M1L7m.txt

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Now we are really ready to see how can we
improve on the shot noise.
It seems it's-- unless we do something special,
we will always get square root n.
I've already shown you at the beginning
that the shot noise limit is not fundamental.
Just take this thought experiment
that you take the n photons, through a highly non-linear
process, you create one photon of frequency n omega
and then you have more precision of this measurement.
So there must be a way to have a precision which scales 1 over n
and not 1 over square root n.
The mathematical argument related
to our treatment of the interferometer
is the following.
Our signal was a dega a minus b dega b.
Well, if one of the input mode dominates,
a dega a is the photon number.
So this sort of looks like the photon number.
But if we now find a scheme-- and I will show you
that this is possible-- that this cos term,
a dega b plus b dega a, which is some form of quantum noise
as we will see, is 0.

Well, what do we get now?
So let's hope that maybe by some squeezing--
I will show you several versions which
shows you how quantum-ness can give us more than shot noise.
One example will be that by squeezing light--
we've learned already that squeezed light can suppress
the noise if you've squeezed light-- something which
comes from the modes where we apply squeezed vacuum has been
used.
That's what we discussed already.
So the best we can do is that the noise term is 0.

So what do we have now?
If we have a sequel which is finite and the noise is 0,
what is our precision of measurement?
Well, it first looks like the signal to noise ratio
is infinite.
But it's not quite that.
Because if the signal is-- our signal is x,
it's the photon number, and this signal
x was sensitive to the phase by cosine phi.
The sensitivity of our measurement of the quantity m
with phase is n times sine phi.
And this is small or equal than n,
where the equal sign is obtained for 90 degrees.
So therefore, we have a photon number n.
We may absolutely know what the initial photon number is,
but now we want to measure whether the phase is
different from pi over 2.
And the smallest change in our signal, which we can resolve,
is that we get one photon less.
So that's the smallest resolvable change
due to the photon nature of our detection.
And that implies that the smallest phase shift
we can resolve is 1 over n.

So this is more thought experiment
now looking at the math and seeing what is the best signal.
a dega a can never be larger than photon numbers
so that's a resource we put in energy in the terms of photons.
And then the best we can do is that we don't have any noise,
and then we are simply limited by the fact
that, when we deviate from the maximum output signal,
because, say, it's a phase shifting at the interferometer,
our sensitivity comes in grains.
It's grainy by the photon number.
So this is more-- I'm not telling you
how we achieve to get y equals 0,
the question is only that at least the math seems
to make it possible.
So now we ask how to achieve sub shot noise precision.
So here we know this is at least mathematically possible.

So we want to improve this interferometer,
and the question is-- we have to change something.
I mean, this interferometer operating with single photons
or coherent state is not giving us any better result.
So we have three options now.
We can do something fancy in the input state--
use entangled state-- we can use some very fancy beam splitters,
or we can do something special to the way
how we read out the interferometer.
And all of those three possibilities,
we can put in some extra quantum-ness
into each of the three steps.

[INAUDIBLE]

The question is, can we do something with the phase shift?
[INAUDIBLE]
I think if you go do something to the phase shift
we would do-- we would change apples with oranges.
Because we want to compare different interferometers
by measuring the same thing, and maybe
as an experimentalist say, what I want to be able to measure,
I take a very, very thin glass plate
which just makes a very small phase shift
and I'm now comparing different interferometers.
And when I put the glass plate in and pull it out,
I want the person who reads out the interferometer
to see a change.
So that's how I want to compare the interferometers.

Actually, the question you are raising was, in another way,
also discussed at lunch with three
of my wonderful colleagues.
Namely, what we discussed was some recent paper-- one
of them published in Nature-- which claims precision
better than Heisenberg.
What I'm sort of indicating to you is Heisenberg
should be the best which can be achieved.
How can we do even better than Heisenberg?
Well-- and this was actually related,
Nancy, to your question-- if you want to measure magnetization
and you use some non-linear physics where
the photons, the number where the magnetization involved
effects the photon field by some higher
power of what you measure, you really change what you measure.
Maybe I'm not expressing it clearly.
If you measure magnetization and you have a certain quantum
limit in measuring the magnetization that's one thing.
But if you now bring another material close
to your magnetization and this material goes through a phase
shift, you sort of amplify by a physical process
by nonlinear Hamiltonian what you want to measure.
And then of course you can, depending
on what kind of nonlinear process you are using,
you can get a signal which scales tremendously
with a photon with a number of photons
you put into your system.
So in other words, there are loop holes
like this in some of them led to very fleshy papers.
And our lunch discussion was that some of it
is really completely trivial, and some papers
who claim that they have done-- that they have seen
a scaling of the precision with photon number which is better
than Heisenberg-- not 1 over n, maybe 1 over n
squared-- that some of those papers were purely classical
and the only quantum character of this paper
was the name of Heisenberg in the title.
It's not related to any uncertainty relation.
But that's not what we want to discuss.
What I want to use now is I want to replace the beam
splitter by something which involves Bell states.
So as a beam splitter we do a massive creation
of Bell states.
It's our entangler.
And our second beam splitter is a Bell analyzer.
It is a disentangler.
So what I mean is the following.
We will actually just put one photon into this input beam
and we will only read out one channel.
So all this here are only auxiliary modes.
This is how I make a special quantum beam splitter.
And what we need for this description
is, essentially, two gates.
We need a single qubit.
The single qubit is the Hadamard gate which we have already
discussed, and in that dual rail representation,
it's photon can be in one of the two modes.
The Hadamard gate is simply connecting
the two modes in this way.
And we discussed that the Hadamard gate
can be described by a beam splitter with a phase shift.
So in other words, we need one element
which is a beam splitter with a phase shift.
But this is only acting on one qubit.
And now we want to connect qubits
with a controlled NOT gate.
To remind you, controlled NOT gate
is something where the photons stay where they are.
But if the control beat is 1, it flips the target beat.
If the control beat is 0, nothing
happens to the target beat.
And we discussed already that we can realize one qubit with one
interferometer.
These are the dual rails.
We always have one photon in two states, this mode or that mode.
And now the other qubit, one of the rails
can go through the nonlinear Kerr medium.
If the beat is in c, through the phase shift in the nonlinear
Kerr medium, it exerts, it flips the beats down there,
and this is the controlled NOT.
So now we want to use two qubits.
So I'm talking about, in our fancy entangler,
I'm just talking about the first two rails here.
So we have those two rails, the Hadamard gate
puts us into a superposition of the logic 0 and 1.
So what we have here is the product state of 0
plus 1 upstairs and 0 downstairs.
But if we have a 1, we flip the 0 to 1.
So therefore, what we have now is this state, 0 0 plus 1 1
over square root 2.
And then we apply our phase shifter
and what we get out of this state
is 0 0 plus e to the 2 i phi times 1 1 over square root 2.
By using two of those, I suddenly
have multiplied the phase by two so something is now
sensitive to 2 phi.
And if I use n such more and more entanglement,
I will show you-- no class next week,
but in the following week-- that we suddenly
have a term in our quantum state which is e to the n phi.