M1L5o2.txt

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So with that introduction, we want
to talk about single photons.
And I want to introduce for you the singular photon qubit.
Now if you talk about singular photons,
you automatically have one qubit.
0 is no photon.
1 is the single photon state.
We make an addition later on.
This is not the most robust qubit
because, due to unavoidable losses are limited quantum
efficiency, you have one photon and you lose it,
you think the [? bit ?] has been flipped.
So you don't want to really use that,
but we will use, in a moment, a similar photon
in two different quantum states.
And then if you detect 0, you know you've lost it.
We will talk about one photon, which is either here or there.
And this is 0 of our qubit, and this will be 1 of our qubit.
But we'll come to that later.
I first need the Hilbert space of 01 to introduce a few
things.
And then we use what we have learned
to define the [? dual rails ?] in a photon state.
OK, but for now, we don't have to make this distinction.
The question that you want to answer
is, what happens when those states pass
through optical components?
And I want to show you how beam splitters and phase
shifters can be used as very simple optical components.
They can be realized, they can be
used to realize an arbitrary single qubit operation.
And then we introduce non-linear interferometers,
and we get two qubit operations.
So let's start out really simple and talk about phase shifters
and beam splitters.
And this is pretty much how far we
will get today in this chapter.
Well, if we have a single photon and time passes on,
the single photon gets phase shifted.
If you pass it through a dispersive medium,
it will have another phase shift.
So this is how phase shift can be realized.
But there is one caveat, and that immediately tells us
we can't get away with just a single photon in one fiber
and phase shifting it, because everything phase
shifts a photon.
On Time, everything which is in the beam pass.
So whenever we talk about a phase, we need a reference.
So therefore, we're immediately [? drawn ?] that one mode
and one photon is not enough, even if you are interested
in [? one ?] photon gates, we need two modes.
So that means now we just think about two waveguides or two
optical fibers.

We call them mode a and b.
If we put one photon on in, and this is now
the symbol for a phase shifter, a little box with the letter
phi, that we get a phase shift of the state 1.
And the second mode a acts as a reference mode
to detect the shifter phase, let's
say by interfering with the two photons.
So in general, we have an input state
and we obtain an output state, which has a phase shift.
We'll come back to phase shifters later.
So we have two modes.
The next thing we can do is we can combine modes.
And this is done with a beam splitter.
So we've talked about beam splitters
in the context of [? homodyne ?] detection.
We've talked about beam splitters
most of the first hour with a clicker question.
And I was mainly referring to the beam splitter to you
before, in a rather non-formal way, if you have a 50/50 beam
splitter and you have two inputs,
you get a symmetric and antisymmetric combination
of the two.
If it's a 50/50 beam splitter, what else can you get?
Or, I used the concept of just reflection and transmission
coefficients.
I think that's all important to get an intuitive
feel for the beam splitter.
But now I use a more vigorous approach.
I have a beam splitter, which is characterized
by an angle theta.
We will later find out what theta is.
And we have an input mode a and a second input mode b.
And those modes get mixed now.
We have output modes b prime, and output modes a prime.
And we can use a Schrodinger picture
and talk about the photon states in those modes,
but I prefer to use the Heisenberg picture.
So when I put a, a prime, and b and b prime in,
just think about annihilation operators, which annihilate
photons in those modes.
So what I postulate is that this system is described
by the following Hamiltonian.

And I'm sort of telling you this Hamiltonian describes this beam
splitter.
You say why and [? how. ?]
Well, I will tell you what this Hamiltonian does to the modes.
And then we have the quantum description
of the beam splitter.
And you will then see in your homework
that this quantum description of the beam splitter, that's
exactly what it is expected to do to cohere in states.
It just splits a laser beam into two
with a reflection and transmission
coefficient, which are cosine square theta
and sine square theta.
And I will also show you that this beam splitter is doing
the same, splitting with cosine square
and sine square theta probabilities a single photon.
So in essence, I give you the Hamiltonian
and I discuss that the transfer matrix, which
is built upon this Hamiltonian, that's
everything [? that you ?] ever want a beam splitter to do.
And therefore you can say this defines the beams splitter.
OK.
If [? you're in ?] Hamiltonian that means,
after propagating for the Hamiltonian,
we have the following transfer.
You may be familiar by putting here the unitary time evolution
and the time which has elapsed.
But we are always using a fixed time
after the light has transformed the beam splitter.
So you can say the time t is absorbed
in the definition of 8, or absorbed
in the definition of [? hc ?] Hamiltonian,
and b is really the unitary time evolution
from beam before and beam after the beam spit up.
So the transformation, which transforms the modes
at this beam splitter, has now the following unitary operator.
And we have modes a, b, mode 1 and 2 before the beam splitter.
After the beam splitter, they have been transformed
to a prime and b prime.
And we obtain that by taking the input modes
and doing, well, unitary time evolution.
You do simply the operator transformation
for unitary transformation.
So we can now transform the operators a and b.
I'm not going with you, through the exercise,
how you transform an operator.
They are, I think in undergraduate quantum mechanics
classes, you do the Baker-Campbell-Hausdorff
formula, how to deal with operators in the exponent.
And if you do that, you find, not surprising,
a beam splitter is a linear element, that what you get
is a linear combination of a and b.

So if you don't like me postulating an operator,
I'm saying, believe me, this is a beam splitter
you can reverse engineer.
You can say this really looks like the modes,
the operators a, b, get linearly superposed
with cosine and sine.
This is what a beam splitter does, is it mixes two modes.
And then you can reverse engineer and figure out
that it's exactly the Hamiltonian above which
does it.
So your homework will show that this happens exactly
for coherent states.
So if you have a coherent state with alpha,
you get a coherent state with alpha cosine theta in one arm
and alpha sine theta in the other arm.
And you will see that everything works out beautifully.
I have to use [? one ?] convention now for a 50/50 beam
splitter.
And I use the convention that I chose theta to be pi over,
to be 90 degrees.
And we want to use symbolic language
because we want to draw diagrams of multiple beam
splitters and such.
So this tilted square, this [INAUDIBLE]
is the symbol for beam splitter.
We have input ports a and b on the left side.
We have output ports, which are now b prime and a prime.
And for the 50-50 beam splitter, the modes
are symmetric and antisymmetric superpositions.
So this is the symbol for the unitary transformation B.
And since one output port has a minus sign,
[? it is ?] antisymmetric and one is positive,
we usually put a dot there, which distinguishes the two
different ports.
So this is the operator B. We have already
used for the unitary transformation the operator B
dega.
So for the operator B dega, using the equations
above, we have input modes a, b.
But what happens now is, for the operator B dega,
the output modes are interchanged.
So we have a plus b over square root
a, a minus b over square root of 2,
and therefore the symbol for B dega has the dot here.
I just want to warn you that there
are other phase conventions in the literature which
are equivalent.
But sometimes if you read a paper and all the minus signs
appear somewhere else, well, people
have a different phase convention.
You can postulate that you have a beam splitter
without having the i here.
And everything works similarly.
It's just that, for instance, in this equation,
you get an i here.
So imaginary units and plus-minus signs
[? in that business ?] are phase conventions.
If you would say, hey, but if you have another definition,
don't you have another beam splitter?
Yes, it is another beam splitter.
But I can always get it out of my beam splitter, which
now is our beam splitter, by putting a phase
shifter afterwards.
So it doesn't limit the generality of the discussion