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M1L4j.txt
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M1L4j.txt
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#
# File: content-mit-8422-1x-captions/M1L4j.txt
#
# Captions for 8.422x module
#
# This file has 123 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
When I learned about the beam splitter and its underlying
physics, there was one thing which really fascinated me.
And this is the most simple element you can think.
I mean what is simpler than it being split?
A beam splitter has two inputs, two outputs.
The simplest optical element is just an attenuator.
Putting a window in your laser beam,
and you lose 4% of your power per surface.
Or put in just a little bit of dirty optics
and you lose a few percent.
So what I want to discuss with you now
is what is really an attenuator, quantum mechanically?
Well, if classically, the attenuator
would do the following.
An attenuator is a device which has a transmission, which
is a transmission coefficient square, which
is smaller than unity.
And in a classical system if you have a coherent state,
you would simply assume that the coherent state
gets multiplied by the transmission coefficient.
In other words, that you have your original state,
described by this phase, or alpha.
And then the action of the attenuator
would simply be to scale everything down
by a transmission factor of alpha.
So the picture you should have is a photo.
You have a coherent state, it gets
attenuated by the transmission coefficient.
But if there are fluctuations about the coherent state,
also the fluctuations get attenuated because everything
gets attenuated by this attenuator.
So if you look at that, you should immediately say, no.
This is quantum mechanically forbidden.
Because a coherent state, with the minimum uncertainty state,
the shaded area cannot be smaller than one half.
But here it has become smaller.
So what I've shown to you here is
it's a violation of quantum mechanics.
It would actually mean-- I mean, let
me just give you the example, it would mean that if you-- yeah,
if you had the cohesion the state is quantum limited.
And if you calculate what is the fluctuations in the photon
number, it's just shot noise.
So just to give you a sort of simple intuitive example,
if you have 10,000 photons plus minus 100,
this is shot noise square root n.
If you could know attenuate it by a factor of 100,
and you would go from 10,000, plus/minus
100, to 100 plus/minus 1.
That's much better than the shot noise.
I mean this is what I'm just telling you,
what a simple minded attenuator will do,
and you would immediately say, that's too good to be true.
I cannot get sub shot noise light.
So what is the wrong?
Impossible, not allowed.
Well, we have just tried to formulate something intuitively
and we have to be careful.
Well, we know all ready, one way how
we can attenuate an input beam-- and maybe we
should go back to the situation and analyze it.
We know that we can attenuate it with a beam splitter,
and this beam splitter has a transmission coefficient of t.
And then we get our transmitted coherent light,
there is something getting reflected,
but now you realize that this beam splitter is not
just a device which has one input, it has another input.
And you may say, well, I don't care about the other input.
I don't want to use it.
Well, if you don't want to use the input
it has the vacuum state.
So therefore, if you would realize the attenuator
with a beam splitter, it would mean that in addition--
and this is what the math really shows-- in addition
to the attenuated coherent state, which mathematically
is also attenuating the fluctuations,
you have to add something which is the reflection
coefficient times the vacuum state, which is this.
And if you now correctly do the math,
if you add the two together, you find
you get a coherent state, which has an amplitude of t alpha.
But has the correct Poisson fluctuation,
is again in minimum uncertainty state.
So the disk of your attenuated state
has exactly the same area as the unattenuated state.
So I've shown you the physical part.
I have shown you the graphical solution.
The math is very simple.
But I really want you to do the math yourself.
This is a new homework problem we designed
to elucidate the physics.
But what it tells you is the following.
If you take a neutral density filter out of the lab pool
and say, this is not a beam splitter,
this is an attenuator.
Sorry, you cannot simply attenuate a quantum mechanical
mold.
This is not a unitary time evolution.
What your attenuator does is, without you knowing it,
it couples the electromagnetic wave to whatever,
to the heat path of the [? black ?] [? pane, ?] which
is in your attenuator.
I don't even want to describe it.
But you're not circumventing the limitations of a beam splitter.
Whenever you attenuate, whenever you have a laser beam,
and it undergoes losses, when you
send a laser beam for the atmosphere
and it undergoes some losses by knows,
Rayleigh scattering through the air or something like this.
That means you get an attenuated coherent state,
but you couple in the fluctuations
of the vacuum and this establishes shot noise
now, at a lower level of intensity.
So your attenuator is not a single state device.
Dissipation, attenuation, really means
you connect with other parts of Hilbert space.
You cannot attenuate in a small part of Hilbert space.
This is impossible.