## Periodic Reporting for period 1 - QUANTUM E-LEAPS (Toward new era of quantum electrical measurements through phase slips)

Reporting period: 2020-01-01 to 2020-12-31

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.

During the first year of Quantum e-leaps our focus has been in particular on developing the individual technologies that the project partners start to combine together already in 2021. We have developed theory, low-dimensional superconducting technologies, sample fabrication, and electromagnetic environments for superconducting nanowires. Preliminary results show, e.g. that we can make superconducting devices in twisted graphene, where the superconductivity is tunable with voltages on gate electrodes. An important joint effort was to design a semiconductor-based tuneable electromagnetic environment for superconducting nanowires. Multiple designs allow integration compatibility with all nanowire technologies of the consortium, and the samples will be available to the consortium partners early 2021.

The main results of the first year are 4 published and a 3 submitted scientific papers. A paper by E. Mykkänen et al. demonstrates a new fabrication method for disordered superconducting films and nanowires by amorphisation with a focussed ion beam. A paper by S. E. de Graaf develops theory for the Aharonov-Casher effect in SNWs, which is dual to the Aharonov-Bohm effect Josephson junction systems. A better understanding of the Aharonov-Casher effect is important it can yield destructive interference that is problematic for CQPS. A paper by I. V. Antonov et al. demonstrates a superconducting twin qubit that is protected against magnetic field fluctuations. A paper by E. Ilin et al. studies an array of superconducting islands that have different critical temperatures, which provides new insight into the physics of the superconductor-insulator transition in strongly disordered superconductors. A preprint by Bäuml et al. demonstrates hybrid 1D-2D superconductivity in a sample that consists of a 1D carbon nanotube and 2D superconducting NbSe2.

The main results of the first year are 4 published and a 3 submitted scientific papers. A paper by E. Mykkänen et al. demonstrates a new fabrication method for disordered superconducting films and nanowires by amorphisation with a focussed ion beam. A paper by S. E. de Graaf develops theory for the Aharonov-Casher effect in SNWs, which is dual to the Aharonov-Bohm effect Josephson junction systems. A better understanding of the Aharonov-Casher effect is important it can yield destructive interference that is problematic for CQPS. A paper by I. V. Antonov et al. demonstrates a superconducting twin qubit that is protected against magnetic field fluctuations. A paper by E. Ilin et al. studies an array of superconducting islands that have different critical temperatures, which provides new insight into the physics of the superconductor-insulator transition in strongly disordered superconductors. A preprint by Bäuml et al. demonstrates hybrid 1D-2D superconductivity in a sample that consists of a 1D carbon nanotube and 2D superconducting NbSe2.

We expect that the tuneability of CQPS in SNWs made from 2D nanowires combined with tuneable electromagnetic environments allow to find the optimal operation regimes of CQPS. This will lay the foundation of dual superconducting electronics, where SNW becomes a standard circuit element like Josephson junction is in conventional superconducting electronics.

A quantum current standard based on CQPS will be compatible with the Josephson voltage standard, which will allow an integrated, universal quantum 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. Therefore, they can in the future have an impact that is analogous to the GPS navigation system where atomic clocks, the fundamental standards of frequency, are utilized directly to the benefit of ordinary citizens. The key to such impact is the user-friendliness, automation, and integration of primary standards, which enables direct applications.

A quantum current standard based on CQPS will be compatible with the Josephson voltage standard, which will allow an integrated, universal quantum 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. Therefore, they can in the future have an impact that is analogous to the GPS navigation system where atomic clocks, the fundamental standards of frequency, are utilized directly to the benefit of ordinary citizens. The key to such impact is the user-friendliness, automation, and integration of primary standards, which enables direct applications.