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PROFESSOR: Good afternoon.
So in the last week of this semester,
we will be finishing up the chapter on coherence.
What we want to continue to explore today
is the presence of a dark state.
If you have a three level system, and that's
what we went through last week, we
will always find one state which is dark,
which means we fully illuminate the atom with laser light,
but there is one state which is not
coupled to the excited state which does not scatter light,
and therefore is dark.
I showed you that for very general conditions, if you have
two laser beams exciting the two states,
there will always be a coherent superposition of the two
states, which is the dark state.
The dark state is the novel feature of three level systems,
and I want to show you different perspectives of it.
We started out by talking about just the existence
of the dark state.
Then we talked about dark state transfer.
You can regard, for detuned light, the dark state
as the lowest energy state of the system.
And then an adiabatic theorem tells you
that you can keep particles in the dark state
even if you change the laser parameters
and change what state is now the dark state.
This is the basis of coherent population transfer,
or the famous STIRAP method.
There is another aspect of the dark state
which gives us a possibility of lasing without inversion.
As I reminded you, lasing has a threshold,
which is inversion that you need more atoms in the excited state
than in the ground state because the atoms in the ground state
absorb the laser light and you want a net gain.
However, if you have a dark state,
you have a situation where the atoms do not absorb the laser
light, and therefore, the conditions for net gain
have changed.
And indeed, lasing without inversion becomes possible.
I started to explain to you the way
how lasing without inversion could come about.
It relies on the fact that in a three level system,
absorption destructively interferes
but stimulated emission does not destructively interfere.
And therefore, you can have lasing
without inversion in such three level systems.
I showed you one possible realization,
and this is about hydrogen in a DC electric field.
If we mix the 2S and 2P states with an electric field,
we have this three level structure.
And for a laser tuned right in the middle between the two
states, the two amplitudes for excitation cancel exactly,
and therefore, you have a zero absorption feature.
However, if you put now a little bit of population into,
let's say, the upper state, this upper state
in the wings of its profile has still
gained for stimulated emission, and what we get
is lasing without inversion.
Now you say, OK, but that's a hydrogen atom.
Which atom is really degenerate between two levels,
and you can split regeneracy with a static electric field.
Well, you're already an expert at this point in the course.
If the 2S and 2P state are widely separated,
well, you add a photon, and the photon,
which is in resonance with the 2S and 2P state,
creates, in the dressed atom picture, degeneracy.
Maybe I should have shown the P state higher.
The 2S state with one more photon and the 2P
state with one photon less have the same energy,
and then you create exactly this situation.
So therefore, the way you can realize
that in atoms other than hydrogen
is use an AC electric field to mix S and P states.
And I'll show you in five minutes a little bit more
in detail what I mean by that.
There is a trivial realization which
I want to mention for lasing without inversion,
and this would be if you have a three level
system with an excited state, two levels, g
and f in the ground state.
You may have an inversion for the e to g transition.
And therefore, you can get lasing
because the population in this state f is not coupled.
This is pretty trivial, but the more subtle part, of course,
is that we can realize it using a driven system using a control
laser by creating the same situation with the bright
and the dark state, and population in the dark state
is hidden from the light and does not absorb light.
Let me just indicate that in both those states,
we would have no absorption, dark state.
Those examples may raise the question whether whenever
you have lasing without inversion,
if you can find a basis where you have inversion again
between the two levels which are relevant
and the extra population is just hidden.
I want to make two comments about it.
This question is sometimes discussed
in the literature, sometimes in a semi-controversial way.
There are two comments about it.
One is when you start dressing up
your system with laser beams, you have strong control lasers.
You have two lasers, omega 1, omega 2.
One is often a strong control laser
and the other one is the weak laser
where we want to have lasing.
You have actually a time dependent system driven
by time dependent fields, and once you have a time dependent
system, it's no longer clear what the eigenstates are,
what the populations are, and what are the coherences.
There is no longer a unique way to distinguish
what are the eigenstates because every state is, so to speak,
time dependent almost by definition.
On the other hand, I think the example
I gave you with atomic hydrogen where you just
a little bit of mixing with an electric field
is an example where you genuinely
have less population in the excited state
than you have in the ground state,
and at least the equation tells you, even without inversion,
you have a net gain.
So my own understanding of that situation
is that in many situations, you can actually reduce it
to a simple picture where you have simply hidden population
in a dark state, but without any sort of unnatural definitions,
you may not find that in some other systems.
Questions about lasing without inversion?
Nancy?
STUDENT: Is lasing without inversion
important in lab, as opposed to lasing with inversion,
or is it more about teaching us [INAUDIBLE]??
PROFESSOR: Well, lasing without inversion
has definitely been touted as a way
to get lasing deeper in the UV, to get
lasing for very blue transitions because when
you want to create inversion--
this actually has been the problem in creating
x-ray lasers in atomic systems.
If you have larger and larger energy separation,
spontaneous emission scales with omega [? cube. ?]
And so therefore, it becomes harder and harder
to fulfill the ordinary gain equation.
STUDENT: So in those cases, even these last two
methods [INAUDIBLE] because [INAUDIBLE]??
PROFESSOR: Well, lasing without inversion
alleviates the requirement to build a laser,
and so people have discussed that where
it's really hard to create a laser in the conventional way,
deep, deep in the blue of the UV or in the x-ray regime,
that lasing without inversion may help.
I'm not aware that any practical development
has emerged from that because there is a price to be paid,
and that is usually in the form of coherence.
You need a certain degree of coherence
in your system to be able to do that.
It's an idea which is powerful, but as far as I know,
there was no killer application of it.
The importance of dark states is definitely
in slow light, manipulation, quantum computation,
and concepts of storing light, and this is
what we want to discuss next.
Actually not next, but after next.
What I first want to discuss is another aspect
of the dark state, and this goes by the name
EIT, electromagnetically induced transparency.
I can introduce this topic by a question to Radio Yerevan,
is it possible to send a laser beam through a brick wall?
And the answer of Radio Yerevan, if you know the joke,
is always, in principle yes, but you need another very powerful
laser.
So in an incoherent way, of course,
a very powerful laser can drill a hole into the wall
and then the next laser can go through the wall
without absorption, but you can be smarter.
If the very powerful laser through coherence
puts all the atoms in the brick in the beam pass of the laser
into a coherent superposition state,
and then they become a dark state,
then your laser can go through a brick wall.
So can a laser beam penetrate an optically thick medium?
And the answer is yes, with the help of another laser.
Original ideas along those lines were formulated by Steve
Harris, who has really pioneered this field,
and he first considered special auto-ionizing excited states
which couple to--
there were two pathways of coupling into the continuum,
but later work has shown that it can
be realized in a lambda system.
Let me talk about this conceptionally simpler
realization in a three level lambda system.
Let's assume we have again our normal three level system
gf and an excited state which has a width, gamma.
And we want to send a probe laser through a dense medium,
and it would be completely absorbed by the resonance
to the excited state.
But now we can have a strong coupling laser
with a rapid frequency, omega c, and so if we drive
the system very strongly, we can create a situation where
the coupling laser does, if it's strong enough, complete
mixing between the excited state and the ground state,
and that means if you have two levels which are completely
mixed, they are split by the energy or the frequency
of the coupling.
So in other words, what we obtain is we
have now two states, e plus f and e minus f.
You know the how you should read is the following.
You can just assume for a second the state
f will degenerate with gamma and then I put in a very strong mix
in between the two.
This is exactly the example we had with hydrogen
in an electric field, and then we
get two states which have both width gamma over 2.
They are strongly mixed, and the splitting
is nothing else than the matrix element of the electric field.
But if you don't have two degenerate states
and we add a photon, then the photon--
I mentioned it again and again in addressed atom picture--
creates a degeneracy.
You can just add a photon, draw a dashed line,
and this is your virtual state if you want.
Or if you look at Schrodinger's equation,
you have something which oscillates
at the frequency of the state, f,
but now you multiply it with an electric field which oscillates
at the resonance frequency, and then
you have something which oscillates
at the sum of the two frequencies,
and this is exactly what I indicated with a dashed line.
So that's how you create, so to speak,
a degeneracy by using the frequency of the laser
to overcome the energy splitting,
and the result is that you have created exactly
this excited state level structure.
And if you now look at the ground state,
and our photon is tuned right between those two continua,
then we have a dark resonance.
And in order to accomplish this, I'm
not going into any calculations here,
but you need a sufficiently strong coupling
laser who can accomplish that.
For instance, coupling laser which
is stronger than the spontaneous emission rate or decoherence
rate gamma in the excited state.
So if we now scan the probe laser
and we look for transmission, let
me assume for simplicity that there is no relaxation
between the two ground states.
If the coupling laser strength is 0,
then we have a broad feature, which
is simply the single photon absorption of the probe light.
If we have an infinitesimal coupling laser strength,
we get a very, very sharp feature,
and if the coupling laser is stronger and stronger,
we get a window of transparency, and the width is given.
The width delta omega of this central feature
is given by the Rabi frequency of the coupling laser.
So what are possible applications?
One is you can design non-linear materials.
Usually, when you want to have frequency conversion processes
or optical switches at one laser beam
affects another laser beam, you want
a very strong non-linear response of your medium,
you usually get that if you go near resonance,
but near resonance, you have strong absorption.
But now, using the concept of EIT, you can have both.
You can have the strong non-linear response
of your medium near resonance but you suppress absorption,
you get a window of transparency using EIT.
So you can have near resonant materials without absorption.
Again, in principle, I'm not aware
that this has really taken off in a bigger time.
One would be if you want to do very sensitive spectroscopy.
Assume you want to measure one isotope which
has a tiny little abundance but you
have to observe it against the background of a very
strong isotope.
If you could switch off the absorption of the background,
the absorption of the strong isotope,
you could still see a small amount of a trace isotope
in the presence of a strong absorption line.
So sensitive detection of trace elements.
Questions about that?
I've given you qualitative pictures
for lasing without inversion and for EIT.
I want to give it a little bit more quantitative touch,
not by going through to the optical Bloch equations which
would be necessary to describe all features of it,
but at least I want to give you one picture where
you can derive and discuss things
in a more quantitative way, and this is the eigenstate picture.
I've also done here what I've said several times,
that when we have splittings between the levels,
we can actually focus on what really matters,
namely the detunings, by absorbing the laser
frequency into the definition of our levels,
and that's what I've done here.
Instead of using levels g, f, and e, I have levels with g, f,
e, but this is the photon number in laser field one
and the photon number in laser field two.
So if all of the photons, the laser field one and laser
field two, would be resonance with their respective
transitions, then all those three levels
would be degenerate.
But now in the three level system,
they're not degenerate only because we
have a relative detuning delta, detuning small delta
from the Raman resonance, and we have a detuning peak delta,
which is sort the common detuning of the Raman laser
from the excited state levels.
So therefore, if I did define the Rabi frequencies,
the Rabi frequencies over 2 are gain coefficient,
and the Rabi frequencies are proportional
to the electric field.
Therefore, they scale with the photon numbers n
and m in the two laser beams.
If I do that, I have a really very simple Hamiltonian.
On the diagonal, we simply have the detuning
of both laser beams form the excited state.
Here we have the Raman detuning, and we have two couplings
to the excited state.
One is laser field one, g1, with n photons,
and the other one is laser field g2 with m photons.
Any questions about?
Just setting up the simple equations
which we have done a few times.
Let me focus first on the simple case
that everything is on resonance.
Then, if everything is on resonance,
we have the structure which I've shown here.
You can sort of say you have three levels, which
are degenerate without any Rabi frequency,
because the detunings are all zero,
or all the diagonal is zero.
And then the off diagonal matrix elements, the Rabi frequencies,
are just spreading the three levels apart,
and the general structure of this matrix
is that in the middle, you always
have a state which is just a superposition of g and f.
So it's a dark state.
It has no contribution in the excited state.
And the two outer states have equal contribution
of the excited state.
So the excited state has been distributed over the two
outer states, and the widths of those levels
is therefore gamma over 2.
And we know when we had two levels
and we were driving them with a Rabi frequency resonantly,
we had splittings which were just
given by the Rabi frequency, and now
the splitting between the outer level and the dark state
is the quadrature sum of the two Rabi frequencies.
PROFESSOR: So that's a very general structure.
I want to go back to the situation
where one laser is a probe laser.
Let's assume this is our laser beam which
wants to go through the brick wall,
and the other laser has to prepare the system.
Let me discuss the limit where the photon number becomes very
small in laser one, the photon number n goes to unity,
and this is much smaller than m.
So we have the situation of a weak probe field.
If we have this limit, then the dark state
has much, much more amplitude in state g,
and the state g is almost decoupled.
It's in a trivial way.
The dark state and the laser beam
is very strongly mixing the state f and the excited state.
So in this limit, we have a nice physical situation.
I should actually point out that you
can solve most of those situations for three level
systems analytically.
It's just those explanations get long and are not
very transparent.
So what I'm trying here in the classroom,
I try to pick certain examples--
weak probe field, resonance--
where we can easily understand the new features which
happen in the system.
So the situation we have now prepared
is we have our dark state, which is level g,
we said we have only one photon in our probe beam,
and we have lots and lots of photons in the coupling laser
which couples the other ground state, f, to the excited state,
e.
So we have the structure that we have now two states which
have half of the widths of the excited state,
and they are both bright.
We can call one the bright state 1
and the other the bright state 2.
The splitting is related to the Rabi frequency.
Let me just call it delta bar.
This is now the level structure which we have,
and what I want to now emphasize in this picture
is the phenomenon of interference.
We've talked about interference of amplitudes,
but now I want to take the system
and show you how we get now interference
when we send one probe photon through the system.
What we can now formulate is a scattering problem.
We have one photon in our probe beam in a special mode,
but then, all the other modes are unoccupied.
And when we are asking, does the probe photon get absorbed?
Does the weak laser beam get stuck in the brick wall?
We are actually are asking if it is possible
that we scatter the photon out of the mode
and it gets absorbed.
But you know absorption is actually always
a two photon process involves spontaneous emission,
and we have emitted the photon into another mode.
The scattering problem is a two photon process,
and the matrix element needs an intermediate state,
but now we have two.
We start with one photon, we have the light atom coupling,
we can go through bright state 1.
From bright state 1, we have the light atom coupling again,
and eventually, we go back to the ground state
without four photon.
And here we have a detuning.
Let's assume we are halfway detuned
between the two bright states.
And then we have a second amplitude,
and it's indistinguishable.
We have a Feynman double slit experiment,
and everything here is the same except that we
scatter through bright state 2.
This matrix element, when it vanishes,
this is now the condition of electromagnetically
induced transparency.
But we want to now understand what happens
when we detune the probe laser.
So we have set up the system with a strong control laser,
we have completely mixed the ground state f with the excited
state, and now we want to ask, can a weak probe laser
go through the brick wall How much of the probe laser
is absorbed?
So what we want to understand now,
and this is a new feature I want to discuss now,
what happens when we detune the probe laser by delta?
Well, it's clear, and I just wanted to show you the formula.
We had two detunings here, and if we detune the probe laser
by delta, that would mean now that those denominators are
no longer opposite but equal.
In one case, we add delta.
In the other case, we subtract delta.
And therefore, we have no longer the cancellation
of the two amplitudes and we have
the scattering of the photon.
This is sort of the framework, and I
want to show you now several examples.
I want to show you examples of a probe absorption spectra.
It's more a little bit of show and tell.
I want to show you the result if you would work that out.
It's pretty interesting.
So first, I want to discuss the case where the coupling
laser is really in resonance.
We are near the one photon resonance.
Let me assume first that the Rabi frequency of the coupling
laser is much, much larger than gamma.
Then if you look at the probe transmission,
we have the situation that we know
when we are right here in the middle,
we have a window of transparency,
but if we detune by the Rabi frequency over two,
we hit the bright state 1, and if we detune in the opposite,
we hit the bright state 2.
And the splitting in this situation, delta bar,
is nothing else than the Rabi frequency omega 2.
So what we have is we have the detuning in between,
and we know already here is our special point,
and that's what we have discussed
for long, electromagnetically induced transparency.
The two bright states are at plus and minus
half the Rabi frequency of the coupling laser.
We know that we have very strong absorption here.
What you get is sort of broad feature.
But this was a situation when we drive the system very strongly.
I now want to discuss the case that the coupling laser is
much weaker than gamma.
Then the splitting between the two bright states
was the Rabi frequency, but the width is gamma,
so then the two bright states pretty much merge
into one continuum feature of width gamma.
For that situation, we have a broad feature of absorption,
which is on the order of gamma, or if you
have an opaque medium, of course,
you put the absorption coefficient
into the exponent of an exponential function
and you get a blackout which is wider than gamma.
But then we have our phenomenon of EIT,
but the width of this feature is now much smaller than gamma.
So what we have here in those two situations
where the lasers are close to resonance with the excited
level, we have the situation that the strong absorption
feature due to the bright states,
either one strong feature of width gamma or two features
here, those broad, similar photon absorption features,
they really overlap with our window of transparency.
And what I find very insightful is now to discuss the situation
where we separate the two, and I want to show you what happens.
It gives a very interesting profile.
So what we want to discuss now are the famous Fano profiles,
and I want to discuss two photon absorption features.
We have discussed the case where the one photon detuning.
We want to go now to a large one photon detuning delta 2,
and the new feature now is that this
will separate the window of electromagnetically
induced transparency from the broad absorption features.
Let me just draw a diagram of the states.
As usual, we have our two states in a lambda transition, g
and f.
We have this continuum of the excited state,
but now, and this is what often is done in the experiment,
you're not using Raman lasers which are in resonance
with the excited state.
You're using Raman lasers--
here is the strong coupling lasers--
which are far detuned.
So here we have a detuning for laser two called gamma 2.
The Rabi frequency is omega 2.
Here we have a weak probe laser omega 1, and the detuning.
Let's call it gamma 1.
So in order to keep the situation simple,
we use a weak probe laser.
Omega 1 is small.
And our Raman detuning delta is the difference
between the two single photons detuning,
capital delta 2 minus delta 1.
For these situations, there are nice analytic expressions,
and together with the class notes,
I will post a wonderfully clear paper
by [? Lunis ?] and [INAUDIBLE] where the two authors discuss
this situation.
Let's just figure out what are the features in the system.
Let's just go through different situations.
One is if the Raman detuning is zero,
we should always get the phenomenon
of electromagnetically induced transparency.
If we don't have any coupling, then
we simply have a two level system,
and if we tune omega 1 into resonance,
we get simple single photon absorption.
So if we look at the system of the three levels
and we are asking what are now the relevant processes,
one limit is, of course, the trivial limit
that we have single photon absorption.
This is, of course, a trivial case.
Let's now go to the more interesting case
that we have a coupling laser.
What happens now?
We have our two states.
The excited state is coupled with the laser
and Rabi frequency omega 2.
Now we know that the laser omega 2, if it is not on resonance,
will give us an AC Stark shift.
This AC Stark shift, we've gone through that several times.
It's the matrix element or Rabi frequency squared
divided by the detuning.
If we now bring in the probe laser,
what are the features we expect?
Well, there are two features.
One is we've just discussed the trivial case above.
If we tune the probe laser into resonance with the excited
state, we have single photon absorption.
We get a broad feature.
It's almost like in the two level system.
But in contrast to the case I just discussed above,
the excited state level e has now an AC Stark shift
so the resonance is shifted.
That's now becoming a four photon process
because we need two photons going
up and down with a coupling laser
to create the AC Stark shift.
And now we have a laser from the probe beam
and the photon is scattered, so it's a four photon process
and it will give us a broad resonance which
is now AC stack shifted.
But in addition, we have a resonance,
which is the Raman resonance.
When the Raman detuning is zero, then we
absorb from the probe laser and we
emit in a stimulated way with the coupling laser,
and we have a stimulated two photon transition.
Now, what is the width of this stimulated two photon
transition?
Well, we go from a stable ground state to--
I wanted to say another stable ground state,
but this other stable ground state
is now scattering photons.
So because of the presence of a strong coupling laser,
you have broadened this level f by photon scattering.
You interrupt the coherent time evolution
by scattering photons, and the photon scattering
happens in perturbation theory by the amplitude to be
in the excited state squared times gamma.
The scattering rate, gamma scattering,
is Rabi frequency divided by detuning.
This is the amplitude to be in the excited state.
We square that, and then we multiply with gamma,
and if I can trust my notes, it's
a factor of two, which I don't want to discuss further.
This is a quantitative argument.
The analytic expressions are in the reference
I've given to you.
So the situation which we have right now
can be in a very powerful way summarized as follows.
We have our ground state, we have
two continua we can couple to.
We can couple to the excited state which
has a width, gamma, through a single photon,
but there is the AC Stark shift.
Or we can couple through a two photon Raman transition
to the state f, but the width is much, much smaller
because it is only the scattering rate due
to the off resonant coupling laser.
Let me just write that down because I said a lot of things.
So g couples now.
When we detune the probe laser, we
can be in resonance with this feature
and we can be in resonance with that feature
to a narrow and wide excited state.
One excited state, of course, is the state f,
but the coupling laser puts some character of the excited
states into the state f.
And the most important thing now is the following.
This is the theme I've emphasized again and again when
we discussed three level systems.
Those two states have a width, and the width
means they spontaneously emit light,
but they emit light into the same continuum.
So if you start in the ground state,
your probe laser has a photon and the photon gets scattered.
You do not know through which channel it has been scattered.
So in general, and this is as far
as I want to push it in this class, in three level system,
we have now the interference between those two continua.
One is narrow and one is wide.
Let me just write that down because this is important,
but both excited states couple to the same continuum.
And by continuum, I mean the vacuum
of all empty states where photons can be scattered,
and this is the condition for interference.
And this is, of course, what gives rise
to electromagnetically induced transparency.
I'll take your questions, but let
me first give you a drawing which
may illustrate, or summarize, what I've just said.
So what we try to understand is what happens
when we detune the probe laser.
Until now, we had always the EIT feature, the EIT window,
was completely overlapping with the one photon resonance,
but now, because the coupling laser has
a detuning of gamma 2, we have to use with the probe laser.
I have to trace back how the detunings are defined.
The Raman detuning was big delta 2 minus big delta 1.
If the detuning delta is chosen to be delta 2,
we have the simple situation that we go from the ground
state right to the excited state,
and we have the feature which has a width of gamma.
This is what you would call the single photon resonance.
It is the single photon resonance.
It is due by resonantly coupling into this continuum,
and the only feature of the coupling laser
is that there's an AC Stark shift to it.
Single normal resonance, almost like in a two level system,
but the only addition is the AC Stark effect.
Now we have a second feature, which can be very sharp.
This is when we do the two photon Raman process
into the other ground state.
Due to photon scattering, this resonance
has a width of gamma scattering, and the position
is not at zero at the naked Raman resonance.
It also has an AC Stark shift because the coupling laser
does an AC Stark shift to both the excited and the ground
state.
The coupling laser, the AC Stark shift,
pushes ground and excited state in opposite directions.
So therefore, we find that at the Rabi frequency
of the coupling laser divided by gamma over 2.
The name of this feature is the two photon Raman resonance
plus the AC Stark shift, which actually means
it's a four photon process.
The question is now, where is the electromagnetically induced
transparency?
We have introduced now specifically the absorption
feature.
We have identified a one photon absorption feature
plus AC Stark shift, a two photon absorption
feature plus AC Stark shift, but where is electromagnetically
introduced transparency?
Here.
Electromagnetically induced transparency
is always at delta equals 0.
You have to fulfill the Raman resonance,
and this resonance is not affected by any AC Stark shift.
It's always at delta equals 0.
So therefore, what that means is you have two absorption
features, and you would think these are two Lorentzians,
but they interfere and they go to exactly 0 at delta equals 0,
and this is our EIT feature.
So there is an interference effect between a broad feature
and a narrow feature, and this is
found in many different parts of spectroscopy,
but you also find it in nuclear physics
whenever you have a narrow feature embedded
into continuum.
What we have here is a narrow feature and a broader feature,
but a narrow feature and something
which is broader or continuum is called a Fano resonance.
You actually have the same situation
when you look at scattering of atoms.
Many of you are familiar with Feshbach resonances.
Well, a lot of people call it Fano Feshbach resonance,
and what you have in a Fano Feshbach resonance is two atoms
can scatter off each other.
This is the continuum.
This would be your broad feature.
But then they can also scatter through a molecular state,
and this is a narrow feature.
And what we've identified now for electromagnetically
induced transparency are the two features.
One is the single photon absorption,
one is the Ramon resonance.
But in general, the concept is much more general.
You have a narrow feature, you have a broader feature.
It's responsible for scattering two atoms
or it's responsible for scattering a photon,
and once the photon or the atoms have been scattered,
you have no way of telling which intermediate state was
involved.
And therefore, you get interference.
And what I just said, what is EIT for light
is the zero crossing of a scattering length
that the atoms do not scatter off each other
because the two different processes completely
destructively interfere.
So what I've shown here is two Lorentzian, two absorption
features.
Let me know re-plot it and plot the index of refraction
minus 1.
Here was EIT, zero detuning.
We have a sharp feature here at the two photon resonance,
we have a broad feature here at the single photon resonance,
and if I now transform the Lorentzian
into a dispersive feature.
I use freehand, so this is a dispersive feature
for the broad transition.
The narrow transition has a much, much sharper
dispersive feature, and the important part
is now at the EIT, at the detuning delta
is zero where we have electromagnetically
induced transparency, we have an index of refraction
which is exactly one because it's a dark state,
you have no absorption, you have no light scattering,
you have no reaction to the light.
And therefore, the index of refraction of the material
is like the index of refraction of the vacuum.
So you have n equals 1.
It looks like a vacuum in terms of index of refraction.
It looks like a vacuum because you have no absorption.
But what you have is you have a large derivative
of the index of refraction with the frequency detuning,
and that affects, and that's what I want to tell you now,
the group velocity of light.
So anyway, this is maybe as far as I want to push it,
and I was actually wondering if this is a little bit too
complicated to present in class.
But on the other hand, I think it sort of also
wraps up the course.
We have a three level system, and we
find a lot of things we have studied separately before--
two photon Raman feature, single photon light scattering--
but now they act together and they
interfere and have this additional feature
of electromagnetically induced transparency.
Colin?
STUDENT: So the way you drew the level diagram, the f, I think,
[INAUDIBLE] state [INAUDIBLE]?
PROFESSOR: Yes.
Actually, I emphasized that we have an AC Stark shift,
and what I didn't say when I discussed it here
that the AC Stark shift pushes this level down
and pushes the other level up.
But since we are talking about a very broad resonance
in the excited state, for all practical matters,
the AC Stark shift doesn't matter,
whereas for the narrow Raman resonance,
the AC Stark shift is important.
STUDENT: Also, you showed on the plot
that the shift of the excited state [INAUDIBLE]..
PROFESSOR: Sorry.
Thanks.
So the AC Stark shift for that detuning
would shift this level up and would shift this level down.
What's the second question?
STUDENT: You drew on the plot--
PROFESSOR: This one?
STUDENT: That one.
That the shift of the excited state
was much higher than the shift of the ground state.
PROFESSOR: You mean those two shifts?
STUDENT: Yeah.
PROFESSOR: We have to now do the bookkeeping.
We have assumed that the coupling laser has
a large detuning, the coupling laser
is very far away from resonance.
And if you want to hit the excited state,
we know we need a Raman resonance, which is delta 2,
but the Raman resonance, capital delta 2,
means that we are smack on the single photon
resonance for the probe laser.
So this feature is pretty much we take the ground state
and go exactly to the excited state with a single photon.
There is an AC Stark shift involved
but it's not relevant here, whereas the other feature is
the two photon Raman feature.
And the one thing I wanted to point out in this context
is that there is actually a small energy
splitting between the two photon Raman feature by the AC Stark
effect, whereas the EIT feature always
happens at Raman resonance delta equals 0.
Because, just to emphasize that, delta
equals 0 is really you induce a coherence between g and f.
It's a dark state, and when you have
a dark state in that situation, you
don't have an AC Stark effect.
So the EIT feature is at delta equals 0,
whereas the photon scattering features, they suffer,
or they experience--
it may not be negative to suffer an AC Stark shift,
but they experience an AC Stark shift.
Further questions?
Yes?
STUDENT: Can we somehow relate this
to Doppler free spectroscopy?
PROFESSOR: Can we relate that to Doppler free spectroscopy?
Actually, I don't think so because I
would say for the whole discussion here,
let's assume we have an atom which has infinite mass, which
is not moving at all.
We're really talking about internal coherences.
However, and that's where it becomes related,
if you look at very, very narrow features
as a function of detuning then, of course, Doppler shifts
play a role, and if you have very, very, very
narrow features, you become sensitive to very, very, very
small velocities, and therefore, you
have an opportunity to cool.
So if you can distinguish spectroscopically
an atom which moves a tiny little bit and an atom which
stands still, if you can, by a narrow line,
distinguish the two, then you can actually
laser cool this atom.
The EIT feature can give you extremely high resolution.
I'm not discussing it here.
I've discussed the phenomenon of EIT, which is
coherent population trapping.
But there is an extension, which is VSCPT, Velocity Selective
Coherent Population Trapping, and VSCPT was a powerful method
to cool atoms below the recoil limit,
but I've not connected anything of coherent population trapping
with the Doppler shift.
So for all this discussion, please
assume the atom is not moving.
Other questions?
Then let me just say a few words about the fact
that we have a large derivative of the index of refraction,
and this, of course, is used for generating
what is called slow light.
The group velocity of light is the speed of light,
but then it has a denominator which
is the derivative of the index of refraction
with respect to frequency.
Towards the end of the last century,
there were predictions that electromagnetically induced
transparency would give you very sharp features which can
be used for very slow light.
And what eventually triggered major developments in the field
was this landmark paper by Lene Hau where
she used the Bose-Einstein condensate to eliminate
all kinds of Doppler broadening.
There are other tricks how you can eliminate it,
but this was the most powerful way
to just take a Bose-Einstein condensate where
atoms have no thermal velocity and, in this research,
she was able to show that light propagated
at the speed of a bicycle.
So it was a dramatic reduction of the speed of light,
and this showed the true potential of EIT.
There have been other demonstrations
before where light has been slowed by a factor of 100
or a few hundred, but eventually,
combining that with a Doppler-less feature
because the atoms don't move in a BEC
created a dramatic effect.
So we have now two ways how we can get
a large derivative, dn d omega.
I've discussed here the general case
that we have a narrow feature, a broad feature, because we
have the coupling light detuned from the excited state,
but let me just point out that even if you have the coupling
light on resonance, depends what you really want,
but you can get an even stronger feature in the index
of refraction versus frequency.
This is now the situation where we have the strong absorption
feature, but then we have the EIT window.
So we have this superposition of a positive Lorentzian
and a negative Lorentzian, and if I now
run it through my Kramers-Kronig calculator
and I take the dispersive shape, I
can sort of do it for the broad feature in this way
and for the narrow feature in this way,
and now you have to add up the two.
And what you realize is at this point,
you have a huge dn d omega.
What I'm plotting here is on the left side, the absorption
of the Lorentzian, and you can regard this sharp notch
as a second Lorentzian.
So you have the positive Lorentzian, negative
Lorentzian, and then you take the dispersive features
and you add them up with the correct sign.
So whether you're realizing now for quite a general situation
where you have single photon detuning, which I discussed
before, or whether you're on the single photon resonance,
you can have extremely sharp features.
So now you can take it to the next level.
You have a light pulse which enters a medium,
and now the light pulse slowly moves through the medium.
But while the light pulse moves through the medium,
you reduce the strength of the coupling laser.
What happens?
So you do now an adiabatic change of your system.
You do an adiabatic change of the control field
omega 2 while the probe pulse is in the medium.
Well, that means that under idealized assumptions which
we've discussed, this feature gets narrower
and narrower and narrower.
If omega 2 goes to 0, the strength of the control field,
this feature becomes infinitesimally narrow.
And therefore, this feature becomes infinitely sharp,
and that means that the group velocity goes to zero.
This is now in the popular press,
it's called stopped light or frozen light
because the light has come to a standstill.
What really happens is the following.
We have our coupling laser, omega 2,
and we have our probe laser, omega 1.
When we do what I just said is that omega 2 goes through zero,
then the dark state originally, for very strong omega 2,
remember the dark state was g?
But now if we let omega 2 go to zero,
the dark state will become f.
And that means that in a way, every photon in the probe pulse
has now pumped an atom from g through two photon Raman
process into f.
So therefore, what it means to stop light
or to freeze light means simply that the photons of the laser
have turned into an atomic excitation where the excitation
is now the state, f.
In other words, you have written the photon.
The photon has now put the atom into a different hyperfine
state.
So if this is done adiabatically,
and I can't do full justice in this course,
but this means that the light is coherently
converted into the atomic--
when I say "coherently" say "into population," I
mean all the quantum phases, everything which
was in the quantum nature of the light has now been converted,
has been written into the state, f.
This is often called, because g and f are hyperfine states,
this means that you have coherently
converted the photon or the electromagnetic wave
in the probe beam into a spin wave or a magnon.
Anyway, I just want to show you the analogy.
The fact that you can put the quantum information of light
into an atomic state and back and forth,
we've discussed that when we had the situation of cavity QED.
We prepared a superposition of ground and excited state
and exactly the same quantum state which we had in the atom
we later found in the cavity as a superposition
of the zero photon and the one photon state.
So from those general concepts, it should be clear to you
that it is possible to coherently
transform a quantum state from light to atoms
and back to light.
And here you see the different realization.
We have a quantum state of the photon in the probe laser,
and we can now describe the excitations
in the system in a parametrized way.
What it means is for the strong probe laser,
for the strong coupling laser, the excitation in the system
travels as a photon in the field one,
but when you reduce the coupling in the coupling laser,
the excitation becomes less and less photon-like.
It becomes more and more magnon-like, spin wave like.
And the moment you reduce the power in the coupling laser
to zero, what used to be an excitation
in the electromagnetic field has now been termed adiabatically
into a spin excitation.
Coherence has been written into the hyperfine states
or your atoms, g and f.
All this is done coherently, and therefore reversibly.
You can read out the information by simply time reversing
the process.
You ramp up again the coupling laser,
and that adiabatically turns the spin excitation back
into an excitation of the electromagnetic field.
Any questions?
Well, we have three minutes left,
but I'm not getting started with superradiance.
We have one more topic left, and this
will be the topic on Wednesday, decay superradiance,
and this is when we discuss the phenomenon of coherence where
we have coherence not only in one atom between two or three
levels.
We will then discuss on Wednesday
if we have coherence between many atoms,
and this at the heart of superradiance.
OK.
See you on Wednesday.