M1L4h.txt

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So the question now is how to measure squeezing.
So the situation we are facing is the following.
Let's assume we have done a nice squeezing job.
And that means that we have a sharp value.
We have created a narrow distribution
of the cosine omega [? 2 ?] coefficients
so the cosine omega [? 2 ?] motion is rather sharp.
But we also know that the electric field itself
is sharp at this moment.
But since this ellipse rotates, the electric field
will have enormous uncertainty a quarter period later.
So if we want to take advantage that we have squeezed
the electromagnetic field, there are a couple
of ideas which we can use.
One is we could just measure the electric field
stroboscopically.
We would just make a set up where
we look at the system in a [? quantum ?]
measurement process.
We only measure the electric field
when the ellipse is like this or is like this.
And therefore, we have a sharp value of the electric field.

But instead of doing a stroboscopic measurement,
we can do something else.
Remember, we have a distribution of cosine omega t and sine
omega t, and we have a distribution of coefficient c
and s.
And we know we are interested in the cosine omega t.
So how to pick that out has actually
been solved in early radios.
You do a homodyne detection.
In other words, you take a reference oscillator
which is strong.
B of t is B0 times cosine omega 0 plus delta.
And if you mix-- maybe before I use the word "mixing,"
I should simply mean multiply.
If you multiply the two signals, your signal
you are interested in-- or at least
you're interested in one component-- you multiply it
with your huge local oscillator, and then
you integrate over time.
Then, of course, when you pick the phase delta to be zero,
cosine omega t times cosine omega Bt gives cosine
square omega t.
It averages to 1/2 so you would time average
where its cosine omega t times sine omega t averages to zero.
So for delta equals zero, you project out
the cosine component.
And for delta equal to 90 degree,
you pick out the sine component.
So, therefore, you can have a measurement.
It's a phase sensitive measurement
by multiplying your signal with a local oscillator where
you are only sensitive to the component you have squeezed.
And therefore, your measurement uncertainty
has now been reduced by the squeezing factor.

This only should be the frequency-related one?
Yes, actually. homodyne means we use the same frequency.
Heterodyning would mean we use two different frequencies,
but I'm not talking about that.

So we have to use exactly the same frequency here.
So it means that this reference oscillator, you'd also
use a laser to [? mix it? ?]
Yes.
So to address your question, [INAUDIBLE],
what usually happens is you start with one laser
in those experiments.
You frequency double the laser.

If you wanted to do some squeezing,
you remember that we need a parametric oscillator
where one energetic photon gives us two photons.
So what you do is you start with a laser.
Often it's a YAG laser at 1,064 nanometers.
You frequency double it to a green laser.
The green laser pumps your parametric oscillator.
And then you get-- through down conversion,
you get squeezed light at 1,064.
But this is because you first doubled the laser,
and then you break the photon into two pieces.
It has exactly the same frequency as your laser
you started with.
And this laser is then the local oscillator
or the reference clock for your whole experiment.
So everything in your experiment--
the doubled laser, the parametrically down-converted
beams-- everything is related to the single,
one laser you started with, and everything is phase coherent.
So that's how usually the experiment is done.

Before I tell you what we're doing quantum mechanically,
let me just also get another question
out of the system, which I've been asked several times.
People ask me, well, the problem is that the ellipse rotates
like this.
Isn't there a way-- now I need my hand--
that we can have an ellipse rotating like this.
That would be great, but this is sort of unnatural
because the harmonic oscillator does that.
So if you wanted to do that, you need
an operator, which is really-- at every cycle
of the electromagnetic field is when the light wants
to do this-- no.
Always push it back.
And this is impractical.
You need really an oscillator, which would completely
change the quadrature components of your harmonic oscillator
in every single cycle of the electromagnetic field.
But what homodyne detection is, instead
of now forcing the light to stay aligned, to sort of do
this, which is very unnatural, we
allow the light to freely evolve.
But we have now an observer, our local oscillator,
which is rotating synchronously with the ellipse.
So we have a local oscillator, which is cosine omega t.
It does, so to speak, exactly what the ellipse is doing.
So in that sense, the local oscillator
allows us now to observe the ellipse always
from it's narrow side because the local oscillator is
co-rotating.
But the mathematics is pretty much the Fourier transform.
The mathematics is the Fourier transform.
The physics is the physics of a [? login ?] detector.
OK, now the only question that remains is how do we mix?
How do we get a product of our signal
and the strong local oscillator?
In an old radio, it's done by an element, maybe a diode,
which has a nonlinear circuit.
If you drive a nonlinear element with two input sources,