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M1L2e.txt
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#
# File: content-mit-8-421-1x-subtitles/M1L2e.txt
#
# Captions for 8.421x module
#
# This file has 48 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
What happens when we have an off-resonant rotating field?
When we have an off-resonant rotating field,
we go to the rotating frame, but of course the rotating frame
we go to is now not rotating at the Larmor frequency,
because the purpose of going to rotating frame
is get rid of the time dependence of the rotating
field.
So we go to rotating frame, which
rotates at the frequency of the rotating field.
So if the rotating field rotates at a frequency omega,
we have a fictitious field, which is omega over gamma.
Gamma is the gyromagnetic ratio.
For the resonant case, we were just completely
canceling the static field in the z direction,
but for the off resonant case, when
omega is larger or smaller, in both cases
this one here is no longer zero.
And our total effective field is now
the quadrature sum of what we have in the z direction,
and what we have in the x direction,
or x prime direction.
This is b1.
So in the z direction, we have the static field
minus the fictitious field.
And then the two are added up.
On resonance, the angle theta is 90 degrees,
but for the off resonant case, the angle
is different, given by the simple geometric result.
And the effective field is the quadrature sum
of b1 squared plus b0 minus.
And the fact is this adds something
to the effective field in the rotating frame
whether we drive it above and below resonance.
So therefore, the magnetic moment
precesses at what is called the generalized Rabi
frequency, which I now call the generalized Rabi
frequency, which is gamma times b effective,
and this is the quadrature sum of the detuning
plus the capital omega.
The capital letter omega, omega R,
is the Rabi frequency at resonance.
And this is nothing else than a measure for the drive
field for the strengths of the drive field b1 in frequencies.
So therefore the generalized Rabi frequency
is the resonant Rabi frequency added in quadrature
with the detuning squared.