M1L7k.txt

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OK.
Let's now obtain the standard quantum limit
from a real measurement device, because this
is what we want to generalize for entangles,
[INAUDIBLE] state, and all the special things.
So an interferometer is the following.
Consists of two beams splitters, after the second beam splitter
we have two detectors, and the quantity we will measure
will be the difference of the two photocarbons.
And if you don't do any phase shift,
we know the two beam splitters are just the identity
transformation, but now we put in a phase shift
and the question is, what is the smallest phase
shift we can measure?
What is the [INAUDIBLE] measure in this phase?
We have discussed at length all the elements.
So we have a beam splitter, a phase shift, and a second beam
splitter.
You know that the phase shifter is a rotation in z,
the beam splitter is a rotation-- was it x or y?
Looks like I think y because of the i's.
Just a warning here, in this section I use a beam splitter
which uses a different phase convention but it's-- they are
all equal apart from some phases.
So by multiplying the three matrices,
we get the transfer matrix for the Mach-Zehnder
interferometer.
So the output cd is this matrix here
which is a simple rotation times the input state.
And our measurement is the difference
in photon numbers in the two output modes.
So it's d dega d minus c dega c.
So what we can do now is we can obtain c and d
from the inputs-- from the input modes a and b.
So therefore, we can now express our signal
in terms of the input modes.
When we know what we put into the interferometer,
because we know all its element, we know what we can get out.
And this is done here.
And the phase which appeared in the rotation matrix,
the phase shift in the interferometer,
appears now as a cosine phi and sine phi contribution.
And we have-- the cosine and sine phi
have two operators as a prefactor.
In one case it's a dega a minus b dega b.
In the other case, it's a cost term, a dega b plus b dega a.
Now, we will be more specific what is signal and noise.
I just want to tell you, in the standard way of operating
the interferometer, you have only one beam in the mode a.
You take a beam, you split it, you recombine it.
b is nothing, or the vacuum.
It may introduce some noise.
But if you have a lot of intensity in the beam a,
it is this term which dominates, and you find fringes
as a function of the phase shift cosine phi.
Whereas, what comes here, is sort of more the vacuum mode.
It gives rise to noise terms.
So if you ask, what is the expectation value
for x-- this x operator-- it varies [? cosinusoidally. ?]
There is a special point for a phase shift of 90 degree when
we have the steepest slope and the highest sensitivity.
This is all just setting the stage.
What we are interested in is, if we now
have an input state of light, and as you know,
there's all this noise-- there's the fundamental noise
of the vacuum, or coherent state,
of Heisenberg's uncertainty-- this means, when
we do repeated measurements we will have
a variance in the phase, and we want
to know what is the fundamental limit on the standard deviation
in the measurement of the phase.
Well, the standard deviation in the phase
is nothing else than the standard deviation
of what we measure M divided by how sensitive M
is to the phase.
So by taking this expression for M,
M is an operator x times cosine phi plus
something times sine phi, we can now evaluate this expression.
This will actually appear several times today
and it's also a key question to one of your homework problems.
So this is what defines-- what defines the [INAUDIBLE]
of the interferometer.
And what we have here are expectation values
of operators.
And now we can-- that's what we will
do for the rest of this lecture-- we
will look at this expression for different input states.
Coherent state, single photons, [INAUDIBLE], entangled state.
So that's what you want to do.
So if you take the derivative of dm d phi,
the cosine becomes a sine-- a minus sine--
the sine becomes a cosine.
And if we specialize to the situation,
which is where the slope is very steep-- a 90 degree phase shift
around this point-- our phase sensitivity is given
by the expectation value of-- the variance of the operator
y divided by the operator x.
So now we are ready to plug things in.
The first thing is, of course, the coherent state.
For the coherent state, our input to the beam splitter,
one is a coherent state, the other one
is a vacuum in mode and b.
We had expressions-- let me just scroll back-- we
had expressions for x and y expressed by the operators
a, b, a dega, b dega.
So therefore, we can calculate that now.
And x is nothing else than the number of photons
in the coherent state.
And this is our signal, and y is 0.
This was the noise term.
And the variance in y, just to [INAUDIBLE] us, is given by n.
So therefore, if we calculate square root of y square over x
for the coherent state input, we take the square root
of y square, which is square root n, we divide by n,
we obtain the standard shot noise limit.
Sure.
What else would we expect?
So now we know how to use the formalism.
We can now apply it to-- we can go
from the most classical state of light, the coherent state,
to what I regard, at least for single mode, the most quantum
state, namely a single photon.
Remember when we talked about the G2 function.
G2 function can only be smaller than 1
for non-classical states, and for the single photon it's 0.
It's the biggest violation of the classical equation
that G2 has to be larger or equal than 1.
So now we're really dealing with a quantum system,
and the question is, what will we get for the single photon?
OK.
So for the single photon, we want to use the dual rail
representation.
We want to use the powerful formalism we have used.
After the beam splitter, the photon
can be in one mode or the other mode.
We call this, in the dual rail representation, the logical 0
or the logical 1.

So in this representation, when we
start with a photon in one input mode, this is the logical 0.

The beam splitter is directing the photons
into one mode or the other mode.
But this is a single qubit rotation.

The phase shifter is another thing single qubit operation,
the second beam splitter is another operation.
So by just using the rotation we'll
find out what is the output state.
So we start with the logical 0 with the photon in one mode,
the beam splitter creates a superposition state-- remember,
we have the interferometer and in the lower arm
we put in a phase, so therefore the lower arm
gets multiplied with a phase shift--
and then it goes again through a beam splitter
which is just another rotation.
And with that we obtain the output state.
And so the output state, in-- the output state
is now a superposition of the logical 0 and the logical 1
and we've picked up a phase shift.

So now we ask-- we have the beam splitter, phase shift,
recombiner-- and now we are asking,
what is the probability to detect the photon in one
of the two output modes?
We just put a counter there.
And this probability is nothing else than the probability
to have a logical 1, because a logical 1 in the dual rail
representation means the photon is in one mode.
The logical 0 would be the other mode.

I felt if I would spend three or four times as much time on it
you wouldn't-- it wouldn't increase your understanding.
You may have to sit down and look through it.
But it's just really putting matrices together,
single photons, mapping it.
Every step is trivial and is what
we have done in a different context before.
So the result is that the probability
of finding the single photon in our detector
has a cosine phi factor.
So that's how we measure phase.
Our counter, the probability that photons arrive,
if phi is 0, the probability is 1, if phi is 180 degrees,
all the photons go to the other mode.
That's what you expect an interferometer to do.
But now comes the interesting question.
How precise can be we measure the phase?

Remember, in the coherent state, the coherent state
had fluctuations in the photon number of square root n.
And this gave rise to the standard quantum limit.
A signal photon is a single photon.
There is no noise.
We always measure single photon if it would simply
detect the number of photons in the input state.
But we're not doing that.
We have sent the photon through an interferometer in order
to measure the phase, and now we have--
and this is the result of this, I would say,
trivial calculation, that this is now the probability
to observe a photon.
Well, there is not a whole lot we can learn from one photon,
so we run the experiment n times.
And what we get is it a binomial distribution.
Just a coin toss with probability p.
What is the variance in p?
It's not a [INAUDIBLE] distribution,
it is an binomial distribution.
Of course, if the probability is 1, you detect all your photons,
there is no uncertainty.
If the probability is 0, you detect 0 with 0 uncertainty.
So therefore, the expression for the variance
of the binomial distribution is p times 1 minus p.
If not, I admit I had to refresh my memory
about the binomial distribution yesterday evening.
So therefore, when we repeat the experiment many times
and we have sent into our interferometer n photons
and we measure n clicks, and n over m
is our measurement for the probability,
this measurement of the probability has noise.
And the noise is the variance of the standard deviation
of the binomial distribution.
This expression for p, put into the variance
of the binomial distribution and a little bit
of trigonomic manipulation, gives us that result
on the right hand side.

Well, we want to know what is the uncertainty in the phase.
I mean, this is what we want to measure.
Well, the uncertainty in the phase
is the uncertainty in our measurement, which is delta p.
And then we have to divide by how sensitive the probability
is with respect to the phase.
So we take this expression of the probability
as a function of the phase, take a derivative,
and then by doing the ratio of this over that,
we find 1 over square root n.
So it doesn't make any difference
if you put the n photons into a coherent beam
and run our interferometer or if we
go through the great pain of preparing
each photon in a non-classical flux state
and just operating the interferometer
with single photons.
In both cases do we obtain the standard quantum