Rabi Oscillations

November 2020

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As part of McGill Physics' annual Hackathon, my friends (Christian Bye, Cheng Lin, Su Goh) and I developed a tool to aid visualization of states in time-dependent potentials. We were motivated by the fact that solutions to physical problems with time-dependent potentials are quite rare. In most cases, we resort to time-dependent perturbation theory to obtain an approximate answer, valid only at early times and for small potentials. One of the few exactly solvable systems for which we wish to gain some intuition is the two-level spin system coupled to an oscillating magnetic field.

Visualization of the states as a function of time

The first tool shows how a given initial state varies with time. It allows us to vary the many input parameters (mass, charge, field strength and frequency). Try tweaking the parameters for yourself, and see if you can find the resonant frequency!

Bloch sphere

A useful tool to analyze two-level systems is the so-called Bloch sphere. A vector in this sphere points in the direction of the |+n> state, that is |+n> = cos(theta/2)|+z> + sin(theta/2)*exp(i*phi)|-z>. For a constant magnetic field, we expect the spin to precess about the field axis. If the field is varying in time, however, then the axis of precession is itself moving! So, our state undergoes a wobbly motion known as Rabi oscillations. Here, you can see the oscillations for the initial |+z> (spin-up) state at non-resonant and resonant frequency (left and right, respectively). Notice the resonant frequency visualization can have the vector pointing down, i.e: in the |-z> state, even though it started off in the |+z> state!