So this is our last lecture. We will discuss at this lecture probably the most fascinating

phenomenon in quantum optics, namely entanglement. And then in the end I will tell you about

some experiments on Bell inequalities violation. And that will be concluding this course.

So why is it important? Because it is something very fundamentally important. Quantum mechanics

is probabilistic in its basis. So it's fundamentally probabilistic. And the concept of entanglement

shows that even a single quantum particle or two particles, quantum particles, behave

statistically if you want to make a measurement on this. And not just, they behave probabilistically.

And not just an ensemble of particles behaves probabilistically just from the viewpoint

of classical statistics. And that was the main argument at the beginning of the quantum

theory, when the quantum theory was just being created. So the first event historically that

I want to mention was 1935. And that was the so-called EPR paradox. So EPR means Einstein,

Podolsky, and Rosen. The three scientists wrote a paper claiming that the quantum mechanics,

the quantum theory, is incomplete. And what was their argument? The argument was that

imagine two particles, A and B, particle A and particle B, and they're just created at

the same time moment. They are created and they start to propagate in different directions

along the x-axis. So at the moment when the particles are created, their coordinates are

equal. So xB is equal to xA, but the momentums, so they start to fly from each other with

opposite momentums. And so PA is equal to minus PB. And because quantum mechanics stated

that a particle can be in a position state or in a momentum state, and for instance by

measurement we can put a particle A in the momentum state. For instance, we measure this

PA, and then the coordinate, the position becomes uncertain. But then from the viewpoint

of this situation, then we do the measurement on particle A at some point. And this puts

the particle B into the state with a given momentum, because we measure the momentum

of the particle A, and then we measure the momentum of the particle B. But they can be

very far from each other. They can be one at the Moon and the other at Mars, for instance.

And so this puts some paradox, because how did we really change the state of particle

B if we did the measurement on particle A? Indeed, it looks strange. Moreover, the same

authors were writing moreover, if we measure for particle A position, and so we know xA,

and for particle B we measure momentum PB, it means that for the same particle basically

we know the point at which it was created, and we know the momentum. And this is impossible

because of the uncertainty relation. So this was the argument. And there was a long discussion

between Niels Bohr and Einstein, mainly, Rosen and Podolsky were his PhD students, as far

as I understand. So this discussion was mainly philosophical, because the arguments were

purely Gedanken experiments and not real experiments. It was impossible to do this experiment at

that time. But now I'm going to show you that now, today, we can do experiments mimicking

this paradox, mimicking this situation. So imagine a paramedic down conversion that we

discussed as PDC, spontaneous paramedic down conversion, and we pump a crystal with K2

with some strong pump, and sometimes particles, photons, are generated. And for simplicity,

I assume that they are generated with the same frequency, or we don't care about the

frequency, but they are generated in different directions. Ka is the wave vector, or the

momentum equivalent to the momentum of photon A, and Kb is the K vector of photon B. And

what I will be interested in is the x direction, so the transverse direction. The pump, I will

draw it with some blue color. The pump has some waste, and this waste is rather broad,

so the photons can be generated along x position at different points of the crystal, and they

can be generated also with different projections of the K vector on the x direction. So this

is, I will be interested in the Ka x and Kb x, and also at the positions where the photons

are created. And we know that there is phase matching condition, so the K vector of the

pump should be equal to the K vector of photon A plus K vector of photon B. And of course,

if I consider just x direction, and the pump propagates orthogonally to the x direction,

then I can write that Ka x plus Kb x is 0. And this is exactly equivalent to the situation