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Deutsche Physikalische Gessellschaft IOP Institute of Physics

Quantum carpets woven by Wigner functions

O M Friesch-+, I Marzoli++ and W P Schleich-+

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Video sequence 1
In this movie we present the time evolution of the probability density $\vert\psi$(x, t)|2 to find the particle at time t at position x in a box of width L. The initial wave function is a Gaussian wave packet, centred around $\obar{x}$ = L /4, having width $\sigma$ = L /20 and an average momentum $\obar{p}$ = 15$\hbar\pi$/L. To illustrate the regularity in the wave packet dynamics, we have placed two arrows, moving with constant velocity, underneath the position axis. The left-hand arrow always points to a minimum in the probability density (canal), while the right-hand arrow is sometimes at a minimum and sometimes at a maximum (chopped ridge). The rectangular bar on the left-hand side indicates time. We have marked on it important fractions of the revival time T, at which we can observe the fractional revivals of the wave packet. Especially remarkable are the revivals occurring at T /6 and T /4, characterized, respectively, by the appearance of three and two replicas of the original wave packet.

Video sequence 2
The upper part of this animation shows the time evolution of the Wigner function, and the lower one shows the resulting probability distribution of finding the particle in the box. The Wigner function is plotted in phase space, where the horizontal axis corresponds to position and the vertical axis, on which we have marked the zero, corresponds to momentum. Positive values are displayed in orange, and negative ones are shown in blue. The movie starts from the situation depicted in figure 4 and follows the dynamics up to t = T /2, when the wave packet reshapes itself. Also, the time evolution of the Wigner function is characterized by fractional revivals culminating in the full revival at T /2.

Video sequence 3
We concentrate on the enlarged view of the central part of the Wigner function, around momentum p = 0. This magnified portion encompasses interference terms with momentum ranging from -3p1 to 3p1. In the lower diagram, we follow the simultaneous weaving of the quantum carpet. To make clear the correspondence between the interference terms of the Wigner function and the world lines in the probability distribution, we mark, with the same colour, the components of the Wigner function with momenta ±p1 and the corresponding structures in the carpet.

Video sequence 4
We show the same sequence as in the previous animation, but this time we highlight the terms in the Wigner function with momenta ±2 p1 and the corresponding zig-zag patterns in the quantum carpet.



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