This exercise is the final part of the quantum control workshop that we hold for all new students and interns. At the end of this exercise, students are asked to write a report on their findings.

The exercise asks students to solve the problem of exciting a system from the ground state of a potential to the first excited state of a potential via shaking the potential in 1D. The systems that students work with are:

- A single particle in a harmonic or anharmonic oscillator potential
- A Bose-Einstein Condensate (BEC, with nonlinear interactions determined by the Gross-Pitaevskii equation) in a harmonic or anharmonic oscillator potential

Students can explore this control problem by manipulating points on a graph directly, adjusting parameters in an analytic formula, or using the GRAPE algorithm. In each case, students are manipulating the potential modulation (shaking) function.

Note that in the case of BEC control in an anharmonic potential, students are solving the same control problem as players of the Shake Up level in our citizen science game Quantum Moves 2. Therefore, we encourage students to explore the game as well!

Because there are so many different avenues a student can explore, we leave it up to the student to choose what to study. Until now, students have provided reports on aspects like:

- Analyzing strategies of how they personally adjust parameters to optimize curves
- Comparing quantum speed limits for different values of the nonlinearity in the BEC problem
- Understanding how controllable the harmonic system is for various values of the nonlinearity interaction.

This exercise can be modified to incorporate other challenges or personalize it for different scenarios and classrooms, please just contact us. LaTeX files for the exercise PDF and report templates are available as well upon request. Contact Carrie at cweidner-at-phys.au.dk for more information.

We have also developed a nice introduction to numerical quantum calculations and quantum optimal control that you can find here.

Key words and phrases: quantum control, harmonic and anharmonic control, particle and BEC excitation, nonlinearity