Trajectories in coupled waveguides: an application to a recent experiment and Hiley's lessons on the falsification of the Bohmian model

TL;DR

The paper examines claims that the de Broglie–Bohm (dBB) interpretation can be falsified in experiments involving tunneling between coupled waveguides. It demonstrates, first with a 1D double-well model and then via a full 2D Schrödinger treatment, that Bohmian trajectories exhibit nonzero velocities in the tunneling region and reproduce standard quantum predictions, provided the dynamics are applied correctly. A key message is the contextual nature of Bohmian trajectories: stationary states in a closed system do not imply observable stationary behavior once measurements or complex dynamics are considered. The work reinforces Basil Hiley’s stance that apparent falsifications often arise from improper application and highlights the nontrivial link between measurement, dynamics, and observed Bohmian trajectories, while noting that this contextuality leaves the ontology of the pilot wave empirically unmapped by current experiments.

Abstract

From "surreal" trajectories to which-way measurements, Basil Hiley had a lesson: claims of falsifying the Bohmian model do not withstand scrutiny provided the model is applied correctly. In this work we compute de Broglie-Bohm trajectories for particles tunneling in coupled waveguides relevant to a recent experiment having claimed to challenge the Bohmian model. We show that the Bohmian model - correctly applied - gives results identical to the standard quantum approach, first by working out a simple one-dimensional model, and then by computing Bohmian trajectories for the full two-dimensional problem representing a quantum particle propagating inside coupled waveguides. We further recall the contextual nature of the Bohmian trajectories whereby the trajectories of a closed system differ from the ones observed when an interaction with a measurement apparatus takes places.

Figures (4)

• Figure 1: Main and auxiliary waveguides displayed with a height map representing the potential $V(x,y)$ (the height scale corresponds to the numerical computations of Sec. \ref{['sec:2D']}). The step potential (in the region $x > 0$) and the barrier potential along $y$ between the two waveguides are clearly visible.
• Figure 2: Schematics showing the two lowest energy levels in a double well represented with the dashed lines: fundamental level in blue, first excited level in orange, with a sinusoidal behaviour inside the wells, and an exponential behaviour outside (atomic units are used; the left scale refers to the wavefunction amplitudes).
• Figure 3: Time evolution of the probability density $|\psi(x,y,t)|^2$. At time $t=0\,\mathrm{a.u.}$ (a) the wavepacket is centered around $x_0=-12.5\,\mathrm{a.u.}$ and $y_0=10.5\,\mathrm{a.u.}$, with width $\sigma=0.5\,\mathrm{a.u.}$, momentum $p_0=12\,\mathrm{a.u.}$ along $x$ and no initial momentum along $y$. At time $t=1\,\mathrm{a.u.}$ (b) the wavepacket is reflected and transmitted at the step potential (at $x=0\,\mathrm{a.u.}$). At time $t=2\,\mathrm{a.u.}$ (c) the transmitted part of the wavepacket starts to tunnel to the auxiliary waveguide. We use a logarithmic color scale to enhance the visibility of the small transmitted density. The step potential (at $x = 0\,\mathrm{a.u.}$) is indicated by the dashed line, and the barrier potential (between the two waveguides) is indicated by the dotted lines. Atomic units ($\mathrm{a.u.}$) are used throughout ($\hbar=m=1$).
• Figure 4: Bohmian trajectories computed for the 2D potential of Eq. (\ref{['pot-def']}). Two sets of trajectories are shown: the first set (in blue) corresponds to particles that are initially located around the initial probability density, and that reflect off the step potential. The second set (in red) corresponds to particles that are taken to be regularly spaced in the auxiliary waveguide at $t=t_f=5\,\mathrm{a.u.}$. The step potential (at $x = 0\,\mathrm{a.u.}$) is indicated by the dashed line, and the barrier potential (between the two waveguides) is indicated by the dotted lines. Atomic units ($\mathrm{a.u.}$) are used throughout ($\hbar=m=1$).