Interference, Reduced Action, and Trajectories

Edward R. Floyd

Edward R. Floyd

Instead of investigating the interference between two stationary, rectilinear wave functions in a trajectory representation by examining the trajectories of the two rectilinear wave functions individually, we examine a dichromatic wave function that is synthesized from the two interfering wave functions. The physics of interference is contained in the reduced action for the dichromatic wave function. As this reduced action is a generator of the motion for the dichromatic wave function, it determines the dichromatic wave function’s trajectory. The quantum effective mass renders insight into the behavior of the trajectory. The trajectory in turn renders insight into quantum nonlocality.

Shohei Imai, Atsushi Ono, Naoto Tsuji

We present an effective theory for describing electron dynamics driven by an optical external field in a Schrödinger's cat state. We show that the reduced electron density matrix evolves as an average over trajectories $\{ρ_α\}$ weighted by the Sudarshan--Glauber $P$ distribution $P(α)$ in the weak light--matter coupling regime. Each trajectory obeys an equation of motion, $\mathrm{i} \partial_tρ_α=\mathcal{H}_α ρ_α-ρ_α\mathcal{H}_α$, where an effective Hamiltonian $\mathcal{H}_α$ becomes non-Hermitian due to quantum interference of light. The optical quantum interference is transferred to electrons through the asymmetric action between the ket and bra state vectors in $ρ_α$. This non-Hermitian dynamics differs from the conventional one observed in open quantum systems, described by $\mathrm{i} \partial_tρ=\mathcal{H}ρ-ρ\mathcal{H}^\dagger$, which has complex conjugation in the second term. We confirm that the reduced, trajectory-resolved effective theory agrees with full electron-photon simulations for the few-electron Dicke model, thereby validating the interferential non-Hermitian description in the weak-coupling regime.

Markus Kindermann

John Ewan Inglesfield

Floyd, E R

The double slit problem is idealized by simplifying each slit by a point source. A composite reduced action for the two correlated point sources is developed. Contours of the reduced action and trajectories are developed in the region near the two point sources. The trajectory through any point in configuration space also passes simultaneously through both point sources.