M1L4i.txt

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So the principle of homodyne detection
is now that we want to mix light at the beam splitter.
So the device which does all that for us
is, after using so many words, I would say it's
disgustingly simple.
It's really a half [INAUDIBLE].
We'll talk about beam splitters a lot
here, because beam splitters perform
wonderful unitary transformations.
And we will exploit them for many purposes.
For the purpose of this lecture, I simply
assume that we have a 50-50 beam splitter.
So what I'm telling you now about the beam splitter
will be generalized either later today,
or in the lecture on Wednesday.
So a beam splitter has two input parts, and two output ports.
We have light impinging on the beam splitter.
We call those modes a and b.
And after the beam splitter, we have two output modes,
and let me call them a0 and b0.
And what we measure is the output of the beam splitter.
So we have two photo detectors, and we measure the output.
OK, the output modes, a0 and b0 are simply
obtained by taking the input modes and propagating them.
We have two modes a, b, sort of, you
can see what goes vertical is a, before it was a, we call it a0.
Here it's b, it becomes b0.
And what we have to do is, we have
to transform the operator a0.
I'll just make a comment.
I mean, sometimes in the beam splitter, you want to think,
you know, a quantum state comes and is transformed.
But instead of transforming this state of a photon,
I can also transform the operator
which creates a photon.
One is the Heisenberg picture, one is a Schrodinger picture.
So right now I use the Heisenberg picture.
I have two operators, a, b.
They transform into, before they hit the beam splitter,
you say I have two operators, a0, b0.
Afterwards, I have two other operators.
I have operators a, b, afterwards
I have operators a0, b0.
And if you do the time evolution,
it is a unitary transformation.
And for operators, we have to multiply the operators
from the left and the right hand side
with the unitary operator, u and u [? dega. ?]
And we really talk about the beams splitter
in its full beauty in a short while.
It will be a special case of what we discuss later,
but I think it's pretty obvious that a 50-50 beam splitter is
simply creating two modes-- one are the symmetric combination,
and one are the antisymmetric combination.
OK, so in our homodyne detector we
are now measuring the number of photons in the mode a0.

We measure with the upper detector the number
of photons in the mode b0.
But what we are then doing is we run it for a different circuit.
You want to cancel certain noise, as you see in a moment.
This is why we obtain the different signal, which
we call i minus.
So let's now calculate what i minus is.
Well, the number of photons in the mode a0
is a0 [? dega ?] a0.
That's the operator for the intensity of,
for the number of photons.
We subtract b0 [? dega ?] b0.
And as you can immediately see is that, because of the beam
splitter, this is now involving a product of a and b.
[? ab dega ?] plus a [? dega ?] b.
So in other words, when you ask yourself, how can we remember?
I said, we pick out the [? cosine ?] omega t component
by multiplying our signal with the local oscillator.
And you would say, how do you multiply two modes
of the electromagnetic field?
Well, just send it to a beam splitter.
In a beam splitter, you create sum of them.
But then your photodetector is a [? squalar ?] detector.
You measure the electric field squared,
or you have to measure the operator a [? dega ?] a.
And now you get a cross term between a b [? dega ?]
and [? a dega ?] b.

So this is how we multiply to operators,
how we get a signal which is proportional to you know,
a, or [? a dega ?] times b or b [? dega. ?]

By the way, if you wouldn't take the difference,
we get terms of a, you know a a [? dega, ?] b [? b dega. ?]
Just the mode a, or the mode b by themselves,
and we want to get rid of them.
And by taking the difference between the two photodetectors,
those parts of the signal are [? common ?] mode
and are subtracted out.
So therefore, this is called balanced homodyne,
because we have a balanced beam splitter.
We measure two signals, and then we
take the difference of the two signals.
There's one more thing we have to add,
and then we can find we understand
the balanced homodyne detection.
We want to exploit that our mode b, remember
we want to measure mode a, a has squeezed,
a has quantum properties, a has only single photons.
And b is just a trick to project out the [? cosine ?] omega t.
And we do that by choosing, for b, a strong coherent state,
described by coherent state parameter beta.
Of course, whenever we have a strong coherent state,
that means that we can replace b by beta, and b [? dega ?]
by beta star.
This is sort of the classical field limit
of a strong coherent state.
And now we can ask, what is our different signal, i minus.
Well, the coherent state depends on the phase angle theta,
so it will be phase sensitive.
It will depend on the angle theta.
But now there is one more thing.
And that is, if you use this stronger coherent state,
of course, most of our outputs go up in proportion to beta.
So therefore, we want to do a normalization
by dividing by beta.
So this is now our normalized output.
So this is now simply-- I'm looking at this.
So this is our operator for the signal i minus.
And it is a-- b has been replaced by,
b [? dega ?] has been replaced b star.
That gives us a times e to the-- because
of the complex conjugation-- minus i-theta, plus e
[? dega ?] times e to the i-theta, divided by 2.
So therefore, if we choose the phase to be 0,
we measure the symmetric combination,
a plus a [? dega ?] over 2.
And this is simply the a1 quadrature operator
divided by square root 2.
And if you put a phase shifter, just
dispersive element into this long local oscillator,
and shift the phase by pi over 2 by delaying the light pulse
by a quarter wavelengths.
What we are now projecting out is the other quadrature
component.
So therefore, using a beam splitter
on the local oscillator, we can now measure the expectation
value for a1 and for a2.

I've posted a few papers on the website, which
shows some of the pioneering experiments where
people did exactly that.
So then when they observed, let's
say they observed the quadrature component, which was squeezed.
They usually did it for a squeezed vacuum,
so a1 and a2 had zero expectation value.
But the interesting part is how much noise was there.
If you don't squeeze, you find that your normalized noise
simply corresponds to the classical photon short noise.
But if you are squeezed, you find
that one quadrature component has
smaller noise and short noise.
So if you take the n photons in your different signal,
and your noise is less than square root n,
whereas, in the other quadrature component, it's larger.
So you can look up on the website papers which,
where people slowly varied the phase of the local oscillator.
And you see, this is short noise,
how the noise is below short noise, exceeds short noise,
is below short noise, and exceeds short noise.
And this was the first evidence for the generation
of squeezed light.
OK so, we have discussed the detection
of squeezed light using balanced homodyne detector.
Take And balanced means that the beam splitter was 50-50.
So now we are ready to discuss the unbalanced case.
So let's get our unbalanced beam splitter.
And just for the sake of the argument,
let's say it has a really good transmission of,
OK t is the consumption coefficient.
T-squared tells you what fraction of the power
is transmitted.
And let's just assume, for the sake of the argument,
99% are transmitted.
So therefore, if we start with our signal
a, which may be an interesting squeezed quantum
state, number state, you name it, a0 is pretty much the same
as a.
We haven't lost so much.
But now we use our local oscillator,
which is very strong.
Only a very small fraction of it will be reflected,
but we can always compensate for this small reflection
by making b even stronger.
Let's see what we get.
So the mode a0, the output the mode,
is now the transmission coefficient times a.
a and a0-- the operators are sort of amplitudes,
so we have to use if we have a 99% beam splitter,
we have to take the square root off 0.99.
This is the transmission co-efficient.
And now we have the transmission coefficient
for this strong local oscillator, which
we will, again, approximate by its eigenvalue, coherent state
eigenvalue.
And the reflection coefficient is 1 minus t squared.
So if we make the approximation that t is approximately 1,
we can take it out of the parentheses.
And we obtain that, if t is close to 1, we can neglect it.
And what we see now is, what we have obtained,
is actually the original quantum state a,
or I should say the operator, I'm in the Heisenberg picture.
Is the mode operator for the input
of the unbalanced homodyne detector.
And what we have simply done is we have displaced the operator.
We have displaced the mode operator a.
And the displacement operator has an argument
of 1 minus t squared beta, and this
is the Hermitian conjugate.
1 minus t squared beta.
So what I've shown you here is that local oscillator
and the beam splitter is a realizational implementation
of the displacement operator.
So in the limit that t goes to unity,
we are not losing anything of our quantum state.
But by reflecting in the amplitude
of a strong coherent state, we simply take our quantum state,
and we displace it in this two-dimensional plane.
So that's what this beam splitter does for us.
Any questions?

In the next few classes, we really
take advantage of those elements.
We know now that the displacement operator is just
a beam splitter.
We know when we have squeezed some light
by using balanced homodyne detection,
we can just do measurement of one quadrature component
of the other.
So what I hope for those of you who
haven't heard about it, what you take away from that
is that these are extremely simple elements,
but by combining them, we can really
realize very sophisticated schemes of quantum optics.
To some extent, when I heard about for the first time,
I think the mathematics looked so fancy.
I couldn't believe that such simple elements can actually
realize what those operators describe.
So I gave you the example, for the displacement operator.