We show that, in the framework of quantum circuit electrodynamics in the Heisenberg picture, a finite-length transmission line can be described as a two-port lumped element of Th\'evenin type. Each port consists of a resistor connected in series to a controlled voltage source. The resistance of the resistors is equal to the characteristic impedance of the line. The controlled voltage sources are governed by linear equations with delay that take into account the reflections at the line ends. We apply this model to a transmission line capacitively coupled to two lumped circuits and obtain the reduced system of Heisenberg equations that governs them. Then, we show these equations can be reformulated as a pair of quantum Langevin-like equations that are coupled through the controlled voltage sources. Finally, we apply our approach to an analytically solvable network. This approach may be useful for the modeling of quantum links between superconducting circuits.
We show that, in the framework of quantum circuit electrodynamics in the Heisenberg picture, a finite-length transmission line can be described as a two-port lumped element of Th'evenin type. Each port consists of a resistor connected in series to a controlled voltage source. The resistance of the resistors is equal to the characteristic impedance of the line. The controlled voltage sources are governed by linear equations with delay that take into account the reflections at the line ends. We apply this model to a transmission line capacitively coupled to two lumped circuits and obtain the reduced system of Heisenberg equations that governs them. Then, we show these equations can be reformulated as a pair of quantum Langevin-like equations that are coupled through the controlled voltage sources. Finally, we apply our approach to an analytically solvable network. This approach may be useful for the modeling of quantum links between superconducting circuits.
Abstract The quantization of systems composed of transmission lines connected to lumped circuits poses significant challenges, arising from the interaction between continuous and discrete degrees of freedom. A widely adopted strategy, based on the pioneering work of Yurke and Denker, entails representing the lumped circuit contributions using Lagrangian densities that incorporate Dirac δ-functions. However, this approach introduces complications, as highlighted in the recent literature, including divergent momentum densities, necessitating the use of regularization techniques. In this work, we introduce a δ-free Lagrangian formulation for a transmission line capacitively coupled to a lumped circuit without the need for a discretization of the transmission line or mode expansions. This is achieved by explicitly enforcing boundary conditions at the line ends in the principle of least action. In this framework, the quantization and the derivation of the Heisenberg equations of the network are straightforward. We obtain a reduced model for the lumped circuit in the quantum Langevin form, which is valid for any coupling strength between the line and the lumped circuit. We apply our approach to an analytically solvable network consisting of a semi-infinite transmission line capacitively coupled to an LC circuit and study the behavior of the network as the coupling strength varies.
We study the emergence of continuum, discrete-multimode, and single-mode regimes in finite-length transmission lines capacitively coupled to transmon qubits. We show that the appropriate description is selected by the hierarchy among the qubit frequency ωq, the characteristic transmission line frequency ωTL, and the characteristic coupling frequency ωg. In the long-line continuum limit, the transmission line acts as a structured reservoir described by a Drude--Lorentz spectral density; in the short-line limit, it reduces to an effective single-mode resonator; and, between these limits, it behaves as a discrete multimode coupler. This provides a unified cQED picture of the dynamical regimes of finite-length transmission lines in superconducting-circuit architectures.
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.
Superconducting quantum circuit design is reliant on accurately mapping design parameters to a quantum Hamiltonian. Designers typically rely on computationally intensive finite-element electromagnetic simulations. However, effective circuit-level models can in principle capture much of the relevant physics for these systems, reducing the need for finite-element simulations. A barrier to implementing these simpler models has been the prevalence of distributed elements such as coplanar waveguides resonators in device designs. Techniques for modeling these distributed elements have been developed, but may be difficult to scale, reduce intuition, and give inaccurate results under strong coupling. In this work we describe a simple effective circuit-level model that faithfully reproduces the scattering parameters, and subsequently can predict certain Hamiltonian parameters, for a coupled, distributed element within a broader two-port network even up to strong coupling. Our approach does not explicitly require an electromagnetic simulation. Our model, along with other common lumped models used in black box quantization, is publicly available to researchers via an open-source code package, simpleLOMs.
Currently, circuit quantum electrodynamics architectures have emerged as one of the most popular approaches to implement practical quantum information processing hardware. Although significant progress has been made, many technical issues remain that limit the performance of fabricated devices. One approach to accelerate progress in the engineering design of these devices is to develop improved numerical modeling methods. Current modeling methods generally rely on approximate lumped element circuit models to describe the complex network of microwave transmission lines required to operate a circuit quantum electrodynamics device. This reduction in complexity in the theoretical model is valuable for gaining insight into the operation of a device, but does limit the opportunity for using these models to optimize the performance of practical devices. To develop rigorous numerical modeling methods, it is necessary to move toward theoretical descriptions of circuit quantum electrodynamics devices that retain the full details of the three-dimensional vector electromagnetic fields that exist in these systems. In this work, we present details on such a theoretical model for one of the most commonly used circuit quantum electrodynamics systems, a transmon qubit coupled to microwave transmission lines. We discuss the quantization of our new model and show that by adopting relevant approximations our model can be reduced to the same lumped element descriptions commonly used in the literature. We also discuss the derivation of quantum equations of motion for the coupled field-transmon system, which can be used as the starting point for developing full-wave numerical solvers for these circuit quantum electrodynamics systems in the future.
In superconducting circuit quantum information technologies, classical microwave pulses are applied to control and measure the qubit states. Currently, the design of these microwave pulses use simple theoretical or numerical models that do not account for the self-consistent interactions of how the qubit state modifies the applied microwave pulse. In this work, we present the formulation and finite element time domain discretization of a semiclassical Maxwell-Schrödinger method for describing these self-consistent dynamics for the case of a superconducting qubit capacitively coupled to a general transmission line network. We validate the proposed method by characterizing key effects related to common control and measurement approaches for transmon and fluxonium qubits in systems that are amenable to theoretical analysis. Our numerical results also highlight scenarios where including the self-consistent interactions are essential. By treating the microwaves classically, our method is substantially more efficient than fully-quantum methods for the many situations where the quantum statistics of the microwaves are not needed. Further, our approach does not require any reformulations when the transmission line system is modified. In the future, our method can be used to rapidly explore broader design spaces to search for more effective control and measurement protocols for superconducting qubits.