In the last two decades, a spectacular development in the superconducting qubits has been accomplished experimentally as well as theoretically. The main reason for that is the realization that superconducting qubits can play a crucial role in the emergent field of quantum information processing and quantum computing. The interaction of superconducting qubits with the quantized electromagnetic field leads to make the circuit quantum electrodynamics, inspired by cavity quantum electrodynamics. The field of superconducting qubits and circuit QED is fast growing and branching to diverse applications in multidisciplinary fields. This makes it harder for first-time readers to follow-up. Therefore, this article is targeting to introduce in an intuitive way the topic of superconducting qubit and circuit QED by giving all the mathematical background, Lagrangian formalism for electric circuits, interaction Hamiltonian, and contextual physical interpretation essential to anew scientists and engineers planning to be specialized in this field as well as portraying a wide vantage for higher studies through the suggested recommendation resulted from the simulated examples and written code based on Python packages and Qiskit software. In this article, the superconducting qubits, transmon and fluxonium, are reviewed after introducing the abstract definition of a quantum bit. Moreover, light-matter interaction in fluxonium qubit and quantization of transmission line resonator are discussed briefly. Some of the simulated systems such as the energy levels of fluxonium as a function of offset flux are presented. Finally, applications of superconducting qubits in quantum computing and other fields such as quantum transistors, quantum machines and metamaterials are discussed. Index Terms Superconducting qubits, circuit QED, quantum logic gate, fluxonium, transmon.
In the last two decades, a spectacular development in the superconducting qubits has been accomplished experimentally as well as theoretically. The main reason for that is the realization that superconducting qubits can play a crucial role in the emergent field of quantum information processing and quantum computing. The interaction of superconducting qubits with the quantized electromagnetic field leads to make the circuit quantum electrodynamics, inspired by cavity quantum electrodynamics. The field of superconducting qubits and circuit QED is fast growing and branching to diverse applications in multidisciplinary fields. This makes it harder for first-time readers to follow-up. Therefore, this article is targeting to introduce in an intuitive way the topic of superconducting qubit and circuit QED by giving all the mathematical background, Lagrangian formalism for electric circuits, interaction Hamiltonian, and contextual physical interpretation essential to anew scientists and engineers planning to be specialized in this field as well as portraying a wide vantage for higher studies through the suggested recommendation resulted from the simulated examples and written code based on Python packages and Qiskit software. In this article, the superconducting qubits, transmon and fluxonium, are reviewed after introducing the abstract definition of a quantum bit. Moreover, light-matter interaction in fluxonium qubit and quantization of transmission line resonator are discussed briefly. Some of the simulated systems such as the energy levels of fluxonium as a function of offset flux are presented. Finally, applications of superconducting qubits in quantum computing and other fields such as quantum transistors, quantum machines and metamaterials are discussed. Index Terms Superconducting qubits, circuit QED, quantum logic gate, fluxonium, transmon.
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.
Superconducting qubits, realized by incorporating Josephson junctions into superconducting circuits, behave as artificial atoms with anharmonic energy spectra and can be precisely controlled and measured using microwave cavities within the framework of circuit quantum electrodynamics (cQED). Since its emergence in the early 2000s, cQED has established superconducting qubits as leading candidates for scalable quantum devices and has enabled the exploration of hybrid quantum systems that integrate disparate physical platformsThis review surveys superconducting hybrid quantum electromechanical systems in which mechanical resonators are coupled to superconducting qubits, with a focus on two widely used qubit platforms: the transmon and the fluxonium. We provide an overview of the underlying coupling mechanisms arising from interactions through the phase and charge degrees of freedom of the qubit, and discuss how these mechanisms give rise to both longitudinal and transverse qubit-mechanical interactions. We further review extensions of electromechanical platforms to electro-optomechanical architectures, in which optical cavities are integrated to enable coherent interfacing between superconducting circuits and optical photons. This review aims to present a unified framework and perspective on qubit-mechanical and qubit-mechanical-optical hybrid systems in superconducting quantum technologies and applications related to sensors.
In the last two decades, the field of circuit quantum electrodynamics, that studies the interaction between superconducting qubits and 1-dimensional waveguides, has been of great interest. It provides a great potential to build quantum devices, which are important for quantum computing, quantum communication and quantum information. The restriction to one dimension decreases losses and information can be transferred efficiently. Superconducting qubits are artificial atoms that consist of a non-linear Josephson element and work in the microwave regime. These superconducting qubits make on-chip tunable quantum experiments possible.In the appended paper, we investigate the spontaneous emission of an initially excited artificial atom (superconducting transmon qubit) which is capacitively coupled to a semi-infinite transmission line (atom in front of a mirror). We can choose the distance to the mirror arbitrarily so the interaction with the reflected field is delayed if the qubit is far away from the mirror and we have to take time-delay effects into account. We derive equations of motion for the transmon by circuit quantization and solve them semi-classically. In this thesis we give an introduction to circuit quantization, transmission lines and superconducting qubits. Then we discuss the methods and results of the appended paper which are based on the topics introduced.
Superconducting quantum bits (qubits) and circuits are the leading candidate for the implementation of solid-state quantum computation. They have also been widely used in a variety of studies of quantum physics, atomic physics, quantum optics, and quantum simulation. In this article, we will present an overview of the basic principles of the superconducting qubits, including the phase, flux, charge, and transmon (Xmon) qubits, and the progress achieved so far concerning the improvements of the device design and quantum coherence property. Experimental studies in various research fields using the superconducting qubits and circuits will be briefly reviewed.
In a quantum computer, the information carriers, which are bits in ordinary computers, are implemented as devices that exhibit coherent superpositions of physical states and entanglement. Such components, known as quantum bits or qubits, can be realized with various different types of two-state quantum systems. Quantum computers will be built for computational speed, with hoped for applications especially in cryptography and in other tasks where classical computers remain inefficient. Circuit quantum electrodynamics (cQED) is a quantum-computer architecture which employs superconducting electronic components and microwave photon fields as building blocks. Compared to cavity quantum electrodynamics (CQED), where atoms are trapped in physical cavities, cQED is more attractive in that its qubits are tunable and conveniently integrable with the electronics already in use. This architecture has shown some of the most promising qubit designs, despite their coherence times reaching tens of microseconds, are still below the state of the art with spin qubits, which reach milliseconds. Coherence times are historically the most relevant parameters describing the fitness of a qubit, although these days not necessarily the limiting factor. This thesis presents a comprehensive set of theoretical and experimental methods for measuring the characteristic parameters of superconducting qubits. We especially study transmission-line-shunted plasma oscillation qubits, or transmons, and presents experimental results for a single sample. Transmon capacitively couples a superconducting quantum interference device (SQUID) with a coplanar waveguide (CPW) resonator, often with added frequency tunability utilizing an external magnet. The number of superconducting charge carriers tunnelled through a junction in the SQUID are used as qubit degrees of freedom. Readout of the qubit state is carried out by measuring transmission through the CPW. A cryogenic setup is employed with measurement and driving pulses delivered from microwave sources. Steady-state spectroscopy is employed to determine the resonance frequencies of the qubit and the resonator, qubit-resonator coupling constants, and the energy parameters of the qubit. Pulse-modulated measurements are employed to determine the coherence times of the qubit. The related analysis- and simulation programs and scripts are collected togithub.com/patomaki.
Superconducting quantum computing is a solid state quantum computing technology based on Josephson junction circuits. It becomes one of the most promising candidates for realizing practical quantum computer/simulator, because of its scalability, easy to coupling and control, compatibility to semiconductor technologies, etc. As the complicated electromagnetic environment in solid state metal and insulator, superconducting quantum bits (qubits) suffered from shot coherence time since its early development. In this review, we described some recently developed superconducting qubit structures, including Transmon/Xmon, Fluxonium and C-shunt flux qubit. Although these new qubit structures have very different circuit parameters and energy scales, they all find their way to effectively suppress environmental noises. As a result, their coherence time were improved significantly. Transmon, a transmission-line shunted plasma oscillation qubit, was now among the most popular structures for building a multi-qubit superconducting quantum circuit. It introduced a relatively large shunt capacitor to remarkably reduce the charge dispersion. Although the non-linearity is also decreased, by operating in a properly chosen EJ/Ec, transmon can greatly reduce the charge noise while only sacrificing a small amount of non-linearity. Xmon can be considered as a modified version of transmon. It is designed and optimized for multi-qubit operation and measurement with separate X/Y and Z control lines for each qubit, and a readout bus (a transmission line or a low-Q bandpass filter) for simultaneous readout of all the qubits. Fluxonium provide another solution for suppressing the charge noise in Cooper-pair box (CPB). It introduces a large shunt inductance to filter low-frequency offset charge in the small capacitor of the phase-slip junction (the small junction). To keep the qubit operating in the charge region, the shunt inductor should be large enough to keep the impedance larger than the superconducting impedance quantum RQ~1 kΩ. As a result, the inductance can only be provided by a large series of Josephson junctions. Fluxonium keeps the advantage of high anharmonicity of CPB. In addition, it introduces another degree of freedom: external flux Φext. Quasiparticles (electrons or holes) tunneling at the small junction add coherently to produce a ″1+cosφ″ term. As a result, when Φext= Φ0/2, dissipation caused by quasiparticle tunneling is suppressed and the energy relaxation time (T1) increases drastically. C-shunt flux qubit (CSFQ) decreases α and EJ to suppress flux noise in the qubit. While the charge noise becomes the dominant noise source, a large shunt capacitor is added to suppress the charge noise, similar to that in a transmon qubit. It is worth to notice that in CSFQ, the α decreases to below but near 1/2. As a result, the double-well potential no longer exists. The energy structure become similar to transmon, with the anharmonicity be opposite. CSFQ can get a much improved coherence time, while its anharmonicity can be larger than that of transmon. Those new superconducting qubit structures provide different solutions to improve the coherence time. Although it looks much different in circuit models and parameters, they have ideas in common, and may act as guides to other novel designs with longer coherence time. We can summarize as below. (1) The qubit circuit should be free of small superconducting islands. Charge fluctuation in small islands is a major source of decoherence. As a result, increase the area of the islands, that is, increase the capacitance becomes a straight forward way to suppress the charge noise. In Fluxonium, the small junction is connected with a large series of large junctions that there is also no small islands in the whole loop. (2) Remove superconducting loops in the qubit circuit, or minimize the loop size and loop current to suppress flux noise. (3) Bias the phase difference of the small junction to π, where dissipation caused by quasiparticle tunneling can be suppressed coherently. (4) Decrease the coupling bandwidth of qubit to the environment. In a circuit-QED readout scheme, a high-Q readout resonator that dispersively coupled to the qubit protects the qubit from exposed to the electromagnetic environment of the readout circuit. A ″Purcell filter″ can also be added to suppress such decoherence further. In conclusion, we overviewed some qubit engineering techniques or ideas to improve the coherence time, which is a very important figure for building practical quantum computer/simulator. We hope that someone can combine and optimize these ideas, while interact with developments in materials and micro-fabrication technologies, to develop novel new generation superconducting qubit structures in the near future.