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IEEE International Symposium on Circuits and Systems proceedings

Probabilistic particle modeling of quantum wave propagation with reflection, transmission, and coupling

Nobuyuki Hirami, Hisato Fujisaka, Takeshi Kamio

Published Jun 1, 2014• 4 citations
Publisher

Abstract

Quantum particles possess both wave and particle natures. When quantum effect devices exploiting the wave nature and the devices described from the view point of particle nature coexist in a system, the following advantages are gained by representing former devices in the same way as we describe the latter devices: First, it is not necessary to compute probability distribution from wave functions and sample device states in accordance with the probability distribution. Secondly, the particle models of the devices are suitable models for conventional circuit simulators. In this paper, simple particle models of two quantum systems operating as wave filters are constructed by applying Nelson's stochastic quantization theory. The filter characteristics obtained from the wave functions and from the behavior of the particle models were almost equal.

DOI
10.1109/iscas.2014.6865176

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Quantum particles possess both wave and particle natures. When quantum effect devices exploiting the wave nature and the devices described from the view point of particle nature coexist in a system, the following advantages are gained by representing former devices in the same way as we describe the latter devices: First, it is not necessary to compute probability distribution from wave functions and sample device states in accordance with the probability distribution. Secondly, the particle models of the devices are suitable models for conventional circuit simulators. In this paper, simple particle models of two quantum systems operating as wave filters are constructed by applying Nelson's stochastic quantization theory. The filter characteristics obtained from the wave functions and from the behavior of the particle models were almost equal.
Quantum particles possess both wave and particle natures. When quantum effect devices exploiting the wave nature and the devices described from the view point of particle nature coexist in a system, the following advantages are gained by representing former devices in the same way as we describe the latter devices: First, it is not necessary to compute probability distribution from wave functions and sample device states in accordance with the probability distribution. Secondly, the particle models of the devices are suitable models for conventional circuit simulators. In this paper, simple particle models of an electromagnetic wave detector and an electron lens are constructed by applying Nelson's stochastic quantization theory. The device characteristics obtained from the wave functions and from the behavior of the particle models were almost equal.
Electrons possess both wave and particle natures. In this paper, we represent an electron on multi-stage coupled electron waveguides both by a wave function and by a probabilistic particle. We show first that time evolution of the wave function can be obtained by the combined use of the tight-binding method and two-port circuit theory. The wave function is in variable-separable form when wave propagation on the coupled electron waveguides is single mode. We present secondly that the variable separation simplifies the general Langevin equation describing behavior of the probabilistic particle. According to these two schemes, wave functions and sample particle trajectories on single and two-stage coupled electron waveguides were exemplified.
A phase-space formulation of the filtering process of a quantum state is developed that models both quantum interference and delayed-choice experiments without reference to the controversial wave-particle duality concept and without any explicit model of the interaction between the particles and the measuring apparatus. Quantum particles are considered as localized and indivisible concentrations of energy and/or mass, their probability amplitude in phase space being described by the Wigner distribution function. The "wave" or "particle" nature emerges in experiments in which the interference term of the Wigner distribution function is present or absent, respectively. The Wigner distribution function is modified by filtering devices that act on the quantum wave function throughout the setup, from its generation to its final detection.
Classical transport equations with probabilistic initial conditions can be viewed as quantum systems. In a discrete version they are probabilistic automata. The time-local probabilistic information is encoded in a classical wave function. Its unitary evolution obeys a Schrödinger equation. Statistical observables measure properties of the classical probability distribution. In the quantum formalism they are represented by operators which do not commute with the ones associated to classical observables. Momentum, quantum energy and angular momentum or charges yield conserved quantities which constrain the evolution of the classical probability distribution. The characteristic features of quantum mechanics, as the superposition of wave functions, interference, the importance of phases, non-commuting operators or a unitary time evolution, are realized by probabilistic classical transport equations. Stochastic transport equations are described by a quantum system with a stochastic Hamiltonian.
This paper presents a method of computing sample trajectories of particle models of electrons on symmetric electron-wave stub-filters. The time-independent Schrödinger equation describing the electron-wave filters is transformed to a system of linear algebraic equations. The Fourier series expansion transforms the eigenvalues of the system to continuous and differentiable eigenfunctions. A electron wave packet is constructed from the obtained smooth eigenfunctions and a nonlinear stochastic ordinary differential equation (NSODE) which governs the behavior of the electron particle model is established by applying the stochastic quantization to the wave packet. Sample trajectories of the particles are computed by integrating the NSODE. Numerical experiments have shown that the statistical behavior of the particles is almost equal to that of the wave packets.
When talking about noise in quantum devices two issues must be faced: how to model the evolution of an electronic system with scattering and how this noise is practically computed in a quantum device simulator. In the present paper, we address both problems from a practical and computational point of view. In particular, as the electronic quantum subsystem is an open (normally far from equilibrium) system, we use the notion of conditional wave function, the wave function of a subsystem in Bohmian mechanics, an alternative version of quantum mechanics which along with the wave function posited definite positions for the particles. This allows us to define an effective equation in several physical situations, ranging from the simple tunneling barrier to the interaction with a bath of phonons. Finally, we present how this development can be used to compute quantum noise in a quantum device simulator.