I was surfing Internet trying to understand why most books i read simply considers that the stimulated photon emission has same properties as the stimulating photon, and treats this simply as an "take as it is".

I liked it becuase it explains how Einstein thought about this in his original paper, but It states that actually there is no sufficiently satisfactory proof for that even in quantum mechanics!!:

so although we using Lasers every day we don't understand how they really works?is that true?

This is the point where I stopped reading. Pretty much every part of these sentences is wrong. That guy is really daring to cite the Mandel/Wolf, while at the same time ignoring pretty much everything which is written in it. Writing that a full quantum treatment is not sophisticated enough is funny in some strange way.

In "fundamental" treatments of stimulated emission one rather goes the other way round. You start by dividing the emission into distinguishable and indistinguishable parts and then show that the indistinguishable part can create a stimulated enhancement. Basically one can backtrack stimulated emission to the spin statistics theorem and the properties of bosons as opposed to fermions.

In short there is a significant enhancement for processes which create indistinguishable bosons like stimulated emission or bosonic final state stimulation. This is basically a consequence of interference of indistinguishable probability amplitudes.

Yes I agree with you, that sentence seems to be very suspicious.

Can you please point out to some book to read more about this matter? becuase I couldn't find anything else on this matter on Internet (even wikipedia), and the most books of general course of atomic physics gives it as an axiom, and my knowledge of QM still very shallow to investigate this by my self.

The reason why that topic is not covered in too many books lies in my opinion in the fact that stimulated emission can be explained pretty well classically in full analogy to a classical driven dipole and it is only spontaneous emission which needs a quantum treatment.

Therefore there are not many who really bother with setting up a complete first principle quantum description of stimulated emission.

For suggested reading, many books on quantum optics will cover this topic. "Optical coherence and quantum optics" by Mandel and Wolf is the bible in this field. However, I must warn you that although it is well written, it takes a lot of time to read and digest it. For a more pedagogical approach to the problem, you might start by reading about the Hong-Ou-Mandel effect (the original citation is C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987), but you will find many easier descriptions using google or books on quantum optics) or the Hanbury Brown-Twiss effect (R. Hanbury-Brown and R. W. Twiss, Nature 177, 27 (1956), but again there are summarized descriptions all over the web).
These give a good first impression of the link between statistical properties and indistinguishability. A thorough theoretical description of these links has been given by Glauber, for example in: R. J. Glauber, in Quantum Optics and Electronics (Les Houches Lectures), p.63, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965), but also here understanding takes some time.

For experimental tests you can check F. W. Sun et al., "Stimulated Emission as a Result of Multiphoton Interference", Phys. Rev. Lett. 99, 043601 (2007). Maybe following this paper and the references therein is an easier way to get the topic. A free version under a similar name is available on the ArXiv in case you do not have access to PRL.

I checked Hong–Ou–Mandel effect on wikipedia and "Stimulated emission of two photons in parametric ampliﬁcation and its interpretation as multi-photon interference" paper from arXiv, now it's much clearer how interference of indistinguishable photons plays it's game (it's nice to understand things deeper), even so I need to know more QM to understand the math of that paper.

anyway, just wanted to point, that even in that paper the authors said at the beginning that:

so the author of the article I posted in the first post was right about this point, even so he mentioned that in a ambiguous way.

The relationship between treating the electromagnetic field classically and quantum mechanically is also discussed in https://www.amazon.com/Introductory-Quantum-Optics-Christopher-Gerry/dp/052152735X, section 4.3, "Interaction of an atom with a quantized field". On p83, they write "(n=0) ... This is spontaneous emission and it has no semiclassical counterpart. If n>0, the emission of an additional photon is called stimulated emission.". (A free account will allow you to view search results.)

Well, the effects on non-classical states have indeed not been investigated too much, but common lasing was well understood. Actually, as atyy pointed out correctly, the semi-classical J-C model is also sufficient for most problems (and the next topic to read about if you are interested in the field).

I wouldn't rely on Wikipedia for things. Even though when you read an article and you see it well explained, doesn't make it true. Anyone could have written that, and if you're not well knowledged in the area you won't see errors if there are any.

Going on the same theme... I followed to OP's link and then amputated the URL to find out who the author is. I won't make further comments one way or another, but for a good "WTF moment," check out this guy's CV: http://www.sjsu.edu/faculty/watkins/resume2.htm

Again, this is the guy critiquing the sophistication of quantum mechanics.....