M5L22g.txt

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Let me just point out one important aspect
about coherence in the two-level system.
And this is related to, well, something
very mundane, the precession of spin.
And where it's a two-level system,
it's spin-1/2 in the magnetic field.
In other words, I just want to quickly remind you in a few
minutes that, for any two-level system,
we can always map it on spin-1/2.
I mean, I was really emphasizing this message
throughout the whole course.
But for spin-1/2, if you think of spin up and down
in a magnetic field, there is a very clear visualization
of the coherence.
If you have a coherence of procession of spin up and down,
the face of the superposition decides whether the spin
points in x or y.
So the precession of a spin in the transverse xy plane
is actually the manifestation of coherence.
And it's not just the [? spatial ?] coherence
of spin-1/2 in a magnetic field.
Because all two-level systems are isomorphic to that,
you can always use it as an intuitive visualization
of what coherences are.
So what I just want to point you out
is the relation to the quantum mechanical or classic
precession of spin in a magnetic field.
That it is simply an effect of coherence,
coherence within one atom, coherence
between two levels in an atom.
So if spin points in the x direction,
it is a coherent superposition of plus e and minus e,
spin up and spin down.
But this is a situation at time t equals zero.
If you let time evolve, spin up and spin down evolve
with the Larmor frequency, actually a factor of 1/2,
but with opposite phases.
Because one has plus the Larmor, h bar omega Larmor over 2.
The other is minus h bar omega Larmor over 2 as energy.
Or in other words, if you look at the relative phase,
it's a beat note with the same [? unsplitting ?]
between spin up and spin down.
But that means, now, due to this coherent time
evolution of the two amplitudes, that the spin
precesses in the xy plane.

The statistical operator for the spin-1/2,
we have 50-50 population in spin up and spin down.
But now the face here precesses as e to the minus
and e to the plus i omega l t.
Which means, if you use the statistical operator
and find the expectation value for the x spin,
so it means we take the statistical operator describing
actually the pure state of a two-level system.
We multiply with the power matrix in x.
And this is the prescription to get the expectation
value for sigma x.
And we find it's cosine omega l t.
So the spin is precessing the x component
change is cosinusoidal. e.
That would mean the y component change is sinusoidal e.
Let me just contrast it to the case of no coherence.
And this would mean off-diagonal matrix elements are zero.
Then if you have a statistical operator where
the off-diagonal matrix element are zero, in one minute
you can show that then any expectation
value for the x or y component of the spin vanishes.
So in that case, if you have a statistical mixture
between spin up and down, of course,
there is no face determined.
And it is the face of the superposition state
which tells you where between zero