Walking in the Quantum World

Michal Karski, Artur Widera, and Dieter Meschede (Left to Right)

[This is an invited article based on recent works of the authors -- 2Physics.com]

Authors: Artur Widera, Michal Karski and Dieter Meschede

Affiliation: Institut für Angewandte Physik der Universität Bonn, Germany

While the random motion of classical particles is well understood and such random walks have found their way into most fields of modern science, quantum particles are expected to behave differently. The intriguing new properties of these quantum walks may lead to novel applications in quantum information science as quantum search algorithms, for example, or yield insight into the transition from the classical to the quantum regime. A quantum walk in position space has recently been observed with single Caesium (Cs) atoms by fluorescence microscopy [1].

Imagine a walker, e.g. a particle, which can move stepwise on a line. In each time step, let the walker now move randomly to the right or to the left, just as the diffusive Brownian motion of a particle. The probability of finding the particle at a certain position is given by a binomial distribution, with high probability at the initial position and a width that scales with the square-root of the number of steps taken. This well known scaling serves as the basis for numerous models in modern science, for example to estimate the speed of searching algorithms.

In the quantum world, two effects change the particle’s motion drastically: First, quantum particles can be in so-called coherent superpositions, for example, of moving to the left and to the right. This sounds weird, but atoms in coherent superpositions are routinely used, for instance, in atomic clocks where the atoms are in a superposition of two spin states. As a consequence of these coherent superpositions, the quantum particle is delocalized over two lattice sites as it moves simultaneously to the left and to the right. If this delocalization is successively repeated for more and more steps, the particle delocalizes over more and more sites of the line. At certain positions, two parts of the delocalized atom can be re-combined at a common site. Here, the second quantum effect becomes important:
Quantum mechanical objects are described by wave functions and as such they can interfere. Depending on their respective path, they can amplify or extinguish each other. This leads to a drastically changed probability distribution of finding a particle at a certain position. In particular, for a quantum walk it is unlikely to find the particle at the initial position. Its distribution rather shows pronounced peaks with large probability at the outermost edges. The width of the resulting distribution scales linearly with the number of steps. This ballistic scaling is envisioned to speed up search algorithms in quantum search devices or quantum computers.

Figure 1: (a) A single atom is trapped at an initial site of an optical lattice and prepared in a coherent superposition of two states, red and blue. (b) The two states are selectively shifted into opposite directions along the lattice, delocalizing the atom (c) over two sites. (d) After another step of coherent superposition and state-dependent shifting, two parts of the atomic wave function are re-combined, giving rise to matter wave interference.

Experimentally, we realized a quantum walk using single Cs atoms. In an ultra-high vacuum, the atoms were cooled by laser light to approximately 10 µK and then trapped in a so-called optical lattice. This is generated by two counter propagating laser beams forming a standing wave which provides a periodic intensity pattern in space. The Cs atoms are trapped in the intensity maxima of this standing wave. To create coherent superpositions, we used microwave radiation which allows us manipulating the internal states of the atom, similar to those used in atomic clocks. The superposition created is then transferred to position space by using the fact that the optical lattice can be state-selectively moved [2].

This means one of the two internal states is moved to the left, the other to the right. Experimentally this is realized by controlling the polarization of the counter propagating laser beams. After a shifting step, each part of the wave function is again brought into a coherent superposition before a next shifting step and so forth. Finally, after a certain number of steps the system is illuminated and imaged onto a CCD camera [3]. Due to the measurement, the delocalized wave function collapses to one position where the Cs atom is detected. To reconstruct the distribution, hundreds of identical measurements were performed.

Figure 2: Reconstructed wave function of a single atom in the optical lattice. (a) The atom is localized at a lattice site. (b) The atom has performed a 24 step random walk. (c) The atom has performed a 24 step quantum walk.

From the measurements we find that a particle performing a quantum walk shows the expected linear spreading. If the coherence of the process is intentionally destroyed, the classical random walk behaviour is recovered. Our system shows the quantum regime for approximately ten steps of the walk, where the particle is delocalized over more than twenty lattice sites. Then, imperfections, noise and uncontrolled interaction with the environment turns the quantum walk gradually into a random walk.

The quantum walk not only illustrates the mind-boggling laws of quantum mechanics; it might serve as a first step towards the development of novel search algorithms exploiting the properties of quantum mechanics and as a precursor for quantum information processing devices, such as quantum cellular automata [4-6]. Moreover, it can yield deeper insight into the transition from the microscopic quantum world to our every-day classical world.

References
[1] M. Karski, L. Förster, J. Choi, A. Steffen, W. Alt, D. Meschede, and A. Widera, "Quantum Walk in Position Space with Single Optically Trapped Atoms", Science 325, 174 (2009). Abstract.
[2] O. Mandel, M. Greiner, A. Widera, T. Rom, T.W. Hänsch, and I. Bloch, "Coherent transport of neutral atoms in spin-dependent optical lattice potentials", Phys. Rev. Lett. 91, 010407 (2003). Abstract.
[3] M. Karski, L. Förster, J. Choi, W. Alt, A. Widera, and D. Meschede, "Nearest-Neighbor Detection of Atoms in a 1D Optical Lattice by Fluorescence Imaging", Phys. Rev. Lett. 102, 053001 (2009). Abstract.
[4] R. Raussendorf, "Quantum cellular automaton for universal quantum computation", Phys. Rev. A 72, 022301 (2005). Abstract.
[5] D. J. Shepherd, T. Franz, and R. F. Werner, "Universally Programmable Quantum Cellular Automaton", Phys. Rev. Lett. 97, 020502 (2006). Abstract.
[6] K. G. H. Vollbrecht and J. I. Cirac, "Reversible universal quantum computation within translation-invariant systems", Phys. Rev. A 73, 012324 (2006). Abstract.