This is another of our favorite exercises. It introduces the concept of a correlation amplitude, or how much a state differs from its original state as it evolves over time. This is nicely linked later to a fidelity when one teaches quantum control.

In this exercise, students explore how the correlation amplitude changes over time when one time-evolves the displaced ground state of a harmonic potential. In particular, students are asked to understand how the collapse and revival times of the correlation amplitude change as a function of the state and potential parameters. Students then explore what happens when the potential is changed to be anharmonic.

Key words and phrases: quantum harmonic oscillator, correlation amplitude, displaced ground state of harmonic well, time evolution, anharmonic well