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Canadian Journal of Physics

Quantum description of microwave passive circuits

Nicolae Marinescu, Rudolf Nistor

Published Oct 1, 1990• 7 citations
Publisher

Abstract

The paper gives a formal analogy between the distribution of the electromagnetic field in a wave guide and microwave cavities and the quantum-mechanical probabilities distribution. We show that the wave guide of the cutoff frequency ωc acts on an electromagnetic wave as a quantum potential barrier [Formula: see text]. We also establish a nonhabitual time-independent Schrödinger equation that replaces Maxwell's equations in describing guided wave propagation.

DOI
10.1139/p90-157

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The paper gives a formal analogy between the distribution of the electromagnetic field in a wave guide and microwave cavities and the quantum-mechanical probabilities distribution. We show that the wave guide of the cutoff frequency ωc acts on an electromagnetic wave as a quantum potential barrier [Formula: see text]. We also establish a nonhabitual time-independent Schrödinger equation that replaces Maxwell's equations in describing guided wave propagation.
Expressions are derived for the electromagnetic fields of guided waves which are analogous to the quantum-mechanical equations representing barrier tunneling. This analogy is achieved by comparing the propagation constant of the Schrödinger time-independent wave equation with that of the electromagnetic wave equation in waveguide and by comparing the de Broglie wavelength of a particle with the wavelength of the waves that propagate the energy. This results in an expression relating the form of an arbitrary one-dimensional energy barrier to the physical dimensions of a section of waveguide. The analogy is tested by the propagation of energy in the TE10 mode at both the 3- and 6-cm bands for the cases of rectangular and hyperbolic barriers. Although evanescent modes are present at the discontinuous regions, the analog for the rectangular barrier, which is considered to be the worse case, is verified when an effective barrier length leff of about 1.2l is used. This experimental verification demonstrates the possibility of waveguide simulation of quantum-mechanical energy barriers and the practicality of utilizing an electromagnetic analog for demonstrating the tunneling phenomenon and provides a method for measurement of the transmission coefficient through an arbitrarily shaped barrier.
In this chapter, we shall expose several analogies between oscillatory phenomena in mecahanics and optics.The main subject will be the analogy between propagation of electromagnetic waves in dielectrics and of electrons in various time-independent potentials.The basis of this analogy is the fact that both wave equations for electromagnetic monoenergetic waves (i.e. with well-defined frequency), obtained directly from the Maxwell equations, and the time-independent Schrodinger equation are Helmholtz equations; when specific restrictions -like behaviour at infinity and boundary conditions -are imposed, they generate similar eigenvalues problems, with similar solutions.The benefit of such analogies is twofold.First, it could help a researcher, specialized in a specific field, to better understand a new one.For instance, they might efficiently explain the fiber-optics properties to people already familiar with quantum mechanics.Also, even if such researchers work frequently with the quantum mechanical wave function, the electromagnetic modal field may provide an interesting vizualization of quantum probability density field [1].Second, it provides the opportunity of cross-fertilization between (for instance) electromagnetism and optoelectronics, through the development of ballistic electron optics in two dimensional (2D) electron systems (2DESs); transferring concepts, models of devices and experiments from one field to another stimulate the progress in both domains.Even if in modern times the analogies are not credited as most creative approaches in physics, in the early days of developement of science the perception was quite different."Men's labour ... should be turned to the investigation and observation of the resemblances and analogies of things... for these it is which detect the unity of nature, and lay the fundation for the constitution of the sciences.",considers Francis Bacon, quoted by [1].Some two centuries later, Goethe was looking for the "ultimate fact" -the Urphänomenon -specific to every scientific discipline, from botanics to optics [2], and, in this investigation, attributed to analogies a central role.However, if analogies cannot be considered anymore as central for the scientific investigation, thay could still be pedagogically useful and also inspiring for active scientific research.Let us describe now shortly the structure of this chapter.The next two sections are devoted to a general description of the two fields to be hereafter investigated: metallic and dielectric waveguides (Section 2) and 2DESs (Section 3).The importance of these topics for the development of optical fibers, integrated optics, optoelectronics, transport Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts
An electromagnetic model of a “potential barrier” may be constructed from three sectors of waveguide, two outer sectors being filled with dielectric, the short inner sector air-filled. Using the mathematical analogy between the propagation data of the De Broglie wave and the wave inside a waveguide—above and below cutoff—it is demonstrated that, for a frequency of operation between the cutoff frequencies of the air-filled and the dielectric-filled waveguide, the air-filled section represents the potential barrier region. Thus the described apparatus, the construction of which has been carried out in full detail, enables the modeling of the tunnel effect experimentally and in a quantitative fashion.
Mesoscopic electric circuits can exhibit a large number of effects relevant to quantum mechanics, quantum electrodynamics and quantum statistics. The topic presented here can be identified as an analog of conventional cavity quantum electrodynamics, where the quantum objects are multilevel atoms and quantized optical fields (photons), namely, a theoretical subject called “cavity-circuit quantum electrodynamics”, where the atoms are replaced with quantum mesoscopic circuits, will be developed. We will study the quantum characteristics of the interaction between the quantum mesoscopic circuits and the quantized electromagnetic field at microwave frequencies. These issues include quantum entangled eigenstates of two coupled mesoscopic circuits, time evolution of the circuit energy quanta (governed by the time-dependent Schrödinger equation) and entanglement transfer between external photons and mesoscopic circuit energy quanta. Since there is inevitable electromagnetic interaction such as mutual capacitance and inductance coupling between two neighboring circuits, the quantum effects resulting from the aforementioned quantized circuit coupling would unavoidably affect the relevant processes in quantum computing devices and hence they deserve consideration in some issues of quantum information.