What is the "quantum walk"?
The "quantum walk" is the movement of a quantum particle, i.e. of a particle obeying the laws of quantum mechanics.

What is the difference between the quantum and the classical random walk?
In a classical random walk in each time step a coin is tossed. Depending on the random result, a particle moves to the right or to the left.
A quantum particle can move to the right and to the left simultaneously. It is delocalized over two positions. This is called a "coherent superposition" of right and left. After several steps the particle is delocalized over many positions. In this case, different parts of the atom can be combined at a common position. They can amplify or extinguish each other. This is called "matter wave interference".

Why is it interesting to study the quantum walk?
There are several reasons: On the one hand the quantum walk forms the basis for quantum search algorithms which are faster than all known classical methods. One exploits the fact that a quantum particle can sample many entries of a data base simultaneously. In contrast, for the classical case, every element in a data base has to be sampled in a separate step.
On the other hand the quantum walk allows studying the transition from the quantum me-chanical behaviour of the microscopic world to the classical behaviour of the everyday, macroscopic world.

What is a "coherent superposition"?
A quantum mechanical system with two states, for example red or green, zero or one, left or right, can be simultaneously in both states. This is in contrast to classical systems of our eve-ryday life. There a bit of information, for instance, can have the value zero or one, respec-tively.

What is "matter wave interference"?
Quantum objects have to be described by waves – or more precisely by a wave function – just like light waves. And just as with light this description implies the possibility of two waves interfering: Depending on their relative phase they can amplify or extinguish each other. In our experiment this means that matter and matter at the same position can be more matter – or no matter at all.

How was the quantum walk experimentally realized?
The principle of the first two steps of the experimental realization of a quantum walk is shown in the figure below:

(a) The walker in this case is a single cesium atom. The atom is trapped in a lattice made of light, similar to optical tweezers used in biology The atom has two states, visualized as "red" and "green". 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 green lattice, where the red state of the atom experiences only the red lattice, and the green state of the atom experiences only the green lattice. If these two lattices are moved in opposite directions, the red part of the atom moves to one side, the green 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.

How can we see where the atom is?
We use a high-resolution microscope to detect the fluorescence light emitted by the atom. We take images before and after the quantum walk sequence. The position of the atom can be inferred from the images up to a few ten nanometer. Thus, the distance the atom has moved can also be obtained with high precision.

How do the results differ for quantum and random walk?
We measure how often the particle has moved a certain distance. For the classical case this is well known: The walker performs a random motion as in Brownian motion or in any diffusive process. The probability to find the particle at a certain position follows a so-called binomial distribution, which has a large probability close to the initial position of the particle.
For the quantum case, however, the probability distribution is significantly altered by matter-wave interference: The particle has moved most probably a large distance, whereas it is found only rarely close to the initial position.

Can both cases be studied experimentally?
Yes. Ideally, we obtain results which agree well with the expectations for a quantum mechanical walk. However, if we intentionally destroy the coherent superposition in each step, we find results as expected for a classical case.