# Rabi Flopping Oscillations

Since its introduction in 1937, the mechanism of flopping oscillation was one of the most fundamental. Today there are many applications that are a consequence of the the mechanism but the most useful one is certainly the Magnetic Resonance Imaging (MRI) where the measurement of Rabi oscillations allows to make the most precise images of the human body.

There are many models that can be used to illustrate this mechanism. In fact, this oscillation is so fundamental in atomic physics, that this is often used as an example in most theories. We used the model of magnetic resonance. Any other model or approach will lead to the same concept. We hope that this will not be to confusing.

The thick black segment represents a Bloch Vector. This is a vector that moves in the three dimensional space. The box around the figure shows  the Cartesian coordinate system.  The vertical axis has a red segment that represents the projection of the vector on the vertical axis.  This is done so that we can visualize in real time the longitudinal component of the Bloch vector. We also see, plotted in red the Rabi oscillation which is the amplitude of the red segment as a function of the time, but for now this curve is not related to the rest of the drawing, so lets forget about the red sine function.

As we just mentioned the Bloch vector can be seen as three components, one longitudinal, ie. the projection on the vertical  axis, and the two others whose the sum is in the horizontal plane. We call this one the transverse component. Now, a close look at the movement reveals that the system goes from longitudinal to transverse to longitudinal  again and so on... This is what we call Rabi oscillations. Before we understand what causes this movement, we should explain what causes the Bloch vector to move.

In order to have a behavior like what we observe,  the Bloch vector needs to pocess two fundamental properties: First, is has a dipole moment.  This can be represented by a magnet. If you apply a magnetic field then the moments in your system tend to align to this field.  This is how we apply forces on our Bloch vector: Using a dipole moment.  We see it as a small magnet subjected to TWO separate fields.

The first field is not shown on the picture. It is vertical (pointing up for example), very large and does NOT vary in time, ie. it is static. Since this field is large and according to what we just said, the Bloch vector should become vertical and stop there. This is not what we see, therefore we need to know the second property: angular momentum. This is the property for example of a spinning wheel. If you apply a torque on it you get a rotation around the axis normal to the plane in which lie the torque and the angular momentum.  This is the so enigmatic precession movement, familiar to the "young" and "not so young" physicist watching a spinning top.  Well, in order to see some Rabi oscillations, wee need the concept of angular momentum in our model, otherwise the Bloch vector will align parallel to the field and there will never be any oscillation.  In fact, most of the theories in physics and chemistry are based upon rotations and momentum!

Now we have completed our Bloch vector model:  just like a small magnet that spins like a top. The rotation axis and pole alignment are parallel and represented by the Bloch vector. I recall that the magnetic field is vertical and very large.  Just like a top, the Bloch vector will start to rotate around the vertical axis.  This movement is called precession and the frequency at which it rotates is called the Larmor frequency. It is proportional to the amplitude of the magnetic field  and a constant that depends on the system itself. A tiny electron  might rotate faster than a bigger neutron for example, therefore we need this constant that we call gyromagnetic factor.

In order to have a Larmor precession, the Bloch vector needs to be away from the vertical axis (a similar behavior exist for a top as well). If the  field produces no torque then there will no movement induced. In order to create a torque then the field needs to be at right angle from the Bloch vector.  This brings back the concept of longitudinal and transverse components introduced earlier.  If the Bloch vector is aligned on the magnetic field (vertical), then nothing will happen, just like when the top is exactly vertical. On the other hand if the Bloch vector is parallel to the horizontal axis, then it will precess forever  in the plane. We call those two states longitudinal alignment and transverse alignment respectively.  Of course, the top gradually looses speed and eventually falls on the ground, but the nucleus and the electrons that are in an atom are more remote from friction forces. Therefore, we suppose that there are no relaxation effects , and the movement is not scrambled and will last forever.

In order to create some Rabi oscillations, we MUST start with a longitudinal state, ie. when  the vector is vertical. Then, we apply a second magnetic field. It is represented on the figure by the RED/GREEN arrow. It is a rotating field this time. Its frequency MUST be the same as the Larmor frequency. The amplitude is usually very small, compared to the other field. Since the rotating field is perpendicular to the Bloch vector, then the vector will start to precess around it. But since it rotates at the speed as the Larmor frequency, the Bloch vector always remains at right angle angle with it, and the longitudinal orientation is slowly converted into a transverse orientation... then to longitudinal... then transverse again...  The frequency at which this occurs (the Rabi frequency!) is proportional to the amplitude of the rotating field times the gyromagnetic constant.  In the animation that you see above, we have set to 12 the ratio between both, but in real experiments, it is usually ranges from one thousand to one million.