M1L4d.txt

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So after being familiar with this operator,
we want to know what is this operator doing.
I can describe now what this operator
does in a Schrodinger picture or in a Heisenberg picture.
I pick whatever is more convenient
and, for now, this is the Heisenberg picture.
So in the Heisenberg picture, what is changing
are the operators.
So therefore, in the Heisenberg picture,
this unitary transformation transforms
the operators and we can study what happens when
we transform the operator x.
The operator x is transformed by multiplying
from the left side with s from the right hand side,
with s dega.
And you are familiar with expressions like this
and how to disentangle them.
If you have e to the i alpha, e to the minus
i alpha, if you could move the alpha past x-- so
if a and x commute, i a minus i a would just give unity,
so therefore, this expression is just
x unless you have non- vanishing commutators between a and x.
And I think you have solved in your basic quantum mechanics
course many such problems which involve
identities of that form.
Then the higher order commutator,
the commutator of a with the commutator of a x,
and unless one of those commutator
vanishes, you can get an infinite series.
Now our operator a is nothing else than the annihilation
operator, a square, minus the creation operator, a dega
square.
So we can express everything in terms of a and a dega.
And the position operator in our harmonic oscillator
can also be expressed by a and a dega.
So by doing elementary manipulations on the right hand
side and recouping terms, we find immediately
that the transformation, the unitary transformation,
of the Heisenberg operator x gives you an x operator back,
but multiply it with an exponential e to the a.
And if we would do the same to the momentum operator, which
is a minus a dega over square root 2,
we will find that the unitary transformation of the momentum
operator is de-amplifying the momentum operator
by an exponential factor.

So if we would assume that we have a vacuum
state in the harmonic oscillator, and the vacuum
state, well, classically, it would
be at x equal 0 p equals 0.
Quantum mechanically, you have zero point noise
in x and zero point noise in p.
Then you would find that the squeezing operator is
amplifying the quantum noise in x,
but it squeezes or reduces the noise in p.
So if we apply this squeezing operator to the vacuum state,
we obtain what is usually called squeezed vacuum.
And it means that in this quasi probability diagram,
the action of the squeezing operator
is turning the vacuum state into an ellipse.

What happens to energy here?
The vacuum state is the lowest energy state.

If you now act with a squeezing operator to it,
we obtain a state which has the same energy--
is it energy conserving?
Or very high energy?

Yes.
Why?
It's no longer [INAUDIBLE].
Pardon?
It's no longer [INAUDIBLE].
Yeah, don't let you fool-- it's a vacuum state.
We act on the vacuum state, but we
get a state which is no longer the vacuum state.
Now can you maybe-- the reason why we have extra energy,
the squeezed vacuum is very, very energetic
because the squeezing operator had a dega squared--
a dega square e square.
Well, e square, the annihilation operator acting on the vacuum,
gives 0.
But what we are creating now, we are acting on the vacuum
and we are creating passive photons.
So we are adding, really, energy to the system and the energy,
of course, comes from the drive laser,
from the laser to omega naught, which
delivers the energy in forms of photons
which are split into half and they go into our quantum field.

Now in the limit of infinite squeezing,
I will show it to you mathematically,
but it's nice to discuss already here.
In the limit of infinity squeezing,
what is the state we are getting?

Eigenstate--
Eigenstate of momentum.
So we get the p equals 0 eigenstate.

So what is the energy of the p equals 0 eigenstate?

It has to contain all number sets.
Contains all number sets?
OK, you just think quantum mechanic--
you think immediately into number sets, which is great,
but in a more pedestrian way, the p
equals 0 state has no kinetic energy.
But if a state is localized in momentum p equals 0,
it has to be infinitely smeared out on the x-axis.
And don't forget, we have an harmonic oscillator potential.
If you have a particle which is completely localized in x,
it has infinite potential energy at the wings.
So therefore, in the limit of extreme squeezing,
we involve an extreme number of number states-- actually,
I want to be more specific of photon pass,
we have states with 2 n and n can be arbitrarily large,
but you also see in the classical picture,
that photon seen in the classical picture, what
we get here when we squeeze it is we
get the p equals 0 eigenstate which has infinite energy
due to the harmonic oscillator potential.

If we would allow this system now
after we have squeezed it to evolve for a quarter
period in the harmonic oscillator,
then the ellipse would turn into an vertical ellipse.
So this is now an eigenstate of x.
It's the x equals 0 eigenstate, but the x
equals 0 eigenstate has also infinite energy
because, due to Heisenberg's uncertainty relation,
it involves momentum states of infinite momentum.

Questions?
So this is a photon field, right?
So p is roughly the electrical field, right?
Yes.
So, it's kind of the electric field has 0
and x is kind of the a.
[INAUDIBLE] So the electrical field is the squeezed?
Yes.
It means we have no electrical field?
We come to that in a moment.
I will actually-- I wanted to do a little bit more math to show
you what is-- I wanted to derive for you an expression
of this squeeze state in terms of the number basis and such,
but your question is absolutely correct.
Your question, let me mention, something
which is absolutely correct.
By squeezing that, we have now-- the p
x is this electric field axis.
So now we have actually in the limit of infinite squeezing,
we have an electric field which is no uncertainty anymore.
By squeezing the coherent state into momentum eigenstate,
we have created a sharp value for the electric field.
We have created an electric field eigenstate.
Value would say it's pretty boring because the only
electric fields that we've created
is electric field e equals 0, but in the next half hour,
we want to discuss the displacement operator
and I will tell you what it is that we can now
move the ellipsis and move the circles anywhere where we want.
So once we have an electric field
state which has a sharp value of the electric field of e
equals 0, we can just translate it.
But before you get too excited about having
an eigenstate of the electric field,
I want you to think about it what
happened after one quarter period of the harmonic
oscillator frequency.
It turns upside down and your electric field
has an infinite variance.

So that's what quantum mechanics tells us.
We can create electric fields which are very precise,
but only for a short moment.
So in other words, this electric field state
which we have created would have a sharp value,
a moment later, it would be very smeared out,
and it has a sharp value again, and then
it's smeared out again.
I mean that's what squeeze states are.
Other questions?
Is that why we need [INAUDIBLE]?
That's why we need homodyne detection, yes, exactly.
If we have squeezed something which is sort of narrow,
that's great for measurement.
Now we can do a measurement of maybe a [INAUDIBLE]
measurement for cavitational raised with higher precision
because we have a more precise value in our quantum state.
But we have to look at it at the right time.
We have to look at it synchronized
with the harmonic motion.
And homodyne detection means we look only at the sine component
or at the cosine component.
Or if I want to simplify it, what
you want to do is if you have a state like this,
you want to measure the electric field, so to speak,
stroboscopically You want to look at your system
always when the ellipse is like this.
And the stroboscopic measurement is, as I will show you,
in essence a [INAUDIBLE] measurement,
which is phase sensitive and this
will be homodyne detection.
So we can only take advantage of the squeezing
of having less uncertainty in quadrature component
if we do phase sensitive detection, which
is homodyne detection.

OK.
Back to basics.

We want to explicitly calculate now
how does squeezed vacuum look like.
So we are now-- I actually want to do it twice
because it's useful.
We have to see it in two different basis set.
One is I want to write down the squeezed vacuum for you
in a number representation and then in the coherent state
representation.
So this squeezing operator is an exponential function involving
a square and a dega square.
And of course, we are now using the Taylor expansion of that.
We are acting on the vacuum state
and I will not do the calculation.
It's, again, elementary.
You have n factorial, you have terms with if a dega acts on 0,
you pay two photons in it.
If acts again, it adds two more photons
in the matrix element of a dega acting on n
is square root n plus 1.
You're just sort of rearrange the terms and what you find
is what I will write you down in the next line.
But the important thing you should immediately realize
is that the squeeze state has something very special.
It is a superposition of number states,
but all number states are even because our squeezing operator
creates passive photons.
This is what the parametric down conversion does.
We inject photons into the vacuum,
but always exactly in pass.
And therefore, it's not sort of a random state,
it's a highly correlated state with a very special properties.
OK.
If you do the calculation and regroup the terms,
you get factorials, you get 2 to the n,
you get another factorial, you get hyperbolic tangent.
Solve it with a power n.
And the normalization is done by the square root
of the core's function.

And the more we squeeze, the larger
the amplitudes at higher and higher in.
But this is also obvious from the graphic representation
I've shown you.
Let me add the coherent state representation.

So the coherent states are related to the number states
in that way.
So if you transform now from number states
to coherent states, the straightforward calculation
gives now superposition over coherent states.
Coherent states require an integral e to the minus--
e to the a over 2 divided by-- Anyway,
all these expressions they may not
be in its general form too illuminating,
but those things can be done analytically.
I just want to mention the interesting limiting case
of infinite squeezing.
The limiting case, of course, is quite interesting.
If r goes to infinity, you can show
that this is simply the integral d alpha over coherent states.
So we have discussed graphically the situation
where we had-- so these are quasi probabilities.
So in that case of infinite squeezing,
we have the momentum eigenstate p equals 0.
This is the limit of the infinitely squeezed vacuum.
And in a coherent state representation,
it is the integral over coherent state alpha--
I'm pretty sure that alpha is real here, so seeing that now.
There is a second limit which happens simply and say
by rotation or by time evolution which is the x
equals 0 eigenstate.
And this is proportional to the integral over alpha
when we take the coherent state i alpha
and we integrate from minus to plus infinity.
OK, so we have connected our squeeze states,
the squeezed vacuum with number states, with coherent states.

Now we need one more thing.
So far, we've only squeezed the vacuum.
And we have defined the squeezing operator
that it takes a vacuum state and elongates it.
In order to generate more general states,
we want to get away from the origin