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
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
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.
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
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:
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
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
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
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
those two states longitudinal
alignment and transverse
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
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.
© 2004-2009, Alain Michaud