M1L3g.txt

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I promised you, and this is what we
want to do now, is we 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 to the Heisenberg
Uncertainty Principle, which says that the width in p
and the width 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,
symmetric and antisymmetric combinations--
so let me just call those prefectors q0 and this one p0.

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 on minus a.
If we act with a on alpha, we get alpha,
because alpha-- the coherent state--
is an eigenstate of alpha.
With a dagger-- well, 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 all of it
just in powers of alpha, alpha square, alpha star, and such.
So what we want is, we want to find out
what are the fluctuations delta q, delta p.
Just 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.
So-- 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 two is Heisenberg uncertainty limit.
So therefore, coherent states-- 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.
We want to find the minimum uncertainty states.
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 two uncertainties--
the area of the shaded region delta p times delta q--
is h bar over 2.
Or, if I want to re-write that in the real part of alpha,
times the uncertainty in the imaginary part of alpha,
for the quasi-probabilities, then this is 1/4.
OK, 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 lets do that.
The fluctuations of the intensity
are usually expressed by the second order temporal coherence
function.
And that's what we want to introduce now.

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 classical description first.
So I want to first tell you what is a second order coherence
function for classical light, which
has a classical 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-- well, you're just sitting here.
You perceive 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 one.
Well, pretty much what you show is for tau equals 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 one.
It can be smaller than 1.
And that's actually an interesting, you could say,
litmus test for the quantumness if you generate
states of the electromagnetic field-- Fock
states, photon number states, we'll 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-- an intensity
of the electromagnetic field-- with that photon state,
because whenever you can associate a classical intensity
with that, 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 that 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 tomography, 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.
Yeah?
So in the definition of g2 classical,
the expectations values are averaging over time, not
over ensembles of i of t?
Ah, OK.
Yeah, you can average over ensemble,
so you can have 1,000 light bulbs, switch them all on,
and then measure, at a certain time, that.
But what we assume here is that the light bulb is
on continuously, so things don't really depend on t.
So you can set-- I mean, I've implicitly
taken care of 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 an ensemble in steady state.
But, I mean, it's sort of the same story again and again.
In classical physics, you can determine an ensemble average
by taking a [INAUDIBLE] 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 the 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 you can measure the intensity of light
is with a photo multiplier.
It makes click, the photon is absorbed.
And this is not fully described by the electric field.
Just as soon as 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 two photons.
So you start with an initial state.
You annihilate two photons, and then you have a final state.
But if you are only interested in what
is the probability you want to characterize
your initial state, you may want to sum over all final states.
And this is your total probability
that you can absorb two photons out of an initial state.
But since you sum now over all final states,
this turns into-- now I could put for you in, as
in the final state as 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 one photon, which is a dagger a expectation
value squared.
I just try to be a little bit sort of 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 photo multipliers, that's what you measure.
But I wanted to show you how it is
related to the intensity correlation function, defined
classically.
Cory?
But this click is constant in tau, which doesn't really
make sense intuitively to me.
Because, at least classically, it
shouldn't be constant at tau.
OK.
That's the next thing I wanted to say,
is I've swept here-- I mean, I just
wanted to give you the structure of the operators,
and not get sort of 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-- I mean,
what is a single mode?
It's just a sine wave, and nothing
happens as a function of time.
It's constant.
So both modes coincide?

You know, I'm squeezing a textbook of quantum optics
into two classes.
And I want to give you sort of the ideas and the concepts.
What I've sort of mixed up here-- deliberately--
is I've given you the classical definition of an intensity
correlation function, which is the famous [INAUDIBLE]
experiment.
And if you have correlations as a function of time,
then I have motivated how this should be defined
for quantum states of light.
But when I transition 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 a 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
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'll find everything you want for coherence time.
Coherence time is the time for two modes to get out of phase.
But if you have one mode, there is no coherence time.
And when you find, for classical light,
that the g2 function has a peak which decays with time,
it takes 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 classical constraint
that g2 has to be larger than 1.
We find now g2 functions which are smaller than 1.
And this sort of tells us now, where
do we find the most non-classical behavior?
Maybe 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.

And it's related to an average and 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.

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.