Introduction

  1. Justification
  2. Reading guide
  3. References

Justification

This thesis contains a broad selection of work. The connection between the topics is not glaringly obvious, but definitely exists. I have two justifications for the choice of topics and inclusions in the thesis; allow me to explain.

The first justification is a little boring but is simply that these are the topics that I happened to work on during my doctorate. From the beginning, it was clear that my Ph.D. work would have a dual nature since I would work at the ITF for 1 year and then at imec for another 3. The chapters of the thesis represent — roughly chronologically — this split: the first chapter on active matter, the second on zig-zag dynamics for electrons, and then on to superconducting devices.

It may then come as a surprise that there is a second justification for the topics presented, and that is the following. The first chapter will introduce the concept of active matter, which involves particles which are motile in the sense that they generate motion by using energy won from some metabolism. These particles are characterized by their ballistic motion at small scales, yet they behave diffusively at large scales. One prominent example of such an active particle is the run-and-tumble particle (RTP), which moves in approximately straight lines (runs) until, at a random time, it tumbles, and picks a new direction. This model is based on a simplification of the motion of the bacterium E. coli, which can use its flagella to generate either propulsive forward motion or chaotic rotational motion; that is, it either runs or tumbles.

At the end of the first chapter, a model is presented in which it is shown to be possible to create a bound state out of a priori purely repulsive interactions between RTPs in a passive thermal medium. When we were preparing this work for publication in 2022, we could not resist the temptation of analogy. This model system was not the only bound state between otherwise mutually repulsive particles, where the attraction was mediated by a thermal bath, that we knew of. This begs for an analogy to Cooper pairing!

The analogy would not be so amusing if that was all to it — indeed, there is more in Chapter 3. There we introduce Bohmian mechanics, a theory of quantum mechanics that assigns positions to particles, allowing for discussion of trajectories and positions. The fun begins when we realize that the Dirac equation in the Weyl representation is formally identical to the master equation for a two-state RTP. Thus, we propose to view the electron (constantly tumbling between left- and right-handed chiralities) as a run-and-tumble particle. Then, of course, the analogy comes to its full right: the bound RTPs can be viewed as electrons, and the density fluctuations in the underlying medium as phonons! You may call it far-fetched, but it is far from the only analogy between active systems and superconductivity [1], [2], [3], [4], and although not immediately useful, such similarities may yet teach us much about the inner structure of both systems.

This brings us from Chapter 2 and Chapter 3 straight to superconductivity, which will occupy the remainder of the thesis. At imec, I was brought into an environment that cares deeply about practical implementations of scientific ideas, which shocked me — who was until then very theoretically minded — deeply. From this practical point of view, I was tasked with investigating the effect of noise and impurities on the functioning of superconducting circuits and qubits. To this end, it became necessary to dive into the microscopic theory of superconductivity and to create numerical models of the superconducting circuit element: the Josephson junction.

Chapter 4 details the famous BCS theory of superconductivity and introduces the Bogoliubov–de Gennes formalism, which will be instrumental to the modeling of Josephson junctions in later chapters. Chapter 4 is also the only chapter that does not contain original research material.

After that, Chapter 5 zooms in on the Josephson junction and specifically on the superconducting–normal metal–superconducting (SNS) junction. The SNS junction is the natural starting point for modeling in the BdG formalism and can also accommodate other junction types, once the numerical model has been set up. Due to the so-called Andreev reflections, which can take place at the SN boundaries, the SNS junction hosts the Andreev bound states, which play an instrumental role in mediating the Josephson current.

The idea is to set up the numerical model of the SNS junction, manipulate it by adding impurities and irregularities, and analyze the effect of said impurities on the circuit in which the junction is embedded. While collecting results for this project we noticed that our model did not conserve charge: current seemed to appear and disappear out of and into the superconducting leads around the junction! This turned out to not only be the case for our model, but for a whole class of models which has since long been embraced as the standard model of SNS junctions. The solution to this issue is to ensure that the current in the superconducting leads exactly matches the current in the junction by carefully treating the superconducting phase gradient. This work is included in the final section of Chapter 5.

Chapter 6 then discusses the role of superconducting circuits in quantum computing and specifically zooms in on the sensitivity of such circuits to modification of the Josephson element. The final section of Chapter 6 contains some results obtained before we stumbled upon the issue of current conservation in microscopic models of Josephson junctions.

Reading guide

Early in the process of writing this thesis, I made the conscious decision to explain a lot of the physics used such that a physicist who is not specialized in the field may read the whole text and not be surprised at any equations or concepts appearing in the published sections. However, this means that you, dear reader, may not want to read the entire text, depending on your background. If you are well aware of the physics of active matter and nonequilibrium physics, you should by all means skip §§2.1–2.4. If you are well versed in Bohmian mechanics and the measurement problem, there will not be much to learn for you in §§3.1–3.2. Chapter 4 is where the theory of superconductivity is introduced. The only part that is relied upon later is the BdG formulation of superconductivity introduced in §4.3.2; the rest is introduction. Similarly in Chapter 5, the first sections §§5.1–5.2 are introductory, with the final section §5.5 a research article. Finally, §§6.1–6.3 comprise an introduction to circuit quantum electrodynamics (cQED), while §6.4 contains some research results.

The sections of this thesis have been categorized as follows:

Even as I recommend you to skip as you must, I hope that you might flip through the pages and that your eye may be caught by one of the carefully prepared figures. I hope you enjoy reading as much as I enjoyed writing.

Active Matter chapter coming soon.

References