Time-Modulated Hamiltonian for Interpreting Delayed-Choice Experiments via Mach–Zehnder Interferometers

Many delayed-choice experiments based on Mach–Zehnder interferometers (MZI) have been considered and made to address the fundamental problem of wave-particle duality. Conventional wisdom long holds that by inserting or removing the second beam splitter (BS2) in a controllable way, microscopic particles (photons, electrons, etc.) transporting within the MZI can lie in the quantum superposition of the wave and particle state as \psi =a_\rm w \psi _\rm wave +a_\rm p \psi _\rm particle. Here we present an alternative interpretation to these delayed-choice experiments. We notice that as the BS2 is purely classical, the inserting and removing operation of the BS2 imposes a time-modulated Hamiltonian H_\bmod (t)=a(t)H_\rm in +b(t)H_\rm out, instead of a quantum superposition of H_\rm in and H_\rm out as H=a_\rm w H_\rm in +a_\rm p H_\rm out, to act upon the incident wave function. Solution of this quantum scattering problem, rather than the long held quantum eigen-problem yields a synchronically time-modulated output wave function as \psi _\bmod (t)=a(t)\psi _\rm wave +b(t)\psi _\rm particle, instead of the stationary quantum superposition state \psi =a_\rm w \psi _\rm wave +a_\rm p \psi _\rm particle. As a result, the probability of particle output from the MZI behaves as if they are in the superposition of the wave and particle state when many events over time accumulation are counted and averaged. We expect that these elementary but insightful analyses will shed a new light on exploring basic physics beyond the long-held wisdom of wave-particle duality and the principle of complementarity.