Abstract Superconducting circuits are one of the leading quantum platforms for quantum technologies. With growing system complexity, it is of crucial importance to develop scalable circuit models that contain the minimum information required to predict the behaviour of the physical system. Based on microwave engineering methods, divergent and non-divergent Hamiltonian models in circuit quantum electrodynamics have been proposed to explain the dynamics of superconducting quantum networks coupled to infinite-dimensional systems, such as transmission lines and general impedance environments. Here, we study systematically common linear coupling configurations between networks and infinite-dimensional systems. The main result is that the simple Lagrangian models for these configurations present an intrinsic natural length that provides a natural ultraviolet cutoff. This length is due to the unavoidable dressing of the environment modes by the network. In this manner, the coupling parameters between their components correctly manifest their natural decoupling at high frequencies. Furthermore, we show the requirements to correctly separate infinite-dimensional coupled systems in local bases. We also compare our analytical results with other analytical and approximate methods available in the literature. Finally, we propose several applications of these general methods to analogue quantum simulation of multi-spin-boson models in non-perturbative coupling regimes.
Abstract Superconducting circuits are one of the leading quantum platforms for quantum technologies. With growing system complexity, it is of crucial importance to develop scalable circuit models that contain the minimum information required to predict the behaviour of the physical system. Based on microwave engineering methods, divergent and non-divergent Hamiltonian models in circuit quantum electrodynamics have been proposed to explain the dynamics of superconducting quantum networks coupled to infinite-dimensional systems, such as transmission lines and general impedance environments. Here, we study systematically common linear coupling configurations between networks and infinite-dimensional systems. The main result is that the simple Lagrangian models for these configurations present an intrinsic natural length that provides a natural ultraviolet cutoff. This length is due to the unavoidable dressing of the environment modes by the network. In this manner, the coupling parameters between their components correctly manifest their natural decoupling at high frequencies. Furthermore, we show the requirements to correctly separate infinite-dimensional coupled systems in local bases. We also compare our analytical results with other analytical and approximate methods available in the literature. Finally, we propose several applications of these general methods to analogue quantum simulation of multi-spin-boson models in non-perturbative coupling regimes.
Circuit quantum electrodynamics (QED) studies the interaction of artificial atoms, open transmission lines, and electromagnetic resonators fabricated from superconducting electronics. While the theory of an artificial atom coupled to one mode of a resonator is well studied, considering multiple modes leads to divergences which are not well understood. Here, we introduce a first-principles model of a multimode resonator coupled to a Josephson junction atom. Studying the model in the absence of any cutoff, in which the coupling rate to mode number n scales as n for n up to $\ensuremath{\infty}$, we find that quantities such as the Lamb shift do not diverge due to a natural rescaling of the bare atomic parameters that arises directly from the circuit analysis. Introducing a cutoff in the coupling from a nonzero capacitance of the Josephson junction, we provide a physical interpretation of the decoupling of higher modes in the context of circuit analysis. In addition to explaining the convergence of the quantum Rabi model with no cutoff, our work also provides a useful framework for analyzing the ultrastrong coupling regime of a multimode circuit QED.
Waveguide circuit quantum electrodynamics (waveguide circuit QED) studies light-matter interaction with superconducting circuits in one dimension. In circuit QED, natural atoms are replaced by superconducting qubits consisting of a non-linear Josephson junction, resulting in an anharmonic energy spectrum just like real atoms. With superconducting qubits, it is possible to study quantum optical phenomena and reach new regimes hard to achieve with real atoms due to weak coupling to the electromagnetic field. The reduction to one dimension in waveguide QED increases the electromagnetic field's directionality, which results in reduced losses. In this thesis, we first introduce circuit quantisation, giving the basis for the next part, where we investigate a transmon, a charge-insensitive artificial atom, coupled to a semi-infinite transmission line. An atom coupled to a semi-infinite waveguide is referred to as an atom in front of a mirror and is the subject of all appended papers. We proceed by summarising Paper I and III's main results: in Paper I, we investigate the spontaneous emission of a transmon coupled to a semi-infinite transmission line, where we take time-delay effects into account. We find that the system dynamics strongly depend on the coupling strength to the transmission line and the atom's position with respect to the electromagnetic field, leading to the Purcell effect or the convergence to a dark state with finite excitation probability. In the high-impedance regime, which we investigate in Paper III, the properties of the transmon coupled to the high-impedance transmission line change drastically. It becomes highly reflective and creates its own cavity with the mirror, resulting in the emergence of cavity modes and vacuum Rabi oscillations in the spontaneous emission dynamics. In the next chapter of the thesis, we demonstrate how to quantise an electromagnetic field and derive a light-matter interaction Hamiltonian within dipole approximation. We then give an introduction to open quantum systems and derive the quantum-optical master equation in Lindblad form. Furthermore, we introduce the dressed state picture, where the interaction of light and matter is so strong that the individual energy levels of light and matter are no longer separable. Both the quantum optical master equation and the dressed state picture are relevant in Paper II and IV. In Paper II, an experimental collaboration, we perform several experiments to characterise and discriminate different decay rates of a superconducting qubit coupled to the end of a transmission line. One experiment measured the atomic fluorescence spectral density, which shows an asymmetry for off-resonant driving, resulting from pure dephasing: an effect that we explain in more detail in this thesis and Paper II. In Paper IV, we theoretically investigate amplification mechanisms realised by different set-ups of an atom coupled to a semi-infinite waveguide. In the considered systems, the amplification of a probe field happens either due to population inversion between the pure states or dressed states or multi-photon processes. We find that compared to an open waveguide, we can achieve a higher gain in the amplification with a semi-infinite waveguide.
In the quest to produce quantum technology, superconducting networks, working at temperatures just above absolute zero, have arisen as one of the most promising physical implementations. The precise analysis and synthesis of such circuits have required merging the fields of physics, engineering, and mathematics. In this dissertation, we develop mathematically consistent and precise Hamiltonian models to describe ideal superconducting networks made of an arbitrary number of lumped elements, such as capacitors, inductors, Josephson and phase-slip junctions, gyrators, etc., and distributed ones like transmission lines. We give formal proofs for the decoupling at high and low frequencies of lumped degrees of freedom from infinite-dimensional systems in different coupling configurations in models based on the effective Kirchhoff's laws. We extend the standard theory to quantize circuits that include ideal nonreciprocal elements all the way to their Hamiltonian descriptions in a systematic way. Finally, we pave the way on how to quantize general frequency-dependent gyrators and circulators coupled to both transmission lines and other lumped-element networks. We have explicitly shown, that these models, albeit ideal, are finite and present no divergence issues. We explain and dispel misunderstandings from the previous literature. Furthermore, we have demonstrated the usefulness of a redundant basis for performing separation of variables of the transmission line (1D) fields in the presence of point-like (lumped-element) couplings by time-reversal symmetry-breaking terms, i.e. nonreciprocal elements.
Devices built using superconducting circuits are a popular approach being pursued to develop quantum information processing hardware. These systems involve the quantized interactions of electromagnetic fields with large coherent collections of charges in superconductors, and so are typically referred to as macroscopic circuit quantum electrodynamics (QED) devices or simply circuit QED devices. Although significant progress has been made recently, substantial engineering efforts are required to improve performance and scale these devices so they can address problems of practical interest. Performing these engineering efforts is difficult due to the absence of rigorous numerical modeling approaches. This work begins to address this by providing a new mathematical description of a commonly used circuit QED system, a transmon qubit coupled to microwave transmission lines. Expressed in terms of three-dimensional vector fields, our model is better suited to developing rigorous numerical solvers than approximate lumped element circuit descriptions used in the literature. We present details on the quantization of our model, and derive equations of motion for the coupled field-transmon system. These results can be used in developing full-wave solvers in the future. To make this work more accessible, we assume a limited amount of training in quantum physics and provide many background details.
Building on the first-order circuit quantization method [arXiv:2304.12252, arXiv:2401.09120], we provide simple formulas to construct exact Hamiltonians for Josephson-junction-based superconducting qudits capacitively, inductively, or galvanically coupled to passive linear environments. These environments may be multiport, multimode, discrete or continuous, reciprocal or nonreciprocal, and are characterized directly by their impedance or admittance matrices. In the weak-coupling regime, we further derive \emph{divergence-free} dispersive Hamiltonians for mode-resolved environments and transition-resolved weak-coupling master equations for dissipative continua. Mode structure, frequency renormalizations, environment-mediated interactions, decay rates, and directional cross couplings then follow from the same causal immittance response, while spurious Lamb-shift divergences arising from uncontrolled approximations in previous treatments are made explicit and avoided. We apply the theory to a set of illustrative circuits comprising a discrete resonator filter, finite-band metamaterial environments, nonreciprocal waveguide-QED systems, and superconducting giant atoms, for which analytical response matrices can be obtained, although the method is particularly well suited to numerical responses from electromagnetic solvers or experimental characterization. We thereby extend the black-box quantization framework to multiport, dissipative, and nonreciprocal settings, establishing a simple and scalable route toward optimized and automated electromagnetic design of large-scale superconducting quantum hardware.
The precise engineering of electromagnetic couplings is paramount for constructing scalable and high-fidelity superconducting quantum processors. While essential for orchestrating qubit operations, these couplings also present significant design challenges, including the mitigation of crosstalk and the management of environmental decoherence. A clear and unified theoretical framework is therefore crucial for the design, simulation, and analysis of these complex quantum circuits. This paper presents a comprehensive theoretical treatment of the fundamental electromagnetic coupling mechanisms in superconducting devices. Starting from first principles, we formulate the equations of motion and derive the input-output relations for canonical systems, including a single resonator coupled to a multi-port microwave network, interacting resonators, and coupled transmission lines. We review rigorous definitions for key parameters such as the energy decay rate (κ) and the dimensionless coupling coefficient (ζ) and connect these formalisms to practical methods of parameter extraction from electromagnetic simulations. This work provides a rigorous and pedagogical foundation for understanding and modeling linear electromagnetic interactions, serving as a vital resource for the development of advanced superconducting quantum hardware.