Abstract This chapter introduces the basic concepts of quantum information processing with superconducting quantum circuits. It starts with a demonstration of how,in these electrodynamic systems, both the superconducting gap and the long-range Coulomb interactions contribute to the existence of collective modesthat have extremely low dissipationand that can be quantized. The chapter also gives in-depth coverage of quantum harmonic oscillators in all their glory from various perspectives. Superconducting qubits are then introduced, starting with the fundamental theory of the Josephson effect and moving on to a presentation of the variety of existing superconducting qubits. Particular attention is given to the dispersive readout of qubits through their interaction with a cavity into which flying modes can be scattered. Finally, the opportunities offered for quantum information processing by employing the cavity as a quantum bus are presented.
Abstract This chapter introduces the basic concepts of quantum information processing with superconducting quantum circuits. It starts with a demonstration of how,in these electrodynamic systems, both the superconducting gap and the long-range Coulomb interactions contribute to the existence of collective modesthat have extremely low dissipationand that can be quantized. The chapter also gives in-depth coverage of quantum harmonic oscillators in all their glory from various perspectives. Superconducting qubits are then introduced, starting with the fundamental theory of the Josephson effect and moving on to a presentation of the variety of existing superconducting qubits. Particular attention is given to the dispersive readout of qubits through their interaction with a cavity into which flying modes can be scattered. Finally, the opportunities offered for quantum information processing by employing the cavity as a quantum bus are presented.
In this paper, the coherent interaction of individual microwave photons with superconducting qubits is presented both for quantum information processing and for novel quantum optics experiments in the circuit quantum electrodynamics (QED) architecture. In the resonant regime we observe the vacuum Rabi mode splitting. In the non-resonant regime we use the dispersive coupling between the qubit and the cavity field to perform high visibility quantum non-demolition (QND) measurements of the qubit state. In the strong dispersive limit we also perform measurements of the cavity state from which the photon number statistics of the field can be extracted. Generally, the circuit QED architecture is attractive for quantum information processing and as an interface in hybrid quantum systems.
As superconducting circuits emerge as a leading platform for scalable quantum information processing, building comprehensive bridges from the foundational principles of macroscopic quantum phenomena to the architecture of modern quantum devices is increasingly essential for introducing new researchers to the field. This tutorial provides a self-contained, pedagogical introduction to superconducting quantum circuits at the undergraduate level. Beginning with an overview of superconductivity and the Josephson effect, the tutorial systematically develops the quantization of microwave circuits into the framework of circuit quantum electrodynamics (cQED). The transmon qubit is then introduced as a state-of-the-art application, with a detailed derivation of its Hamiltonian and its interaction with control and readout circuitry. The theoretical formalism is consolidated through a numerical simulation of vacuum Rabi oscillations in a driven transmon-resonator system, a canonical experiment that demonstrates the coherent energy exchange characteristic of the strong coupling regime. This work serves as a foundational guide and first point of contact, equipping students and researchers with the conceptual and mathematical tools necessary to understand and engineer superconducting quantum hardware.
In this thesis, experimentally relevant aspects and open questions of quantum information processing and measurement in circuit quantum electrodynamics have been investigated theoretically. Circuit quantum electrodynamics is a relatively young field combining superconducting transmission line resonators on-chip with superconducting quantum bits which serve as artificial atoms. Remarkable experiments have underlined the prospects of circuit QED as a possible architecture for quantum information processing as well as a framework within which quantum optics experiments can be performed. In contrast to their optical counterparts, these experiments reach the ultra-strong coupling limit and allow e.g. the generation of Fock states, the study of decoherence and the observation of quantum jumps. We present the physics of superconducting circuits based on the Josephson junction as a non-linear circuit element. We discuss the various types of quantum bits (qubits) before turning to an introduction on cavity QED and conclude the introduction with a brief presentation of the most relevant experiments and a short introduction to the principles and prospects of quantum information processing. Building on the pioneering theoretical and experimental work in which up to three qubits have been integrated on a chip and successfully coupled via a cavity mode, we propose a scheme to overcome the natural limits concerning scalability in these systems. Usually, scalability is severely limited by the resonance width of the qubit transitions and the attainable frequency range for both qubit transition and cavity resonance frequencies. This limits the present setups to at most ten qubits. As a possible solution, we propose and investigate a cross-bar grid layout of cavities with qubits at the intersections. It turns out that this setup, allowing for up to about 1000 qubits, offers significant advantages over e.g. nearest-neighbor coupling schemes. In addition, it can serve as a building block for a truly scalable scheme for fault-tolerant quantum information processing with superconducting circuits by combining 7x7 arrays in a staircase manner. Furthermore, we discuss and simulate different possibilities to improve upon the suggested scheme by e.g. using resonant gates and optimal control theory. Another focus of the present thesis is measurement physics. In order to investigate e.g. the possibilities of using a circuit QED system to dispersively detect single microwave photons or to generate multi-qubit entangled states, we introduce in a pedagogical manner the basics of stochastic master equations which generate quantum trajectories before turning to concrete applications. Quantum trajectories have proven to be a state-of-the-art tool to analyze measurement situations in a very realistic manner by giving access to both the measurement record as well as the internal quantum dynamics conditioned on this record. Using the method of quantum trajectories as generated by stochastic master equations, we discuss a scheme to efficiently generate multi-qubit entangled states in a very flexible way. The method is based on a collective, dispersive qubit readout. By this measurement, a suitable product state of N qubits can be reliably converted into e.g. an N-qubit Greenberger-Horne-Zeilinger or W state. We propose and discuss the scheme and investigate the effects of decoherence and parameter spread as they would be encountered in experiment. Last but not least, we propose and fully analyze a scheme to dispersively detect single itinerant microwave photons. While experiments to generate single microwave photons have already been successfully performed, their single-shot quantum non-demolition detection has not been possible so far. The scheme presented in this thesis closes this gap in the circuit QED toolbox and reaches detection efficiencies of 30% only limited by fundamental quantum mechanics in this case the Quantum Zeno effect. This detection scheme could be used in quantum communication or quantum cryptography applications.
Controlling Photons in Superconducting Electrical Circuits Blake Robert Johnson 2011 Circuit quantum electrodynamics (circuit QED) is a system that allows for strong coupling between microwave photons in transmission line cavities and superconducting qubits or artificial atoms. While circuit QED is often studied in the context of quantum information processing, it also provides an attractive platform for performing quantum optics experiments on-a-chip, because the level of control and coupling strengths available in circuit QED opens a vast array of possibilities for the creation, manipulation, and detection of quantum states of light. In this thesis, the extension of circuit QED to two cavities is examined, including design issues for cavities with very different Q-factors, and a new qubit design is proposed that couples to both cavities. The qubit-cavity interaction, while providing much of the utility of circuit QED, also introduces additional qubit relaxation. A powerful formalism for calculating this energy decay due to the classical admittance of the electromagnetic environment is presented in the context of circuit QED. Measurements of a wide range of samples validate this theory as providing an effective model for relaxation. A new circuit element, called the ‘Purcell filter’, is introduced and demonstrated to decouple the relationship between cavity Q and qubit relaxation. Finally, a new method for performing quantum nondemolition measurements of microwave photons is demonstrated. Controlling Photons in Superconducting Electrical Circuits A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy by Blake Robert Johnson Dissertation Director: Professor Robert J. Schoelkopf
In the last decade, the field of Quantum Optics has been enriched with a completely new platform: superconducting quantum circuits, also known as circuit-QED. This platform builds on the original designs of superconducting qubits, that is few-level systems made of superconducting islands and Josephson junctions, but adds to it a new ingredient, microwave photons. In this talk I will review circuit-QED with a focus on two complementary lines of research: ultrastrong light-matter coupling and strongly correlated propagating photons. The first topic deals with novel designs of superconducting qubits, engineering their coupling to the microwave radiation field to reach Rabi frequencies that approach the qubit and photon frequencies. In this regime, known as ”ultrastrong coupling”, the rotating wave approximation from Quantum Optics is no longer valid, the vacuum may spontaneously polarize, producing photons that get bound to the qubit, and novel physics is expected to emerge. I will review recent experiments on this topic, together with various theoretical proposals to tame this ultrastrong coupling. The second topic revolves around the notion of ”propagating” photons in these circuits. Most experiments in circuit-QED so far adopted the typical configuration of artificial atoms (qubits) interacting with artificial cavities (LC resonators). This proved useful in order to reach the strong coupling regime, demonstrating Rabi oscillations and developing a variety of protocols for quantum information, tomograhy, entanglement, etc. However, recent experiments have shown that it is possible to also work with propagating photons, replacing the LC resonators with open transmission lines, and studying the scattering properties of those artificial atoms. In this context we have worked on developing new technology to detect and control these photons, engineering single-photon detectors, quantum metamaterials and photonic crystals, or even new nonlinear media based on coupled cavities.
Hybrid circuit quantum electrodynamics (QED) involves the study of coherent quantum physics in solid state systems via their interactions with superconducting microwave circuits.Here we present a crucial step in the implementation of a hybrid superconducting qubit that employs a carbon nanotube as a Josephson junction.We realise the junction by contacting a carbon nanotube with a superconducting Pd/Al bi-layer, and implement voltage tunability of the quantum circuit's frequency using a local electrostatic gate.We demonstrate a strong dispersive coupling to a coplanar waveguide resonator by investigating the gate-tunable resonator frequency.We extract qubit parameters from spectroscopy using dispersive readout and find qubit relaxation and coherence times in the range of 10 -200 ns.