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

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 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 . 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 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 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.