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arXiv:1109.5285

Field theoretical approach to quantum transmission in time-dependent potentials

Diakonos, Fotis K., P. A. Kalozoumis, A. I. Karanikas, Nektarios Manifavas, Peter Schmelcher

Published Sep 24, 2011• 0 citations
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Abstract

We develop a field theoretical approach based on the temporary basis description as a tool to investigate the transmission properties of a time-driven quantum device. It employs a perturbative scheme for the calculation of the transmission of a monochromatic beam of particles through the time-dependent set-up. The main advantage of the proposed treatment is that it permits the use of the particle picture for the calculation of the scattering matrix and the transmission coefficient. Therefore the elementary physical processes contributing to the transmission can be identified and interpreted in a transparent way. We apply the method to the simple but prototype problem of transmission through an one-dimensional oscillating delta potential and we demonstrate how it enables a deep understanding of the underlying physical processes.

DOI
10.48550/arxiv.1109.5285

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We develop a field theoretical approach based on the temporary basis description as a tool to investigate the transmission properties of a time-driven quantum device. It employs a perturbative scheme for the calculation of the transmission of a monochromatic beam of particles through the time-dependent set-up. The main advantage of the proposed treatment is that it permits the use of the particle picture for the calculation of the scattering matrix and the transmission coefficient. Therefore the elementary physical processes contributing to the transmission can be identified and interpreted in a transparent way. We apply the method to the simple but prototype problem of transmission through an one-dimensional oscillating delta potential and we demonstrate how it enables a deep understanding of the underlying physical processes.
A field-theoretical approach to the scattering off an oscillating quantum system is developed. As a key ingredient it employs the adiabatic eigenstate basis and consists of a perturbative scheme for the calculation of the geometric phases influencing the transmission through the time-dependent potential landscape. The main advantage is the identification of basic diagrams which allow for an immediate interpretation of the underlying elementary physical processes contributing to the scattering and transmission behavior. We apply our method to the simple, but prototypical, problem of transmission through an one-dimensional oscillating $\ensuremath{\delta}$ potential and demonstrate how it enables a deeper understanding of the relevant physical processes.
The aim of this thesis is to investigate the effect of strong ultrashort electric fields on scattering of a particle by a short range potential, using a simple one-dimensional model. The interaction potential has the form of a simple potential step. The transmission amplitude is calculated using Fourier analysis of the scattered wavefunction. The scat- tered wavefunction is obtained by solving numerically the time-dependent Schrödinger equation combining the finite difference spatial representation with the Crank Nicholson method to approximate the evolution operator. We validate and test the method on the analytically solvable problem of scattering by a potential barrier without the presence of the external field. Finally, we apply the method to calculation of laser assisted trans- mission through the potential barrier and find that transmission through the barrier can be strongly enhanced or suppressed depending on the choice of the field parameters. We provide elementary clues on interpretation of our findings. 1
We give a pedagogical introduction to time-independent scattering theory in one dimension focusing on the basic properties and recent applications of transfer matrices. In particular, we begin surveying some basic notions of potential scattering such as transfer matrix and its analyticity, multi-delta-function and locally periodic potentials, Jost solutions, spectral singularities and their time-reversal, and unidirectional reflectionlessness and invisibility. We then offer a simple derivation of the Lippmann-Schwinger equation and Born series, and discuss the Born approximation. Next, we outline a recently developed dynamical formulation of time-independent scattering theory in one dimension. This formulation relates the transfer matrix and therefore the solution of the scattering problem for a given potential to the solution of the time-dependent Schr\"odinger equation for an effective non-unitary two-level quantum system. We provide a self-contained treatment of this formulation and some of its most important applications. Specifically, we use it to devise a powerful alternative to the Born series and Born approximation, derive dynamical equations for the reflection and transmission amplitudes, discuss their application in constructing exact tunable unidirectionally invisible potentials, and use them to provide an exact solution for single-mode inverse scattering problems. The latter, which has important applications in designing optical devices with a variety of functionalities, amounts to providing an explicit construction for a finite-range complex potential whose reflection and transmission amplitudes take arbitrary prescribed values at any given wavenumber.
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Abstract This review presents time-independent perturbative methods for solving the one-dimensional Schrödinger equation, highlighting representative cases that reveal key aspects of the theory. The focus is on their relevance to quantum computing applications, particularly in systems with finite-dimensional state spaces. The main link between these methods and quantum processing emerges when we consider systems with only a finite number of well-defined energy levels, e.g., two levels corresponding to the ‘zero’ and ‘one’ states of a quantum bit (qubit), for which the Schrödinger equation describes a quantum algorithm corresponding to the unitary evolution of the qubits. Perturbative techniques can then be applied to this evolution to gain deeper insight into the algorithm or to simplify it in a controlled manner, as described in the last sections of this review. We initially discuss representative applications using benchmark cases that exemplify the essential features of perturbation theory while avoiding unnecessary technical complications. The methods presented involve direct approaches to solutions for quantum systems with small potential perturbations, with consideration of indirect and approximation techniques. Key cases examined include the particle in a potential well, the harmonic oscillator, and systems with polynomial or Dirac delta distribution perturbations, each chosen to highlight unique aspects of perturbative analysis. Starting with an overview of the unperturbed Schrödinger equation, we derive canonical solutions and then use perturbative techniques to approximate energy shifts and wavefunction corrections in both non-degenerate and degenerate settings. By combining theoretical discussion with case studies, this review underscores how perturbative methods are applied across a range of elementary quantum mechanical problems especially for applications in the qubit model, which, in many cases of interest, can be reduced to the explicit calculations shown here.