Okay, good morning. Let's start for today. Today's lecture will be about quantization

of the electromagnetic field. But before I start, I wanted to ask you if someone made

the home task, someone did the problem, please give me the solutions before the end of the

lecture, because in the end of the lecture I want to explain how the problem should be

solved because one of you asked me some questions and I want to give the, not the solution,

but at least to explain to all of you. Okay? So, we, during the last lectures, we came

to several, not really paradoxes, but they become paradoxes if you take into account

experimental data. So, we derived certain conditions from this classical approach to

light. During the three lectures we considered light classically. At the fourth lecture we

still considered light classically, but atom quantum mechanically. And we, it was mentioned

that from time to time that something will not hold true in experiment. And I will list

the conditions that we obtained. So, classical, classical approach led to first, when we spoke

of photocounts, number of photocounts m, I mentioned that, and I derived it from classical

concept, that the number of photocounts always should manifest superpoissonian or poissonian

behavior. So, delta m squared, the variance, should not be lower than mean value. It means

that the situation where m, delta m squared is less than mean value, and it's called,

by the way, subpoissonian statistics of photocounts, is impossible in classical optics. This can

be violated because there will be experiments, and we will discuss such experiments, where

you have light for which the number of photocounts shows superpoissonian behavior. Second, anti-bunching.

So I mentioned to you that, of course, there is G2, the second order correlation function,

and we derived classically that G2 is, again, larger than 1. It means that in classical

optics only bunching is possible, or the absence of any bunching, but anti-bunching is impossible

in classical optics. And yet it happens. It happens, for instance, if you have a single

atom, it emits a single photon. You put two detectors, try to look for coincidences, never

happened, because an atom can emit just one photon at a time. It needs time to be very

excited. But we will speak about it at a separate lecture. Third thing, in classical optics,

it's possible that the intensity is just constant. It means that if you send a beam on a beam

splitter and then put two detectors here and register some intensities, intensity 1, intensity

2, using, for instance, photocurrent. And then if it is a 50-50 beam splitter, then

the mean value of intensity 1 is equal to the mean value of intensity 2. But also, classically,

the variance of intensity 1 minus intensity 2 will be also 0, because what you send some

classical signal, you split it exactly into, if it fluctuates with time, the two signals

will fluctuate exactly the same way, and here they will be always 0, so it will be 0. And

so it doesn't predict so-called short noise. In reality, light consists of photons, and

we will come to this at today's lecture. In reality, there are single photons here, and

they are split irregularly by the beam splitter. So on the average, the same number of photons

goes here, the same number of photons goes there. But short noise leads to the fact that

variance of n1 minus n2, and now I write photon numbers, is not 0. And actually, it's given

by the mean values of the numbers of photons, but it's a separate story. I also want to

say, I only want to say that classical optics does not predict the short noise. So if you

split a beam on a beam splitter, there will be exactly equal, classically exactly equal,

than two detectors. And moreover, if we go further in quantum optics, and we'll go further

in quantum optics, there are some sources of light that have this, that have 0 here.

But these are very special sources, very special sources, and this effect is called two-mode

squeezing or photon number correlation, and so on. So I'm now going a little bit ahead

of the material of today's lecture. At today's lecture, probably you'll have an idea that

there is short noise. Number four, during the last lecture, we described transitions

of an atom in an electromagnetic field. So in the presence of light, an atom can go from

level one to level two, and it can go also the other way. And the probability of transition

up is called W, the rate of transitions from level one to level two is called W12, and