M1L7r.txt

#
# File:   content-mit-8422-1x-captions/M1L7r.txt
#
# Captions for 8.422x module
#
# This file has 198 caption lines.
#
# Do not add or delete any lines.  If there is text missing at the end, please add it to the last line.
#
#----------------------------------------

There is another state which you have
encountered in your homework.
And this is a superposition not of and 0 n and n 0.
It's a superposition of n minus 1 n and n n minus 1.
This state goes by the person, I think, who invented it,
the Yurke state.
And you showed in your homework that with that, you also
reach Heisenberg limited interferometry
where the phase case is 1/n.

What I want to add here to it is how one
can create such a Yurke state.
And again, I want to use the example
with an atomic Bose-Einstein condensate.
And here's the reference where this was very nicely discussed.
So let's assume we can create two Bose-Einstein condensates,
and they have exactly n atoms.
And I would actually refer to [INAUDIBLE] questions,
how to make them.

Have 2n atoms in a trap, and then make
a double-well potential.
Deform the harmonic oscillator potential
to a double-well potential.
Then for strong repulsive interactions,
the condensate will symmetrically
split into two Fock states, each of which has n atoms.
So now how can we create the Yurke state from that?
Well, we simply have--
we leak out atoms.
We leak atoms out of the trap.
My group demonstrated an RF beamsplitter,
how you can just split--
in a very controlled way, start rotating
the cloud on non-trapped state, and then atoms slowly leak out.
Well, when you can measure, of course,
you can take an atom detector and measure
that an atom has been out-coupled, that an atom has
leaked out of the trap.
If you don't like RF rotation, you
can also think that there is a tunneling barrier and atoms
slowly leak out by whatever mechanism.
And the moment you detect an atom, you then
project the state in the trap to n minus 1 atom,
because you have measured that one atom has come out.
But now you use a beamsplitter.
And a beamsplitter could simply be a focused laser beam.
And the atoms have a 50% probability
of being reflected and of tunneling through.
So therefore, if you have now a detector
which measures the atoms on one side
and the atoms on the other side, then you
don't know anymore when the detector makes click from which
atom trap the atom came.
Or more formally, a beamsplitter transforms the two input
modes a b into a plus b and a minus b,
normalized by square root 2.
So therefore, if this detector clicks,
then you project the remaining atoms into the symmetric state.
Here you detect one atom in the mode a plus b.
And that means that the remaining atoms have been
projected into the Yurke state.
If the lower atom detector would click,
well, you get a minus sign here.
So that's one way how, at least in a conceptually simple
situation, you can prepare this highly non-classical state
by starting with a dual Fock state of Bose-Einstein
condensates, and then using--
and this is an ongoing theme here--
by using a measurement, and then the post-measurement state
is the non-classical state you wanted to prepare.
Question?
Yes.
[INAUDIBLE] only one atom leaks through [INAUDIBLE]?

The idea is that we leak atoms out very, very slowly.
And then we have a detector which we assume
has 100% quantum efficiency.
So therefore, we simply wait until the detector tells us
that one atom has leaked out.
And in an idealized experiment, we
know the atoms either have been measured by the detector
or they are still in the trap.

So is there a property that [INAUDIBLE] you
detect atoms in one detector [INAUDIBLE]?
In principle, yes.
But the idea here is if you have a very slow leakage
process, the probability that you
detect two atoms at the same time is really zero.
You leak them continuously and slowly,
but then quantum mechanically, that
would mean for most of the time, you measure nothing.
That means the leakage hasn't taken place,
that the quantum mechanical system has not developed yet.
But the moment you perform a measurement, you project--
it's really the same if you say you have n radioactive atoms.
You have a detector.
And when the detector makes click,
you know yo have n minus 1 radioactive atoms left.
It's just applied here to two atom tracks.
Other questions?
[INAUDIBLE]
No.
And maybe I tell you now why not.
We have discussed the NOON state,
and we have discussed here highly
non-classical superposition state.
Let's just go back to the NOON state.
We have n atoms here, zero here.
Or n atoms-- or the reverse.
But now assume a single atom is lost, is lost from your trap,
by some [INAUDIBLE] gas collisions.
And you have surrounded the trap with a detector.
So if you have the NOON state, the symmetric superposition
state, all atoms here and all atoms there,
but by a background process, by an inelastic collision,
you lose one atom, and you detect it,
you could set up your detectors that you know from which--
that you figure out from which trap was the particle lost.
So therefore, a single particle loss,
if you localize form which trap the particle is lost,
would immediately project the NOON state into a state
where you know I have n or n minus 1 atoms in one
well and zero in the other well.

So I've already told you with the attenuator,
you can never assume an attenuator is just
attenuating a beam.
An attenuator can always regard it as a beamsplitter,
and you can do measurements at both arms of the beamsplitter.
Or you have to consider the vacuum
noise which enters through the other part of the beamsplitter.
And if you now add that those atoms, n atoms in a trap,
have some natural loss by inelastic collision,
or [INAUDIBLE] gas scattering, the loss
is actually like a beamsplitter that particles
don't stay in the trap, but go out through the other part.
And then you can measure it.
So in other words, every loss process
should be regarded as a possible measurement.
And it doesn't matter whether you
perform the measurement or not.
And I think it's just obvious that the NOON state, the moment
one particle is lost--
and you would use this particle to figure out
if n particles are here or n particles are there--
the whole superposition state is lost if you just
lose one single atom out of n.

So the lifetime of a NOON state is then
not your usual trap lifetime, where
you lose half of the atoms, or 1/e of the atoms.
It is n times faster, because it is the first atom lost
which is already completely removing
the entanglement of your state.
To say it more specifically, the limitation is loss.
When you have a fully entangled state, maximally entangled
state, usually a loss of one particle
immediately removes the entanglement.
We had the situa--
no, that's not a good example.
But for the most entangled state, usually--
and for the NOON state, it's trivial to see--
a single particle lost allows you
to measure on which side of the potential barrier
all the atoms are, and all the beauty
of the non-classical state is lost.

So if you assume that in a time window
you have an infinite [INAUDIBLE] loss,
a loss of epsilon, what usually happens
is if you have an entanglement of n particles,
then you lose your entanglement.
If you expand 1 minus epsilon to n,
it becomes 1 minus n times epsilon.
So that's one reason why people have not scaled up
those schemes to a large number of photons
or a large number of atoms, because the larger n is,
the more sensitive you are to even very, very small losses.
[INAUDIBLE]
The super-beamsplitter would create the NOON state.
I've not explained to you what it physically
would be for photons, but I gave you
the example for atoms to be [INAUDIBLE]
in a double-well potential with attractive interaction.
So to start with the condensate in--

[INAUDIBLE]
If you start with a double-well potential,
and you put n atoms initially in,
but then you switch on tunnel coupling,
then you would create the NOON state
if the interactions are attractive
and if everything is idealized, that you
have a completely symmetric double-well potential.