SUPERCONDUCTING QUANTUM PHYSICS AND COMPUTATION

Module SUPERCONDUCTING QUANTUM PHYSICS

Expected Learning Outcomes

The aim of the course is to provide students with advanced knowledge of superconductivity, ranging from its fundamental theoretical foundations to its applications in quantum technologies.

Knowledge and Understanding

Students will acquire a critical understanding of advanced developments in modern physics, encompassing both theoretical and experimental aspects and their interconnections, including in interdisciplinary contexts. They will develop an appropriate command of advanced mathematical and computational tools commonly used in fundamental and applied research, together with a strong understanding of the scientific method and of the nature and practice of physics research.

The course will cover experimental phenomena and theoretical models of superconductivity and its applications, with reference to modern experiments and innovative theoretical interpretations. Appropriate software packages for modelling and simulating state-of-the-art superconducting qubits will also be introduced.

Applying Knowledge and Understanding

Students will develop the ability to identify the essential features of a physical phenomenon in terms of relevant orders of magnitude and the level of approximation required, and to apply the appropriate approximations. They will also develop the ability to use analogies to transfer known solutions to new problems and to apply effective problem-solving strategies.

In presenting the phenomenology and theoretical aspects of superconductivity, particular emphasis will be placed on the physical quantities and mechanisms that are most relevant to each phenomenon, while additional effects and contributions will be introduced through progressively refined approximations.

Making Judgements

Students will develop the ability to formulate and critically discuss their own interpretations of physical phenomena, including through interaction and discussion within working groups. They will also develop a sense of responsibility and autonomy through the selection of elective courses and the choice of a Master's thesis topic.

Throughout the course and during the final examination, connections will be highlighted with other courses within the degree programme, including elective courses, as well as with possible research topics for Master's theses, encompassing both theoretical and experimental approaches.

Communication Skills

Students will develop the ability to communicate scientific concepts and results clearly and effectively in both Italian and English, particularly in advanced areas of physics. They will acquire the ability to present and discuss physical phenomena, theoretical models, computational approaches, and research results using appropriate scientific terminology.

Learning Skills

Students will acquire the methodological and conceptual tools required for the continuous development and updating of their knowledge. They will develop the ability to access and critically assess specialised scientific literature both in their chosen field and in closely related areas.

Students will learn to use bibliographic databases and scientific information resources to identify relevant information and research perspectives, and to use these resources to frame, develop, and deepen their own study and research activities. They will also develop the ability to acquire, through independent study, knowledge in new and emerging areas of science.

Throughout the course, students will be regularly introduced to scientific review and research articles, thereby developing familiarity with the current literature and with the practice of engaging with state-of-the-art research.

Lectures. Should the course be delivered in a blended or fully online format, the teaching methods may be appropriately adjusted, as necessary, to ensure that the planned course content and learning objectives outlined in the syllabus are fully covered.

Knowledge of advanced quantum mechanics, solid state physics, preferably also of the theory of many-body systems.

Course Contents

1. Phenomenology and Phenomenological Theories of Superconductivity (8 hours of lecture)

Characteristic properties of superconductors: zero resistance, the Meissner effect, and flux quantization. London theory of the electrodynamics of superconductors. Ginzburg–Landau theory.

2. Microscopic Theory of Superconductivity (10 hours of lecture)

Cooper instability and the origin of the attractive interaction. Formation of Cooper pairs and s-wave pairing. BCS theory: ground state and excitation properties. Band structure, superconducting gap, and density of states. Finite-temperature properties and critical temperature. Penetration depth. Connection between BCS theory and Ginzburg–Landau theory.

3. Fundamental Aspects of Superconductivity (4 hours of lecture)

Macroscopic quantum coherence. Phase coherence and off-diagonal long-range order. Spontaneous symmetry breaking. Phase-number uncertainty relation.

4. Electron Tunneling and Interface Phenomena (4 hours of lecture)

Electron tunneling and quasiparticle tunneling. Charging effects. Josephson effect. Proximity effect. Andreev tunneling.

5. Macroscopic Quantum Physics (8 hours of lecture)

Classical dynamics of Josephson circuits. Gauge invariance and the Josephson effect in the presence of magnetic flux. Superconducting Quantum Interference Devices (SQUIDs). Phase quantization. Quantum dynamics of Josephson systems. Secondary quantum effects.

6. Superconducting Qubits (12 hours of lecture)

Superconducting devices for quantum computation. Quantum circuits and derivation of the Hamiltonian of circuits containing qubits. Phase, charge, and flux qubits. Transmon and fluxonium qubits. Capacitive and inductive coupling between qubits. Resonator–superconducting qubit coupling. Dispersive regime and quantum non-demolition (QND) measurement. Qubit state readout techniques and Josephson amplifiers. Coherent control of qubits: Rabi oscillations and Ramsey fringes. Relaxation and coherence times.

7. Computer-Assisted Analysis and Design of Superconducting Qubits (12 hours of lecture)

Numerical tools for the modeling and simulation of superconducting quantum architectures. Numerical diagonalization methods in Python and with the scqubits library, and their applications. Calculation and analysis of the spectral properties of transmon and fluxonium qubits. Dependence of qubit properties on circuit parameters.

8. Quantum Computing with Superconducting Qubits (9 hours of lecture)

Computational tools for the analysis of qubit dynamics and for the design of quantum-computing protocols (quantum gates). Introduction to optimal-control and artificial-intelligence methods for the design of physical systems and the control of coherent dynamics.

M. Tinkham, Introduction to Superconductivity, Dover (2004).

Yuli V. Nazarov and J. Danon, Advanced Quantum Mechanics: a practical guide, Cambridge (2013)

James F. Annet, Superconductivity, superfluids and condensates, Oxford University press (2003)

Steven M. Girvin, Kun Yang, Modern Condensed Matter Physics, Cambridge University Press (2019)

A. O. Caldeira, An introduction to Macroscopic Quantum Phenomena and Quantum Dissipation,

Cambridge University Press (2014)

P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver “A quantum engineer's guide to superconducting qubits”, Appl. Phys. Rev. 6, 021318 (2019); https://doi.org/10.1063/1.5089550

https://scqubits.readthedocs.io/

https://qutip.org/

Assessment Methods

Assessment of learning consists of a final oral examination structured in two parts. The first part consists of the presentation of a topic agreed upon in advance with the instructor. The second part consists of the presentation of a topic selected by the candidate at the time of the examination from three topics proposed by the instructor and characterised by different levels of difficulty.

Passing the examination is conditional upon passing the first part of the assessment, while the second part contributes to the final grade. At the student's request, the first part of the examination may be replaced by the preparation and discussion of a written project involving an analytical and/or numerical analysis or calculation. The project must be developed independently by the student, with guidance from the instructor, based on the recommended textbooks and any review articles suggested during the course.

The final grade will take into account the relevance and completeness of the answers to the questions posed, the level of understanding and mastery of the course content, the accuracy in performing and presenting calculations, and the ability to establish connections with other topics covered in the course or in previous courses and to provide appropriate examples. The use of appropriate scientific terminology and the clarity and coherence of the presentation will also be assessed.

If circumstances require it, the assessment may also be conducted remotely.

The questions listed below are intended solely as examples and do not constitute an exhaustive list of the topics that may be covered in the examination. Examples include the phenomenology of superconducting materials; the formulation and main implications of Ginzburg–Landau phenomenological theory; the distinction between Type-I and Type-II superconductors; the assumptions underlying BCS theory and the structure of its ground state; the description of excited states in a superconductor; the Josephson effect; electron tunneling in SN and SS junctions; and the mechanisms and consequences of spontaneous symmetry breaking.