## Periodic Reporting for period 1 - QUANTUM HARVEST (Harvesting Non-Classical Fluctuations with Thermal Machines)

Reporting period: 2018-07-01 to 2020-06-30

Quantum mechanics, the theory describing the very small and very cold, makes fascinating predictions with puzzling interpretations. A popular example is provided by a quantum bit (qubit), which unlike a classical bit cannot only encode either 0 or 1 but may be in what is called a superposition of both. Interestingly, when observing the qubit, the outcome is still either 0 or 1. Which of those outcomes occurs is, according to quantum theory, completely random. Only the probability for observing 0 or 1 can be known in advance. What exactly happens during a measurement, and how to interpret it, remains a topic of debate even more than a century after the introduction of quantum mechanics. Such quantum features are not only an academic curiosity but may result in technological advances in, e.g. the fields of information theory and cryptography. For instance, a quantum computer based on qubits may outperform a classical computer in certain tasks.

In many areas of physics, it is still an open question if such a quantum advantage exists, i.e. if quantum theory can be leveraged to design technological devices that perform better than what is allowed by classical theories. A promising framework for addressing the question of a potential quantum advantage is provided by thermodynamics. Thermodynamics investigates concepts such as heat, work, and temperature and it played a pivotal role in the industrial revolution, which brought about immensely useful machines such as steam engines, paving the way for modern devices ranging from refrigerators to solar cells. Currently, thermodynamics is actively being investigated in the quantum regime. In contrast to conventional thermal machines such as steam engines, for quantum features to become relevant, quantum thermal machines are usually extremely small. One important consequence of their small size is that fluctuations matter. Let us take a thermoelectric generator as an example. Such devices use heat to generate electricity and find applications in many areas ranging from the Mars Perseverance Rover to the automobile industry. A macroscopic thermoelectric generator, which may be centimeters in size, produces a relatively stable electric current. Nevertheless, the current will exhibit fluctuations, being sometimes larger, sometimes smaller. Scaling the generator down to the nano-scale (1 nanometer is one billionth of a meter), the produced electric current will become smaller and become comparable to its fluctuations.

While these fluctuations may seem detrimental, they can actually contain a lot of interesting information about the physical properties of the system under investigation. To appreciate this, it is illustrative to consider possible sources of fluctuations. For instance, an electric current may fluctuate because the electrons that carry the current bounce off obstacles on their way in an unpredictible fashion. This is a classical source of fluctuations. Intriguingly, quantum mechanics offers sources of fluctuations that are absent in classical theory. These are related to the fundamental randomness in measurement outcomes, as discussed above using the example of a qubit.

In this project, we investigated fluctuations of thermodynamic observables, such as work and heat with the aim to improve quantum machines, i.e. to harvest the fluctuations.

Our results provide a considerable step forward in understanding the fluctuations of thermodynamic observables. In particular, we found that the first law of thermodynamics, which splits energy changes into heat and work, may break down because of quantum fluctuations. This provides a new perspective on the fundamental concepts of heat and work with a large potential impact on upcoming quantum technologies. Furthermore our results provide crucial insight into the connection between information and thermodynamics, a topic that is of crucial importance in downsizing information technology.

In many areas of physics, it is still an open question if such a quantum advantage exists, i.e. if quantum theory can be leveraged to design technological devices that perform better than what is allowed by classical theories. A promising framework for addressing the question of a potential quantum advantage is provided by thermodynamics. Thermodynamics investigates concepts such as heat, work, and temperature and it played a pivotal role in the industrial revolution, which brought about immensely useful machines such as steam engines, paving the way for modern devices ranging from refrigerators to solar cells. Currently, thermodynamics is actively being investigated in the quantum regime. In contrast to conventional thermal machines such as steam engines, for quantum features to become relevant, quantum thermal machines are usually extremely small. One important consequence of their small size is that fluctuations matter. Let us take a thermoelectric generator as an example. Such devices use heat to generate electricity and find applications in many areas ranging from the Mars Perseverance Rover to the automobile industry. A macroscopic thermoelectric generator, which may be centimeters in size, produces a relatively stable electric current. Nevertheless, the current will exhibit fluctuations, being sometimes larger, sometimes smaller. Scaling the generator down to the nano-scale (1 nanometer is one billionth of a meter), the produced electric current will become smaller and become comparable to its fluctuations.

While these fluctuations may seem detrimental, they can actually contain a lot of interesting information about the physical properties of the system under investigation. To appreciate this, it is illustrative to consider possible sources of fluctuations. For instance, an electric current may fluctuate because the electrons that carry the current bounce off obstacles on their way in an unpredictible fashion. This is a classical source of fluctuations. Intriguingly, quantum mechanics offers sources of fluctuations that are absent in classical theory. These are related to the fundamental randomness in measurement outcomes, as discussed above using the example of a qubit.

In this project, we investigated fluctuations of thermodynamic observables, such as work and heat with the aim to improve quantum machines, i.e. to harvest the fluctuations.

Our results provide a considerable step forward in understanding the fluctuations of thermodynamic observables. In particular, we found that the first law of thermodynamics, which splits energy changes into heat and work, may break down because of quantum fluctuations. This provides a new perspective on the fundamental concepts of heat and work with a large potential impact on upcoming quantum technologies. Furthermore our results provide crucial insight into the connection between information and thermodynamics, a topic that is of crucial importance in downsizing information technology.

To achieve a deeper understanding of fluctuations in quantum thermal machines, three different but interrelated research questions were addressed:

1. How can we distinguish quantum from classical behavior from measured data alone?

To address this question, we considered the scenario sketched in Fig. 1, where different detectors are used to measure different physical observables. Making a number of natural assumptions on the detectors, we were then able to derive an inequality that any classical system has to obey. A quantum system may however violate the inequality. As the inequality only depends on measurable quantities, our inequality provides an experimental test for non-classical behavior. This research was published in the renowned journal Physical Review Letters, where it was featured on the cover.

2. How do fluctuations behave in concrete examples of quantum thermal machines?

To address this question, we focused on a quantum thermoelectric generator that is based on a superconducting circuit as illustrated in Fig. 2. In this device, a heat current from a hot to a cold bath drives a charge current against a voltage bias. This could be used to charge a battery. In the action, we performed a detailed study of heat and work fluctuations in this quantum heat engine. We made the important discovery that the first law of thermodynamics, the law that splits energy changes into heat and work, may not be applicable in quantum systems. The underlying reason for this is that in quantum systems, different observables may not be known at the same time, in the spirit of Heisenberg's famous uncertainty relation. How this breakdown of the first law can be exploited in practice remains however an open question at this point, as the first law still holds for average values. At the time of writing, these results are under review.

3. How can fluctuations be exploited using measurement and feedback scenarios?

When performing measurement and feedback, processes that are otherwise forbidden by the laws of thermodynamics can be enabled. One can understand this as exploiting fluctuations: whenever we detect the system to do something favorable by accident, we interfere to make use of it. In this way, work can be extracted from a single heat bath, a process otherwise forbidden by the second law of thermodynamics. Consequentially, the laws of thermodynamics need to be refined in order to include the information obtained from the measurement. In a series of publications, we contributed to the understanding of these refined laws of thermodynamics and the role that fluctuations play in those.

1. How can we distinguish quantum from classical behavior from measured data alone?

To address this question, we considered the scenario sketched in Fig. 1, where different detectors are used to measure different physical observables. Making a number of natural assumptions on the detectors, we were then able to derive an inequality that any classical system has to obey. A quantum system may however violate the inequality. As the inequality only depends on measurable quantities, our inequality provides an experimental test for non-classical behavior. This research was published in the renowned journal Physical Review Letters, where it was featured on the cover.

2. How do fluctuations behave in concrete examples of quantum thermal machines?

To address this question, we focused on a quantum thermoelectric generator that is based on a superconducting circuit as illustrated in Fig. 2. In this device, a heat current from a hot to a cold bath drives a charge current against a voltage bias. This could be used to charge a battery. In the action, we performed a detailed study of heat and work fluctuations in this quantum heat engine. We made the important discovery that the first law of thermodynamics, the law that splits energy changes into heat and work, may not be applicable in quantum systems. The underlying reason for this is that in quantum systems, different observables may not be known at the same time, in the spirit of Heisenberg's famous uncertainty relation. How this breakdown of the first law can be exploited in practice remains however an open question at this point, as the first law still holds for average values. At the time of writing, these results are under review.

3. How can fluctuations be exploited using measurement and feedback scenarios?

When performing measurement and feedback, processes that are otherwise forbidden by the laws of thermodynamics can be enabled. One can understand this as exploiting fluctuations: whenever we detect the system to do something favorable by accident, we interfere to make use of it. In this way, work can be extracted from a single heat bath, a process otherwise forbidden by the second law of thermodynamics. Consequentially, the laws of thermodynamics need to be refined in order to include the information obtained from the measurement. In a series of publications, we contributed to the understanding of these refined laws of thermodynamics and the role that fluctuations play in those.

Our results considerably advance the state of the art and may have a strong impact on emerging quantum technologies. Our inequality for determining non-classical behavior will help to determine when quantum behavior can be exploited for technology. Our results on the breakdown of the first law in the presence of quantum fluctuations provides a new perspective on the first law of thermodynamics and may result in a paradigm shift. The understanding generated thereby may result in higher energy-efficiency of quantum technological devices, contributing to the reduction of our ecological footprint. Finally, the understanding on the role of information in thermodynamics generated from our results may have impact on future information-processing devices with a potential benefit on the generation and management of waste heat.