Fig. 2: Populations |ψ_m,a|² as a function of energy Δ and position x.

a, Measurement example of the particle densities along the waveguide axis in the main waveguide |ψ_m|² (blue) and auxiliary waveguide |ψ_a|² (green) in the classically allowed parameter regime \((\varDelta > {\hbar }{J}_{0})\). The fast periodic modulations on top of the population curves stem from a residual reflection at the end of the coupled waveguide system. b, Particle densities in the classically forbidden regime. c, Decay constant κ of the population in the main waveguide for different energies Δ (see the  Methods for further details). The observed decay constant closely follows \({\kappa }=\sqrt{2m|\,\varDelta \,|}/{\hbar }\). Error bars indicate the standard error of the mean. d, Examples of measured relative populations in the auxiliary waveguide ρ_a = |ψ_a|²/(|ψ_m|² + |ψ_a|²) for classically allowed states. A parabolic fit ρ_a = (J₀x/v)² is used to determine the speed of the motion. e, Relative populations for evanescent states in which the step is fully reflective. In these states, particles enter the step potential and eventually exit it with their direction of motion reversed. As in the case of the propagating states, a parabolic fit ρ_a = (J₀x/v)² is used to determine the speed of the motion. All data shown in this figure, except those in the inset of c, are derived from single data-taking events, each comprising 20 optical excitations of the system.

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