M1L7q.txt

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The next one goes by the name super-beamsplitter method.
It's actually a very fancy beamsplitter,
where you have two parts of your beamsplitter.
One has zero and one has n photons.
And behind the beamsplitter, you have two options,
either all the n photons are in one state
or the n photons are in the other state.
You'll never have any other mixture.
Of course, you know your standard half-silvered mirror
will not do that for you, because it will split the n
photon state with some Poissonian statistics
or whatnot into the two arms and such,
and you can calculate exactly what a normal beamsplitter does
to that.
It's very, very different.
This here is a very, very special beamsplitter.
Just to demystify this beamsplitter a little bit,
I want to show you that, at least conceptually, there
is an easy implementation with atoms.
I really liken this section to grab an example from atoms
and to grab an example for photons,
because it really brings out that the language we
develop equally applies to atoms and photons.
So if I take now an easy example,
if I take an example from the Bose-Einstein condensate with n
atoms, let's assume we have a double well.
But now we have attractive interactions.
This is not your standard BEC.
Your standard Bose-Einstein condensate
has repulsive interaction.
With attractive interaction, if you
go beyond a certain size, a certain atom number,
the condensate will collapse.
So let's assume we have a Bose-Einstein condensate with n
atoms, there are some attractive interactions,
but we stay within the stability diagram
that those atoms do not collapse.

So now, if the atoms are attractive,
they all want to be together, because then they
lower the energy of each other.
So if you have something with attractive interaction,
you want to have n atoms together.
But if your double-well potential
is absolutely symmetric, there is no symmetry breaking,
and you have an equal amplitude for all the atoms
to be in the other well.

So therefore, under the very idealized situation
I described here, you will actually
create a superposition state of n atoms
in one well with n atoms in the other well.
And this is exactly what we try to accomplish
with this beamsplitter.
OK.
So if we have this special beamsplitter which
creates that state, then when we add a phase shift to one arm
off the interferometer, we obtain,
which looks very promising, the phase shift
phi multiplied by n.
And if we read it out, we have now,
if you pass it through the other super-beamsplitter,
we create, again, a superposition of n, 0 and 0, n.
But now, because of the phase shift
with cosine and sine factors, which involve n times
the phase shift.
For obvious reasons, those states also
go by the name NOON states.
That's the name which everybody uses for those states,
because if you just put N zero zero N, it gives NOON.
So this is the famous NOON state.
So it smells already right, because it
has a phase shift of n phi.
Let me just convince you that this is indeed the case.
So the probability of reading out
zero photons in one arm of the interferometer scales
now with the cosine square, if you do one single measurement,
the binomial distribution has-- it's the same as before,
but with a factor of n, the deliberative
has those factors of n.
And that means that the variance in the phase measurement
is now 1/n.
If you want a reference, I put it down here.
As far as I know, the experiment has
been done with three photons, but not
with larger photon number.
Questions?
Colin.
The picture with the double well, isn't there
going to be-- some people would say
there's some-- you spontaneously break the symmetry,
and the BEC either ends up in one side or the other?
Well, Bose-Einstein condensation with attractive interactions
was observed in 1995.
And now, 18 years later, nobody has done this seemingly simple
experiment.
And what happens is really that you
have to be very, very careful against
any experimental imperfection.
If the two double-well potentials are not
exactly identical, the bosons always
want to go to the lowest quantum state.
Well, that's their job, so to speak.
That's their job description.
And if you have a minuscule difference between the two--
the depths of the two wells, you will not
get a superposition state.
You will simply populate one state.
And how to make it experimentally robust,
this is a really big challenge.
Tim?
[INAUDIBLE] I have question.
Say you were somehow able to make the double-well potential
perfect, but if we didn't have attractive interactions, then
would we just get a big product state of
like each atom being in either well?
OK.
If you're a non-interacting system,
the ground state of a double-well potential
is a symmetric state, 1 plus 2 over square root 2.
And for non-interactive BEC, you figure out
what is the ground state for one particle,
and then take it to the power n.
This is the non-interacting BEC.
So then entanglement is because of the interaction, right?
If you have strong repulsive interaction,
you have something that should remind you
of [INAUDIBLE] insulator.
You have n atoms, and n/2 atoms go to one well,
n/2 go to the other well.
Because any form of number fluctuations would be costly.
It will cause the additional repulsive interaction.
So therefore, the condensate wants to break up
into two equal parts.
So that's actually a way how you would
create another non-classical state, the dual flux
state of n/2 n/2 particle.
And for attractive interactions, well, you
should create the NOON state.