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PROFESSOR: OK.
Back to cavity QED, back to the fully quantized radiation
field, back to vacuum Rabi oscillation.
Let me just recapitulate and sort of make the transition
from this intense discussion about homework
to the intellectually stimulating discussion
about atoms and photons.
So in the semiclassical description
of the electromagnetic field, photons
can only be emitted because we have
a Hamiltonian with the semiclassical electric field.
So if you don't drive the system with an electric field,
you cannot stimulate the emission of photons.
But we know this is not what happens.
Photons are emitted into empty space,
photons are emitted into a vacuum.
And for that we needed a quantized description
of the electromagnetic field.
We did field quantization, and we
have now our quantized Hamiltonian.
And on Monday I started to discuss
what is sort of the paradigmatic situation,
the paradigmatic example, for how you should think
about the vacuum and how you should think about emission
of photons into the vacuum.
And these are the vacuum Rabi oscillation
described by the Jaynes-Cummings model.
So the situation which I have in mind,
or which you should have in mind,
is an idealized situation, but it
has been realized experimentally.
And some of those idealized experiments
were recognized with the Nobel Prize
research of Haroche and Dave Wineland.
So the situation is we have an atom,
but it only talks to one mode of the electromagnetic field,
and we make sure that the atom only
talks to one mode of the electromagnetic field
not by eliminating other modes; they exist.
I mean, an atom can emit upwards and downwards.
But we surround it with a cavity which has such a small mode
volume, it has such a small volume, that the single photon
Rabi frequency is huge, and therefore
the emission into this one single mode
dominates over the emission into all other modes.
So this is a condition that the single photon Rabi frequency
has to be larger than gamma.
And, of course, we also have to make sure
that the system is idealized so the loss of photons because
of losses in the mirror, or finite reflectivity
in the mirror, also has to be smaller.
So that means for several Rabi periods
we have a system which has only two parts, a two-level atom
and one single mode of the cavity.
So that's the system we have in mind,
and we discussed the Hamiltonian.
We saw that the Hilbert space of the atom
is excited in ground state, the Hilbert space of the photons
is spent by the [? flux ?] states,
but what happens is-- so there's an infinite number of states,
because of the infinite number of states of the photon field--
but what happens is the Hamiltonian couples only
an excited state with n photons to a ground state with n
plus 1 photons.
So the whole Hilbert space is segmented now
into just pairs of states labeled by the index n.
So after so much work, we are back to a two-level system.
And here is our two-level Hamiltonian.
And, well, a two-level system does
oscillations between the two levels.
Rabi oscillations, no surprise.
And this is what I want to discuss now.
But the new feature is that these are really, well,
these are now really two levels.
Each of them is the combined state
of the atom and the quantized radiation field.
So now we have included in our two-level description
the quantum state of the electromagnetic field.
So first you should realize that this Hamiltonian is absolutely
identical to spn 1/2 in magnetic fields.
And you can recognize by [? comparing ?]
this Hamiltonian, this matrix, to the matrices
we discussed for spn 1/2 in the magnetic field,
that this corresponds to the situation
where this spn 1/2 had a transverse field
in the x direction which caused a precession from spn
up to spn down.
And this x component of the field
corresponds now to the single photon Rabi frequency times
n plus 1.
That's the off diagonal matrix element in this matrix.
The thing which we have to discuss,
and I will focus later, is that it depends on n.
So for each pairs of state labeled by n, the photon
number, we have a different off diagonal matrix element.
But let's discuss first the most important and simplest case.
Let's assume we are on resonance,
and we want to assume that we have a vacuum.
Then our Hamiltonian is simply this.
And when we prepare the system in an initial state, which
is an excited state with no photon in the vacuum,
then we'll have oscillations to the ground
state with one photon.
These oscillations are exactly the oscillations
we saw on the spn 1/2 system.
We can just map the solution.
I'm not really writing anything here.
So what we obtain is the famous vacuum Rabi oscillations.
where the probability to be in the excited state
oscillates with the single photon Rabi frequency omega 1.
I think there's a little bit of an ambiguity in language.
Is it the single photon Rabi frequency?
or is it the vacuum Rabi frequency?
Because there's always the question about plus
minus one photon because we start in the excited
state without photon so you want to say
it's a vacuum Rabi frequency.
But then you have the ground state with one photon,
and this photon is reabsorbed and then
you may want to call it the one photon Rabi frequency.
So I leave it to you, but it's called vacuum Rabi oscillation
and this Rabi frequency is usually referred to as the one
photon Rabi frequency because we obtained the Rabi
frequency by calculating the electric field
of a single photon.
So the Rabi oscillations which we are observing now
correspond to the periodic spontaneous emission
and re-absorption of the same photon.
There's only one photon which is spontaneously
emitted and reabsorbed in a completely reversible
coherent way, and the time evolution is unitary.
So it's a periodic spontaneous emission
and re-absorption of the same photon.
This has been experimentally observed.
Actually, let me back up.
Experiments are done in the microwave regime.
The leading groups are, well, in the older days,
Dan Kleppner, Herbert [? Weidner, ?]
and Serge Haroche.
And this involves Rydberg atoms.
Rydberg atoms in superconducting high q cavities.
And those Rydberg atoms, because things
scale with n and n squared, the principal quantum number,
have a fantastically strong coupling
to the electromagnetic field.
And there is a homework assignment
on Rydberg atoms in such cavities.
The other example is in the optical domain.
And this really involves the D line of alkali atoms.
You drive them on the D line.
rubidium and caesium are often used,
and the work is enabled by the development
of so-called supermirrors which have an extremely
high reflectivity, and you can realize an excellent q factor.
And the leaders in this field are Jeff Kimble and Gerhard
Rempe at the Max Planck Institute.
So let me just discuss an example
taken from the optical domain.
So the generic situation is that you
have two mirrors which define a single mode cavity.
Usually, you have a stream of atoms.
Traditionally in atomic beams, then
in some experiments in slowed atomic beams, more recently
in atoms which are falling out off the mode,
and only very recently single atoms with the help
of other laser beams that are trapped inside the cavity.
So they are streamed in such a way
that only one or a few atoms are in the mode volume interact
with a single mode of the cavity at a given time.
And then you want to figure out what is now happening,
and you have the probe laser, you send it through the cavity,
and then you record the transmission with a photodiode.
Yanosh?
AUDIENCE: What is the mirror made of for the [INAUDIBLE]?
PROFESSOR: The mirror is made of a glass substrate,
but then you would [INAUDIBLE] the coating.
And the mastering is really to put coatings
on which are very pure, but then also I
think using ion sputtering, you make
sure that the coating is extremely smooth
and does not have any surface irregularities which
would scatter a tiny fraction of the light.
I know there are some people in Ike's group and [INAUDIBLE]
group who work with high q mirrors.
What is a typical example for the reflectivity?
Or the q factor you can reach?
AUDIENCE: 5 [INAUDIBLE] and a finesse of, maybe, 500,000.
And they're called superpolished mirrors.
PROFESSOR: So finesse of about a million, and that
means the mirrors have 99.9999% reflectivity.
And the superpolishing, I think, that was the last step.
People had controlled the materials,
but then they found ways to make a super polish
and avoid these even one part per million
scattering by surface roughness.
OK.
So if you do that experiment, what would you expect?
Well, it's a [INAUDIBLE] experiment
so if you would scan the probe laser,
and there is nothing in it, what you
would expect is you would just expect a transmission
peak at the cavity resonance.
And if you tune much further, you
get the next peak at the free spectral range.
Let me just indicate that.
So this is a case for 0 atoms in the cavity.
If you put 1 photon in the cavity,
you no longer-- sorry, 1 atom in the cavity,
you're no longer probing a cavity,
you're really probing a system, which
is no longer the cavity by itself.
It's an atom-photon system.
It's a couple system.
And we know it's described by our two-by-two Hamiltonian,
and this Hamiltonian has two solutions.
And the two solutions are split by the one photon Rabi
frequency.
So the two eigenvalues of our Hamiltonian
are at plus minus omega 1 photon.
So therefore, for n equals 1, we have a situation
that we have two peaks split by the single photon Rabi
frequency.
Of course, I have assumed that great care has
been spent to make sure that the cavity resonance is right
where the atomic resonance is.
So this is now for 1.
If you have 10 atoms, remember the two-by-two Hamiltonian
looks the same, but it has the square root n plus 1 factor.
So neglecting the 1 roughly when we have 10 atoms in the cavity,
it's square root 10 larger Rabi frequency.
And therefore, we would expect that we
have now a splitting of the two modes, which
is square root 10 plus 1 larger.
Actually, I didn't-- sorry, I have to collect myself now.
I showed you that the Rabi frequency
scales with square root n plus 1 in the photon field.
But you should realize that everything
is [? isometric ?] between photons and atoms.
It's the complete coupling between photons and atoms.
And if you would now look-- but I don't want to do it now--
if you would now look what happens if several atoms are
present in the mode volume, you would also
get a scaling which is n plus 1 in the atom number,
because the atom coupled coherently.
It is actually an effect of super-radiance,
which we'll discussed later.
So just take my word.
You have the same scaling with the atom number.
But I have to give you my word now, because in the experiment
this is what people varied.
AUDIENCE: If they had varied power or number of photons
instead, like, we couldn't have drawn these same diagrams,
right?
Because then the top part of the Lorentzian changes.
If you're changing the photon number then Lorentzians change.
PROFESSOR: Say again?
AUDIENCE: So, like, right now, yes, we
are varying the number of atoms so we
can talk about the splitting.
But if we were varying the power or the number of photons
instead, then each of the Lorentzians,
their height would change.
How would you draw this observation
if you were changing the photon numbers?
PROFESSOR: You know, I don't want to go into line shape.
I would probably be a Lorentzian.
I mean, all I want to discuss here
is that we have a two-by-two Hamiltonian, which is split.
And if we have one atom and one photon,
it is split by the single photon Rabi frequency.
If we have one atom and 10 photons,
the atom can of course absorb and emit only one.
As I derived on the previous page,
we would have now a Rabi splitting, which is square root
n plus 1, n being the number of photons.
But if you would start in an empty cavity
with 10 atoms in the excited state,
because all the atoms are identical,
they would spontaneously emit together,
and then you would have 10 atoms in the ground state,
and then you would have 10 photons.
And so maybe this helps you.
If you start with 10 atoms in the excited state,
they do everything together.
If you have 10 atoms in the ground state with 10 photons,
and now you have 10 photons and it's clear
the 10 photons lead to a Rabi frequency,
which is proportional to the square root of 10
or to the square root of 11.
So therefore, what you will observe
is you will now observe a splitting of the single mode
of the cavity which goes by the square root of n plus 1.
I don't want to discuss the line shape and the [? strings, ?]
I just want to sort of discuss, in a way, the eigenvalues
of the Hamiltonian, and the eigenvalues
are the positions of the transmission
peaks with a cavity.
And that has been observed.
I mentioned the two leaders of the field
are Gerhard Rempe and Jeff Kimble.
Well, Gerhard Rempe, he did his Ph.D.
In the same group at the same time
as I did so I know him very well.
Then he went and did post doc work with Jeff Kimble,
and now is the Director of the Max Planck Institute.
He has the world leading group in cavity QED.
But this is sort of here the two leaders have
a joint paper, which is the first observation of the vacuum
Rabi splitting in an optical cavity.
Of course, you can easily observe it
if you have a strong atomic bean with many atoms,
because then you have a good signal.
And secondly, the splitting is large and easily resolved.
So what they managed to do is they managed to throttle down
the atomic beam that fewer and fewer atoms at the given time
were in the cavity.
And eventually they came down to the limit of one atom.
That was an historic experiment.
Of course, it's not perfect in the sense
that you do not see the deep cut between the two peaks simply
because, when on average you have one atom in the cavity,
sometimes you have to atom in the cavity,
and then you have a peak in the middle.
So those experiments in those days
were done only with average atom numbers
and not with trapped atoms where you know for sure there's
exactly one atom in the cavity.
OK.
So I don't show you an experiment,
but let me just state that this sort of single photon Rabi
flopping has been observed.
You start with the cavity in the vacuum field,
and you sort of see this oscillation
to the ground state with one photon.
But what I want to discuss now is the situation
that we are not starting with an empty cavity.
We are starting with a coherent field.
You can also start with a thermal field
so there are different experiments you can do.
What would we expect now?
So now the initial photon state is not
the vacuum state, but the thermal state.
If you have a microwave cavity and you heat it
up a little bit, you have to cool it down to below 1 Kelvin.
People use either helium-free chrio stats or dilution
refrigerators, but if you warm it up a little bit,
you have a few microwave photons in the cavity.
Or, that's even more controlled, you can make the cavity ice
cold, but then you inject a few photons from your synthesizer
into it-- from your microwave generator--
and then you have a weak coherent field.
But a thermal state or a coherent state.
So what we then have is, OK, we would expect now
a Rabi oscillation; however, the frequency for the Rabi flopping
is now proportional to n plus 1.
And we have our photon field in the superposition
of flux states.
So the fact that we have a superposition state implies now
that the Rabi oscillations have a different oscillation
frequency for the different [? tablets ?] of states labeled
by n.
And that leads to a dephasing.
So that would mean that if you would look at the probability
to be in the excited state-- just think about it.
You have a wave function where the atom starts in the excited
state, and the photon field is in a superposition.
So now you have a two-component wave function
which has different parts, and each part
has a specific Rabi frequency.
So you would have oscillations.
Let's say there is a certain probability
that the cavity is in the vacuum,
and then that means that there is a component which oscillates
at the vacuum Rabi oscillation frequency.
But if you have a component in your coherent
or thermal state which has two photons in it,
then you have Rabi oscillations which are faster.
And now you have to superimpose them all.
And if you all superimpose them, and you
find that very soon there is a damping and maybe
a little bit of vigor, but you see
at damping of the population in the excited state.
Q [? 2 ?] dephasing.
I'm just hesitating.
I think I took this plot out of my notes,
but I would expect now the damping should actually
lead to a probability to be in the excited state of 1/2.
So let me just try to correct that.
So there is a little bit of oscillation,
but then there is sort of a damping.
And eventually, if you have only a small number
of photon states, then there will
be a time where you have sort of at least
a partial commensurability.
You have maybe five frequencies.
You know, square root 5, square root 4, square root 3,
square root 2.
But then there is sort of a time where
all these different frequencies have done an integer
number of oscillations each, and then you
get what is called a revival.
And if you go to a large photon number,
you have square root 100, square root 99, square root 88,
the revival will happen at a later and later time
and eventually at infinite times if you use a microscopic field.
But for small coherent states, or thermal states,
which only involve a few photons,
you will get a revival phenomenon.
And this has indeed been observed.
This was actually the PhD thesis of Gerhard Rempe,
and it shows the probability in the excited state.
They had previously observed the Rabi oscillations
at early times, but now the experiment
had to be adjusted, I think by using slower atoms,
to observe the longer time.
And here, well, 1987 for the first time revivals
have been seen.
Let me dwell on that, or first are there
any questions about what happens now?
Atoms in the cavity to Rabi oscillations?
And if the photon field is a superposition
of only a few states due to this pseudo commensurability,
you find times where you have revivals.
I just worked out something this morning
which I think is nice, because it will highlight
how you should think about spontaneous emission.
So let me discuss.
It doesn't really matter, but I want
to give you a specific example that we have a coherent state.
A lot of you know what a coherent photon state is.
For those who don't, it doesn't really
matter for what I want to explain, and recover
that in [? 8.4.22. ?]
But if you have a laser or if you have a microwave generator,
what comes out is a field which has a normalized amplitude
of alpha, but your field is in a superposition
state or [? flux ?] states.
With these prefactors, I just wanted to give you an example.
What I really just mean is that we
have a coherent superposition of number states.
We have prepared that.
So now we have one atom in the excited state,
it enters the cavity which has been prepared
with the short pulse for a laser or microwave
synthesizer in these state alpha.
And now we want to discuss-- so this is at t equals 0--
and now I want to discuss what happens as a function of time.
Well, we know that if you have one tablet,
n, we have Rabi oscillations between the atoms
in the excited state, and we have n photons.
Or it has emitted the photon, and then we
have n plus 1 photon and the cavity.
But now, we have a superposition state,
and we have amplitudes an.
So I mean, that's what we get.
And this includes everything.
It includes everything a two-level atom
does in a single mode of a cavity.
And this is spontaneous emission, stimulated emission,
and reabsorption.
But I want to use that now to discuss
with you the misconceptions about spontaneous emission.
Colin?
AUDIENCE: We're talking about just spontaneous emissions
into the cavity?
PROFESSOR: OK.
I've singled out a single mode.
But what happens is-- and you're just two minutes, 30 seconds,
ahead of me-- that we had discussed vacuum Rabi
oscillations or Rabi oscillations when
we have n photons in the cavity.
This was our two-level system, our Hamiltonian, and all
we get is Rabi oscillations with the Rabi frequency omega n.
And now we have to sort of do averaging.
I'm now discussing that we have a coherent superposition
of number states.
Let's say, a pulse of coherent radiation, a coherent state,
and this is what we get.
You can now, if you want, put in a [? zillion ?] of other modes,
have another sum over all the other modes you want.
So I'm just doing the first step in discussing
with you what will happen, but adding more and more
modes will actually not change the structure of the answer
and will be, of course, quantitatively a mess
but conceptually not more complicated.
So I want you to really look at that
and realize where is the spontaneity
of spontaneous emission.
Where do you see any form of randomness associated
with spontaneous emission in this expression?
I don't see it.
This is a wave function, and this time evolution is unitary.
Everything is deterministic, and depending now
how we choose our coefficient, there is even a revival.
It's not dissipative that a photon is spontaneously
emitted, and it's done.
We saw in the single photon Rabi oscillation
it can be reabsorbed, we saw in a slightly more complicated
situation that there are at least partial revivals,
and it now depends how long we wait
whether revivals will take place or whether they
will be complete revivals or partial revivals.
But we don't need a revival in a coherent evolution,
the coherent evolution can just go to a complicated wave
function and it's still a single coherent wave function
fully deterministically obtained form the Hamilton operator.
Sometimes it pops into our eyes through a reversible
oscillation or through revival, but we don't need that.
So let me write it down but then explain you something.
So it's unitary.
There is no spontaneity at all.
However, eventually we want to retrieve the classical limit.
So if we would go to this situation
that the average photon number is much smaller than 1,
then the fluctuation in the photon field
around the mean number are very small.
For the coherent state the fluctuations are square root n.
And then, we retrieve the limit of semiclassical Rabi
flopping with the Rabi frequency omega
r, which is-- I'm not consistent here with lower and uppercase
[INAUDIBLE].
So it's uppercase or lowercase omega n,
and this is square root n times the single photon Rabi
frequency.
And, of course, for a large number of photons,
we can always make the approximation
that we do not have to distinguish
between n and n plus 1.
So this is the ultimate limit if we
would work in the limit of large photon numbers.
So the way how you should look at it is the following.
This system undergoes a time evolution
to a state which is rather complicated.
But if you make the number n large,
this becomes approximately a state
where you have simply-- you know what
the rate of the semiclassical limit?
In the semiclassical limit, we have a constant laser beam
with constant electric field amplitude, e,
and then we have driven Rabi oscillations
between ground and excited state.
So therefore, I don't want to show you mathematically,
but in the limit of large n, you can approximate
this complicated entangled wave function
by the product of Rabi oscillations
between ground and excited state times a coherent photon field.
And the correction between what I just said
and this complicated wave function is like 1 over n,
because it's sort of a 1 over n approximation
where we have neglected terms which
the relative importance of them is 1 over n.
So therefore, there are people who will say
and who will tell you when we have
an interaction of an atom with a coherent state,
and let's just think in the number of n being large,
that n times out of n plus 1, we have a coherent state.
The atom does Rabi oscillation and what
it does is it just emits photon into the coherent field
and takes it back, like in semiclassical physics.
But in one case out of n cases, or the rate
1 over n of the rate of the wave function is sort of fuzzy.
It's not a coherent state; it's something
much more complicated.
And if you do not keep track of this complicated nature
of the wave function and just do some simple measurement by,
let's say, just measuring the phase
of the electromagnetic wave by projecting
onto a coherent state, then you would
find that with the probability of n
the system was just staying in a coherent state.
And with a probability which is one part out of n,
something else has happened, and your detector cannot capture
the entanglement of that state.
And this last part is what some people
associate with spontaneous emission.
I don't know.
That's my view where the spontaneity in this process is.
It's not a spontaneity in the time evolution.
It's more a spontaneity if you do not
care to detect this complexity, but map it back
to a coherent state.
And then with a precision which is 1 over n,
you retrieve the semiclassical limit,
but the difference between the semiclassical limit
and the entangled wave function, this is what some people say
is spontaneous because it's not captured by a single picture.
I'm actually expecting some people to disagree with me,
but this is sort of my view, what
I'm sort of learning from the simple examples I've
given to you.
Since Ike is an expert on it, maybe, Ike,
can I ask you the question is there actually a simple way
to show that if I go to a large n limit
that you can sort of really show that n parts out of n plus 1
is really described fully by the semiclassical limit
and there is only a 1 over n fraction
where we have to look at the more complicated wave function?
AUDIENCE: I don't think there's a simple way to do it,
but one can look at the equivalence
of a [? and a factor ?] state, and they're only
different by one photon number.
PROFESSOR: It's sort of clear.
I mean, everything is if you approximate n by n plus 1.
If you don't care about the small difference,
everything falls into place and is simple.
But I was just wondering if one could
show sort of in a more direct or more intuitive or maybe
more quantitatively what is really
the extra part beyond stimulated emission
absorption into the coherent state.
So what sort of really the nature
of what people call the spontaneously emitted photon?
AUDIENCE: I think that I don't the question, because I still
argue that it's purely [? unitary ?] evolution even--
PROFESSOR: OK.
AUDIENCE: For that system, and therefore, it's
purely [INAUDIBLE] and nothing spontaneous
is happening at all.
PROFESSOR: OK.
All right.
Good.
OK.
Fine.
What is next?
I think this finishes our discussion on vacuum Rabi
oscillations and revivals.
I have now two topics in light atom
interaction which you may not find in many textbooks,
but it's my experience that they're really relevant.
One is very conceptual.
It's about the rotating wave approximation.
And the other one is just the opposite, very technical.
It's not really a new concept, but this
is about saturation intensities and cross-section
of an atom for absorption.
The last things, cross-section for absorption and saturation
intensity, that's what you need when
you talk to atoms in the laboratory.
These are the quantities in which we think intuitively
about light atom interaction.
So it's not involving any concept.
I want to spend 20 minutes in introducing for you saturation,
saturation parameter, cross-section,
what's different between monochromatic light
and broadband light.
But before I do that, I have a few minutes
on the rotating wave approximation.
So let's call it rotating wave approximation revisited.
Again, rotating wave approximation.
And what I want to discuss can be discussed in
the fully quantized picture, but also
in the semiclassical picture.
In the fully quantized picture, just a reminder,
what we discussed earlier was that when
we have the atomic raising and lowering
operator and the photonic raising and blowing operator,
we got four terms.
And two of the terms are co-rotating,
two are counter-rotating.
But I can get exactly the same number
of four terms in this semiclassical picture,
and I want you to see both.
But in the quantized picture, it's actually easier,
because when you see a and [? a dega, ?] you
know immediately one is absorption one is emission.
So therefore let me explain to you
what I want to tell you about the rotating wave
approximation using the semiclassical picture,
because then you immediately know
how to apply it to the quantized picture.
So what I want to bring in the here in addition to what we
have discussed about light atom interaction,
we had sort of a dipole Hamiltonian,
is the fact that we have circular polarized light,
left-handed and right-handed light,
and I want to sort of use that and combine it
with angular momentum selection roles, which as you remember
we discussed after our discussion of dipole,
quadrupole, and magnetic dipole positions.
So I put now all those parts together
and revisit the rotating wave approximation.
So what I hope for is it tells you a little bit
how selection roles, angular momentum,
circular polarization, and semiclassical field
which rotate in one direction how they are all connected.
So I have to set up the situation
by saying that we use as a quantization axis the direction
k, which is either the direction of the propagation
of the light beam or, in general,
it's orthogonal to the polarization
of the electric and magnetic field.
And I can talk about an electric field driving
an electric dipole transition.
I can talk about a magnetic field
driving a magnetic transition.
It doesn't really matter.
I will use [? Bsc ?] amplitude, but you can also
immediately think electric dipole,
and this field is linearly polarized.
But I want to immediately decompose this field
into right-handed and left-handed field.
Or a linearly polarized field can
be regarded as a superposition of a field which circulates
this way plus one which circulates the other way.
And ultimately, the message we will see
is that if you have linearly polarized light,
we always get counter-rotating term, we always
have a [INAUDIBLE] shift and such.
But if you use rotating fields or circularly polarized light,
selection roles may actually lead to the result
that there is no counter-rotating term at all.
So this is eventually what I'm aiming for,
and this will be the final point of the discussion.
So the field which rotates in the right-handed direction
where the rotating field is a superposition of x and y,
or i and j.
And one the rotates has a cosine omega t
and one has sine omega t.
I don't need to write down the left-handed part,
because there's just a minus sign.
Or this will actually become very handy.
I do all the discussion for the right-handed part,
but I can always obtain the expression
for the left-handed part by replacing omega by minus omega.
Which will mean that some emission process
by the right-handed part will be an absorption
process by the left-hand part.
Be we'll see.
You can change angular momentum by plus 1
by absorbing a right-handed photon or-- you'll see.
We'll get there.
So anyway, those signs will become important.
Let me now take the above expression
for the right-handed part and replace cosine omega t
and sine omega t by e to the i omega t.
So this was the i component, this was the j component.
We divide by 2.
Just to avoid confusion, I want to emphasize
I've started with a real field.
So I'm not using, as you often do in e and m complex field
and the real fields are the real part,
I have not started out by adding,
you know, imaginary parts to the field.
I've started out with a linearly polarized field
in the x direction cosine omega t,
and I've decomposed it into two real fields.
One is right-handed, one is left-handed.
Complex numbers only come because I
want to use a complex exponential to replace
cosine omega t and sine omega t.
We are almost done with the decomposition of the field.
I just wanted to-- we have now four terms,
and I want to [? recoup ?] them.
i minus imaginary unit j.
i plus imaginary unit j.
This is e to the i omega t.
And this is e to the minus i omega t.
So what have we done?
Well, we've just started with linearly repolarized light,
and I've rewritten the expression twice,
and now we are looking only at one of the circular components,
and in the end what we have is four terms.
Well, that's also what we had in the fully quantized
Hamiltonian, and we now want to identify what those four terms.
Two will be co-rotating, two will be counter-rotating,
but it's very helpful to analyze those terms.
But there are two things we have to look at now.
One is we have an e to the i omega t.
And, well, it's probably a sign convention,
but trust me, if you put that in the Schrodinger equation,
it mean that you increase the energy of the atom
if you drive it with e to the i omega t.
You take it from a ground state to an excited state, which
differs in frequency by omega, and therefore this
means you have increased the energy of the system,
and this corresponds to absorption.
Whereas this one here means we take an atom
from an excited state to a ground state,
and this is the situation of stimulated emission.
Remember, in selection roles we take our field
and we multiply it with a dipole moment,
electric, magnetic dipole moment whatever.
But now we want to use also the spherical tensor decomposition
of those dipole moments.
It's a complicated word, but what
it means is those terms are dotted with the dipole moment,
and if you do it now component-by-component,
we retrieve selection roles because this peaks out the--
let me just write it down-- the x plus y
tensor component of the matrix element.
And this corresponds to delta m equals plus 1.
We change the angular momentum by one unit,
and of course this term is then delta m equals minus 1.
So we have done the work.
What I want to do now is just map those terms into
and energy level diagram.
I like sort of pictorial representations,
and each term becomes now a graphical [INAUDIBLE].
So let us assume we have a system, hydrogen is to p state.
But let's say generally we go from a j equals 0 to j
equals 1 state, which has three components.
Now, I have set it up in such a way that-- oops,
we need a little bit extra space.
I've set it up in such a way that the states here, this
is m equals 0, this is m equals plus 1,
and this is m equals minus 1.
So therefore-- let me just use color
coding now-- this one here is delta m equals plus 1
so this one always moves to the right.
It changes angular momentum by 1 so it can always
move to the right, whereas the other one, delta m
equals minus 1, moves to the bank.
Absorption is e to the i omega t, always moves up.
And stimulated emission moves down.
So with that what happens is this term here
transfers one unit of angular momentum and energy.
So that would mean this term goes up here.
It could go up here, if [? there were a ?] state.
The other term-- let me use a green color-- is driving
the process in the opposite direction.
But now we have to also consider that you can go down here,
and you can go down to a virtual state.
A virtual state is just something
which has the same wave function as a state,
it just has an e to the i omega t, which
is not-- it's a driven system.
You drive it.
You [INAUDIBLE] a state.
You [INAUDIBLE] a state at the drive frequency,
and it just means, in this case, this state
has an oscillation e to the i omega t, which is very, very
different from what a state which is populated would have,
and this is what we call a virtual state.
So in other words, what is possible is
we have our three states, plus minus 1 and 0,
but this is the spatial wave function
including angular momentum.
But we can now drive it by plus omega and minus omega,
and therefore we can have it as virtual states
pretty much at any energy we want.
But this process here is not possible,
because this would require to go to a state which has m
equals 2, which does not exist.
So now what I've shown here is if we
would stock in the m equals 0 state,
I've shown you the four terms, two are co-rotating
and two are counter-rotating.
If you neglect this virtual state which
has a detuning of about 2 omega, or 2 resonance
frequency of the atom, this is the rotating wave
approximation.
One term is responsible for absorption;
the other term is responsible for a stimulated emission.
But if I don't make the rotating wave approximation,
I have those two extra terms.
So this is only the right-handed light,
and I want to sort of play a little bit with this concept.
If I would take the left-handed light,
I would add sort of four more arrows.
Two more here and two more here.
But let's just keep the situation
as simple as possible.
But I really sort of like that you
write down right-handed, left-handed side,
decompose it into its components,
and each component is now in this diagram connected
to an arrow where one direction is angular momentum,
the other one is energy.
So let me now talk about other energy diagrams.
And this will lead to the answer.
Well, can we create a situation where
we have only two terms, which would be the simplest
two-level system, can be directly realized
without any rotating wave approximation
a two-level system?
So if we had two levels, which have only m
equals 0 and m equals 1.
So this would be the situation I just
discussed with those two levels.
So the only way how I can fit in this arrow is this one,
and the diagonally downward arrow is that.
So in this case, rotating wave approximation
is not an approximation, it is exact.
But some purists will actually say, hey,
you can never realize that when you have an n
equals plus 1 state.
Then you always have an m equals minus 1 state.
And then you have a virtual state down there,
and then you get two more terms, which
are the counter-rotating terms which I just showed above.
So whole I would say if you have a neutron star which
makes an infinitely high magnetic field,
you can have a huge [INAUDIBLE] splitting between m
equals plus 1 and m equals minus 1
and completely move one of the angular momentum
states out of the picture.
But, of course, in the rotating wave approximation
we are neglecting off resonant terms at 2 omega, omega
being an electronic excitation energy,
so I'm really talking about Zeeman shifts
here to eliminate the other state which
may be comparable to electronic energies.
So in principle, I can say this is my Hilbert space,
and in this Hilbert space no rotating wave approximations
is needed.
But it's maybe an artificial Hilbert space.
When I had a discussion with other people,
we came up with the possibility of some forbidden transition.
If you go from a doublet s to a doublet s state
so all you have is a spin system which
has 1/2 angular momentum plus 1/2 minus 1/2.
And then you realize that the only way how
you can fit in the orange arrow is
in this way, and the green arrow in this way.
So here you would have a situation
where the rotating wave approximation is exact.
But, of course, it's not an electric dipole transition;
it's some sort of weaker conversation, which
may be forbidden.
I need two more minutes.
I have discussed the case where we have quantized
along a direction, I called it the k direction,
and the polarization of the electromagnetic field
was [? i and j ?] was perpendicular to it.
So let me now discuss a case where we quantize along
the polarization of the electromagnetic field,
and you remember from our discussion on selection roles
that this is pi light.
So in this case, our magnetic or electric field
is polarized along the i direction,
and the real cosine omega t gets decomposed into e to the plus
e to the minus i omega t.
And we know already one term is absorption one is emission.
And now, if I take my j equals 0 to j equals 1 system,
pi light has a selection role of delta m equals 0.
So now I have an arrow, which I want to be orange,
which goes up.
And a green arrow-- this is a great program.
The only thing is you have to be very careful when
you change color and press carefully.
That's why sometimes the colors are not doing what I want.
But here is green.
But now, of course, with linearly polarized light
we can always go down to a virtual state.
We have now four terms.
Two are rotating, two are counter-rotating.
So therefore the quick conclusion
of the last ten minutes is that there is the possibility
that counter-rotating terms can be
0 for sigma plus sigma minus light due to angular momentum
selection roles.
But what we have also learned is if you have the m plus 1 state
in there's an m minus state, if you have circularly polarized
light and we drive a transition between two m states,
the counter-rotating term does not
come from the same set of two states, m equals 1.
It involves m equals minus 1.
So it's the other state which is maybe degenerate or only
slightly [? split ?] by a magnetic field which
is responsible for the counter-rotating terms.
Anyway we have talked so much about rotating wave
approximation and those terms, I just
wanted to show you how it is modified
if you use degeneracy p states and angular momentum.
Any question?
OK.