M1L6f.txt

#
# File:   content-mit-8422-1x-captions/M1L6f.txt
#
# Captions for 8.422x module
#
# This file has 274 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 talk now about how we can create
entangled states for atoms.
Maybe let me say the following.
By now we are convinced, entangled states are great
and we want to create them.
And for photons, I showed you that some simple element-- beam
splitters, [INAUDIBLE] medium-- can create entanglement.
It's much harder to do that for atoms.
Now, we want to do it with atoms because atoms, in contrast
to light, they pretty much stand still whereas photons always
move at the speed of light.
And the only way to make photons stand still
is you put them in a cavity and then they
bounce back and forth.
But even in super cavities with the highest reflectivity
mirror, you get-- what are the longest [INAUDIBLE]
downtimes you get?
Fraction of a second, milliseconds,
depending in which domain you work,
microwave domain, optical domain.
Whereas atoms, you can hold on to your qubits for long time.
So therefore, if you want to use entanglement
as a resource for certain protocols,
you want to have entangled atoms where the entanglement would
live for a long time.
So the question is now, for atoms
we do not have perfect beam splitters and perfect care
mediums.
Also, we can control interactions
between atoms for [INAUDIBLE] atoms using [INAUDIBLE] lenses.
But that's another story.
But let me now address the question,
how do we entangle atoms?
And I want to first show you that, if you
had the right system, things can be fairly simple.
This is a suggestion which was made almost 20 years ago
and it goes like follows.
If you have a diatomic molecule of two identical atoms,
in this case mercury, and mercury doesn't have any--
this molecule doesn't have any electron spin
but it has-- but mercury has a nuclear spin.
And if you now photodissociate mercury and the two mercury
atoms fly away, then you have separated a spin singlet
into two parts and now you have created the Bell state up down
minus down up.

So you could say this is sort of the example-- this realizes
very closely the example I gave you with the helium atom
where I said you have an electron in spin up
and spin down.
But there was no way to separate the electrons.
Therefore, you can say it is a state which
has entanglement but it's not entanglement as a resource.
But right now here it becomes a resource.
Once you have found a method to separate the two
parts of the wave function that you can now give one to Alice,
give one to Bob, and they can perform the operations on it.
Well, if it looks so simple why don't we
have entangled atoms everywhere?
This experiment has been suggested 20 years ago
but nobody has done it.
Also-- several groups have worked on it.
You need a very-- you need a molecule with suitable states,
you don't want any electron spin which interferes with it.
All you want to have is two nuclear spins.
You want a singlet state here.
So those requirements are not easily fulfilled
with real atoms.
Of course we liked-- and also, who wants to work with mercury?
Mercury has transitions in the ultraviolet.
They are-- I think there's only one
group in the world who has operated a magneto [INAUDIBLE]
with mercury.
So it's not your tabletop atom.
So therefore let's now talk about a method
how we can entangle atoms by using light.
So if you take atoms, maybe we can
use the light which has been emitted from the atoms,
perform a measurement on the light,
and then depending on the outcome of the measurement,
we know the atoms have been-- are entangled.
So Professor Schwan calls this the poor man's entangler.
I don't know why poor man's.
You still need quite a bit of equipment to do it,
but at least it seems the poor man's solution
if you can't make the above experiment work.
So this addresses the question that for-- just
for technical reasons-- matter is more difficult to entangle
whereas photons are easy.

So you can say, the idea is related
to the purification scheme.
We won't have a system which is--
we can't take a system of two atoms, atom 1 and atom 2,
which are unentangled, we shine some laser
light on them, excite them, then they emit photons.
So all we can do is we can only talk
to the atoms with the photons.
So the only thing we can do now is
we can measure the two photons.
And then the situation will be similar, as in the purification
scheme, where Alice and Bob did a measurement.
If Alice and Bob said, both of our target qubits are 1,
then what was left behind was in a purely-- was
in a pure entangled Bell state.
And similarly, what you want to do here
is we have two atoms-- there was nothing special about them
but they scatter light-- and if you now perform a measurement
on the photons and the outcome of the measurement is positive,
then we know for sure what has been left behind is entangled.
That's the idea.
So it shares with the purification scheme
that it is a probabilistic entanglement.
You run your experiment many times, you do a measurement,
if the measurement is positive, you say,
now I have an entangled state, and maybe then you
can move on to measure the entanglement,
you can move on to be teleportation,
to do teleportation, other things you want to do
within entanglement states.
But if your measurement says, no, bad luck.
The probability it hasn't worked out this time,
you're just press the button again, scatter light again
of your two atoms and-- or your two ions
and hope that the next outcome is positive.
So the idea is we want to introduce now
a probabilistic method.
It's based on two atoms emitting light
and the result is with a certain probability
that we get entangled atoms.
You know, I prepare the notes and everything is clear to me
but then I want to explain it to you and say, hey,
I have to motivate you.
If I just go for a few lines, you
wonder what it leads to, so let me maybe first
give you the explanation I would have given you a little bit
later.
What is an entangled state is if the atoms are maybe
in one of two states.
One atom is in one ground state, the other one
is in the other one, or it is flipped.
So we know one atom is in the ground state one,
one atom is in the ground state two,
but we don't know which one is in which.
It is in the superposition state.
So what we need now in this scheme is the following.
If the atoms scatter light, they have--
they can go to two different ground states.
And we know to which ground state
they have gone because due to selection rules,
they reach one ground state was polarization one,
they reach one ground state with polarization two.
So therefore, when we had two atoms they emit light
and we would measure the polarization of the light,
we would know in which state they are.
But if you know atom 1 is in state one
and atom 2 is in state two, this is not entangled.
So what we have to do is, when the two atoms have
emitted photons, we have to mix the photon at the beam
splitter.
And after the beam splitter, when
we measure that the photon is polarized,
we know that one of the atoms is in one of the ground states
but we don't know which.
So therefore we can now use-- the photon carries
through the polarization the information in which ground
state the atom is.
But now we have to perform operations to the photons
that we, for fundamental reasons, fundamental quantum
measurement, we know there is one photon in one state
but we have no way to ever figure out which photon-- which
atom has emitted the photon.
So that's the idea.
And the protocol I want to show you now
is, what we have to do to the two photons to make sure
that we never know that we can't find out which
atom has emitted the photon?
And we have to do a little bit more tricks also
to make sure that when we do a measurement on the photon
we know the atoms are in a Bell state.
Questions about that?

So therefore I need-- we need beam splitter.
So we can come back what we have already
introduced in the last section.
So each atom will emit a photon and we'll-- I will actually
show you at the end of the class that people have entangled with
that scheme to ions which were in two different ion traps.
So you have two distance atoms, they emit light,
and after the measurement process
you know they are entangled.
That's pretty cool.
So the situation how we do it, that we have two photons which
come, but the first thing we have to make sure is we
have to scramble the photons.
We have to make sure that we can't find out from which
atom the photon has come.
And this is done with a beam splitter.
So there is one aspect of beam splitters
into photons which I have to explain to you now,
this is the famous HOM-- Hong-Ou-Mandel--
this is the Hong-Ou-Mandel interference.
It's a very famous effect and it's actually
very, very special.
So let me explain what happens when
we have two photons in the same state, two identical photons--
same frequency, same polarization--
coming to beam splitter.
So we know-- we have already all the tools.
So we have a beam splitter characterized
by this angle theta, which through cosing theta sine theta
determines what the beam splitter is doing.
And all we want to do it, we apply it now to the state 1 1.
So what we will find is that there is a probability
to get one photon each.
Then there is the probability to get two photons in one output.
And it will actually be the same probability with a minus sign
in the amplitude to get 0 2.
The matrix which acts on each of the photon
has cosines and sines.
So what we get is products of trigonometric functions.
And here for the-- for getting two photons in one arm,
it's square root 2 cosine theta sine theta.
And the spectacular thing here is,
what happens when we have a balanced beam splitter.

If you have a 50-50 beam splitter--
and this is called the Hong-Ou-Mandel interference--
you have the situation that you have a beam splitter,
you have one photon, you have two photons,
and it is absolutely impossible that afterwards you
have one photon in each arm.
So you send two photons on a beam splitter,
and after the beam splitter, have
either two photons coming out here
or two photons coming out there.

You can say it's Poissonic stimulation,
photons and bosons, that all plays a role.
If one photon goes one pass, the other photons follow suit.
This is now very powerful because it only
happens, of course, when the two photons are identical.
We had to use, in this formalism,
the photon is in the same mode on each part.
But it means now the following.
If you set up photo-detectors here,
and each photo-detector makes click,
you know you had one photon each.
And that tells you that the two photons were not identical.
For instance, because they have different polarizations.
So if you want to know where the atoms get at Bell state
where atoms decay-- one atom decays to state 1,
one atom decays to state 2-- the signature of that
would be that we have one atom each.
And if you're-- if you do it right,
one atom each is an ingredient for 1 2 plus 2 1 for Bell
state.
We can detect that we have one photon each because it's
such a beam splitter.
It's only in this situation that we
can get one photon after each beam splitter
if we start with two non-identical photons.
Of course, by the way, experimentally there
are quite some challenges.
Even if you have identical photons
of the same polarization, if they arrive
as a nanosecond parlance and they don't arrive exactly
at the same time at the beam splitter,
then, well, you first split one photon and then the next photon
and the two photons can not influence each other.
So there are a lot of experimental requirements
to realize this ideal experiment.