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M1L4f.txt
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M1L4f.txt
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#
# File: content-mit-8422-1x-captions/M1L4f.txt
#
# Captions for 8.422x module
#
# This file has 86 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
I want to discuss now the electric field
of squeezed states.
So just a reminder, we can discuss the electric field
by using the causal probability representation.
And the electric field is the protection
of the causal probabilities on the vertical axis.
And then the time evolution is that everything
rotates with omega in this complex plane.
So we discussed it already for coherent state.
We have a circle which rotates, therefore
the project the fuzziness of the electric field
is always the same.
And as time goes by, we have a sinusoidal varying
electric field.
Let me just make one comment, if you read the literature,
look into the literature.
Some people actually say the electric field is a projection
on the horizontal axis.
So there are people who say the electric field is
given by the x coordinate of the harmonic oscillator.
Whereas I'm telling you it's a p coordinate.
Well, if you think one person is wrong,
I would suggest you just wait a quarter period of the harmonic
period of the harmonic oscillator,
then the other person is right.
It's really just a phase convention.
What do you assume to be t = 0.
It's really arbitrary.
But here in this course, I will use the projection
on the vertical axis.
OK, if you project the number state,
we get always zero electric field with a large uncertainty.
So that's just a reminder.
But now we have a squeezed state.
It's a displaced squeezed state.
And if you project it onto the y axis,
we have first some large uncertainty.
I think this plot assumes that you rotate with negative times.
I apologize for that.
But you can just invert time if you want.
So after the quarter period, the ellipse is now horizontal,
and that means the electric field is very sharp.
So as time goes by, you see that the uncertainty
of the electric field is large, small, large, small.
It modulates.
And it can become very extreme.
It can become very extreme when you do extreme squeezing.
So you have an extremely precise value of the electric field
here, but you have a large uncertainty there.
Now sometimes you want to accurately measure the zero
crossing of the electric field.
This may be something which interests
you for an experiment.
In that case, you actually want to squeeze, have an ellipse
which is horizontally squeezed.
Now whenever the electric field is 0,
there is very little noise.
But after quarter period when the electric field reaches
its maximum, you have a lot of noise.
So it's sort of your choice which way
you squeeze, whether you want the electric field
to be precise, have little fluctuations when it goes 0
or when it goes to the maximum.
So what we've done here is we have first
created the squeezed vacuum.
And then we have acted on it with a displacement operator.
Let me just say what I wanted to take from this picture.
The fact that the electric field is precise only
at certain moments means that we can only take advantage of it
when we do a phase sensitive detection.
We only one to sort of measure the electric field when
it's sharp.
Or-- this is equivalent-- we should
regard light is always composed of two quadrature components.
We can see the cosine, the sign oscillation, the x and the p.
And this squeezing is squeezed in one quadrature
but it is elongated in the quadrature.
So therefore, we want to be phase sensitive.
We want to pick out either the cosine omega
t or the sine omega t oscillation.
This is homodyne detection.
We discuss it on Monday.