted junction model. The results indicate an upper limit spread of the critical current and resistance of the Josephson junctions. Specifically, to keep the maximum first Shapiro step above 0.88 mA, the critical current standard deviation, $σ$, should not exceed 25%, and for it to stay above 0.6 mA, the resistance standard deviation should not exceed 1.5%.

arXiv:2506.12989•Jun 15, 2025•0 citations

1 V programmable voltage standards based on SNIS Josephson junction series arrays

V. Lacquaniti, Natascia De Leo, Matteo Fretto +3 more

Binary-divided 1 V arrays were fabricated for programmable Josephson voltage standards in close cooperation between INRIM and PTB. The arrays, consisting of 8192 overdamped Nb/Al-AlO(x)Nb SNIS Josephson junctions, were successfully operated at microwave frequencies around 70 GHz. While the characteristic voltage of the junctions ranged from 0.25 to 0.6 mV, the current margins were larger than 1 mA for the Shapiro step n = 1. In addition, flat Shapiro steps of higher order (n = 3 and 4) were generated at increased microwave power levels due to the large characteristic voltage, demonstrating the small parameter spread of the SNIS junctions. Operation at temperatures up to 6-7 K has also been verified, measuring quantized voltages up to 1.25 V on the n = 1 step.

Superconductor Science and Technology•Jan 27, 2011•37 citations

Overdamped Josephson junctions for applications to precision measurement

V. Lacquaniti, Domenico Andreone, Matteo Fretto +6 more

Overdamped Nb/Al-AlOx/Nb Josephson junctions are an intermediate state between the SIS and SNS Josephson junctions. With a proper choice of fabrication parameters, stable and reproducible nonhysteretic curent-voltage characteristics have been obtained Featuring critical current densities up to 70 kA/cm2and characteristic voltages up to 450 μV at 4.2 K, overdamped junctions are suitable for many aplications ranging from RSFQ electronic circuits to programmable and ac voltage standards. Tests on single junctions arrays of series-connected junctions have shown a suitable behaviour when driven by a microwave with frequencies around 75 GHz. The quantized steps have proven to be stable and flat at nV level. These junctions also shown an improved temperature dependence, which can be interesting in view of the development instrumentation with an integrated quantum voltage reference operated in a closed cycle refrigerator.

International Workshop on Advances in Sensors and Interfaces (IWASI)•Jan 1, 2007•0 citations

Josephson arrays for dc and ac metrology

J. Niemeyer

Large series arrays of underdamped SIS tunnel junctions are the basic element of the primary dc voltage standards used at present. The development of versatile quantum voltmeters for dc and ac metrology requires the preparation of large and perfect series arrays of overdamped Josephson junctions. Overdamped junctions can be realized by externally shunted SIS junctions or by internally shunted SNS or SINIS junctions. Arrays of up to 8000 SINIS junctions or 30000 SNS junctions were successfully operated at the 1 V dc level. In addition to being used in large arrays for voltage metrology and oscillators, the described junction types may become very useful for the preparation of highly integrated single-flux-quantum digital devices.

Superconductor Science and Technology•May 1, 2000•18 citations

Junction Yield Analysis for 10 V Programmable Josephson Voltage Standard Devices

Anna E. Fox, Paul D. Dresselhaus, Alain Rüfenacht +2 more

Analysis of the Josephson junction yield in the National Institute of Standards and Technology 10 V Programmable Josephson Voltage Standard (PJVS) has been performed by fabricating and measuring over 25 million Nb/NbxSi1-x/Nb junctions. Using the 265 116 junctions per PJVS device, it was possible to measure Shapiro steps on current voltage curves and then estimate the number of single-junction defects within these long arrays. By comparing the results of a quantum-based measurement to a model simulating junction defects, the quantity and magnitude of a small number of junction defects can be estimated from an array of many thousands of junctions. Understanding the source of junction defects in the fabrication process is important for maximizing yield and uniformity, both of which directly relate to the current margin of the quantized voltage step. We have recently fabricated 18 imperfect but usable PJVS devices and 16 that were free from defects, many with n = 1 Shapiro step margins approximately equal to 2 mA. The best of these wafers had a junction defect density of around 3 in a million.

IEEE Transactions on Applied Superconductivity•Dec 5, 2014•34 citations

Abstract The dynamics of overdamped Josephson junctions under varying microwave-driving conditions have been studied through numerical simulations using the resistively-shunted junction model, with a focus on primary voltage metrology applications, where a significantly high number of series-connected junctions and stringent uniformity of their electrical parameters are required. The aim is to determine the optimal junction characteristics and external microwave (rf) parameters that maximize the width of quantum voltage levels (Shapiro steps) from order n = 0 to n > 1. Both the rf and dc power requirements, along with the junction parameter spread and power attenuation, are analyzed as key factors that need to be optimized for improved performance of the quantum device. This work aims to advance the development of next-generation programmable Josephson voltage standards with logic architectures that surpass the conventional binary and ternary codifications used in present quantum voltage arrays, while significantly reducing the overall number of junctions as well as the number of sub-arrays and bias lines. Existing technologies exploiting n = 0 and n = ± 1 voltage steps are first discussed and analyzed to verify the validity of the simulation model. They are then further investigated to extend their usability with multi-order quantum steps for n up to 3. From the simulation results, it follows that present junction technologies may be employed with no modifications for the simultaneous operation of quantum steps up to n = 2, although optimal power efficiency would require a retrimming of the junction’s electrical parameters. On the contrary, extending the highest step order to n = 3 strictly requires the junction’s characteristic parameters to be properly adjusted to maintain sustainable power levels as well as acceptable quantum-locking ranges.

Quantum metrology based on Josephson junction array reproduces the most accurate desired voltage by far, therefore being introduced to provide voltage standards worldwide. In this work, we quantitatively analyzed the dependence of the first Shapiro step height of the junction array at 70 GHz on the parameter spread of 8,000 Josephson junctions with an average critical current (Ic) of 3.5 mA and an average normal state resistance (Rn) of 40 mΩ, using numerical simulations based on the resistively shunted junction (RSJ) model. The results indicate an upper limit spread of the Ic and Rn of the Josephson junctions. Specifically, to keep the maximum first Shapiro step above 0.9 mA, the Ic standard deviation, σ, should not exceed 25%, and for it to stay above 0.5 mA, the Rn deviation should not exceed 9%. This analysis specifically clarifies the required level of manufacturing process sophistication for superconducting devices to realize high-temperature superconducting (HTS) quantum-based voltage standards.

Quantum metrology based on Josephson junction array reproduces the most accurate desired voltage by far, therefore being introduced to provide voltage standards worldwide. In this work, we quantitatively analyzed the dependence of the first Shapiro step height of the junction array at 50 GHz on the parameter spread of 10,000 Josephson junctions by numerical simulation with resistively shunted junction model. The results indicate an upper limit spread of the critical current and resistance of the Josephson junctions. Specifically, to keep the maximum first Shapiro step above 0.88 mA, the critical current standard deviation,

σ

, should not exceed 25%, and for it to stay above 0.6 mA, the resistance standard deviation should not exceed 1.5%.

Binary-divided 1 V arrays were fabricated for programmable Josephson voltage standards in close cooperation between INRIM and PTB. The arrays, consisting of 8192 overdamped Nb/Al-AlO(x)Nb SNIS Josephson junctions, were successfully operated at microwave frequencies around 70 GHz. While the characteristic voltage of the junctions ranged from 0.25 to 0.6 mV, the current margins were larger than 1 mA for the Shapiro step n = 1. In addition, flat Shapiro steps of higher order (n = 3 and 4) were generated at increased microwave power levels due to the large characteristic voltage, demonstrating the small parameter spread of the SNIS junctions. Operation at temperatures up to 6-7 K has also been verified, measuring quantized voltages up to 1.25 V on the n = 1 step.

Overdamped Nb/Al-AlOx/Nb Josephson junctions are an intermediate state between the SIS and SNS Josephson junctions. With a proper choice of fabrication parameters, stable and reproducible nonhysteretic curent-voltage characteristics have been obtained Featuring critical current densities up to 70 kA/cm2and characteristic voltages up to 450 μV at 4.2 K, overdamped junctions are suitable for many aplications ranging from RSFQ electronic circuits to programmable and ac voltage standards. Tests on single junctions arrays of series-connected junctions have shown a suitable behaviour when driven by a microwave with frequencies around 75 GHz. The quantized steps have proven to be stable and flat at nV level. These junctions also shown an improved temperature dependence, which can be interesting in view of the development instrumentation with an integrated quantum voltage reference operated in a closed cycle refrigerator.

Large series arrays of underdamped SIS tunnel junctions are the basic element of the primary dc voltage standards used at present. The development of versatile quantum voltmeters for dc and ac metrology requires the preparation of large and perfect series arrays of overdamped Josephson junctions. Overdamped junctions can be realized by externally shunted SIS junctions or by internally shunted SNS or SINIS junctions. Arrays of up to 8000 SINIS junctions or 30000 SNS junctions were successfully operated at the 1 V dc level. In addition to being used in large arrays for voltage metrology and oscillators, the described junction types may become very useful for the preparation of highly integrated single-flux-quantum digital devices.

Analysis of the Josephson junction yield in the National Institute of Standards and Technology 10 V Programmable Josephson Voltage Standard (PJVS) has been performed by fabricating and measuring over 25 million Nb/NbxSi1-x/Nb junctions. Using the 265 116 junctions per PJVS device, it was possible to measure Shapiro steps on current voltage curves and then estimate the number of single-junction defects within these long arrays. By comparing the results of a quantum-based measurement to a model simulating junction defects, the quantity and magnitude of a small number of junction defects can be estimated from an array of many thousands of junctions. Understanding the source of junction defects in the fabrication process is important for maximizing yield and uniformity, both of which directly relate to the current margin of the quantized voltage step. We have recently fabricated 18 imperfect but usable PJVS devices and 16 that were free from defects, many with n = 1 Shapiro step margins approximately equal to 2 mA. The best of these wafers had a junction defect density of around 3 in a million.