Fig. 2

Atomic quantum walks in regular and disordered momentum-space lattices. a–d Nonequilibrium quantum walk dynamics of 1D atomic momentum distributions vs. evolution time for the cases of a uniform tunnelling, b random static tunnelling phases, c random, dynamically varying tunnelling phases characterised by an effective temperature k _B T/t = 0.66(1) and d pseudorandom site energies for Δ/t = 5.9(1). e–h Integrated 1D momentum distributions (populations in arbitrary units; symmetrised about zero momentum) for the same cases as in a–d, after evolution times τ = (2.96(2)ħ/t, 2.51(2)ħ/t, 3.80(3)ħ/t, and an average over the range 5.1(1) to 6.4(1)ħ/t) for e–h. For e, f, we compare to quantum random walk distributions of the form \({P_n} \propto | {J_n}(2\tau t{\rm{/}}\hbar)|^2\), for g we compare to a Gaussian distribution \({P_n} \propto {e^{ - {n^2}/2\sigma _n^2}}\) for \({\sigma _n} = \sqrt {2\tau t{\rm{/}}\hbar } \), and for h we compare to an exponential distribution \({P_n} \propto {e^{ - |n|/\xi }}\). i Annealed disorder realised with tunnelling phases φ(τ) that vary dynamically with time τ. Phases contain N = 50 frequency components ω that sample an ohmic spectrum S(ω), shown here peaked at effective temperature k _B T/t = 1. j Transport under pseudorandom site energies following the form \({\varepsilon _n} = \Delta \,{\rm{cos}}(2\pi bn + \phi )\) of an incommensurate cosine potential (dashed line). As in h, 1D momentum distributions are shown for varying pseudodisorder strengths Δ/t