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Bulletin of the Polish Academy of Sciences Technical Sciences

Quantum simulation of the tunnel effect

Marcin Ostrowski

Published Jun 1, 2015• 5 citations
Publisher

Abstract

Abstract In this paper, we examine whether a quantum computer can efficiently simulate quantum processes such as the tunnel effect. We examine a quantum algorithm that calculates the value of transition and reflection coefficients for the Gaussian wave packet scattered on a rectangular potential. We compare the results obtained in this way with the results of classical simulations and analytical calculations

DOI
10.1515/bpasts-2015-0042

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Abstract In this paper, we examine whether a quantum computer can efficiently simulate quantum processes such as the tunnel effect. We examine a quantum algorithm that calculates the value of transition and reflection coefficients for the Gaussian wave packet scattered on a rectangular potential. We compare the results obtained in this way with the results of classical simulations and analytical calculations
In this paper we examine whether a quantum computer can efficiently simulate time evolution of a one dimensional Schrödinger particle. Two cases are considered: free particle and particle dispersed on a rectangular potential. In order to simulate the Schrödinger particle, we use a quantum algorithm based on the procedure of diagonalisation of time evolution operator. The procedure of diagonalisation is based on the Quantum Fourier Transform (QFT) algorithm. The effects of simulation are presented in the form of figures. We also compare the results obtained from quantum algorithm with the results of classical simulations (Cayley's method).
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
The article presents calculations and visualization of wave packet tunneling through the potentialbarrier of various widths done by using the MATLAB software. The graph of distribution of the probability densityalong barrier width is obtained in two and three dimensions. The article contains materials about quantummechanical tunnel effect such as the reflection of the part of the wave packet from the potential barrier and passingthrough the barrier of the other part. Calculations and visualization of the tunneling process are performed forvarious widths of the barrier. Figures given in the article show the probability of finding the particle at a certain time,i.e. the square of the absolute value of the state vector 2 . The part of the total number of particles tunnels throughbarrier, the other part reflects from it. The probability of tunneling increases with decrease of the barrier width. Thedetailed analysis of the graphs reveals that the wave function exponentially decreases inside the potential barriertherefore the observation of this effect requires a rather small width of the barrier. Students are offered to developprograms for various widths of the potential barrier. Results of experiments are used during the study of the quantummechanics.
Abstract A propagation method for the scattering of a quantum wave packet from a potential surface is presented. It is used to model the quantum reflection of single atoms from a corrugated (metallic) surface. Our numerical procedure works well in two spatial dimensions requiring only reasonable amounts of memory and computing time. The effects of the surface corrugation on the reflectivity are investigated via simulations with a paradigm potential. These indicate that our approach should allow for future tests of realistic, effective potentials obtained from theory in a quantitative comparison to experimental data.