Semi-infinite Transmission Line Research Articles

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

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.

Entanglement properties of superconducting qubits coupled to a semi-infinite transmission line

Quantum entanglement, a key resource in quantum information processing, is reduced by interaction between the quantum system concerned and its unavoidable noisy environment. Therefore it is of particular importance to study the dynamical properties of entanglement in open quantum systems. In this work, we mainly focus on two qubits coupled to an adjustable environment, namely a semi-infinite transmission line. The two qubits’ relaxations, through individual channels or collective channel or both, can be adjusted by the qubits’ transition frequencies. We examine entanglement dynamics in this model system with initial Werner state, and show that the phenomena of entanglement sudden death and revival can be observed. Due to the hardness of preparing the Werner state experimentally, we introduce a new type of entangled state called pseudo-Werner state, which preserves as much entangling property as the Werner state, and more importantly, it is experiment friendly. Furthermore, we provide detailed procedures for generating pseudo-Werner state and studying entanglement dynamics with it, which can be straightforwardly implemented in a superconducting waveguide quantum electrodynamics system.

Landau-Zener-Stückelberg-Majorana interferometry of a superconducting qubit in front of a mirror

We investigate the Landau-Zener-Stuckelberg-Majorana interferometry of a superconducting qubit in a semi-infinite transmission line terminated by a mirror. The transmon-type qubit is at the node of the resonant electromagnetic (EM) field, hiding from the EM field. "Mirror, mirror" briefly describes this system, because the qubit acts as another mirror. We modulate the resonant frequency of the qubit by applying a sinusoidal flux pump. We probe the spectroscopy by measuring the reflection coefficient of a weak probe in the system. Remarkable interference patterns emerge in the spectrum, which can be interpreted as multi-photon resonances in the dressed qubit. Our calculations agree well with the experiments.

Semiclassical analysis of dark-state transient dynamics in waveguide circuit QED

The interaction between superconducting qubits and one-dimensional microwave transmission lines has been studied experimentally and theoretically in the past two decades. In this work, we investigate the spontaneous emission of an initially excited artificial atom which is capacitively coupled to a semi-infinite transmission line, shorted at one end. This configuration can be viewed as an atom in front of a mirror. The distance between the atom and the mirror introduces a time delay in the system, which we take into account fully. When the delay time equals an integer number of atom oscillation periods, the atom converges into a dark state after an initial decay period. The dark state is an effect of destructive interference between the reflected part of the field and the part directly emitted by the atom. Based on circuit quantization, we derive linearized equations of motion for the system and use these for a semiclassical analysis of the transient dynamics. We also make a rigorous connection to the quantum optics system-reservoir approach and compare these two methods to describe the dynamics. We find that both approaches are equivalent for transmission lines with a low characteristic impedance, while they differ when this impedance is higher than the typical impedance of the superconducting artificial atom.

Radiation from free space TEM transmission lines

This work derives exact expressions for the radiation from two conductors non-isolated transverse electromagnetic (TEM) transmission lines of any small electric size cross-section in free space. The cases of infinite, semi infinite and finite transmission lines are covered and it is shown that while an infinite transmission line does not radiate, there is a smooth transition between the radiation from a finite to a semi-infinite transmission line. The present analysis is in the frequency domain and the authors consider transmission lines carrying any combination of forward and backward waves. The analytic results are validated by successful comparison with ANSYS commercial software simulation results, and successful comparisons with other published results.

Optimizing microwave photodetection: input–output theory

High fidelity microwave photon counting is an important tool for various areas from background radiation analysis in astronomy to the implementation of circuit quantum electrodynamic architectures for the realization of a scalable quantum information processor. In this work we describe a microwave photon counter coupled to a semi-infinite transmission line. We employ input–output theory to examine a continuously driven transmission line as well as traveling photon wave packets. Using analytic and numerical methods, we calculate the conditions on the system parameters necessary to optimize measurement and achieve high detection efficiency. With this we can derive a general matching condition depending on the different system rates, under which the measurement process is optimal.

Efficient single-photon frequency conversion in the microwave domain using superconducting quantum circuits

We present an approach to achieve efficient single-photon frequency conversion in the microwave domain based on coherent control in superconducting quantum circuits, which consist of a driven artificial atom coupled to a semi-infinite transmission line. Using the full quantum-mechanical method, we analyze the single-photon scattering process in this system and find that single-photon frequency up- or down-conversion with efficiency close to unity can be achieved by adjusting the parameters of the control field applied to the artificial atom. We further show that our approach is experimentally feasible in currently available superconducting flux qubit circuits.

On-Demand Microwave Generator of Shaped Single Photons

We demonstrate the full functionality of a circuit that generates single microwave photons on demand, with a wave packet that can be modulated with a near-arbitrary shape. We achieve such a high tunability by coupling a superconducting qubit near the end of a semi-infinite transmission line. A dc superconducting quantum interference device shunts the line to ground and is employed to modify the spatial dependence of the electromagnetic mode structure in the transmission line. This control allows us to couple and decouple the qubit from the line, shaping its emission rate on fast time scales. Our decoupling scheme is applicable to all types of superconducting qubits and other solid-state systems and can be generalized to multiple qubits as well as to resonators.

Antibunched Photons from Inelastic Cooper-Pair Tunneling.

We demonstrate theoretically that charge transport across a Josephson junction, voltage-biased through a resistive environment, produces antibunched photons. We develop a continuous-mode description of the emitted radiation field in a semi-infinite transmission line terminated by the Josephson junction. Within a perturbative treatment in powers of the tunneling coupling across the Josephson junction, we capture effects originating in charging dynamics of consecutively tunneling Cooper pairs. We find that within a feasible experimental setup the Coulomb blockade provided by high zero-frequency impedance can be used to create antibunched photons at a very high rate and in a very versatile frequency window ranging from a few GHz to a THz.

A Closed-Form Analytical Transient Response Model for On-Chip Distortionless Interconnect

This paper presents a closed-form analytical transient response model for an on-chip distortionless interconnect considering resistance/capacitance loads via solving a semi-infinite transmission line equation. As verified by the simulation results, this transient response model has high accuracy, which could be used to derive the characteristics of the transmitted signal for facilitating the design of the distortionless interconnect. Based on the model, the steady-state output voltage and the delay related to the interconnect have been analyzed.

Simulation of the spin-boson model with superconducting phase qubit coupled to a transmission line

Based on the rapid experimental developments of circuit QED, we propose a feasible scheme to simulate the spin-boson model with superconducting circuits, which can be used to detect quantum Kosterlitz-Thouless (KT) phase transition. We design the spinboson model by using a superconducting phase qubit coupled to a semi-infinite transmission line, which is regarded as a bosonic reservoir with a continuum spectrum. By tuning the bias current or the coupling capacitance, the quantum KT transition can be directly detected through tomography measurement on the states of the phase qubit. We also estimate the experimental parameters using the numerical renormalization group method.

Approaching perfect microwave photodetection in circuit QED

In this work we propose a microwave photon detector which successfully reaches 100% efficiency with only one absorber. Our design consists of a metastable quantum circuit coupled to a semi-infinite transmission line which performs highly efficient photodetection in a simplified manner as compared to previous proposals. We extensively study the scattering properties of realistic wavepackets against this device, thereby computing the efficiency of the detector. We find that the detector has many operating modes, can detect detuned photons, is robust against design imperfections and can be made broadband by using more than one absorbing element in the design.

Time-Domain Simulation of Power Combining in a Chain of THz Gunn Diodes in a Transmission Line

Solid-state THz devices usually suffer from low power output. This, however, can be overcome by efficient power combining. In this paper, the power combining of a chain of active devices (longitudinal array of N Gunn diodes) in a semi-infinite transmission line has been simulated. Superlinear growth of power P is found in a line with inductive-capacitive elements: P is quadratic in N when P is small and becomes linear when P is large. Intricate auto-modulation due to coupling between the devices is shown to accompany power combining.

Leakage fields from planar semi-infinite transmission lines

The leakage fields radiated by planar semi-infinite transmission-line currents are studied for two canonical structures: the air-gap stripline and the planar strip of current in free space. The first structure represents a general class of printed-circuit structures where leakage occurs into a substrate mode. The second structure is representative of structures for which leakage occurs primarily into space. It is found that the transverse behavior of the leakage fields for a semi-infinite line differs in significant ways from that expected from an infinite line, a result that is at first surprising, but is understandable on physical grounds, and which is explained by the analysis presented here. The exact leakage fields are compared with two asymptotic approximations: a stationary-phase [geometrical optics (GO)] evaluation, and a uniform asymptotic expansion (UAE). The UAE solution is the more accurate one, and accounts for diffraction-like effects that arise due to the semi-infinite nature of the line. One conclusion is that the observation distance must be very large for the true leakage fields to closely resemble the GO fields, although a quasi-GO behavior may still be evident at shorter distances, depending on the value of the leakage angle. Another conclusion is that the transverse behavior of the GO leakage field of the printed-circuit structure is an exponentially increasing field within the leakage region, as expected from a simple ray-optics picture, while it is not for the strip in free space.

Spectral theory of a semi-infinite transmission line over a ground plane

The electromagnetic modeling of wave propagation along a semi-infinite wire with ground return is addressed. External excitation, via plane wave is considered. Both wire and ground plane are assumed to be perfect conductors. Through a spectral approach, a one-port Thevenin model of the structure, valid for any distance between wire and ground, is developed. Such a model allows to evaluate rigorously the limits and the accuracy of the transmission line (TL) equations.

Energy loss and storage in RC-transmission lines

The distribution of energy supplied by a step-voltage generator to a semi-infinite transmission line with uniform series resistance and shunt capacitance is investigated; and explicit (simple integral) representations for both line and generator losses as well as the line storage during an arbitrary time interval are given. A relatively greater loss due to internal resistance of the generator obtains in the initial phase of excitation while that due to resistance of the line predominates in the long run. A larger proportion of energy is stored in the line at the outset and, ultimately, the output of the generator is equally divided between line loss and storage in its internal capacitance.

A theory of a nonuniform transmission line and the structure of signal flow

AbstractThis paper deals principally with the phenomenological description and structural analysis of the signal flow of a nonuniform transmission line and presents a theoretical discussion, including the traditional theories, of the uniform transmission line. Due to the existence of local multiple reflections caused by nonuniformity, the phenomena on the nonuniform transmission line cannot be as clearly described as in the case of the uniform transmission line. The reason for existence of the local multiple reflections, however, is the lack of appropriate selection of the incident and reflected waves from the solutions of the transmission line equation. From such a viewpoint, this paper newly defines the incident and reflected waves as waves obtained by exciting a semiinfinite transmission line in the positive or negative direction. According to this definition, the two waves are not coupled and do not produce the local multiple reflections due to the nonuniformity. Thus, using such incident and reflected waves, a unified analysis can be made for the transmission line, independently of whether it is uniform or nonuniform. At the same time, the signal‐flow structure for such phenomena as reflection, matching and resonance can clearly be described. Using the concept of signal flow, the ultimate expressions for various quantities can easily be reached.

Transient Resistance of Conductors

Practical calculations of transient resistance effects in conductors are reported. Included are the current density distributions in cylindrical and semi-infinite solid conductors for an applied surface step electric field E, and the density in a flat wide conductor. Two analogies are used to yield useful results: thermal diffusion in a cylinder and voltage transients on a semi-infinite transmission line. Results are applied to the calculation of time constants of a transmission line and force build-up in a solenoid.

Reflection and refraction of microwaves at a set of parallel metallic plates

The paper deals with the theory of the propagation of microwaves, passing from a homogeneous and isotropic medium into a set of parallel metallic plates. The plane of separation between the set of plates and the homogeneous medium outside is assumed perpendicular to the plates. The incident wave is plane, and its direction of propagation is in the plane containing the perpendiculars to the plates and to the surface of separation. The electric field is parallel to the plates.The method is relatively simple. The study is based on Maxwell's equations and on periodicity and continuity considerations. An analytic method is introduced for the solution of the infinite set of linear equations with an infinity of unknowns, which arises from the continuity conditions at the boundary plane of the plate edges.Formulae are obtained giving the nature, direction, phase and amplitude of the reflected and transmitted waves. The reflected waves follow the usual laws found in the grating theory. In the case when only one non-evanescent plane wave is reflected, the transmission and reflection power coefficients are those obtained at the junction of two semi-infinite transmission lines corresponding to the free-space and the plate medium respectively. The analogy with transmission lines is more complex when two non-evanescent waves are reflected. For the usual angles of incidence and plate spacings the transmission power coefficient is high (maximum 99.5%), and the phase shift on transmission is small.