Tunnelling photons pose no challenge to Bohmian machanics

1. Problem Statement

The paper addresses a recent claim by Sharoglazova et al. (Ref. [3]) suggesting that experimental data on photon tunneling in coupled microcavities contradicts Bohmian Mechanics (BM).

• The Conflict: Sharoglazova et al. measured two distinct "velocities" in a system where photons tunnel between a main waveguide and an auxiliary waveguide across a potential step.
  1. Semi-classical speed ($v$): Derived from the population ratio $\rho_a = |\psi_a|^2 / (|\psi_a|^2 + |\psi_m|^2)$, assuming a relation $\rho_a \propto (J_0 x / v)^2$.
  2. Bohmian velocity ($v_s$): Derived from the guiding equation $\vec{v}_s = \frac{\hbar}{m}\nabla S$.
• The Discrepancy: In the regime where the energy detuning $\Delta < -\hbar J_0$ (purely evanescent modes), the measured semi-classical speed $v$ was non-zero and proportional to $\sqrt{2|\Delta|/m}$, while the Bohmian velocity $v_s$ was measured/calculated to be zero.
• The Claim: The authors of Ref. [3] argued this mismatch implies the guiding equation fails to capture the temporal dynamics of scattering, casting doubt on BM.

2. Methodology

The authors re-analyze the physical system and the theoretical framework using the following steps:

• System Modeling: They utilize the same coupled Schrödinger equations as Ref. [3] for the main ($\psi_m$) and auxiliary ($\psi_a$) waveguides:
  $$i\hbar\partial_t\psi_{m,a} = -\frac{\hbar^2}{2m}\partial_{xx}\psi_{m,a} + V_0\psi_{m,a} + \hbar J_0(\psi_{a,m} - \psi_{m,a})$$
  where $m$ is the effective photon mass derived from Einstein's mass-energy equivalence.

• Analytical Solution: They solve the coupled equations assuming no back-propagating modes and symmetric distribution of wavevectors $k_\pm$. The solutions are expressed as superpositions of transmission and evanescent modes:
  $$\psi_m \propto (e^{ik_+x} + e^{ik_-x}), \quad \psi_a \propto (e^{ik_+x} - e^{ik_-x})$$
  where $k_\pm = \sqrt{2m(\Delta \mp \hbar J_0)}/\hbar$.

• Taylor Expansion Analysis: The authors perform a Taylor expansion of the auxiliary population $\rho_a$ for small $x$ to derive the correct coefficient for the $x^2$ term, comparing it against the model used in Ref. [3].

• Bohmian Derivation:

  • They derive the Quantum Hamilton-Jacobi equations and Continuity equations specifically for the coupled waveguide system.
  • Crucially, they identify a tunneling current term ($j_0$) in the continuity equation arising from the coupling between waveguides: $j_0 = 2J_0 R_a R_m \sin(S_a - S_m)$.
  • They solve the continuity equation $\partial_x(\rho_{a,B} v_{a,x}) + j_0 = 0$ using the BM-derived velocity $v_{a,x}$ to calculate the predicted population $\rho_{a,B}$.

• Analytic Continuation: For the regime $\Delta < -\hbar J_0$ (where $k$ becomes imaginary), they employ analytic continuation (introducing a small imaginary part $\epsilon$) to handle the evanescent wave solutions rigorously within the BM framework.

3. Key Contributions

1. Refutation of the "Challenge": The authors demonstrate that the apparent discrepancy in Ref. [3] arises from a category error: comparing a semi-classical speed estimate ($v$) with a strictly defined Bohmian velocity ($v_s$). They argue these represent fundamentally different physical quantities.
2. Correction of the Semi-classical Model: They show that the model used in Ref. [3] ($\rho_a = (J_0 x / v)^2$) is incomplete. Their derived Taylor expansion reveals that $\rho_a$ is quadratic in $x$ but with a coefficient that depends on the regime. Specifically, in the regime $|\Delta| < \hbar J_0$, the coefficient becomes independent of $\Delta$, creating a plateau in the speed vs. energy plot, which matches the experimental data perfectly (unlike the Ref. [3] model).
3. Proof of Equivalence: They mathematically prove that solving for the population $\rho_a$ using the standard Schrödinger equation yields identical results to solving the continuity equation with Bohmian velocities and the tunneling current term.
4. Clarification of Zero Velocity: They clarify that $v_s = 0$ in the evanescent regime ($\Delta < -\hbar J_0$) does not imply the absence of tunneling. In BM, particles are stationary along the $x$-axis in this region, yet non-zero probability densities ($|\psi|^2$) persist, allowing tunneling to occur via the continuity of probability flow (mediated by the coupling term $j_0$).

4. Results

• Theoretical Fit: The authors' theoretical fitting (red line in Fig. 1b) exactly reproduces the experimental data from Ref. [3], including the plateau region where $|\Delta| < \hbar J_0$. The previous model failed to predict this plateau.
• Velocity Analysis:
  • Regime $\Delta > \hbar J_0$: Both waveguides have transmission modes; $v_m$ and $v_a$ are identical.
  • Regime $|\Delta| < \hbar J_0$: One mode is evanescent; $v_m$ and $v_a$ differ due to the quantum potential.
  • Regime $\Delta < -\hbar J_0$: Both modes are evanescent. The Bohmian velocity $v_s$ is strictly zero along the $x$-direction.
• Consistency Check: By integrating the continuity equation with the tunneling current, the derived population $\rho_{a,B}$ matches the Schrödinger solution $\rho_a$ exactly (Eq. 23, 24, 28).
• Conclusion on Velocities: Any velocity defined strictly from the probability density $\rho_a$ must satisfy the continuity equation. Since both Copenhagen and BM obey the same Schrödinger equation, they must yield identical predictions for $\rho_a$ and any velocity derived consistently from it.

5. Significance

• Validation of Bohmian Mechanics: The paper conclusively demonstrates that the experiment in Ref. [3] does not challenge Bohmian Mechanics. The perceived contradiction was due to an incorrect comparison of non-equivalent physical quantities and an incomplete theoretical model of the semi-classical speed.
• Interpretational Equivalence: It reinforces the standard view that BM and the Copenhagen interpretation are empirically equivalent for non-relativistic quantum systems, as they share the same fundamental wavefunction evolution and probability conservation laws.
• Physical Insight into Tunneling: The work provides a clearer physical picture of tunneling in coupled systems within the BM framework. It highlights that "zero velocity" in a specific direction (stationary particles) is compatible with "tunneling" (non-zero probability transmission) when coupling terms and quantum potentials are properly accounted for.
• Methodological Correction: The paper corrects the interpretation of experimental data in coupled waveguide systems, emphasizing the necessity of including coupling-induced tunneling currents in continuity equations when analyzing Bohmian trajectories.