M1L4c.txt

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OK, so we know what classical squeezing is.
And what we have learned or so, and this helps me
out a lot to motivate how we squeeze in quantum mechanics,
you've realized that what is really essential here
is to drive at 2 omega 0.
So what we need now to do squeezing in the [? Kringle ?]
domain, if you want to squeeze light,
we need something at 2 omega 0.
So we wanted to discuss a quantum harmonic oscillator.
We want to have some form of parametric drive at 2 omega 0.
And this will result in squeezed states.
Now, what does it require if you're
going to bring in 2 omega 0?
Well, let's not forget our harmonic oscillators
are modes of the electromagnetic field.
If you now want to couple a mode of the electromagnetic field
at 2 omega 0 with our harmonic oscillator at omega 0,
we need a coupling between two electromagnetic fields.
So therefore, we need non-linear interactions between photons.

So, this was a tautology.
We need non-linear physics, which
leads to interactions between photons.
And linear physics means each harmonic oscillator
is independent.
So we need some non-linear process
which will be equivalent to have interactions between photons.

And the device which will provide that
is an optical parametric oscillator.

I could spend a long time explaining to you how
those non-linear crystals work, what is the polarization, what
is the polarizability, how do you drive it,
what is a linearity.
But for the discussion in this class, which
focuses on fundamental concepts, I can actually bypass it
by just saying, assume you have a system--
and this is actually what the optical parametric oscillator
does is you pump it with photons at 2 omega 0.
And then, the crystal generates two photons
at omega 0, which, of course, is consistent with energy
conservation.
And if you fulfill some phase matching condition,
it's also consistent with momentum conservation.
But I don't want to go into phase matching at this point.

Technically, this is done as simple as that.
You have to pick the right crystal.
Actually, a crystal which does mix mixing between three photon
fields cannot have inversion symmetry.
Otherwise, this non-linear term is zero.
But what you need is a special crystal,
KDP is a common choice.
And this crystal will now do for us the following.

You shine a laser light, let's say, at 532 nanometer,
a green light.
And then this photon breaks up into two photons of omega 0.
And this is how it's done in the laboratory.
And the piece of art is you have to pick the right crystal.
It has to be cut at the right angle.
You may have to heat it and make sure
that you select the same picture for which
some form of resonant condition is fulfilled to do that.
But in essence, that's what you do-- one laser beam
put in a crystal and then the fourth
is broke into two equal parts.
And these are our two photons at omega 0.

OK, so we can bypass-- I hope you enjoy the elegance.
We can completely bypass all the material physics
by putting operators on it.
We call this mode B and we call this mode A.
So the whole parametric process, the down conversion process
of one photon into two, is now described
by the following Hamiltonian.
We destroy a photon in mode B at 2 omega 0.
And now, we create two photons-- we
destroy a photon at 2 omega 0, create two photons at omega 0.
And since the Hamiltonian has to be Hermitian,
the opposite, the time reverse process,
has to be possible too.
And that means we destroy two photons at omega 0
and create one photon at 2 omega 0.

So now, we forget about non-linear crystals,
about non-inversion symmetry and materials.
We just take this Hamiltonian and play with it.
So this is how we now discuss-- we [INAUDIBLE]
can discuss, by simply looking at the Hamiltonian, what
is a time evolution of a photon field under this Hamiltonian?
We figure out what happens when you send light for crystal
and what is the output.
And I want to show you now that the output of that
is squeezed light, which is exactly what I promised you
with these quasi-probabilities.
We have a coherent state, which is a nice circle.
We time evolve the coherent state of a nice round
circle with this Hamiltonian.
And what we get is an ellipse.

And if you want intuition, look at the classical example
we did before which really tells you,
in a more intuitive way, what is happening.
OK, but we want to make one simplifying assumption here.
And this is that we pump the crystal at 2 omega
0 with a strong laser beam.
So we assume that the mode B is a powerful laser
beam or, in other words, a strong coherent state.
So we assume that the mode B is in a coherent state described
by-- coherent states are obviously
labeled with a complex parameter which I call beta now.
Well, it's mode B. Therefore, I call it beta.
For more A, I've called it alpha.
And the coherent state has an amplitude, which I call over 2.
And it has a phase.
We know, of course, that the operator B acting on beta
gives us beta times beta because a coherent state is
an eigenstate of the annihilation operator.
And this is now-- but when we look
at the action of the operator B plus the photon creation
operator, well, you know, the coherent state
is not an eigenstate of the creation operator.
It's only an eigenstate of the annihilation operator.
But what sort of happens is the coherent state
is a sum over many, many number of states with n.
And the creation operator goes from n to n plus 1
and has matrix elements which are square root n plus 1.
So in other words, if n is large and if we
don't care about the subtle difference between n
and 1 plus 1, in this limit, the coherent state
is also an eigenstate of the creation operator
with an eigenvalue which is beta star.
So this means that we have a coherent state which is strong.
Strong means it has a large amplitude
of the electric field.
Then, the photon states which are involved n are large
and we don't care whether we have n or n plus 1.
This is actually also, I should mention it here explicitly,
this is sort of the step when we have a quantum
description of light and we replace the operators B and B
dega by a C number.
Then, we really go back to classical physics.
Then, we pretend that we have a classical electric field which
is described by the imaginary part of beta.
So when you have an Hamiltonian where you write down
an electric field and the electric field
is not changing-- you have an external electric field--
this is really the limit of a quantum field
where you've eliminated the operator by a C number.
And this is essentially your electric field.
And we do this approximation here.
Because we are interested in the quantum features of mode A,
A is our quantom mode with single photons
or with a vacuum state and we want to squeeze it.
B is just the [INAUDIBLE] parametric [? drive. ?]
So with this approximation, we have only the A operators.
So this is our operator.
So--
[INAUDIBLE]
So [INAUDIBLE] your question?
Beta [? wouldn't give ?] us a beta star, right?

Yes, thank you.
That means [? it ?] should be a minus sign, yes.
So, I've motivated our discussion
with non-linear crystal, which generates pair of photons.
This is the Hamiltonian which describes it.
And if you want to have a time evolution by this Hamiltonian,
you put this Hamiltonian into a time evolution operator.
In other words, e to the minus iht is the time evolution.
So if we now evolve a quantum state of light for a fixed time
t, we apply the operator to the minus iht
to the quantum state of light.
So, what I've just said is now the motivation
for a definition of this-- the definition
of the squeezing operator.
The squeezing operator s of r is defined
to be the exponent of minus r over 2 a squared
minus a [? dega ?] squared.
So this is related to the discussion above.
You would say, hey, you wanted that time evolution.
Where is the i?
Well, I've just made a choice of phi.
If phi is chosen to be pi over 2,
then the time evolution with the Hamiltonian
above gives me the squeezing operator below.
So with that motivation, we are now
studying what is the squeezing operator doing
to quantum states of light.

Any questions about that?
I know I spent a lot of time on it.
I could have taught this class by just saying, here
is an operator, the squeezing operator.
Trust me, it does wonderful things.
And then we can work out everything.
But I find this unsatisfying, so I
wanted to show you what is really behind this operator
and I wanted you to have a feeling
where does this operator come from and what is it doing.
But in essence, what I've introduced into our description
is now an operator which is creating and destroying
pairs of photons.
And this will do actually wonderful things to our quantum
seats.
So, what are the properties of the squeezing operator?
Well, what is important is it is unitary.
It has a unitary time evolution.
You may not see that immediately, so
let me explain that.
You know from your basic quantum mechanics course
that e to the i operator a is unitary when a is Hermitian.
So the squeezing operator with the definition
above can be written as I factor out two i's over 2
a squared minus a [? dega ?] squared.
And you can immediately verify that this part here
is Hermitian.
If you do the Hermitian, you get a square
turns into a [? dega ?] squared. a [? dega ?] squared turns
into a square.
So we have a problem with a minus sign.
But if you do the complex conjugate of i,
this takes care of the minus sign.
So this part is Hermitian.
We multiply it with i.
Therefore, this whole operator does a unitary transformation