Alberto del Angel Medina, Applied Quantum Physics

Main supervisor: Anton Frisk Kockum, Associate Professor, Applied Quantum Physics, MC2
Assist. supervisor: Nils Johan Engelsen, Assistant Professor, Quantum Technology, MC2
Assist. supervisor: Ricardo Guriérrez Jáuregui, Associate Professor, Universidad Nacional Autonoma de Mexico

Discussion leader: Alexandru Petrescu, Associate Professor, Mines Paris

Examiner: Janine Splettstösser, Professor, Applied Quantum Physics, MC2

Abstract: Quantum electrical circuits extend the limits of conventional quantum optics and serve as a leading platform for quantum computing, thanks to the flexibility of their light-matter interaction schemes and their ease of fabrication. A central idea in circuit quantum electrodynamics (cQED) is the notion of describing matter and its interaction with microwave fields using lumped elements. In such a description, the geometries of the elements composing the circuit and the spatial distribution of the underlying electromagnetic fields are neglected. However, these simplifications mean that the standard circuit theory may miss interesting and potentially important effects.

This seminar presents a theory that consistently describes the interaction between a Josephson junction, the fundamental building block of matter in cQED, and external quantum electromagnetic fields. Using a path-integral approach, we derive the theory for this interaction starting from the minimal coupling between the fields and the electrons in each superconductor that compose the Josephson junction. This approach allows us to systematically derive the form of the coupling terms of the resulting theory and clarify the physical assumptions underlying the model.

Our framework goes beyond circuit theory by producing interaction Hamiltonians in which the junction degrees of freedom explicitly couple to the electromagnetic fields or potentials, depending on the choice of gauge. For example, when working in Coulomb’s gauge, we obtain an expression that closely resembles the minimal coupling Hamiltonian of charged particles in an electromagnetic field via the scalar and vector potentials. In the multipolar gauge, however, the resulting Hamiltonian is written in terms of the junction’s multipole moments and the electric and magnetic fields. Thus, our theory allows us to explore the role of gauge transformations in cQED and the different, yet physically meaningful, representations they lead to.

As an example, we present the case of a single transmon embedded in vacuum, where non-trivial effects arise from the interplay between the polarization structure of the radiated fields and the junction’s structure. By incorporating the vectorial nature of the fields, our results provide a consistent, gauge-invariant extension of the theory of quantum optics with superconducting circuits, with measurable consequences in current experiments.