Fig. 4: Phase gradient (Bohmian) velocity vS.
From: Energy–speed relationship of quantum particles challenges Bohmian mechanics
a, To measure the phase gradient of quantum states in the step potential, we incorporate a Mach–Zehnder interferometer into our experimental setup. A Dove prism is placed in one arm of the interferometer to create a mirrored image of the input state. The wavenumber associated with the observed interference pattern at the camera, ktot = kMZI + kS, has two contributions: a component related to the alignment of the interferometer, kMZI, which can be determined in an independent measurement, and a component associated with the intrinsic phase gradient of the quantum state kS. b, The top row shows sketches of wave functions and their mirror images, ψ and ψM, interfering at the output of the interferometer in the classically allowed (left) and forbidden (right) regimes, respectively. The middle row displays images of experimentally obtained interference patterns. The bottom row shows the corresponding line-integrated densities. From the periodicity of the obtained interference patterns, we extract ktot. This is then converted to the velocity \({v}_{{\rm{S}}}=\hbar {k}_{{\rm{S}}}/m=\hbar ({k}_{{\rm{tot}}}-{k}_{{\rm{MZI}}})/m\). For the two shown cases, we obtain vS ≈ 2,400 km s−1 for Δ ≈ 0.1 meV and vS ≈ (31 ± 42) km s−1 for Δ ≈ −0.08 meV. c Blue markers indicate the velocity vS in the classically forbidden regime, derived using the procedure described above. The green dashed line shows the average value, ⟨vS⟩ = (59 ± 42) km s−1, of all measurements with Δ ≤ −0.05 meV. Error bars indicate the standard error of the mean.
