## Final Report Summary - NEQUFLUX (Nonequilibrium quantum fluctuations in superconducting devices)

With the current and fast development of nano-science and technology, smaller and smaller electronic, optical, as well as mechanical devices, are being investigated and produced. The degree of miniaturization is so high that quantum effects cannot be neglected and actually play an important role in nano-technology. Quantum effects are rather strange effects that defy our every-day comprehension of the world around us. For example, a quantum mechanical particle can be at two distinct places at the same time, this is called quantum superposition. Likewise a quantum bit can be in states 0 and 1 like classical bits, but it can also stay in a superposition of 0 and 1. A computer whose computation units are quantum bits is called a quantum computer. Quantum superposition allows quantum computers to achieve greater computing power as compared to classical computers. Great investments have been pursued in the last decades to develop such quantum computers. The research and development in the field has overcome the academic stage and has entered already into the industrial sector, with major IT companies competing to stay at the edge of the unraveling new quantum technology.

As a result, solid state superconducting devices for quantum information science have undergone a tremendous and fast development in the last decade, making the creation, manipulation and engineering of quantum states and superpositions thereof, a reality. For this reason, these devices constitute one of the most promising tools for the development of quantum technologies. The benefit for European society of being at the forefront of this research field is enormous. It will keep us in a leading position in the global industrial competition.

Similar to the case of quantum computation, quantum effects can have an enhancing power in boosting other nano-device tasks. This project, in particular, addresses the question if it is possible to manage and transduce energy more efficiently than is currently possible, by means of quantum effects. Just like the advancement of thermodynamics was instrumental for the development of the industrial revolution at the turn of the 20th century, so it is now necessary that “quantum thermodynamics” (namely the science that studies energy transformations and exchanges in the quantum world) will further advance, to ease the currently unfolding nano-technology revolution.

This project brought together “quantum thermodynamics” and “superconducting devices for quantum information science”, which are presently in the limelight of vigorous research activity world-wide. The project cross-fertilized the two fields, and contributed keeping European science at the forefront of the current developments in both fields. This reinforced other actions that are being pursued at the european level, see, e.g., the COST Action MP1209 ”Thermodynamics in the quantum regime”, the ESF Networking Programme “Exploring the Physics of Small Devices”, the network “Circuit and Cavity Quantum Electro-Dynamics” funded by the European Union through a Marie Curie Action within the FP7 Initial Training Network, as well as the project “SOLID: Solid State Systems for Quantum Information Processing” funded by the EU the FP7 FET Proactive Initiative “Quantum Information Foundations and Technologies”.

At a more technical level this project contributed to experimentally enabling the measurement of a most crucial quantity in quantum thermodynamics, namely quantum work and heat. Only recently it has become clear, from a theoretical point of view, how work and heat should be understood at the quantum level, and now it is time to put it to use in experiments, and in the design of energetically efficient nano-devices. In particular it is known that heat and work are subject to quantum fluctuations, which are described by mathematical relations called fluctuation relations. The general aim of this project was to propose feasible experiments where quantum fluctuation relations can be tested and applied to enable efficient quantum thermodynamics devices. The general aim of the project has been achieved by pursuing two specific objectives (OBs)

OB1: The first objective is to design experiments based on superconducting devices for the measurement of work/heat statistics in driven quantum systems.

OB2: The second objective regards the application of fluctuation relations to the design of more efficient superconducting devices for charge transport at the nano scale (the so called Cooper pair pump) in presence of thermal noise

Both objectives were achieved by means of individual research as well as collaborative research work with well established research groups in Europe. In particular through this project two designs of nano quantum devices which can be realized experimentally with current technology have been put forward.

One device is a quantum heat engine. Its principle of functioning is not so different from that of the steam engine of the 19th century, but its dimensions are a few microns and it is lithographically patterned on a microchip. Like the steam engine, it takes heat from a hot source (a resistor in our case), use part of it to output work (in the form of photons) and dumps the rest in a cold source (again a resistor). The engine can also work as a quantum refrigerator, that is it can use energy to take heat from the cold source to a hot one. As an application this can be used to cool parts of a micro-chip. The engine works thanks to the employment of special operations called quantum gates, which have been first developed for quantum computers. Here these gates are used to manage heat transport in the chip, rather than information. The envisaged experiment will also allow to measure heat and work and their fluctuations. The device operates at very low-temperature and is based on superconducting circuits.

The other device uses similar superconducting circuits and what it does is to use externally provided energy to transport charge from one place of a superconducting wire patterned on the chip to another. In the superconductor electrons bind in pairs, called Cooper pairs, and the device allows to manipulate and transport such pairs, even if there is thermal noise in the wires which tend to destroy the pairs and makes their manipulation more difficult. The method that has been envisaged allows to single out the best pattern of operations that need to be done in order to achieve the wanted transport. Such operations consists simply in changing certain applied voltages.

In sum, with the present project we have investigated methods of how to use the technology already developed for quantum computation with superconducting circuits, to steer and manage heat and charge in such circuits, rather than information.

It is foreseen that in the future such methods will be further developed and expanded in order to increase the ability to manage energy efficiently in nano-devices, which ultimately means less energy consumption of electronic devices and greener information technology era.

As a result, solid state superconducting devices for quantum information science have undergone a tremendous and fast development in the last decade, making the creation, manipulation and engineering of quantum states and superpositions thereof, a reality. For this reason, these devices constitute one of the most promising tools for the development of quantum technologies. The benefit for European society of being at the forefront of this research field is enormous. It will keep us in a leading position in the global industrial competition.

Similar to the case of quantum computation, quantum effects can have an enhancing power in boosting other nano-device tasks. This project, in particular, addresses the question if it is possible to manage and transduce energy more efficiently than is currently possible, by means of quantum effects. Just like the advancement of thermodynamics was instrumental for the development of the industrial revolution at the turn of the 20th century, so it is now necessary that “quantum thermodynamics” (namely the science that studies energy transformations and exchanges in the quantum world) will further advance, to ease the currently unfolding nano-technology revolution.

This project brought together “quantum thermodynamics” and “superconducting devices for quantum information science”, which are presently in the limelight of vigorous research activity world-wide. The project cross-fertilized the two fields, and contributed keeping European science at the forefront of the current developments in both fields. This reinforced other actions that are being pursued at the european level, see, e.g., the COST Action MP1209 ”Thermodynamics in the quantum regime”, the ESF Networking Programme “Exploring the Physics of Small Devices”, the network “Circuit and Cavity Quantum Electro-Dynamics” funded by the European Union through a Marie Curie Action within the FP7 Initial Training Network, as well as the project “SOLID: Solid State Systems for Quantum Information Processing” funded by the EU the FP7 FET Proactive Initiative “Quantum Information Foundations and Technologies”.

At a more technical level this project contributed to experimentally enabling the measurement of a most crucial quantity in quantum thermodynamics, namely quantum work and heat. Only recently it has become clear, from a theoretical point of view, how work and heat should be understood at the quantum level, and now it is time to put it to use in experiments, and in the design of energetically efficient nano-devices. In particular it is known that heat and work are subject to quantum fluctuations, which are described by mathematical relations called fluctuation relations. The general aim of this project was to propose feasible experiments where quantum fluctuation relations can be tested and applied to enable efficient quantum thermodynamics devices. The general aim of the project has been achieved by pursuing two specific objectives (OBs)

OB1: The first objective is to design experiments based on superconducting devices for the measurement of work/heat statistics in driven quantum systems.

OB2: The second objective regards the application of fluctuation relations to the design of more efficient superconducting devices for charge transport at the nano scale (the so called Cooper pair pump) in presence of thermal noise

Both objectives were achieved by means of individual research as well as collaborative research work with well established research groups in Europe. In particular through this project two designs of nano quantum devices which can be realized experimentally with current technology have been put forward.

One device is a quantum heat engine. Its principle of functioning is not so different from that of the steam engine of the 19th century, but its dimensions are a few microns and it is lithographically patterned on a microchip. Like the steam engine, it takes heat from a hot source (a resistor in our case), use part of it to output work (in the form of photons) and dumps the rest in a cold source (again a resistor). The engine can also work as a quantum refrigerator, that is it can use energy to take heat from the cold source to a hot one. As an application this can be used to cool parts of a micro-chip. The engine works thanks to the employment of special operations called quantum gates, which have been first developed for quantum computers. Here these gates are used to manage heat transport in the chip, rather than information. The envisaged experiment will also allow to measure heat and work and their fluctuations. The device operates at very low-temperature and is based on superconducting circuits.

The other device uses similar superconducting circuits and what it does is to use externally provided energy to transport charge from one place of a superconducting wire patterned on the chip to another. In the superconductor electrons bind in pairs, called Cooper pairs, and the device allows to manipulate and transport such pairs, even if there is thermal noise in the wires which tend to destroy the pairs and makes their manipulation more difficult. The method that has been envisaged allows to single out the best pattern of operations that need to be done in order to achieve the wanted transport. Such operations consists simply in changing certain applied voltages.

In sum, with the present project we have investigated methods of how to use the technology already developed for quantum computation with superconducting circuits, to steer and manage heat and charge in such circuits, rather than information.

It is foreseen that in the future such methods will be further developed and expanded in order to increase the ability to manage energy efficiently in nano-devices, which ultimately means less energy consumption of electronic devices and greener information technology era.