M3L15r.txt

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Often something is simple to write down,
but if there's a lot of physics insight,
we spend some time in discussing it.
And the first thing I want to just point out and discuss
is this interaction term.
We have the product of sigma plus and sigma
minus, the a and a data.
So what we have here is-- we have an interaction term,
and this interaction part has actually four terms,
in a very natural way.
Now let me just write them down.
It's sigma plus with a, sigma minus with a dagger,
sigma plus with a dagger, and sigma minus with a.
OK.
So let's discuss those.
Sigma plus with a dagger.
Sigma plus is actually-- this is an absorption process.
a that uses to photon number by one,
and increases the atomic excitation
from the ground to the excited state.
The other term looks naturally and intuitively like emission.
The a dagger operator takes us from n to n plus 1,
and sigma minus takes us from the excited
state to the current state.
So these are the two terms, which
we would call intuitive terms, because they make sense.
The other terms are somewhat more tricky.
Sigma plus and a dagger means we create a photon,
and we create an excitation.
So in other words, it's not like the other term,
one quantum of excitation disappears from the field,
appears in the atom and vice versa.
Sigma plus a dagger means we have an atom excitation takes us
from the ground to the excited state,
plus we emit a photon at the same time.
And sigma minus a dagger means that we
go from the excited to the ground state.
So we have an atom de-excitation.
Now we would say, well, if the atom is de-excited,
it should emit a photon.
But instead, the photon disappears.
So we have those processes.
These last two are sometimes referred to,
in the theoretical literature, they are off shell.
Under shell is energy conservation.
Off shell means it cannot conserve energy.
But nevertheless, these are terms,
which appear in the operator, but you should be used
to if you have, oftentimes, in the operator,
which cannot drive a resonant transition.
When we looked at the DC Stark effect,
or when we looked at the AC Stark effect for low frequency
photons, those low frequency photons cannot excite an atom
to an excited state, but they are not causing a transition,
but they lead to energy shift in second order integration
theory.
So therefore, those terms, this language now,
cannot drive transitions.
They can only drive transitions to virtual states, which
would mean they can only appear in second order integration
theory, that you go up to a so-called virtual state,
but you immediately go down.
And those terms give only rise to shifts, no transitions,
because you couldn't conserve energy in the transition,
but you can do shifts in second order.
And one example, which we discussed in the clicker
question is, that those shifts are actually Lamb shifts,
and in other places, especially in the context of microwave
fields, they are called glossy Lamb shifts.
And let's just look at one specific state.
And this is the simplest of all.
We have the vacuum, no photons, and the atom
is in the ground state.
If you look at the four possibilities
of the interaction term.
There is only one non vanishing term.
The photon is at the bottom of all possible states.
The atom is at the bottom of the possible states.
So when we act with the four terms on it,
the only term which contributes is
those that are raised, because all the others are zero.
The only non vanishing term is when
we create a virtual atomic excitation, and also
a virtual excitation of the photon field.
And we know that when we have an atom in the ground
state in the vacuum, that the only manifestation
of the electromagnetic field is, of course, not
spontaneous emission, but the Lamb shift.
So therefore, if you would apply this operator, to the ground
state of an electron in an atom, the complicated one
is very function of hydration, and some of this operator,
over all modes of the electromagnetic field,
then you would have done the first principle QED
calculation of the Lamb shift.
I'm not doing it, but you should understand
that this operator, sigma plus a data, this is the operator
for the Lamb shift.

Questions?

Yes.
What effects does this Hamiltonian model?
Does it capture, for example, radiation reaction?

Oh, no, everything is-- if you have a two level
system, this Hamiltonian captures everything which
appears in nature.
If you have a two level system interacting
with the electromagnetic field.
That's it.
Radiation reaction is just something
we can pull out of here.
Stimulated emission, we can pull out of here.
The way whole vacuum fluctuations
create a Lamb shift, or the way whole vacuum fluctuations
affect an atom in the excited state,
everything is included in here.
The question is just, can we solve it?
And the calculations can get involved,
but this is the full QED Hamiltonian photon level
system.
That's a full picture.
That's why I, sort of, said before, be proud of it.
You understand the full picture how two level systems interact
with electromagnetic radiation.
The only complication is, yes, if you put more levels into it,
and such, and things can get richer and richer.

Yes, we have also made the dipole approximation, which--
I'm just wondering how critical it is-- well
we use the electric field-- my gut feeling is
it doesn't really matter.
What we have here is the most genetic term which
can create atoms and photons.
And if we add the a data term-- actually,
I don't know what would happen if you don't
make the dipole approximation.
Now if you have two levels, which
are coupled by making it dipole, then you
have the same situation.
It is just your prefector for senior photon frequency
is now alpha times smaller, because of the smaller dipole
approximate element.
So you cannot, I think, you can pick pretty much any level you
want, and this is why I actually discussed matrix elements
at the beginning of the unit, for pretty much all
of the discussion want to have, it doesn't really matter what
kind of transition you have, as long as the transition creates
or annihilates a photon, and all of the physics
of the multiplicity of the transition, magnetic, dipole,
electric, portal, whatever, just defines what this uniform hobby
frequency is.
The only thing, and you've put me on the spot,
but the only thing which comes to my mind
now is, if you would formulate QED
not in the dipole approximation, but with a p
minus a formulation, then given a square term,
and then we have the possibility that one transition can
emit two photons.
So that's not included here.
So this is like higher order?
This would be something higher order.
On the other hand, we can show with
the canonical transformation that the p minus a formulation,
with the a squared term, is equivalent to dipole
approximation.
So the question, whether you have
a transition which emits two photons simultaneously,
or two photons sequentially, by going for intermediate state,
this is not a fundamental distinction.
You can have one description of your quintum system
where two photons are emitting in one transition.
You have another description of your quintum system,
where photons cannot-- only one photon can be emitted,
and then you have to land in an intermediate state.
And you will say, you've got two photons at once,
or one photon at a time, this is two different kinds of physics.
But we can show that the two pictures
are connected with a canonical transformation.
So therefore, you have two descriptions here.
But anyway, I'm going a little bit beyond my knowledge.
I'm just telling you bits and pieces I know.
But this Hamiltonian is either, generally exact,
I just don't know how to prove it.
But it really captures in all of QED aspects of the system we
want to get into.