A simple understanding of quantum electrodynamics using Bohmian trajectories: detecting non-ontic photons

Abstract

The use of Bohmian mechanics as a practical tool for modeling non-relativistic quantum phenomena of matter provides clear evidence of its success, not only as a way to interpret the foundations of quantum mechanics, but also as a computational framework. In the literature, it is frequently argued that such a realistic view-based on deterministic trajectories cannot account for phenomena involving the "creation" and "annihilation" of photons. In this paper, by revisiting and rehabilitating earlier proposals, we show how quantum optics can be modeled using Bohmian trajectories for electrons in physical space, together with well-defined electromagnetic fields evolving in time. By paying special attention to an experiment demonstrating partition noise for photons, and to how the Born rule emerges in this context, the paper pursues two main goals. First, it vindicates the pedagogical use of this simple Bohmian framework to compute, understand, and visualize quantum electrodynamics phenomena. Second, given that measurements are ultimately indicated on matter pointers, it clarifies what it means to measure photon or electromagnetic-field properties, even when they are considered non-ontic elements.

Figures (5)

• Figure 1: (a) Sketch of the simulated system. An optical cavity containing a single electromagnetic mode of frequency $\omega_c$, with two quantum wells located inside the cavity. (b) Two-level energy diagram of the different components of the system at two instants of the Rabi oscillations. The initial state at time $t_1$, $\ket{001}$, corresponds to the electromagnetic field in the excited state while both electrons are in their ground states. At time $t_2$, the excitation is coherently transferred to the electronic subsystem, producing the superposition $\frac{1}{\sqrt{2}}\left(\ket{100} + \ket{010}\right)$. The excitation is subsequently exchanged periodically between the photonic and electronic degrees of freedom through Rabi oscillations.
• Figure 2: Unitary dynamics of the non-measured system under resonant coherent coupling. Top panel: time evolution of the populations $|c_{nmk}(t)|^2$ within the odd-parity subspace over four Rabi periods $T_R$. The initial photonic excitation $|001\rangle$ is coherently exchanged with the symmetric electronic states $|100\rangle$ and $|010\rangle$, while $|111\rangle$ remains unpopulated. The total probability is conserved at all times. Bottom panel: expectation values of the cavity and electronic energy contributions, showing periodic energy exchange consistent with Rabi oscillations.
• Figure 3: (a) Sketch of the simulated system including the measurement apparatus. An optical cavity containing a single electromagnetic mode of frequency $\omega_c$ interacts with two quantum dots (QD). Each electron is coupled to an independent pointer degree of freedom, represented by the coordinates $y$ and $z$, which act as measurement devices for the electronic energies. (b) Two-level energy diagram illustrating the measurement process during the Rabi dynamics. As in the non-measured case, the initial state at time $t_1$, $\ket{001}= \ket{0}_{x_1}\otimes\ket{0}_{x_2}\otimes\ket{1}_{p}$, evolves coherently into the superposition $\frac{1}{\sqrt{2}}\left(\ket{100} + \ket{010}\right)$ at time $t_2$. When the measurement interaction is activated, the electronic states become correlated with the pointer degrees of freedom. This correlation dynamically separates the corresponding branches in configuration space, producing an effective collapse of the conditional wave function: depending on the Bohmian trajectory, the system evolves either to $\ket{100}$ or to $\ket{010}$, corresponding to the energy being absorbed by electron 1 or electron 2, respectively.
• Figure 4: Measurement-induced effective collapse in the two possible Bohmian outcomes. Left panel: evolution of the conditional state and corresponding expectation values along a trajectory that follows the $y$-pointer branch, leading to $|c_{100}|^2 \to 1$ and $\langle H_{x_1} \rangle \to E_{1,e}$, corresponding to excitation of electron 1. Right panel: evolution along the alternative trajectory that follows the $z$-pointer branch, yielding $|c_{010}|^2 \to 1$ and $\langle H_{x_2} \rangle \to E_{1,e}$, corresponding to excitation of electron 2. In both cases, the total evolution remains unitary, while the effective collapse emerges from the dynamical separation of the pointer wave packets.
• Figure 5: Conditional wave functions and corresponding Bohmian trajectories at three different times. At $t_1$, the system is in the initial state of the simulation, $\ket{\Phi(t_1)}=\ket{001}\otimes \ket{\varphi_y} \otimes \ket{\varphi_z}.$ At $t_2$, corresponding to half a Rabi oscillation, the energy transfer is maximal and the state has evolved into the superposition $\ket{\Phi(t_2)}=\frac{1}{\sqrt{2}}\left(\ket{100} + \ket{010}\right)\otimes \ket{\varphi_y} \otimes \ket{\varphi_z}.$ At the later time $t_3$, the measurement interaction has been completed. In the left panel, the Bohmian trajectory follows the wave packet associated with pointer $y$, indicating that the energy has been absorbed by electron 1. In the right panel, the trajectory follows the wave packet associated with pointer $z$, corresponding to the case in which the energy is absorbed by electron 2.