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arXiv:2311.09897 · Physica Scripta

A δ-free approach to quantization of transmission lines connected to lumped circuits

Carlo Forestiere, Giovanni Miano

Published Mar 14, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1402-4896/ad340b

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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 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.
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.
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.
We develop a systematic procedure to quantize canonically Hamiltonians of light-matter models of transmission lines coupled through lumped linear lossless ideal nonreciprocal elements, that break time-reversal symmetry, in a circuit QED set-up. This is achieved through a description of the distributed subsystems in terms of both flux and charge fields. We prove that this apparent redundancy is required for the general derivation of the Hamiltonian for a wider class of networks. By making use of the electromagnetic duality symmetry in 1+1 dimensions, we provide unambiguous identification of the physical degrees of freedom, separating out the nondynamical parts. This doubled description can also treat the case of other extended lumped interactions in a regular manner that presents no spurious divergences, as we show explicitly in the example of a circulator connected to a Josephson junction through a transmission line. This theory enhances the quantum engineering toolbox to design complex networks with nonreciprocal elements.
The physical constraints underlying the concept of quantum circuit are considered. In particular it is shown that the point of departure for their modeling starts from the interconnection of the components into a classical network, followed by quantization of the latter, and not by the interconnection of already quantized components. The procedure is straightforward for lossless networks but cannot be worked out in presence of resistors for the impossibility of constructing a Lagrangian function. However the difficulty is circumvented by distinguishing thermal from radiative resistors, the former being the usual ones, the latter being realized by semi-infinite LC transmission lines, for which the Lagrangian exists. In the complex plane s = σ + jω the impedance of a thermal resistor is Z(s) = R in the entire plane while that of a radiative resistor is Rsign(σ), so that the latter is lossless and does not dissipate energy but conveys it to the infinity. Comparison with Lindblad approach shows that the resistor fits into it in the RHP. Radiative resistors make evolution reversible, which is not true for a physical system including thermal.
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.