Okay, good morning everybody. First of all I want to give you these things, these little

slips to do feedback. You are supposed to make some feedback about my lectures, but

unfortunately every year the number of people who give feedback is much less than the number

of people who are present at the lectures. Okay. Actually, there is enough chalk. Thank

you. Just one important thing. Today we will have a lecture on parametric amplifiers, parametric

down conversion, follow-with mixing, finishing the subject I started at the last lecture.

Then I will mostly today I will speak about single photonometers and then there will be

only one more lecture and that will be about quantum key distribution. That will finish

the course and unfortunately it cannot be next week because I go I give a colloquium

talk in Mainz and so the next lecture will be in two weeks and that hopefully will be

the last lecture unless there is something that remains. Concerning the problem that

I gave you, I see that there are already solutions, but just in case I will explain briefly how

to solve this problem. The problem is to find the quadrature squeezing, to find the delta

x1 and delta x2, the quadratures. I remind you that at the last lecture we started considering

parametric amplification and the first type of parametric amplifier we considered was

the so-called single mode amplifier and the Hamiltonian looked like i h bar gamma a dagger

squared plus of course hermitian conjugate. Gamma was the constant into which we included

all parameters like pump, amplitude, chi two, the length of the crystal and then we dragged

i h bar. Then we got equations for the quadratures and we got equations for the a operator, for

the photon annihilation operator. The equation was a was u a naught plus v a naught dagger

and then v was hyperbolic sine of gamma t and t was some constant of with the dimensionality

t of time and that was I can say that was like the pulse duration and the total well

I think it was a factor of two here yeah factor of two for this kind of Hamiltonian and this

total parameter I denoted by g and called the parametric gain. So this transformation

for the photon annihilation operator has a name has a title is called the Bogoliubov

transformation. Let me write it like this or sometimes it's written like this and then

from from this or more correctly to say that we derived it from the evolution of the quadratures

and the evolution of the quadratures was simpler so x one was depending on time. There is someone

oh was it locked? Hi. Okay. The evolution of the quadratures was the exponential amplification

and the amplification so x one was e to the power g x one naught where x one naught is

the initial quadrature and x two was e to the minus g x two naught the initial quadrature.

I'm just reminding what what happened at the last lecture and from this you can of course

immediately derive thank you the variance because well the uncertainty of the quadrature

delta x one or two is square root of delta x one or two squared and this is the variance

so this is mean value of x one or two squared minus some anything. Yeah yeah yeah I'm giving

this to let you make feedback for my lectures. Yeah so minus the mean value squared yeah

so this is the definition you can easily calculate that the mean value is zero this will be zero

and then if you use this equation you immediately find that x one or two squared mean value

first of all x one squared will be given by just the square of this equation so it will

be e to the two g x one naught squared and then the accordingly delta x one it will be

e to the g delta x one naught and this is the initial vacuum. This was x one equal to

exponential two g was here. Here? Depending what we call g right if we call g gamma t

I don't remember how I defined it but you understand the the point yeah because now

I will consider two mode parametric amplifier and there will be no factor of two but the

meaning of g is the gain so how the v parameter depends on develops with time right and then

v squared if you probably remember v squared was the mean photon number after the parametric

amplifier. I tell you that's a very important thing you should actually I notice now that

they give you two sided double double printed things so please use just one I didn't notice

it yeah so the meaning of this gain is that remember the square of v is the mean number

of photons and and so it's c sine hyperbolic sine squared of what I call the gain the parametric