Implementing the one-dimensional quantum (Hadamard) walk using a Bose-Einstein condensate

We propose a scheme to implement the simplest and best-studied version of the quantum random walk, the discrete Hadamard walk, in one dimension using a coherent macroscopic sample of ultracold atoms, Bose-Einstein condensate (BEC). Implementation of the quantum walk using a BEC gives access to the familiar quantum phenomena on a macroscopic scale. This paper uses a rf pulse to implement the Hadamard operation (rotation) and stimulated Raman transition technique as a unitary shift operator. The scheme suggests the implementation of the Hadamard operation and unitary shift operator while the BEC is trapped in a long Rayleigh range optical dipole trap. The Hadamard rotation and a unitary shift operator on a BEC prepared in one of the internal states followed by a bit-flip operation, implements one step of the Hadamard walk. To realize a sizable number of steps, the process is iterated without resorting to intermediate measurement. With current dipole trap technology, it should be possible to implement enough steps to experimentally highlight the discrete quantum random walk using a BEC leading to further exploration of quantum random walks and its applications.