Relations between the single-pass and double-pass transition probabilities in quantum systems with two and three states

TL;DR

This paper provides a theoretical analysis of the relationship between single-pass and double-pass transition probabilities in quantum systems, revealing quantum interference effects that cause deviations from classical expectations.

Contribution

It introduces explicit analytic formulas linking single-pass and double-pass probabilities in two- and three-state quantum systems using SU(2) and SU(3) symmetries.

Findings

Quantum probabilities degrade faster than classical predictions in repeated processes.

Double-pass measurements can underestimate actual single-pass efficiencies.

Explicit formulas are derived for common quantum control techniques.

Abstract

In the experimental determination of the population transfer efficiency between discrete states of a coherently driven quantum system it is often inconvenient to measure the population of the target state. Instead, after the interaction that transfers the population from the initial state to the target state, a second interaction is applied which brings the system back to the initial state, the population of which is easy to measure and normalize. If the transition probability is $p$ in the forward process, then classical intuition suggests that the probability to return to the initial state after the backward process should be $p^2$. However, this classical expectation is generally misleading because it neglects interference effects. This paper presents a rigorous theoretical analysis based on the SU(2) and SU(3) symmetries of the propagators describing the evolution of quantum systems with two and three states, resulting in explicit analytic formulas that link the two-step probabilities to the single-step ones. Explicit examples are given with the popular techniques of rapid adiabatic passage and stimulated Raman adiabatic passage. The present results suggest that quantum-mechanical probabilities degrade faster in repeated processes than classical probabilities. Therefore, the actual single-pass efficiencies in various experiments, calculated from double-pass probabilities, might have been greater than the reported values.