Transient quantum beats, Rabi oscillations, and delay time of modulated matter waves

Abstract

Transient phenomena of phase modulated cut-off wavepackets are explored by deriving an exact general solution to Schr\"odinger's equation for finite range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual \textit{self-induced two-level system} with energies $E_{+}$, and $E_{-}$, due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi-oscillations characterized by a frequency $\Omega=(E_{+}-E_{-})/\hbar$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a \textit{quantum beat} effect with a beating frequency, $\Omega$. We also find a regime characterized by a \textit{time-diffraction} phenomenon that allows to measure unambiguously the delay-time, which can be described by an exact analytical formula. It is found that the delay-time agrees with the phase-time only for the case of strictly monochromatic waves.