M1L4g.txt

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Before I continue with the material,
I want to address a question, which
actually came up in discussions with several of the students.
And this is I realize that some people said OK, everything
makes sense.
But what are we plotting?
What is really squeezed?
Are we squeezing in the spatial domain?
Are we squeezing in the temporal domain?
So the plots look wonderful with these ellipses in the circles,
but what is it really we're doing here?
So let me address that.
First of all, we are talking about a single harmonic
oscillator.
We showed that the Maxwell's equations
can be reduced to a bunch of harmonic oscillator
equation, one for each mode.
And now we are talking today and in the previous few classes,
about one single mode, about one harmonic oscillator.
And the harmonic oscillator has canonically variables
of momentum and position, but this is just
to make a connection for you with something
you have learned.
What we're talking about is a single harmonic oscillator,
which is one single mode of the electromagnetic field.
So maybe let me draw a cartoon for that.
So let's assume we have a cavity.
We have an electromagnetic wave.
That is propagation.
There is e to the ikz.
There is transverse confinement.
Maybe there is a gauche in e to the minus
x square plus y square over some beam [INAUDIBLE] parameter.
All that is simply the spatial mode.
And we just take that for given, because we are not
solving the spatial differential equation.
All we are doing it, we are looking at this one mode,
and the two degrees of freedom is
that this mode can have a certain number of photons,
it's the amplitude, and the second one
is you can see the temporal phase.
It can be a cosine omega t, it can be a sine omega t,
it can be a superposition.
But whatever we are talking is in this mode.
There is nothing happening in the spatial domain.
They're just asking what is sort of the oscillation
in this mode.
The whole mode does what it should.
It has a prefector, which is the amplitude.
And it has a temporal effect, which we factor out.
And this is what we are talking about.
Let me be a little bit more specific
and say that we use when we are plotting things,
we are plotting the Q representation, the phase space
representation of the statistical operator,
which is simply describing this single mode
of the harmonic oscillator.
And by performing the diagonal matrix elements,
we obtain the Q distribution.
In that case, we have the vacuum,
we have a displaced vacuum, which is in coherent state.
And our x's are, from the very definition,
we are in the complex plane with the real part
and imaginary part of alpha.
However, we can also define a weaker distribution,
which is another phase space distribution.
It's almost the same as s Q distribution,
it's just this little bit smeared out
by a H bar because of some computators,
but nothing you have to worry about.
In that case, the projection of the W function
on the vertical axis, on the y-axis,
is the momentum wave function squared on the x-axis,
it is the spatial wave function squared
of the harmonic oscillator.
So therefore we may sometimes think
when we have a distribution here, we project it,
and we see what is the momentum distribution?
Or what is this spatial distribution
of the mechanical harmonic oscillator, which is analogous,
which is equivalent to the one mode
of the electromagnetic field we are using.
I know it may help you to some extent
to think about the P and Q, but it may also be misleading
because it gives you the impression
something is moving with the momentum P
in to the [INAUDIBLE] space.
Let me therefore emphasize what are the normalized forms of P
and Q. If I do the symmetric and anti-symmetric combination
of the annihilation and creation operator, I over square root 2,
I call those A1 and A2.
And they are nothing else than Q and P
normalized by the characteristic momentum or spatial coordinate
of the harmonic oscillator.
So what is important here is that A1 and A2-- forget
about PQ now, they're equivalent--
but for the electromagnetic field,
A1 and A2 have a very direct interpretation.
They are called the two quadrature operators.
And what I mean by that becomes clear
when I use the Heisenberg representation
for the electric field.
And I'm here using the formula, which is given in the book,
by Weissbluth in page 175.
Some pages copied from this book have
been posted on the website.
So we have our normalization factor,
which is of related to the electric field
that of a single photon.
We have the polarization factor.
But now we have an expression, which
involves the quadrature operators, A1 and A2.
Just to be specific, we are not in a cavity here.
Therefore, we have propagating waves cosine kr sine kr.
But you can also immediately use similar expression.
For the case of a cavity, let us specify at r equals 0.
And then we realize what the two quadrature operators are.
A2 is the operator, which creates and annihilates
an electromagnetic field, so to speak.
You have photons, which have an electric field, which
oscillates at cosine omega t.
And A1 is the quadrature operator for the sine omega t
component.
So therefore, if you simply analyze the electric field,
what is cosine omega 2 is related to the A2 quadrature
operator.
The sine omega 2 oscillation is related to the A1 quadrature
operator.
So life would be, well, easier but more boring
if you could create a pure cosine or pure sine
oscillation of the electromagnetic field,
but you can't because A2 and A2 do not compute.
And there is an uncertainty relation
that delta A1 times delta A2 is larger or equal than 1/2.
And we have the equal sign for coherent states of--
So therefore, if we look at the electric field,
everything moves around periodically
in r because it's a traveling wave in t,
because it's an oscillating wave.
So let's not confuse things.
Let's simply pick or equal 0.
We've already done that.
But now let's pick t equal 0.
At t equals 0, the sine omega t is 0.
And therefore, the distribution for A2, the expectation value,
and the variance for the quadrature operator A2
can be simply read off by looking at the electric field.
So in other words, at t equals 0,
the electric field, which is obtained
by projecting our cause and abilities on the y-axis,
gives the expectation value for A1
and the variance, delta A1 squared.

Somebody says 2.
At t equals 0, yes.
And now if you want to see what the other quadrature
component is, well, we just wait a quarter period
until this sine, which was 0 is maximum,
and the cosine omega t is 0.
So therefore, at pi over 2 omega,
using the projection on the y-axis gifts
us A1 and delta A1 squared.
Or alternatively, we don't need to wait.
We can do t equals 0, and we can project onto the x-axis.
So therefore, and we just a few things into this diagram.
If we had a classical motion, which would simply
be cosine omega t, then if you had a motion which
where only cosine omega t, yes it
would be a point on the y-axis.
However, classically, we can never
have something which is just cosine omega t.
The point has to be blurred into a circle.
This is a coherent state.
So this coherent state has now-- and we just
called it 1 for the sake of the argument.
The point would be the classical oscillator.
It has just cosine omega t.
Everything is deterministic.
No uncertainty, no nothing.
Of course it means the time evolution, it goes in a circle.
But this is what everything does in an harmonic oscillator
when time evolves into the e to the i omega t factor.
So let's not get confused with that.
Let's just look at t equals 0.

And then for traveling wave r equals 0,
the classical oscillator is one point.
But now we have a spread here that
says that quantum mechanically the amplitude
of the cosine omega t term is not arbitrarily fast.
It's not arbitrarily sharp.
They are fluctuations.
And in addition, we have [INAUDIBLE]
in this direction, which we projected to the x-axis.
And this tells us what the distribution
in our ensemble, in our quantum mechanical ensemble
is for the amplitudes of the sine omega t motion.
So the best we can do is trying to mechanically--
if you want to have something, which is really just cosine
omega t, we have to squeeze it.
That the cosine omega t amplitude
is now extremely sharp, but the sine omega t amplitude
in the ensemble is completely smoothed out.
So this is what we're talking about.

Now, what I think has confused some of you
is what I thought was a wonderful example,
the classical squeezing experiment.
I mean, these are visuals which will be in your head forever.
When you saw Professor Pritchard with a circular pendulum,
he's just pulling.
And then the circle squeezes into an ellipse.
And it seems that something is here squeezed in real space.
But this is actually wrong.
But how you should have looked at the experiment--
and I made a comment about it, but maybe not
emphatically enough, you should have really thought
about a single pendulum.
And this single pendulum, if it has a an arbitrary phase,
it's in a superposition of sine omega t and cosine omega t.
And if you pull on the string, if you
shorten the lengthen in the pendulum, at sine 2 omega t,
you will amplify the prefector in front of sine omega t,
and you will exponentially deamplify the factor
in front of cosine omega t.
So therefore what will happen if this pendulum oscillates--
and let me say with a phase-- well, sine omega t plus delta,
if delta is 90 degrees cosine, if delta is 0, at sine.
And let's say this pendulum oscillates at 45 degrees.
Sine omega t plus 45 degrees.
If you know parametrically drive it,
the squeezing action would now mean
that you-- let's just make it a sine convention-- you deamplify
the cosine, you amplify the sine.
And after a while, instead of oscillating with sine omega
t plus 45 degree, it will oscillate
with an amplified amplitude at sine omega t.
This is what you have done.

And this is the mechanical analogy.
There is of course no squeezing in any way.
Because in a classical pendulum, we
start with one definite value-- if you
prepare the system well-- and then we just change the motion.
We amplify-- the we pick out a phase.
And that's what we we're doing.
Now the true ways how classical squeezing can come in-- one
is if the motion of the pendulum is-- maybe
there is an uncertainty.
Maybe Professor Pritchard did experiments with an ion trap
and actually 20 years ago he published a [INAUDIBLE]
on classical squeezing.
You think how can you publish a [INAUDIBLE]
on classic squeezing.
Well, he had the world's most accurate ion trap,
measuring atomic masses with 10 and 11 digits position.
And what was actually one limiting factor
was, for Kelvin, the thermal distribution
of harmonic oscillator modes.
And so what he had is, he didn't have just one clean amplitude.
The amplitudes had the sine omega
t amplitudes had the spread because
of the thermal distribution he started from.
And so what he then did is, by simply classical squeezing,
by doing classically with the ion trap
exactly what he did with a pendulum,
drive it with sine 2 omega t, he could now
take this classical distribution of-- this
is a classical distribution in one axis it's
a distribution of amplitudes of the cosine motion,
and here it's a classical distribution of the amplitudes
of the sine motion.
And he was squeezing it into this direction.
So he had a narrow definition of the coefficient
for the cosine omega t motion.
And as I will tell you today is you can now
do a homodine measurement, which is reducing the noise.
So essentially, he prepared quote,
unquote, "effectively a [? colder ?]
ensemble by squeezing the uncertainty
in the cosine prefector at the expense of increasing
the uncertainty, the variance in the prefect of the sine omega
t motion."
Finally, you all saw something visually.
You saw how a circular motion became a linear motion.
So what was going on here?
Well, I mentioned to you that the circular pendulum
is actually has two modes.
These are two modes of the harmonic oscillator.
And I'm not talking about two modes
of the harmonic oscillator.
Everything we're discussing here is about one mode
of the harmonic oscillator.
The circular motion of the pendulum
was sort of just a nice visualization
trick that, if the pendulum moves in a circle,
you have a degenerate harmonic oscillator.
One is excited with sine omega t, the other one
is excited with cosine omega t.
And instead of doing two experiments,
if you would start with sine omega t,
and you parametrically drive it, you amplify it.
It will start with cosine omega t,
you could bring the pendulum to a stop.
But instead of doing two experiments,
Professor Pritchard just did one and showed that the sine omega
t motion became larger.
And the orthogonal cosine omega t motion shrunk.
And therefore you saw that the circular motion, which
was a superposition of sine and cosine,
became pure sine motion.
But the fact that there was something
we could see in a spatial domain was simply due to the fact
that we had kind of two experiments at one.
Two versions of the same harmonic oscillator,
one in x and one in y.
And then when we did the experiment,
we saw something visually in the spatial domain.
So that's why we saw squeezing in the spatial domain.
But you should really think about it.
What was the whole action is it's
an interplay of deamplifying prefectors of cosine,
amplifying prefectors of sine.
And if the prefector has a distribution,
by deamplifying it, you also shrink
the bits of the distribution.
And this is what we call squeezing.
Yes, [INAUDIBLE].
Two quick questions, so the operators A1 and A2 here,
you use those instead of A and A dagger
because you used cosine and sine other than either the i omega
t and the minus i omega t.
Because they should contain A dagger, right?
Let me go back to the definition.
They're actually exactly-- they correspond exactly
to position and momentum of the mechanical harmonic oscillator.
So that makes sense.
So technically speaking, we could
call the [INAUDIBLE] electric field e cos kr minus omega t
is maximumly squeezed if it only has a cos component.
Depends how I define squeezing.
So you would now give a definition
of squeezing, which says that the variance in A1
is now unequal to the variance in A2.
So the classical oscillator is a point.
It has 0 variance in A1, 0 variance in A2.
But as I said, you can actually apply all that
to a classic oscillator if you add technical noise
or thermal noise.
Then your system is prepared not with a sharp value,
but with a distribution, which is simply
maybe a [INAUDIBLE] distribution due to the preparation.
So we call it squeezing when the noise in the amplitude
of the sine motion is not equal to the noise in the amplitude
of the cosine motion.
Some people say it a little bit narrower.
They reply squeezing to the situation
that we are uncertainty limited, and then we squeeze.
But of course you can say you can always
reduce the noise in your system by just preparing
the system, by cooling the system,
by selecting the system for measurements until you
reach the quantum limit.
So you can get a smaller delta A1, a smaller delta A2
without squeezing by just better preparation
or by selecting your ensemble.
So squeezing in a narrower sense only makes sense
when you hit the limit what quantum mechanics allows you.
And now you want to distribute the variance unequally
between A 1 and A2 because then you can get something
in delta A1 or delta A2, which is better than 1
over square root 2.
And this is now really quantum mechanically squeezed.
But they're both definitions.
Classical squeezing exist.
It's just not as common as quantum mechanical squeezing.
Other questions?
Yes?
I'm really confused when you said
that in the classical squeezing you
were attenuating one amplitude and you were
amplifying the other amplitude.
So in this picture then shouldn't we
have the ellipse come from down from the circle on the y-axis?
Not just changing the A, but also changing [? e ?] itself.
Let me just get a sketch pad here.
So this is A2, this is A1.
A2 is for the cosine omega t and A1 for the sine omega t.
So just to be specific.
So what do you want to prepare?
You want to prepare an harmonic oscillator, which
is just sine omega t.
This is a point here.
If we are now parametrically-- so
this has a value at t equal 0.
Our A1 is not.
So if you are now squeezing your classical harmonic oscillator,
you would have a situation where each A1 of t
is s0 times e to the plus or minus t, depending whether you
do the parametric drive at sine 2 omega t
or at cosine 2 omega t.
So therefore what would happen is,
this point will be amplified.
That would mean it would just move out on the x-axis.
So this would be for the plus sign.
Or for the other case, you would damp the motion to 0,
and this is a minus sign here.
And you are also changing the variance.
A point does not have variance.
But for an ensemble.
So if you want to build up variance,
you need, let's say three points.
One is the average value, one is the left outlier,
one is a right outlier.
And what happens is now as you amplify the motion,
you would also amplify, magnify the distance
between the points.
And if deamplify it with a minus sign,
that distance between the points would
shrink because all the three points converge to 0.
So pretty much what I just told you for the three point
ensemble, you can now use it and construct any initial condition
you want and see what happens you to squeezing.