Quantum systems which are discrete both in space and time can be realized with pseudo-spin-1⁄2 atoms stored in the spin-dependent optical lattice presented inlecture I: the quantum state evolves in a digital way through finite time steps, with spatial coordinates taking only discrete values on the lattice. At individual sites q, the quantum state is defined by the internal spin (s={↑,↓}) and vibrational degrees of freedom, ǀψ〉 =ǀq〉siteǀn〉vibǀs〉spin. Simultaneously, temporal discreteness is imposed through the application of periodic actions, e.g., transport steps. It was shown by S. Lloyd [1] that systems whose quantum dynamics is ruled by finite-time local operations can be regarded as universal quantum simulators, capable of simulating the evolution of any continuous-time quantum system. Note that this system makes a very large Hilbert space available, which is accessible with well-controlled properties at every position.
System dynamics is implemented with discrete operations. The shift operator is
Ŝ = ǀq+1〉〈qǀsiteǀ↑〉〈↑ǀ+ǀq-1〉〈qǀsiteǀ↓〉〈↓ǀ,
which connects each site with its adjacent sites conditioned on the spin state. So-called coin operations cause spin rotations R (φ,θ) which may or may not be site dependent, Ĉ = R (φ,θ)ǀs⟩⟨sǀ. A quantum walk [2] consists of n-fold applications (Ŵ(n)) of a fixed combination of coin and shift operation Ŵ = Ŝ⨂Ĉ . After n steps, an atom initially positioned at site 0 is coherently delocalized over 2n+1 sites. The final distribution appears as the result of a multi-path interference of the delocalized atom; its details depend on the initial spin state of the atom. Examples of an experiment averaging about 1000 atoms are considered and compared to their theoretical prediction [3]. As a consequence of its unitary character, the rms width of the quantum walk scales as n ("ballistically"), in contrast to the random walk, which shows the well-known √n, or diffusive behavior.
