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PROFESSOR: So good afternoon.
Let's have an on-time departure.
Last week, we talked about quantum states of light.
So we're talking about the photon, the quantum description
of the electromagnetic field.
And I introduced for you the most classical quantum state
of the electromagnetic field, namely the coherent state.
I don't want to go through the formalism again,
but the coherent state has a simple definition,
simple but subtle.
It's an eigenstate of the annihilation operator,
and it has a complex eigenvalue alpha.
And from that, you can pretty much
derive everything you want to know about coherent states.
Colin.
AUDIENCE: What was the purpose of-- whoever
discovered eigenstates-- what was the purpose of trying
to find the eigenvalues of destruction?
PROFESSOR: Yeah, isn't that subtle?
Isn't that weird?
How does somebody want to find the eigenfunctions
of an annihilation operator, because the annihilation
operator is non-Hermitian?
And why the annihilation operator?
Why not the creation operator?
Well, if you try to find the eigenstates of the creation
operator, you run into problems.
The eigenstates are non-normalizable.
It doesn't make any sense.
So here, creation and annihilation are not symmetric.
You have to pick the annihilation operator.
Colin, my gut feeling is that somebody found it
in a different way.
The person who popularized those states was [INAUDIBLE],
and he got amply rewarded for that.
And I think you define coherent state
as some form of superposition state.
Maybe people have figured out if you take a superposition
state of number states that this has special properties,
that this is a minimum uncertainty state, which
is as close as we can go in quantum physics
to obtain a classical electric field.
So I think maybe people found it like that.
And then they say, hey, but there is another operator
definition eigenstate of the annihilation operator.
I'm 99 percent certain.
But maybe you can ask [INAUDIBLE]
when he comes to a CUA seminar if he discovered coherent state
as eigenstates of the annihilation operator.
I don't think so.
Other questions?
So what I want to do is-- what the theme
of the end of last lecture and today
is that the coherent state is a complex number.
The coherent state has a time evolution.
It moves in a circle.
And this is really the phasor of the electric field associated
with it.
So coherent state, the alpha value
is directly related to an electric field.
That's why this state is closely related
to the classical limit of the electromagnetic field.
We discussed some properties of the coherent state.
We looked at the fluctuations and showed
that it's a Poissonian statistic.
And then we use now-- once you define something,
you can use it as a tool.
We are now using the coherent states
to look at any other quantum state
of the electromagnetic field, any statistical operator which
describes photons by forming the diagram matrix
element of the statistical operator with alpha.
If you just want to sort of think about it intuitively,
alpha is like the best description
of an electromagnetic field.
So if you write down this, we ask,
What is the probability for an arbitrary quantum state
described by an arbitrary statistical operator-- what
is the probability that the electric field is
alpha in the complex description?
So that's what those quasi-probabilities
are-- Q of alpha.
And we immediately looked at some examples, of course,
of probabilities.
We showed that the vacuum state is sort of an area.
It's a Gaussian.
And the area is-- it's an area.
I don't know.
I forgot the [? prefect. ?] It's on the order of 1 or 1/2.
We realize that thermal states are also
a Gaussian centered at the origin.
But it is a much wider distribution.
So thermal states have much more uncertainty
in the electric field than the vacuum state.
And then we looked at the coherent state.
And now, of course, when we looked at the coherent state,
we realize that coherent states are maybe not
as wonderful as I tried to make you believe.
They have some nasty properties.
And that is when you ask, What is the probability?
The quasi-probability of the coherent state would be beta.
Now, if you would expect a delta function,
because when you ask, What is the probability of a momentum
state to be p naught?
If you have a momentum state p naught,
it has a delta function at p naught.
But here, it's not a delta function.
It's a Gaussian of the standard deviation.
I think the standard deviation here is-- what is it,
square root 1 over square root 2?
Because the Gaussian has usually 2 times sigma squared.
And since the denominator is unity--
so the standard deviation of 1 over square root 2.
And that showed us that the coherent states are not
orthogonal.
You have to be a little bit careful.
It's not your standard basis set.
It's an overcomplete basis.
So therefore the coherent state is not
a delta function in the causal probability.
It's a little bit blurred circle with an area
on the order of unity.
And you have some homework assignment to look at it.
So in some way, how you should look at
it is that if you see some quasi-probability distribution,
the distance from the origin-- this
is the absolute value of the electric field.
The phase here is almost 45 degrees.
But if you look at this uncertainty,
you would also say, Well, the phase
is uncertain to within this angle.
So you can pretty much read the uncertainty
in the total electric field, the uncertainty in the phase.
You can read pretty much everything
you want from this diagram.
Now, when we look at a number state-- Well,
you often know in quantum mechanics,
number and phase are complimentary.
If the number of photons is fixed,
you know nothing about the phase.
And indeed the quasi-probability of a number state is a ring.
It has no phase.
It has completely random phase over the 2 pi circle.
The energy is sharp of a number state,
since the energy is e squared.
You may have expected a delta function
in the radial coordinate.
But what you get is also something
blurred on the order of unity.
And I want to say something about that in a second.
Finally, we discuss the time dependence.
And the time dependence is very easy.
After all, we're dealing with an harmonic oscillator.
And in an harmonic oscillator, if you
have a plane of x and p, symmetric and anti-symmetric
combination of a and a dagger, in this plane
in an harmonic oscillator, the quantum state
is just rotating circle, a rotation with omega.
And indeed we showed that when we apply the time evolution
operator-- and some of you were right, of course,
with a minus sign-- it moves with e to the minus i omega t.
And therefore everything rotates in a clockwise way.
Now, we discussed the operator of the electric field.
And in this quasi-probability-- Sorry,
we discussed the operator of the electric field.
And I hope you remember that in the analogy
with the harmonic oscillator, the electric field was
a minus a dagger.
And this is the momentum operator.
In those quasi-probabilities-- and we will see more about it--
something which is sharp in momentum
is a sliver parallel to the x-axis.
So therefore, you can regard the vertical axis,
which is the imaginary part of alpha, as the momentum axis.
And you can regard this as-- the horizontal axis--
as the x-axis.
So therefore, since momentum is electric field,
you always get the electric field
by projecting onto the vertical axis.
And if you project this fuzzy ball,
you get a value 0 with some uncertainty.
And if this quasi-probability starts
to rotate due to the time evolution,
we get an oscillating electromagnetic field, almost
classical except for that fuzziness.
So that's where we want to continue.
Any questions about that?
Yes?
AUDIENCE: Why is there no phase fuzziness in the E t?
PROFESSOR: No phase fuzziness.
There is a phrase fuzziness.
For instance, if you would say the phase is determined
by the 0 crossing, you don't know exactly when the 0
crossing happened, and that corresponds
to an uncertainty in the phase.
Trust me, everything is in this diagram.
Now, there are two things we want to continue.
One is I want to show you that the coherent state is
a minimum uncertainty state.
The product of delta x delta p is just--
is it h-bar or h-bar over two?
One of the two.
So it's a minimum uncertainty state.
And therefore, you can never have a quantum state
which is less fuzzy than the coherent state.
So this fuzziness here is the intrinsic uncertainty
of quantum physics.
So that's what we want to discuss today.
But then we will immediately start
with non-classical states.
And that is, well, if this area is determined
by Heisenberg's uncertainty relation, what
can be maybe deform the circle into an ellipse,
and these are three states of light.
So that's an outlook.
That's what you're going to do in the second half
of the class.
But before I do that, I want to be a little bit more
accurate about quasi-probabilities.
And this is almost like a disclaimer now.
Just to give you the bigger picture.
What we want to achieve with those quasi-probabilities,
we want to do what phase space densities in classical physics.
We have a coordinate which is x, a coordinate which is p.
And you can often describe a classical system
if you know the phase space distribution,
if you know the probability that a particle has
position x and momentum p.
So all this is about phase space densities.
Probability of x and p.
Of course, writing it down, you immediately see the problem.
In quantum mechanics, you cannot measure x and p simultaneously.
These are non-commuting variables.
So therefore, what happens is, if you now define a phase space
function, which is done in quantum mechanics textbooks,
you can actually do it in three different ways.
And the three different ways are Q, P, and W.
The definition-- I don't want to go
through the mathematical subtleties-- the definition
of those functions involves the operator definition, a
and a dagger, or, which is equivalent,
the x and p operator.
If you define something in units of x and p or a and a dagger,
you can have a product which is fully symmetric i
in the ordering of x p, which is anti-normal or normal.
So in other words, the order matters.
And you have three choices.
In an operator product, you can have symmetric ordering, which
means not x p, but x p plus p x, then it's over 2.
That's symmetric.
You can have an ordering which is called normal
and one which is called anti-normal.
Ordering of operators in the operator base definition.
Of course, if you have three choices,
you would say, Which one is the best?
Which one is the winner?
But the fact is all three have their advantages
and disadvantages.
So they all have pluses and minuses.
The reason why I picked for the course Q of alpha
is that it's a real probability, it's always positive.
It is a diagonal matrix element of a statistical operator,
and this has to be positive.
So it's a real probability.
The other guys, P of alpha, can be positive or negative.
And also, W of alpha can be positive or negative.
So if you use a P alpha distribution
statistical operator, it can be written like this.
And as a result, the coherent state is now not this Gaussian.
It doesn't have this Gaussian distribution
as a course of probability.
It's what you want-- what maybe some of you
wanted to see-- oh, by the way, it's a delta function.
The probability of the coherent state alpha
has a delta function peak at alpha, which is sort of nice.
And the number state is not a ring of a finite radius.
I just mentioned to you the number state.
You would naively expect the energy is sharp.
The square root of the energy's electric field,
shouldn't it be sharp?
And indeed, it is sharp.
It's actually worse than a delta function.
It's a derivative of a delta function.
But at least here, in the probability P,
which is also called the Glauber-Sudarshan P
representation, you get the delta function,
which may be very natural for certain purposes.
AUDIENCE: Since the [INAUDIBLE] and you can express one
in terms of the other, is that unique?
PROFESSOR: It is unique by some symmetry choice here.
So that's the advantage of it, that by-- I'm not
going into the mathematics.
I'm not giving you the definition
but the way how it's defined.
It's unique.
It can be written in such a way.
And the way how the quasi-probabilities
P-- the P representation-- is defined,
you get a delta function at the coherent state.
And finally, W stands for Wigner distribution.
And the Wigner distribution is something you actually
find in most quantum mechanics textbooks.
The Q and P distribution are more common in quantum optics.
But the Wigner distribution has the advantage
that the projection on the x- and y-axes
are indeed psi of x squared, psi of p squared.
So you get actually the x wave function and the p wave
function.
So now, as a full disclaimer, if you
want the electric field, which is the momentum of the harmonic
oscillator, which is the electromagnetic field, what you
really want to project is the W function, because for the W
function, the Wigner function, the projection
is exactly the momentum distribution
in the electric field.
So the Wigner distribution is closest to the classic phase
space distribution, as close as you
come without violating commutators.
But of course it has a disadvantage
that it has negative values.
And try to explain to your next neighbor what
is a negative probability.
Some people actually see negative probabilities spring
out the non-classical character.
And I know in our field, in AMO physics,
a few years ago, in my lifetime, there
were really attempts to use quantum state homography
and measure for the first time a negative Wigner distribution
and show that to the world.
So that meant something to a lot of people.
Anyway, you can read about it in quantum physics textbooks.
All I want you to know is to know about it,
but then also just relax.
In the bigger picture, all the three distributions
are the same.
It's more sort of on the level of whether something
is a delta function or has widths unity.
So on the small scale, it matters.
But if you map out something on a bigger scale,
they are all related to each other.
And for the rest off today and the next class,
when I show you those phase space distribution, when
I say I project onto the vertical axis
to get the electric field, I'm not
completely rigorous which of the three functions
I've really chosen.
Any questions?
OK.
So I promised you-- and this is what
we're going to do now is we're want to understand
in more depth the fluctuations.
And in particular, I want to show you
that coherent states are minimum uncertainty states.
So by identifying the vertical axis with p,
the horizontal axis with x, we immediately expect
that we find a result related the Heisenberg uncertainty
principle, which says sit the widths in P and the widths in Q
has to be larger than h-bar over 2.
And we are mainly talking with light about the photon creation
and annihilation operator.
But, just as a reminder, the momentum and position operator
are symmetric and anti-symmetric combinations.
So let me just call those [? prefecters ?]
Q naught and this one P naught.
Well, we can then immediately, by just
using elementary commutator, calculate
what are the expectation values in a coherent state for P,
P squared, Q, and Q squared.
For P, it is-- the P operator is a dagger minus a.
If we act with a on alpha, we get alpha,
because alpha-- the coherent state--
is an eigenstate of alpha.
Now, with a dagger we act on the left hand side.
And we get alpha star.
Well, with P squared and Q squared,
you have to use one or two more steps
to get rid of the products.
But ultimately, you can express space all that just
in powers of alpha, alpha squared, alpha star, and such.
So what we want is we want to find out
what are the fluctuations delta Q, delta P. Just as a reminder,
delta Q squared is of course Q squared average minus Q average
squared.
And Q squared average and Q squared
is what we have just calculated.
Actually, let me leave it in as a reminder for you--
Q squared minus Q average squared, the square root of it.
And what we find is that it's square root h-bar over 2 omega
and, for delta P, it is square root h-bar over omega 2.
And what we verify, that indeed, for a coherent state,
the product of the 2 is Heisenberg uncertainty limit.
So therefore, coherent states are minimum uncertainty states.
And maybe, Colin, to address your question,
I could imagine that coherent states were maybe
invented by simply saying, We have an harmonic oscillator
data.
We want to find the minimum uncertainty
states for which the uncertainty in x and p,
when expressed in natural units, is the same.
In other words, the coherent state
is the solution to the following question.
If you plot the quasi-probability distribution,
you give yourself an uncertainty area.
One horizontal uncertainty is delta Q.
The vertical uncertainty is delta P.
And the minimum uncertainty state
means that the area of the 2 uncertainties, the area
of the shaded region, delta P times delta Q is h-bar over 2.
Or if I want to rewrite that in-- Yes,
in the real part of alpha times the uncertainty
in the imaginary part of alpha for the quasi-probability,
then this is 1/4.
OK.
So what we have learned is that-- just one second.
We have learned about one way to characterize uncertainties--
the quantumness of the electromagnetic field--
by looking at the uncertainties in the x and p variable.
And that led us to minimum uncertainty states.
Now, I want to now introduce to you
two other ways of characterizing the uncertainty of quantum
states of light.
One is we can ask, What are the fluctuations in the photon
number?
Right now, we have asked, What are
the fluctuations in the electric field?
But the next question is, What are
the fluctuations in the photon number?
Or we can ask, what are fluctuations
in the intensity when we measure the intensity
of the electromagnetic field?
So let's do that.
The fluctuations-- It seems I've lost my lines.
OK.
So the fluctuations of the intensity
are usually expressed by the second order temporal coherence
function.
That's what we want to introduce now.
Yes, this is the second order temporal correlation
or coherence function.
What I'm always encountering in this course
is I would just like to immediately tell you
how it is defined in terms of a's and a daggers.
It's simple.
It's quantum mechanical.
It's exact.
But I always feel that if you want to really appreciate
the quantum character, you have to know the classic description
first.
So I want to first tell you what is the second order coherence
function for classical light, which
has a classic description.
And then you'll see what is the difference
for quantum states of light.
So the classical description is you
measure the intensity of light.
You're just sitting here.
You receive light from a light bulb.
You measure the intensity at time t.
You measure the intensity at time tau later.
And then you form the product.
And you normalize it by the average intensity squared.
So if tau equals 0, it's nothing else
than the intensity squared average divided
by the average of the intensity squared.
So this is the classical definition, g2 of tau.
And I've left the proof to the homework
to show that the classical g2 of tau is always larger than 1.
But it's pretty much what you show is for tau equal 0,
it means I squared average is larger
than I average squared, which is, of course, always the case.
But you can show that this is also the case for finite tau.
So quantum mechanically, we will see that the g2 function is not
necessarily larger than 1, it can be smaller than 1.
And that's actually an interesting--
you can see-- litmus test for the quantumness.
If you generate states of the electromagnetic field-- Fock
states, photon number states-- we see that in a moment.
And they have a g2 of tau which is smaller than 1.
You know it is not possible in any way
to associate an intensity of the electromagnetic field
with that photon state, because whenever
you can associate a classical intensity with it
and use a classical intensity to calculate the second order
correlation function, you get something
which is larger than 1.
So often therefore again, when people
want to show we really have now non-classical photon states,
they show g2 of tau is smaller than 1.
This is similar to what I said before when
you want to show that you have non-classical light,
you do quantum state homography, measure the Wigner
distribution, and show that the Wigner distribution has
negative quasi-probabilities.
That's again something which is classically not possible.
It's only possible if you have a truly non-classical state.
AUDIENCE: So in the definition of g2 classical,
the expectation values would need averaging over time, not
over ensembles of I of t?
PROFESSOR: OK.
You can average over ensembles.
So you can have 1,000 light bulbs, switch them all on,
and then measure at a certain time z.
But what we assume here is that the light bulb
is on continuously.
So things don't really depend on t.
I've implicitly taken care of it by defining the g2 function
only of tau and not of t and tau.
You assume any time is the same, because it's a distribution,
it's this ensemble in steady state.
But it's sort of the same story again and again.
In classical physics, you can determine an ensemble average
by taking an ergotic system and observing it
at many, many times.
And then the idea is that one system as time
goes by will sample all possible states.
Or you can prepare many, many identical systems
and do more of what is an ensemble average.
So in other words, you would actually
think, if you switch on a light bulb with a stable power
supply, that the light emitted by the light bulb
will go through all possible quantum states as time evolves.
And therefore, the temporal average
is equal to the ensemble average.
So how do we generalize that to quantum mechanics?
Well, one possibility would be that-- OK.
First of all, for quantum mechanics,
we have to use operators.
And one choice would be that, well, you
say the intensity of the electromagnetic wave
is proportional to the electric field squared.
And you want to use the operator for the electric field squared.
Now, there is a little bit of a problem,
because what we really mean quantum mechanically
by g2 of tau is we measure the intensity now
and a little bit later.
But measuring the intensity really means absorbing photons,
because the only way how you can measure the intensity of light
is with a photomultiplier.
It makes click, the photon is absorbed.
And this is not fully described by the electric field.
Just assume you have no photon, you
are in the zero-point state of your harmonic oscillator,
and your E squared has zero-point fluctuations.
So what is more closely related to an experiment how
you measure the correlation function is
you want to look at something else,
namely at the probability of absorbing 2 photons.
So you start with an initial state.
You annihilate 2 photons.
And then you have a final state.
But if you're only interested in what
is the probability you want to characterize
your initial state, you may want to sum over all final state.
And this is your total probability
that you can absorb 2 photons out of an initial state.
But since you're in sum now over all final state,
this turns into-- Now, I could put
for you as in the final state is a complete basis.
But I can also take it out.
So this is what we get.
So this suggests that experiments
where we look at two subsequent clicks of a photomultiplier
where we determine the photon correlation,
that this is measuring a correlation function, which,
for quantum states of light, should be defined
as the expectation value of a dagger, a dagger, a, a.
And now we have to normalize by the probability squared
of absorbing 1 photon, which is a dagger a expectation
value squared.
I just tried to be a little bit too motivated.
In many, many textbooks, the discussion
would start with this expression.
You ask, Where does it come from?
And you realize yes, for some measurement
with photomultipliers, that's what you measure.
But I wanted to show you how it is related to the intensity
correlation function defined classically.
Cody.
AUDIENCE: But this looks like it's
constant in tau, which doesn't really
make sense intuitively to me, because at least classically,
it shouldn't be constant in tau.
PROFESSOR: OK.
That's the next thing I wanted to say
is I've swept here-- I just wanted to give you
the structure of the operators and not
get distracted by discussing time.
I've dropped the time argument here.
But the fact is that as long as we limit ourselves
to a single mode of the electromagnetic field,
a single harmonic oscillator, things are independent of tau.
In other words, g2 of tau equals g2 of 0,
as long as we're dealing with single mode light.
You can actually say-- and that sort of tells you
where the fluctuations come-- if g2 of tau changes,
it comes because you have several modes
of the electromagnetic field, which, as a function of time,
constructively and destructively interfere.
But if you have a single mode, in a single mode--
and what is a single mode?
It's just a sine wave.
And nothing happens as a function of time.
It's constant.
AUDIENCE: So both modes go inside.
PROFESSOR: I'm squeezing a textbook of quantum optics
into two classes.
And I want to give you the ideas and the concepts.
What I've sort of mixed up here deliberately
is I've given you the classic definition of the intensity
correlation function, which is the famous Hanbury Brown Twiss
experiment, and used correlations
as a function of time.
Then I've motivated how this should
be defined for quantum states of light.
But when I transitioned to quantum states of light,
I decided to deal with only one mode of the light.
We should now sum over-- I should
put double or triple indices on all the alphas
for polarization, for spatial modes,
for different frequencies, and we sum over all of them.
But instead what I did is I wanted to just show you
the simple case-- and I think you will be thankful
for that in your homework, that you only
have to deal with the simple case--
that you only have to look at a's and a daggers
in the operator algebra for single mode,
for single harmonic oscillator.
But what we lose, so to speak here,
is there is nothing interesting going on in time.
I've already told you that, for a single mode, all
of these quasi-probabilities, they just rotate in a circle.
So the time evolution of the system
you're describing right now is completely boring.
It's really a rotation.
And if you would rotate your head at omega,
nothing will happen.
And this is exactly what you see here.
So you find everything you want to for coherence time.
Coherence time is the time for 2 modes to get out of phase.
But if you have 1 mode, there is no coherence time.
And when you find, for classical light,
that the g2 function has a peak which decays with the time,
it is the time for modes to get out of phase.
But in a single mode picture, this is absent.
Anyway, what is important now for the discussion of quantum
character of light is really, we find
that in a single mode picture.
So I want to show you now-- or at least give you
the summary of the results, which
can be very easily derived, because the math is very
simple of those operators, that we are now
with that definition.
We have g2 functions, which are no longer following
the classic constraint that g2 has to be larger than 1.
We find g2 functions which are smaller than 1.
And this sort of tells us now where
do we find the most non-classical behavior,
namely when g2 of tau is as small as possible.
OK.
In your homework, you will show immediately
that the g2 function is related to number fluctuations.
It's related to an average and an n-squared average.
Let me just write that down.
It's independent of tau.
And the reason is we have now limited ourselves
to just one mode of the electromagnetic field.
Fano factor.
Just give me one--
OK So we are back to-- We started
with intensity fluctuations.
But for a single mode of the electromagnetic field,
we are back to photon numbers.
So what we are now expressing with the g2 function
are, in other words, just fluctuations of photon numbers.
I want to in a minute draw you a table.
What are the fluctuations in the photon number
for the states we have encountered-- the number
state, the Fock state, the coherent state,
and the thermal state?
And we want to characterize those quantum states of light
by the g2 function.
And actually, I can drop to time argument or set it to 0.
But then we first make a reference
that g2 can now be smaller than 1.
So g2 for a single mode light is nothing else than a function.
When you know what is an average and n squared average,
you know your g2 function.
There is another quantity, which we often
use to characterize the fluctuations in the photon
number.
And this is called the Fano factor.
The Fano factor is-- wants to compare the fluctuations.
n square average minus n average squared,
these are the fluctuations.
The classical fluctuation-- well,
I say classical, classical in the simplest case--
are Poissonian fluctuations.
So maybe we want to normalize the fluctuations
by Poissonian fluctuation.
So for Poissonian statistics, what I just wrote down
would be 1.
And well, if you now subtract 1, we
have the situation that the final factor, which
is positive, is super-Poissonian--
more fluctuation than Poissonian--
and the negative final factor is sub-Poissonian.
So with those definitions, we can now
compare the different states of light we have introduced.
We started out with black-body radiation, thermal radiation.
We defined coherent states.
And the harmonic oscillator description naturally
gave us harmonic oscillator eigenstates,
the number of states, or Fock states.
So there are three ways to characterize it.
They are all useful.
One is we can look at n squared.
We can calculate the Fano factor.
Or we can calculate g2 of 0.
For the coherent state-- remember,
the coherent state is as close as we
can come quantum mechanically to the ideal
of a pure electromagnetic wave.
It has a Fano factor of 0.
This means it's Poissonian.
The g2 function is simply 1, which
is the lowest classical limit.
So those two tell you that a coherent state is sometimes
what you think-- what comes out of a laser is
an ideal electromagnetic wave, which
has no temporal fluctuations in the intensity.
Therefore, g2 is 1.
And the photon number is Poissonian distributed.
And that means that n squared is an average times 1
plus an average.
We actually discussed it earlier.
The thermal state is quite different.
If you use-- kind of pluck together
the results we have obtained, it has a Fano factor of n-bar.
So this is super-Poissonian.
If the occupation number n is large,
you have fluctuations which are much,
much larger than Poissonian fluctuation.
The g2 function is 2.
It's sometimes called thermal light.
Chaotic light has a g2 function of 2.
Laser light or coherent state has a g2 function of one.
And n square average is n-bar 1 plus 2 n-bar.
OK.
But now finally, maybe the most interesting state
from the perspective of non-classical light--
of quantum light-- is the photon number state.
Well, for the photon number state, the number of photons
is an eigenvalue.
Therefore, n squared average is n average squared.
The Fano factor is minus 1.
Sub-Poissonian distribution.
And the g2 function, which classically cannot go below 1,
is now n minus 1 over n.
It is smaller than 1.
And you see immediately that the biggest violation for g2
is to go to minus 1 for the case of a single photon state.
Any questions?
AUDIENCE: Shouldn't it be 0 for [INAUDIBLE]?
PROFESSOR: The g2 function?
No, if you put in-- wait.
Yes.
Gosh.
I'll double-check.
I'm a little bit confused.
But I-- At least it's consistent now.
I'm not sure if it's right, but it's consistent.
By the way, if there is a question,
I sometimes make a question mark in my notes.
And when I post the notes-- maybe not when
I post the notes on the next day,
but a few days later-- the question marks are eliminated.
Just as an example, last class, there
was some relation for which I didn't
remember how the derivation was done.
When I post those notes today, the derivation
is now in the notes.
I'm not announcing it separately.
But whenever I have a question mark in class
and I don't think it's worth an extra announcement,
it is fixed in the posted notes and you
find the information there.
OK.
So this is now drawing our attention to the single photon.
And this is our next subsection.
This shouldn't come unexpected.
If you want to emphasize the difference between quantum
light and classic light, where does it come from?
Well, it comes from the quantization of light.
It comes from the photon character.
It comes from the effect that light is not
a continuous stream of energy, it comes quantized in photons.
So the granularity of light due to the photon character
is, of course, most pronounced for a single photon.
For instance, when we define the g2 function
as a correlation of detecting a photon
and detecting another photon, well,
if you've only one photon, you find one photon,
and for the next photon, there is no photon to be detected.
So the probability of detecting two photons is 0.
And that only happens when you go down to similar photons.
So this is when certain fluctuations
are most pronounced, because the energy is
dependant on a singular photon.
Let me first address one misconception.
You can say, Well, let's just use a coherent state.
And we talk about the attenuator--
the quantum attenuator with all its operator
beauty-- probably not today but it the next lecture.
But let me already sort of prepare you for that.
When you have a coherent state, when you have a laser beam,
you can put an attenuator.
And your laser beam gets weaker and weaker and weaker.
But it stays a coherent state.
And I will prove that to you very soon.
So you can now say that you take your coherent state
and you attenuate it down that there is only one photon left.
Is that a single photon state?
The answer is no.
It is an attenuated coherent state.
Coherent states, as I've just shown you, are very classical.
They've always Poissonian distribution.
They've always a g2 function of 1.
And attenuation is not changing it.
Attenuation is preserving that.
So now I want to show you explicitly
why an attenuated coherent state may have an average photon
number of 1 but it shares nothing else with a quantum
state of 1 photon of n equals 1 Fock state.
So a coherent state with an expectation value of 1 photon
is not a single photon.
And this can be, for instance, expressed
by looking what is actually the probability,
if you have such a coherent state,
to find 1, 2, or 3 photons.
Well, the probability to find 0 photons-- no photon at all--
is actually 1/3.
So the probability to find 1 photon or to find 0 photons
is the same.
The probability to find 2 photons is 0.18.
So here you have a probability of finding a photon
and correlating it with the next photon click.
And you have even, on first sight,
a surprisingly large probability to find 3 and 4 photons.
So in 2 percent of the cases, you will find 4 photons.
Whereas in contrast, the Fock state with quantum number n
equals 1 is an eigenstate of the number operator
beta with eigenvalue n equals 1.
So that tells you, if you want to get n equals 1,
if you want to get a single photon state,
you cannot just use a strong laser beam or a strong light
source and attenuate it.
You have to work with something which genuinely creates only 1
photon without any ambiguity, without any fluctuations,
without any possibility of creating 2 photons.
And I would actually say over the last 10 or 15 years,
the creation of single photons has
been sort of a small cottage industry,
because single photons are often needed for protocols in quantum
computation-- for experiments which really require accurate
quantum state preparation of light--
and in particular, non-classical light.
But of course, there are ways how you can get single photons.
And this is, well, you start with single atoms.
If you have a single atom in the excited state,
it can emit only one photon.
So in other words, you cannot-- Usually,
we don't have the tools to prepare a single photon--
to take a single photon out of many, many photons--
and store it separately.
But what we can control is single atoms.
We can prepare single atoms.
And then we can make sure that single atoms
create single photons.
It's a little bit a way that we cannot control the bullets
which are fired.
But we can control the guns.
And we make sure that each gun can emit exactly one bullet.
So that's a way how we can create
non-classical states of light.
Yes.
So let's now look for those single photons
at the quasi-probability distribution.
So we obtain the quasi-probability distribution
by taking the single photon state
and projecting it on a coherent state.
You remember-- it's higher up in the notes-- what
the coherent state alpha is in the Fock state basis,
in a number basis.
And so we're just picking out n equals 1.
And this was nothing else than alpha
squared times e to the minus alpha squared.
So therefore, in these diagrams with the real part
and imaginary part, if I plot the quasi-probability
distribution, we get a ring.
The ring immediately tells us that there is no phase defined.
All the phases are equally probable.
And that also means, if you don't
have a phase, the average value of the electric field this 0.
Since the equation of single photons
is essential for studying non-classical light,
but also since it's a very active frontier of our field--
actually, one of the leaders in this field
is Professor Vladan Vuletic here at MIT--
let me at least give you a taste of how
to create single photons.
I already gave you the major ingredient.
Namely, it's about-- it involves single atoms.
But it's a little bit more demanding like that.
So you want to take 1 atom home or 1 ion.
But the problem is if the atom or ion emits a light,
it can emit the light into all directions.
And therefore, you have a single photon afterwards.
But you have many, many different spacial modes.
And in any given mode, it will not have a single photon.
So therefore, what you have to add to the single atom
or single ion is-- you have to put it in a cavity.
And then, in the limit of a very, very high finesse cavity,
the probability will be very, very high
that your photon is emitted into the mode of the cavity
and it is not emitted perpendicular into other modes.
But that itself is not yet sufficient,
because you have a single photon,
but you also want to know when does the single photon arrive.
You want to do experiments.
You want single photons on demand.
And so one option is that you prepare your atoms
in the ground state.
You take a pi pulse, which, with 100 percent probability,
excites the atom to the excited state.
And then within an actual lifetime,
or maybe even in resonator-enhanced inverse
natural line widths, the single photon is emitted.
So therefore, you pump the system
and then you know within the next few nanoseconds
your cavity mode will have a single photon.
The problem here is that you're dealing--
You have to prepare single atoms, which is difficult.
You have to couple them to a high finesse cavity.
There is technically another approach,
where you use many atoms.
So you could say that many atoms,
you no longer have a single gun.
So therefore, you can get several photons.
But the singleness in photons comes because the photons
are now heralded-- they are announced.
And the idea is the following.
If you start with the state 1, and you have an excited state,
and you have a pump pulse.
Now, if you have many atoms, you have a much higher efficiency
of pumping atoms comes to the excited state,
because if you have n atoms, it's n times more efficient.
But now you take the following situation.
You wait until there is a Raman transition--
until you detect the photon for Raman transition,
where the excited state decays to the state 2.
At this moment, you know I have now 1 atom in state 2.
So in other words, you're not starting
with single atoms, which is sometimes more demanding.
How can you prepare the system?
It has other disadvantages that n atoms have.
I'm not sure if I should mention it, but with n atoms,
they have a super-radiant factor n with n atoms.
You can get an n-times enhancement of emission
into a single mode.
So there are real, massive advantages
in working with many atoms.
But now you know that one atom is in state 2
when you detect the first photon.
And then you have the same situation which we had above.
You can now take your system-- let me just
redraw the level diagram.
Excited state, state 1, state 2.
You know now that you have 1 single atom here.
And by using a laser pulse, you can excite it
to the excited state.
And then you observe the single photon 2.
So in other words, the observation
off the first photon tells you that your system
is prepared with 1 atom in state 2.
And then you can get a single photon out
of it, which is the photon for the inverse Raman
transition when you pump the system back to state 1.
AUDIENCE: Doesn't this rely on you
being able to reliably prepare a single atom in another quantum
state?
Won't there always be some uncertainty
as to how many atoms you can prepare in one sitting?
PROFESSOR: There are some uncertainties.
And what you're saying is correct.
We don't have perfect single photon sources.
And people characterize the fidelity of the single photon
source.
For instance, if you detect the single photon, you would say,
Now my system is ready to emit a single photon triggered
by this pump pulse.
And if you can get now a single photon
in 90 percent of the cases, you publish a wonderful paper,
because you've set a new record for the fidelity
of a single photon source.
So people are really struggling with some
of those uncertainties.
But to involve, for instance, three levels
is sort of an advantage, because you're not
limited by the preparation of a single atom, for instance.
You can have many atoms.
The atoms are always there.
And the moment one atom is prepared in state 2,
this atom announces itself with a single photon.
So it takes a lot of uncertainty out that the system says,
with the first photon, I'm ready now.
I can emit a single photon.
And then you get your single photon.
Then you can gate your whole experiment
to a time following the detection of the first photon.
And for your gated time afterwards, you
have a very, very high probability
of finding this photon.
Or if you want, you can now do experiments with 2 photons.
If you keep this laser on and pump the atom immediately back,
you have actually a single photon here,
followed by a single photon here.
It's sort of click clack.
And now you can do correlations between two
different single photon states and such.
It's a very rich frontier of our field.
You have another question?
AUDIENCE: Yeah.
Is this also within a cavity now, like the single photon
[? reflection ?] in the first test?
PROFESSOR: Yes.
Actually, these photons-- the wavy lines--
are emitted spontaneously.
Spontaneous emission, pretty much by definition,
goes into all possible spatial modes.
And the only way to control the spatial mode is with a cavity.
What I don't want to go into details--
but those who have an understanding of that--
is the following.
If you n atoms and you prepare one atom here in state 2,
you do not know which of your n atoms is prepared.
So you have n indistinguishable possibilities.
And if you have n indistinguishable probabilities
which atom you have prepared, the emission of the photon
back is n times enhanced.
So therefore, you have actually a system
which has an n times stronger coupling to the cavity.
So having n atoms makes it much, much easier
to construct a high finesse cavity.
You get this super radiance increase
of the strong coupling for free.
And this is why there are many reasons why you do not
want to work with the single ion or a single atom.
It's possible.
For very fundamental experiments,
people have shown that it can be done.
But technologically, it's much easier and much
more robust to work with many atoms.
But then you need the many atoms have
to tell you when 1 atom is prepared to emit now
a single photon.
If you're interested in this subject,
talk to the students in [INAUDIBLE] school.
They're really the world experts in that.
OK.
Finally, we're not really getting to squeeze light.
We start with squeezing the light tomorrow.
The last thing in this major section
is the famous Hanbury Brown Twiss experiment.
This was a landmark experiment done in the 1950s.
And it was the first experiment which
really looked at g2 functions correlations, which one could
say was the beginning of quantum optics
and modern experiments with light.
Probably until then, light was just an electromagnetic wave
people regarded as boring.
But by measuring now correlations,
people realize that it's an interesting object to study.
I want to just go over the basic scheme.
You will look at it a little bit more
closely in your homework assignment.
So the idea behind an Hanbury Brown Twiss experiment
is that you have a light source and you
want to characterize it.
So the light source emits light.
And now, this is important.
Whenever you want to detect 2 photons,
you want to figure out it's 1 photon followed
by another photon.
Technically, you cannot do it with a photomultiplier,
because when a photomultiplier clicks, it needs many,
many nanoseconds for the photomultiplier to read out
the signal and recover to recharge up its electrodes
and be ready to detect the next photon.
So therefore, when you want to find click click--
double clicks in the stream of light--
you have to involve a beam splitter
and involve two photodetectors.
Sure, you now need an adequate description.
But in principle, you can now find
2 photons which are only a few picoseconds apart,
because the first photon is observed by the first detector,
and the second photon is detected
by the second detector.
Sure, you would say, What happens if you're in the beam
splitter, both photons go to detector one?
Well, that's tough luck.
You take 1 chance in 2 that the 2 photons
go to the 2 different detectors.
But the beam splitter has to be part of your experiment,
has to become part of your description.
And this is what you do in homework number one.
So what we are asking here when we measure the g2 function.
Let's assume that tau equals 0.
We are reading out a signal.
And what we are using is a coincidence detector.
We are looking that in a very small temporal window,
2 photons are detected simultaneously.
Well, this is the quantum version.
The classical version is the g2 function
is the product of the intensity at time t times
the product of the intensity at time t plus tau.
So what you would do here in this circuit,
you would take the signal from this detector, the signal
from that, and multiply the two.
This is how you determine I of t times I of t plus tau.
So whether you do it in the classical domain
or whether you do it in the quantum domain,
this is the way how you experimentally
measure the second order correlation function.
In the classic limit, you don't have to worry about it.
The classical limit is always a limit of high intensity.
So at any given time, you have a ton of photons.
And then, at beam splitter, half of the photons go left,
half of the photons go right.
You have an equal splitting.
You have exactly half the intensity
in the left arm and the right arm after the beam splitter.
So there is no problem at all with the quantization,
because this is the classical limit.
So classically you can say-- actually,
I shouldn't say photon, I should actually say intensity.
The intensity splits equally.
And if you do the measurement with the coincidence detector,
you find the difference between the g2 function,
which I discussed earlier.
That the g2 function is 1 for coherent light or laser light,
and the g2 function is 2 for thermal light.
That's actually the only way how you can distinguish
a light bulb from a laser beam.
If you put a light bulb into a cavity or couple the light
from a light bulb into a fiber, the light
becomes spatially a single mode.
If you put a very narrow spectral filter
or Fabry-Perot cavity, the light becomes
single mode in frequency domain.
So when you take a light bulb and spatially and spectrally
filter it that it is single mode, in terms of the mode,
it is single mode as a single mode laser.
But it is the correlation experimental
which shows you that you started with a thermal source.
It has a g2 function of 2, and you can never get rid of it,
whereas the laser beam has a g2 function of 1.
OK.
So this is the classical version.
In the quantum version, especially when
we have a single photon, the photon
can go to only one detector.
And that means, for this extreme case of a single photon,
the g2 function is 0.
Anyway, you will look at those situations in more
detail in you homework assignment.
Yes?
AUDIENCE: Sir, if you know that different
for tau non-zero, then even the single mode in the function
would have a final g2 [INAUDIBLE], because there
would be some probability [INAUDIBLE].
PROFESSOR: We have to be now a little bit careful.
If you have a single mode Fock state, there's only 1 photon.
And once this photon is detected, we have vacuum,
we have no photon left.
So what may be confusing here is, on first sight,
is that the way how the experiment is done and has been
historically done, it's not done we Fock states.
It's done with the light source in a continuous stream
of photons.
So Hanbury Brown Twiss experiment,
he looked at in light bulb, or he
looked at star light and such, and determined the intensity
correlation.
He couldn't do the experiment with lasers,
but now we would with a laser beam.
And we find that g2 is 1.
But in those situations, we actually
do not have the way-- we have actually a beam, which
is a stream of photons.
And this requires a little bit different description.
In other words, when I have a coherent state,
a laser beam is always replenishing
the coherent state.
The laser beam preserves the electric field,
whereas, strictly speaking, the way how I described it
for pedagogical reasons is you have a cavity, which
is filled with a coherent state alpha.
And then you start analyzing that.
But in other words, what we have focused
in the simple description is that we have a quantum
state which is prepared-- it's a closed system.
And now we do our detection.
The experiment how it is done is often
done as in an open system, where you couple the system
to a light source, which is always
replenishing your experiment.
So to be careful, especially with a single photon.
I think, in essence, the experiment
would be done with a single photon light source.
But the single photon light source
has a high repetition rate.
I showed you how to generate single photons.
You have an heralded single photon.
You know the single photon comes now.
You do the experiment.
And then you repeat it again.
And then what you have is you only
look at one single photon bunch at a time.
And then you find indeed, that during that temporal window,
you will never find a second photon.
So this may happen in a few nanoseconds.
Then you wait a microsecond, and the next photon arrives.
Of course, if you would now describe your light source,
what you is the g2 of tau, and tau is a microsecond,
and the microsecond is the time between the first single photon
burst and the next single photon burst,
now you will find that your g2 function is not 0.
But this is then related to the repetition
rate of a single photon and not to the single photon itself.
I think you've got the taste.
Your homework is a really simple.
You just deal with a closed system.
You look at a quantum state at a time.
But if you map it onto a beam experiment,
you have to think about what does it
mean to have a replenishment of the quantum state.
Any question?
I know I have to stop, but.
AUDIENCE: In the beam splitter, the other port
of the beam splitter should do some stuff, right?
PROFESSOR: Yes.
Actually, this will keep us busy for a while, that even if you
do not put in any light, we put in the vacuum state.
And you will find that the description of the beam
splitter in the quantum state is incomplete,
unless you specify that your beam splitter is coupling
in from a different mode the vacuum state.
Yes, this will be very important.
And we will cover it in its full glory.
AUDIENCE: I just have a quick question.
The way we can distinguish from a thermal state
and a coherent state is through the g2.
So from an experimental perspective,
could we do most of experiments if we just
had thermal sources that were very single mode, so to speak?
PROFESSOR: Well, yes.
All the laser cooling, all the [INAUDIBLE], all the absorption
imaging-- all that would work if you had a single mode
thermal source.
Let's face it, we say we use lasers for everything
we're doing.
But the only property which distinguishes
a laser from the thermal light source-- we're
not taking advantage of it.
Of course practically, if you take a thermal source
and filter it down to a single mode,
you will be left with only a few photons.
You cannot create an intense enough single mode light source
unless you use stimulated emission into a single mode.
And that's a laser.
Colin.
AUDIENCE: The correlation function for, for example,
an LED light-- that's not really a thermal source.
It's not described by a Maxwell-Boltzmann distribution.
And that would be closer to something for a laser,
even though there is no cavity?
PROFESSOR: Well, LEDs in some limit are quantum objects.
Actually, do you know what the g2 function of LED is?
Does it become laser-like, or does it
become even antibunched?
Because if it's a relaxation mechanism-- So anyway,
what Colin says is there are actually
more different light sources that just the laser
and the thermal light source.
There are LEDs or semiconductor devices, which provide photons
with interesting statistical properties.
I've heard about it-- that LED light sources have
some special properties, but I don't
remember which was the one.
OK.
We have to stop.
I'll see you on Wednesday.