Using Light to Simulate the Quantum Mechanics of the Simple Pendulum

We present experiments with a type of non-diffracting optical beams that satisfy a form the 2-dimensional Helmholtz equation that is identical to the Schrodinger equation for the simple pendulum. This form is known as the Mathieu equation. As a result the optical beams are in spatial modes that are related to the quantum solutions of the pendulum. The light intensity in the far field is propostional to the probability of the pendulum bob as a function of the angular position. Depending on the parameters of the problem we can select either states that describe the pendulum swinging or rotating. This system also allows us to investigate an array of interesting quantum phenomena: stationary states with non-classical analogs, wavepacket revivals and even cat states.