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arXiv:1310.1897

Circuit QED - Lecture Notes

Nathan K. Langford

Published Oct 7, 2013• 10 citations
arXivHTMLPDF

Abstract

The new and rapidly growing field of circuit QED offers extremely exciting prospects for learning about and exercising intimate control over quantum systems, providing flexible, engineerable design and strong nonlinearities and interactions in systems consisting of microwave radiation fields and fixed artificial "atoms". These notes aim to provide a non-expert introduction to the field of circuit QED, to give a basic appreciation of the promise and challenges of the field, along with a number of key concepts that will hopefully be useful for the reader who is new to the field and beginning to explore the research literature. They were written as a pedagogical text designed to complement a course delivered to third-year undergraduate students. After a introductory section which discusses why studying circuit QED might be worthwhile and interesting, I introduce the basic theory tools from quantum optics and quantum information which are needed to understand the key elements of circuit QED. I also provide a brief overview of superconductivity, focussing on the concepts which are most relevant to operation in the regimes of interest in circuit QED. I then describe the three main types of superconducting qubits, and finally give a basic introduction to decoherence and mixture and how they relate to quantum behaviour in electronic circuits.

DOI
10.48550/arxiv.1310.1897

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The new and rapidly growing field of circuit QED offers extremely exciting prospects for learning about and exercising intimate control over quantum systems, providing flexible, engineerable design and strong nonlinearities and interactions in systems consisting of microwave radiation fields and fixed artificial "atoms". These notes aim to provide a non-expert introduction to the field of circuit QED, to give a basic appreciation of the promise and challenges of the field, along with a number of key concepts that will hopefully be useful for the reader who is new to the field and beginning to explore the research literature. They were written as a pedagogical text designed to complement a course delivered to third-year undergraduate students. After a introductory section which discusses why studying circuit QED might be worthwhile and interesting, I introduce the basic theory tools from quantum optics and quantum information which are needed to understand the key elements of circuit QED. I also provide a brief overview of superconductivity, focussing on the concepts which are most relevant to operation in the regimes of interest in circuit QED. I then describe the three main types of superconducting qubits, and finally give a basic introduction to decoherence and mixture and how they relate to quantum behaviour in electronic circuits.
In the last 25 years a new understanding has evolved of the role of information in quantum mechanics. At the same time there has been tremendous progress in atomic/optical physics and condensed matter physics, and particularly at the interface between these two formerly distinct fields, in developing experimental systems whose quantum states are long-lived and which can be engineered to perform quantum information processing tasks. These lecture notes will present a brief introduction to the basic theoretical concepts behind this `second quantum revolution.' These notes will also provide an introduction to `circuit QED.' In addition to being a novel test bed for non-linear quantum optics of microwave electrical circuits which use superconducting qubits as artificial atoms, circuit QED offers an architecture for constructing quantum information processors.
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Circuit quantum electrodynamics (‘circuit QED’) describes the quantum mechanics and quantum optics of superconducting electrical circuits operating in the microwave regime near absolute zero temperature. It is the analog of cavity QED in quantum optics with the role of the atoms being played by superconducting qubits. The present lecture notes present a brief overview of circuit QED and then focus on some of the novel quantum states that can be produced and measured (via photon number parity and the Wigner function) using the strong coupling between an artificial atom and one or more cavities. Of particular importance are Schrödinger cat states of photons. Despite long being considered exemplars of frail quantum superpositions that quickly decohere, such states have recently been used as the basis for quantum error correction codes which have reached the long-sought goal of enhancing the lifetime of quantum information through active quantum error correction.
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This tutorial aims at giving an introductory treatment of the circuit analysis of superconducting qubits, i.e., two-level systems in superconducting circuits.It also touches upon couplings between such qubits and how microwave driving and these couplings can be used for single-and two-qubit gates, as well as how to include noise when calculating the dynamics of the system.We also discuss higher-dimensional superconducting qudits.The tutorial is intended for new researchers with limited or no experience with the field but should be accessible to anyone with a bachelor's degree in physics.The tutorial introduces the basic methods used in quantum circuit analysis, starting from a circuit diagram and ending with a quantized Hamiltonian, that may be truncated to the lowest levels.We provide examples of all the basic techniques throughout the discussion, while in the last part of the tutorial we discuss several of the most commonly used circuits for quantum-information applications.This includes both worked examples of single qubits and examples of how to analyze the coupling methods that allow multiqubit operations.In several detailed appendices, we provide the interested reader with an introduction to more advanced techniques for handling larger circuit designs.
Scope and aims of this book Quantum optics is one of the liveliest fields in physics at present. While it has been a dominant research field for at least two decades, with much graduate activity, in the past few years it has started to impact the undergraduate curriculum. This book developed from courses we have taught to final year undergraduates and beginning graduate students at Imperial College London and City University of New York. There are plenty of good research monographs in this field, but we felt that there was a genuine need for a straightforward account for senior undergraduates and beginning postgraduates, which stresses basic concepts. This is a field which attracts the brightest students at present, in part because of the extraordinary progress in the field (e.g. the implementation of teleportation, quantum cryptography, Schrödinger cat states, Bell violations of local realism and the like). We hope that this book provides an accessible introduction to this exciting subject. Our aim was to write an elementary book on the essentials of quantum optics directed to an audience of upper-level undergraduates, assumed to have suffered through a course in quantum mechanics, and for first-or second-year graduate students interested in eventually pursuing research in this area. The material we introduce is not simple, and will be a challenge for undergraduates and beginning graduate students, but we have tried to use the most straightforward approaches. Nevertheless, there are parts of the text that the reader will find more challenging than others.