## Periodic Reporting for period 2 - QUANTUM E-LEAPS (Toward new era of quantum electrical measurements through phase slips)

Reporting period: 2021-01-01 to 2022-06-30

The ability to perform more and more accurate and precise measurements is driving the development of new and technologies and the quality of life. At the core of our measurement capabilities lies the SI, systeme international, which sets out the standards and definitions for the fundamental physical units that underpins all measurements. In 2019 something remarkable took place, then the SI was redefined in terms of fundamental constants of nature, eliminating the need for the last remaining artefacts to realise the fundamental SI units. This means that in practice, units no longer need to be realised by a specific method or device, as long as the underlying fundamental physical relation based on constants of natures is realised. In particular, this opens up new possibilities for realising more accurate measurements of electrical currents through novel physics, which one of the base SI units that today is most challenging to disseminate with high accuracy.

In the new SI the unit of electrical current, the ampere, can be disseminated by any device that realised the simple relation I [A] = e*f, where f is a frequency defined through atomic clocks and e is the fixed charge of the electron. The easiest way to think of this definition is that the current is defined in terms of the number of electrons that pass through a device per unit of time, and if we somehow could count the electrons, we would know the current. The challenge is that the fundamental electron charge is a very small number, e=1.602176634*10-19 which means that in order to produce any appreciable currents we would need to be able to count individual electrons at incredibly high rates, several billion counts per second, without a single error. Despite progress with semiconducting electron pumps, the ampere is dramatically more difficult to realize than the volt, which is another unit that is important for all electrical measurements.

The volt is one of the first metrological standards that were realised through a quantum mechanical effect – the so-called Josephson effect – and based entirely on fundamental constants of nature. When a Josephson junction (a very thin piece of insulator sandwiched between two superconducting electrodes) is driven by an ac voltage (microwave radiation) the charge tunnelling across this junction becomes phase locked to the external drive, producing constant voltages at integer multiples of (h/2e) = 2.067834 mV/GHz, V=(h/2e)f. Here h is the Planck constant. This effect is extremely robust, and its universality has been shown down to a record-breaking accuracy of 10-19.

Fundamental theory of quantum mechanics stipulates a duality in superconducting circuits, which implies that, loosely speaking, what can be realised with Josephson junctions and the volt, should have a dual counterpart that yields the ampere. This fundamental process is called Coherent Quantum Phase Slip (CQPS), which is analogous to the tunnelling of a magnetic flux across a superconducting nanowire (SNW), as opposed to the tunnelling of a charge across the insulating barrier of a Josephson junction. Realising devices based on CQPS has been an open challenge, as the requirements put stringent demands on materials and nanofabrication of superconducting devices.

The overall objective of the Quantum e-leaps project is to develop a robust and easy-to-use universal quantum standard of all electrical quantities on a single chip by utilizing the duality of superconducting physics. As a specific step toward the overall goal, we aim to demonstrate a proof-of-concept quantum current standard using coherent quantum phase slips (CQPS). Achieving these objectives requires a wide range of scientific and technical objectives. We take the advantage of a European level effort to utilize, develop and combine the latest advances in nanofabrication, for example 2D superconductors, to yield SNW devices with unprecedented tuneability.

In the new SI the unit of electrical current, the ampere, can be disseminated by any device that realised the simple relation I [A] = e*f, where f is a frequency defined through atomic clocks and e is the fixed charge of the electron. The easiest way to think of this definition is that the current is defined in terms of the number of electrons that pass through a device per unit of time, and if we somehow could count the electrons, we would know the current. The challenge is that the fundamental electron charge is a very small number, e=1.602176634*10-19 which means that in order to produce any appreciable currents we would need to be able to count individual electrons at incredibly high rates, several billion counts per second, without a single error. Despite progress with semiconducting electron pumps, the ampere is dramatically more difficult to realize than the volt, which is another unit that is important for all electrical measurements.

The volt is one of the first metrological standards that were realised through a quantum mechanical effect – the so-called Josephson effect – and based entirely on fundamental constants of nature. When a Josephson junction (a very thin piece of insulator sandwiched between two superconducting electrodes) is driven by an ac voltage (microwave radiation) the charge tunnelling across this junction becomes phase locked to the external drive, producing constant voltages at integer multiples of (h/2e) = 2.067834 mV/GHz, V=(h/2e)f. Here h is the Planck constant. This effect is extremely robust, and its universality has been shown down to a record-breaking accuracy of 10-19.

Fundamental theory of quantum mechanics stipulates a duality in superconducting circuits, which implies that, loosely speaking, what can be realised with Josephson junctions and the volt, should have a dual counterpart that yields the ampere. This fundamental process is called Coherent Quantum Phase Slip (CQPS), which is analogous to the tunnelling of a magnetic flux across a superconducting nanowire (SNW), as opposed to the tunnelling of a charge across the insulating barrier of a Josephson junction. Realising devices based on CQPS has been an open challenge, as the requirements put stringent demands on materials and nanofabrication of superconducting devices.

The overall objective of the Quantum e-leaps project is to develop a robust and easy-to-use universal quantum standard of all electrical quantities on a single chip by utilizing the duality of superconducting physics. As a specific step toward the overall goal, we aim to demonstrate a proof-of-concept quantum current standard using coherent quantum phase slips (CQPS). Achieving these objectives requires a wide range of scientific and technical objectives. We take the advantage of a European level effort to utilize, develop and combine the latest advances in nanofabrication, for example 2D superconductors, to yield SNW devices with unprecedented tuneability.

The Quantum e-leaps project has resulted in significant progress in 2D superconductivity, e.g. the first Josephson junction made from graphene (de Vries et al. 2021), the first SQUID made from graphene (Portoles et al. 2022), and the first supercurrent diode in 2D materials (Bauriedl et al. 2022). Studies of quantum phase slips in 2D materials will be an important focus for the final phase of the project.

A significant effort of the project has been the development of reproducible and scalable fabrication techniques for thin films and nanowires made from disordered superconducting materials. Combined with the development of improved electromagnetic environments, this has allowed the demonstration of dual Shapiro steps based on CQPS (Shaikhaidarov et al. 2022). Importantly, effort on the theory of the interaction between the CQPS element and its electromagnetic environment has allowed the theoretical description of the experimental observation. This demonstration of dual Shapiro steps is a key scientific milestone and forms a central part of the Quantum e-leaps vision that now lies much closer to realisation.

A significant effort of the project has been the development of reproducible and scalable fabrication techniques for thin films and nanowires made from disordered superconducting materials. Combined with the development of improved electromagnetic environments, this has allowed the demonstration of dual Shapiro steps based on CQPS (Shaikhaidarov et al. 2022). Importantly, effort on the theory of the interaction between the CQPS element and its electromagnetic environment has allowed the theoretical description of the experimental observation. This demonstration of dual Shapiro steps is a key scientific milestone and forms a central part of the Quantum e-leaps vision that now lies much closer to realisation.

Despite many significant scientific breakthroughs, CQPS still has relatively low technological readiness and the dual Shapiro steps are still too inaccurate for actual usage in quantum metrology. However, significantly improved understanding of the effect and its limitations is a major improvement beyond the state of the art, thus laying the basis for the wide impact. The same applies also to the breakthroughs in 2D superconductivity. It is likely that the first results to be exploited are technological improvements in the fabrication of disordered superconductors and devices from them.

We expect to achieve improved accuracy and understanding of the physics of dual Shapiro steps during the remaining 18 months of the project, pushing the state of the art even further towards our vision. Likewise, further exploring 2D superconductors and refining materials is likely to enable new devices with capability to control Josephson and phase slip physics towards a deeper understanding of the physics and a practical realisation of dual Shapiro steps for metrology.

Eventually, we expect that the 2D material development will yield entirely new functionalities to superconductivity, which should enable impact beyond our present imagination. The main goal of the project, the development of CQPS elements, may lay the foundation of dual superconducting electronics, where SNW becomes a standard circuit element like Josephson junction is in conventional superconducting electronics. Finally, the specific objective of the project, to achieve a quantum current standard based on CQPS that will be compatible with the Josephson voltage standard, will allow for an integrated, universal quantum electrical standard. The combination of these two standards enable the quantum standards of all electrical quantities on a single chip. Availability of such a standard will also promote dissemination of primary noise thermometry and the quantum mass scale via so-called Kibble balance. In addition to the conventional metrological calibration chain, quantum electrical standards may support and enable other quantum technologies.

We expect to achieve improved accuracy and understanding of the physics of dual Shapiro steps during the remaining 18 months of the project, pushing the state of the art even further towards our vision. Likewise, further exploring 2D superconductors and refining materials is likely to enable new devices with capability to control Josephson and phase slip physics towards a deeper understanding of the physics and a practical realisation of dual Shapiro steps for metrology.

Eventually, we expect that the 2D material development will yield entirely new functionalities to superconductivity, which should enable impact beyond our present imagination. The main goal of the project, the development of CQPS elements, may lay the foundation of dual superconducting electronics, where SNW becomes a standard circuit element like Josephson junction is in conventional superconducting electronics. Finally, the specific objective of the project, to achieve a quantum current standard based on CQPS that will be compatible with the Josephson voltage standard, will allow for an integrated, universal quantum electrical standard. The combination of these two standards enable the quantum standards of all electrical quantities on a single chip. Availability of such a standard will also promote dissemination of primary noise thermometry and the quantum mass scale via so-called Kibble balance. In addition to the conventional metrological calibration chain, quantum electrical standards may support and enable other quantum technologies.