PROFESSOR: Mach-Zehnder--
interferometers.
And we have a beam splitter.
And the beam coming
in, it splits into 2.
A mirror--
another mirror.
The beams are recombined
into another beam splitter.
And then, 2 beams come out.
One to a detector d0--
and a detector d1.
We could put here any kind
of devices in between.
We could put a little
piece of glass,
which is a phase shifter.
We'll discuss it later.
But our story is a
story of a photon
coming in and somehow leaving
through the interferometer.
And we want to describe this
photon in quantum mechanics.
And we know that the
way to describe it
is through a wave function.
But this photon can live
in either of 2 beams.
If a photon was
in 1 beam, I could
have a number that tells me the
probability to be in that beam.
But now, it can be
in either of 2 beams.
Therefore, I will
use two numbers.
And it seems reasonable to
put them in a column vector.
Two complex numbers that give
me the probability amplitudes--
for this photon to be somewhere.
So you could say, oh, look here.
What is the probability that
we'll find this photon over
here?
Well, it may depend on the time.
I mean, when the photon
is gone, it's gone.
But when it's crossing here,
what is the probability?
And I have 2 numbers.
What is the probability
here, here, here, here?
And in fact, you could
even have 1 photon
that is coming in through 2
different channels, as well.
So I have 2 numbers.
And I want, now, to do
things in a normalized way.
So this will be the probability
amplitude to be here.
This is the probability
amplitude to be down.
And therefore, the probability
to be in the upper one--
you do norm squared.
The probability to
be in the bottom one,
you do norm squared.
And you get 1.
Must get 1.
So if you write 2 numbers,
they better satisfy that thing.
Otherwise, you are not
describing probabilities.
On the other hand, I may have
a state that is like this.
Alpha-- oh, I'll
mention other states.
State 1-0 is a photon
in the upper beam.
No probability to be
in the lower beam.
And state 0-1 is a
photon in the lower beam.
So these are states.
And indeed--
think of superposition.
And we have that the
state, alpha beta--
you know how to
manipulate vectors--
can be written as alpha 1-0.
Because the number goes
in and becomes alpha 0.
Plus beta 0-1.
So the state, alpha
beta, is a superposition
with coefficient alpha of
the state in the upper beam
plus the superposition
with coefficient beta
of the state in the lower beam.
We also had this
little device, which
is called the beam
shifter of face delta.
If the probability
amplitude completing
is alpha to the left
of it, it's alpha
e to the i delta to the right
of it, with delta a real number.
So this is a pure phase.
And notice that alpha is
equal to e to the i delta.
The norm of a complex
number doesn't change when
you multiply it by a phase.
The norm of a complex
number times a phase
is the norm of
the complex number
times the norm of the phase.
And the norm of any phase is 1.
So actually, this doesn't
absorb the photon,
doesn't generate more photons.
It preserves the probability
of having a photon there,
but it changes the phase.
How does the beam
splitter work, however?
This is the first thing
we have to model here.
So here is the beam splitter.
And you could have
a beam coming--
A 1-0 beam hitting it.
So nothing coming from below.
And something coming from above.
And then, it would reflect
some and transmit some.
And here is a 1--
is the 1 of the 1-0.
And here's an s and a t.
Which is to mean that this beam
splitter takes the 1-0 photon
and makes it into an st photon.
Because it produces a
beam with s up and t down.
On the other hand, that
same beam splitter-- now,
we don't know what those
numbers s and t are.
That's part of designing
a beam splitter.
You can ask the
engineer what are s
and t for the beam splitter.
But we are going to figure
out what are the constraints.
Because no engineer would be
able to make a beam splitter
with arbitrary s and t.
In particular, you
already see that if 1-0--
if a photon comes in,
probability conservation,
there must still be a photon.
You need that s squared
plus t squared is equal to 1
because that's a photon state.
Now, you may also have a
photon coming from below
and giving you uv.
So this would be a 0-1
photon, giving you uv.
And therefore, we
would say that 0-1--
gives you uv.
And you would have u plus v
norm squared is equal to 1.
So we need,
apparently, 4 numbers
to characterize
the beam splitter.
And let's see how
we can do that.
Well, why do we need,
really, 4 numbers?
Because of linearity.
So let's explore that
a little more clearly.
And suppose that I ask you,
what happens to an alpha beta
state--
alpha beta state if it
enters a beam splitter?
What comes out?
Well, the alpha beta
state, as you know,
is alpha 1-0 plus beta 0-1.
And now, we can use our rules.
Well, this state, the beam
splitter is a linear device.
So it will give you alpha times
what it makes out of the 1-0.
But out of the 1-0 gives you st.
And the beta times 0-1
will give you beta uv.
So this is alpha s plus
beta u times alpha t
plus beta v. And I can write
this, actually, as alpha beta
times the matrix, s u t v.
And you get a very nice thing,
that the effect of the beam
splitter on any photon
state, alpha beta,
is to multiply it by
this matrix, s u t v.
So this is the beam splitter.
The beam splitter acts
on any photon state.
And out comes the matrix
times the photon state.
This is matrix action,
something that is going
to be pretty important for us.
How do we get those numbers?
After all, the beam
splitter is now
determined by these
4 numbers and we
don't have enough information.
So the manufacturer can tell
you that maybe you've got--
you bought a balanced
beam splitter.
Which means that if you have
a beam, half of the intensity
goes through, half of the
intensity gets reflected.
That's a balanced beam splitter.
That simplifies things
because the intensity
here, the probability,
[INAUDIBLE]
must be the same as that.
So each norm squared
must be equal to 1/2,
if you have a balanced--
beam splitter.
And you have s squared equal
t squared equal u squared
equal v squared equal 1/2.
But that's still far from enough
to determine s, t, u, and v. So
rather than determining,
them at this moment,
might as well do a guess.
So can it be that the
beam splitter matrix--
Could it be that the
beam splitter matrix
is 1 over square root of
2, 1 over square root of 2,
1 over the square root of 2,
and 1 over square root of 2.
That certainly satisfies
all of the properties
we've written before.
Now, why could it be wrong?
Because it could be pluses
or minuses or it could be i's
or anything there.
But maybe this is right.
Well, if it is right, the
condition that it be right
is that, if you take a
photon state, 1 photon--
after the beam splitter,
you still have 1 photon.
So conservation of probability.
So if you act on a
normalized photon
state that satisfies this
alpha squared plus beta squared
equal 1, it should still give
you a normalized photon state.
And it should do
it for any state.
And presumably, if you get
any numbers that satisfy that,
some engineer will be able
to build that beam splitter
for you because it
doesn't contradict
any physical principle.
So let's try acting on
this with on this state--
1 over square root of 2,
1 over square root of 2.
Let's see.
This is normalized--
1/2 plus 1/2 is 1.
So I multiply.
I get 1/2 plus 1/2 is 1, and 1.
Sorry, this is not normalized.
1 squared plus 1
squared is 2, not 1.
So this can't be
a beam splitter.
No way.
We try minus 1 over
square root of 2.
Actually, if you try this for
a few examples, it will work.
So how about if we
tried in general.
So if I try it in general,
acting on alpha beta,
I would get 1 over square root
of 2 alpha plus beta and alpha
minus beta.
Then, I would check
the normalization.
So I must do norm
of this 1 squared.
So it's 1/2 alpha
plus beta squared
plus 1/2 alpha minus
beta norm squared.
Well, what is this?
Let me go a little
slow for a second.
[INAUDIBLE] plus beta star.
Plus alpha minus beta.
Alpha star minus beta star.
Well, the cross terms vanish.
And alpha alpha star, alpha
alpha star, beta beta star,
beta beta star add.
So you do get alpha
squared plus beta squared.
And that's 1 by
assumption because you
started with a photon.
So this works.
This is a good beam
splitter matrix.
It does the job.
So actually--
Consider this beam splitters.
Actually, it's not the
unique solution by all means.
But we can have 2 beam splitter
that differ a little bit.
So I'll call beam splitter
1 and beam splitter 2.
Beam splitter or 1
will have this matrix.
And beam splitter 2 will have
the matrix were found here,
which is a 1 1 1 minus 1.
So both of them work, actually.
And both of them are
good beam splitters.
I call this--
beam splitter 1.
And this, beam splitter 2.
And we'll keep that.
And so we're ready, now, to
think about our experiments
with the beam splitter.
