Collapse and transition of a superposition of states under a delta-function pulse in a two-level system

TL;DR

This paper derives exact analytical expressions for state transitions in a two-level quantum system under a delta-function pulse, revealing conditions for wavefunction collapse from superposition to an eigenstate independent of energy gap or phase.

Contribution

It provides a general analytical solution for superposition-to-eigenstate transitions under impulsive perturbations, highlighting collapse scenarios without measurement.

Findings

Transition probabilities are independent of energy gap and phase.

Specific interaction strengths lead to complete collapse into an eigenstate.

Wavefunction collapse can occur instantaneously via a delta-function pulse.

Abstract

Under a time-dependent perturbation it is common to calculate the transition probability in going from from one eigenstate to another eigenstate of a quantum system. In this work we study the transition in going from a \textit{linear superposition of eigenstates} to an eigenstate under a delta-function pulse (which acts at $t=0$). We consider a two-level system with energy levels $E_1$ and $E_2$ and solve the coupled set of first order equations to obtain exact analytical expressions for the coefficients $c_1(t>0)$ and $c_2(t>0)$ of the final state. The expressions for the final coefficients are general in the sense that they are functions of the interaction strength $\beta$ and the coefficients $\alpha_1$ and $\alpha_2$ of the initial superposition state which are free parameters constrained only by $|\alpha_1|^2+ |\alpha_2|^2=1$. This opens up new possibilities and in particular, allows for a ``collapse" scenario. We obtain a general analytical expression for the transition probability $P_{\alpha_1,\alpha_2 \to 2}$ in going from an initial superposition state 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 multiple values of the interaction strength $\beta$, we show that the system ends up in a definite eigenstate i.e. probability of unity. Such a transition can be viewed as a ``collapse" since a superposition of states transitions abruptly to a definite eigenstate. The collapse of the wavefunction is familiar in the context of a measurement. Here it occurs via a delta-function pulse in Schr\"odinger's equation. We discuss how this differs from a collapse due to a measurement.