Quantum Walk in Position Space with Single Optically Trapped Atoms Michal Karski, et al. Science 325, 174 (2009)

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1 Quantum Walk in Position Space with Single Optically Trapped Atoms Michal Karski, et al. Science 325, 174 (2009) abstract: The quantum walk is the quantum analog of the well-known random walk, which forms the basis for models and applications in many realms of science. Its properties are markedly different from the classical counterpart and might lead to extensive applications in quantum information science. In our experiment, we implemented a quantum walk on the line with single neutral atoms by deterministically delocalizing them over the sites of a one-dimensional spin-dependent optical lattice. With the use of site-resolved fluorescence imaging, the final wave function is characterized by local quantum state tomography, and its spatial coherence is demonstrated. Our system allows the observation of the quantum-to-classical transition and paves the way for applications, such as quantum cellular automata.

2 Random Walk The random walk is a simple concept that has been used to describe many real-world systems from stock market prices to the Brownian motion of tiny particles floating on a liquid. It is usually described as a person who dictates his movements by the toss of a coin: get heads, for example, and he moves one step to the right; get tails and he takes a step to the left. After many coin tosses, the person s position is random, but is most likely to be close to the start point.

3 Feynman s Quantum Walk The quantum walk was first proposed by Nobel laureate Richard Feynman. After every toss, a quantum particle moves in both directions simultaneously, and adopts a coherent superposition of right and left. After many steps, the particle becomes blurred or delocalized over many different positions. However, the nature of this process means that, after more than one toss, the new superposition will overlap part of the old one, and will have the effect of either amplifying or removing that position. This is known as matter-wave interference, and means that the eventual position of the particle is most likely to be farthest away from the starting point.

4 Overlapping optical lattices In their experiment, the researchers trap a single, cold cesium atom in two optical lattices that initially overlap. They begin using a laser pulse to prepare the atom in a superposition of two internal states. Next, they move the lattices in opposite directions, which makes the atom simultaneously step both to the right and left. When they repeat this maneuver, the superposition stretches over another step, but the position in the middle then contains two parts of the atom that interfere with each other. After ten steps, the Bonn group used a high-resolution microscope to detect the fluorescence emitted by the atom, and thus cause it to settle in one position. The probability distribution of final positions built up from many experiments was antisymmetric about the start point, which agreed with a computer model of a quantum walk. However, if the researchers destroyed the superposition at every step, the distribution reverted to the classical case in other words, a binomial with the peak around the start point.

5 2 hyperfine states (a) The atom is trapped in a lattice made of light. It has two states, visualized as red and blue. Experimentally, the atom can be brought into a coherent superposition of the two states a sort of quantum coin is tossed. (b) The optical lattice depends on the state of the atom. This can be imagined as a red and a blue lattice, where the red state of the atom experiences only the red lattice, and the blue state of the atom experiences only the blue lattice. If these two lattices are moved in opposite directions, the red part of the atom moves to one side, the blue one to the other. (c) If both lattices overlap again, the atom is delocalized over two lattice sites, it is simultaneously to the left and to the right. The first step of the quantum walk is complete. (d) For the second step, each part of the atom is again brought into a coherent superposition of the two states. After applying the state dependent shifting, the atom is delocalized over three lattice sites. Now, two parts of the atom are located at a common position. At this site, both parts of the atom can amplify or extinguish each other, they can interfere. (Courtesy: Institute for Applied Physics, University of Bonn).

7 coin toss operators 0 ( 0 1)/ 2 1 ( 0 + 1)/ 2

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