Transition of a superposition of states under a delta-function pulse in a two-level system

Abstract

Under a time-dependent perturbation, it is common to calculate the probability of a transition from one eigenstate to another eigenstate of a quantum system. Here we study the transition from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse. We consider a two-level system with energy levels $E_1$ and $E_2$ and obtain exact analytical expressions for the coefficients $c_1$ and $c_2$ of the final state. The expressions are general since the coefficients $α_1$ and $α_2$ of the initial superposition state are free parameters constrained only by $|α_1|^2+ |α_2|^2=1$. This opens up new possibilities and in particular allows for an abrupt transition to a definite eigenstate with unit probability. We obtain a general analytical expression for the probability $P_{α_1,α_2 \to 2}$ of an initial superposition state to transition to the second eigenstate. Armed with this general expression we study some interesting special cases. With a delta-function pulse, the transitions are abrupt/instantaneous and we show that they do not depend on the energy gap $E_2-E_1$ and hence on the relative phase between the two eigenstates. For specific values of the interaction strength $β$, the initial superposition transitions abruptly to a definite eigenstate with unit probability. We discuss the similarities and differences such a transition has with the collapse of the wavefunction familiar in the context of a measurement.