Measurement-based Lorentz-covariant Bohmian trajectories of interacting photons

In a recent article [Nat. Commun. 13, 2 (2022)], we devised a method of constructing the Lorentz-covariant Bohmian trajectories of single photons via weak measurements of the photon's momentum and energy. However, whether such a framework can consistently describe multiparticle interactions remains to be seen. Here we present a nontrivial generalization of our framework to describe the relativistic Bohmian trajectories of two interacting photons (this interaction arising from the symmetrization of the two-particle wave function) exhibiting nonclassical interference due to their indistinguishability. We begin by deriving the average velocity fields of the indistinguishable photons using a conditional weak measurement protocol, with detectors that are agnostic to the identity of the respective photons. We demonstrate a direct correspondence between the operationally derived trajectories and those obtained using a position- and time-symmetrized multiparticle Klein-Gordon wave function, whose dynamics are manifestly Lorentz covariant. We propose a space-time metric that depends nonlocally on the positions of both particles as a curvature-based interpretation of the resulting trajectories. Contrary to prior expectations, our results demonstrate a consistent trajectory-based interpretation of relativistic multiparticle dynamics in quantum theory.

Figure 1

Schematic diagram of the system of interest. Two single-photon light sources are directed towards each other, with the detectors, operating on an equal time slice, performing weak measurements of the detected photons.

Figure 2

Two-photon trajectories for (a) one of the photons at a fixed initial position and (b) symmetric initial conditions. Each pair of trajectories in each color corresponds to a given pair of initial conditions for both photons. To obtain these trajectories, we solved the set of differential equations $v_i({\mathbf{X}}_1,{\mathbf{X}}_2)=\mathrm{d}{x}_i/\mathrm{d}t$ for a pair of initial positions $x_{i0}\equiv x_i\left(t_0\right)$. The time evolution occurs on the equal time slice $t_1=t_2=t$. We have utilized the parameters $k_0/\sigma =20$.

Figure 3

Individual time slices of the trajectories plotted in the $(x_1,x_2)$ parameter space. The respective plots display a random sample of trajectories that are initially distributed according to the probability density ${\rho }_t(t,x_1,x_2)$ at $t=-2$. The three time slices correspond to (a) $t=-1$ and $-0.5$ and (b) $t=0$. We have used $k_0/\sigma =20$ as in Fig. 2.

Figure 4

Two-photon trajectories from a boosted reference frame with $\vartheta =0.4$, corresponding to the initial conditions in (a) Fig. 2 and (b) Fig. 2. Both plots are obtained with a single evolution parameter $t^{\prime }$. The different initial conditions for each trajectory reflects the relativity of simultaneity between the initial conditions in the original frame compared with those in the new frame.

Figure 5

Zoomed-in view of the backward-in-time trajectories for $\vartheta =0.6$.

Figure 6

Individual time slices of the trajectories plotted in the $(x_1^{\prime },x_2^{\prime })$ parameter space. The respective plots display the boosted initial conditions utilized in Fig. 3, for a boost of $\vartheta =0.2$. We have used the wave-packet parameters $k_0/\sigma =20$.

Figure 7

Schematic diagram of the physical setup that would demonstrate the nonlocality of the Bohmian trajectories and hence the guiding metric.