It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true? How fast should photo-detectors be for observing of the interference of beams from two of the "best available" lasers? What is the coherence length of the best-available laser? More specifically, does there exist any (optical single-wavelength) laser with coherence length exceeding 500 meters?
This paper seems relevant to your question. If I'm reading the abstract correctly, the answers to your questions are:
Q: It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true?
Q: How fast should photo-detectors be for observing of the interference of beams from two of the "best available" lasers?
A: 1 millisecond or faster
Q: What is the coherence length of the best-available laser?
A: More than 300 km
Q: More specifically, does there exist any (optical single-wavelength) laser with coherence length exceeding 500 meters?
The abstract in the paper:
Interference fringes produced by a pair of intracavity stabilized diode laser beams, each impinging separately on one aperture of a double slit, are recorded on a linear charge-coupled device array. The peculiar result of the experiment is that the fringe system is found to persist for a time of the order of 1 ms and loses contrast for longer integration times. This implies that the individual linewidths of the two beams from the stabilized lasers are narrower than 1 kHz and that the average drift rates of the central peaks are far less than 0.1 MHz/s. The device was built within the advanced undergraduate electronics laboratory of the department of physics and represents a considerable improvement over previous demonstration apparatuses used to detect interference fringes from independent lasers.
An interesting 1986 review of interference from independent sources:
"Interference between independent photons", Rev. Mod. Phys. 58, 209–231 (1986)
Detectors as fast as 50 GHz can be easily bought (if you have the money :P). This means that if the difference in the frequency of the lasers is smaller then 50 GHz or wavelength difference is smaller than 60 pm then you can detect the beating using these fast detectors. This wavelength difference can be achieved (sorry but I am not in the mood of finding a papers reporting said achievement), you can even phase-lock the lasers (but then I would not call them independent anymore)
personally i work with two pulsed laser source and i managed to use one as master and the other one as slaver succeding in this way to have a perfect sincronization in wavelenght, phase and pulse frequency.. i've never tried to see interference path between them (I use these just for microscopy purpose) but as Tobias was saying I don't see why that should not work as long as a stable sincronyzation is kept.
I really don't know about the 500m coherence length..
I'm assuming from the phrasing of the question that you would like to see a spatial interference pattern-- bright and dark fringes-- not something in the time domain. As noted earlier, you can easily beat two independent lasers together and get a signal that pulses at the difference between the two. You can also do coincidence counting experiments using photons derived from two different lasers and show that they interfere with each other-- I believe that's the "Hong-Ou-Mandel" experiment, but I could have the name wrong.
If what you want to see is a pattern of interference fringes in space, you need a detector that is both fast and position-sensitive, as the interference pattern will shift around rapidly in time. You would need to take a "snapshot" of the pattern at some instant, before it has time to shift position and wash out the fringes from an earlier time. So, what kind of speed are we talking about?
Well, the pattern between the lasers would be steady for a time on the order of the coherence time of the lasers. A good rule of thumb for finding the coherence time for a fairly ordinary laser is that the coherence length of the laser is roughly equal to the length of the cavity. A typical gas laser of the sort you would readily find on the shelves of your local physics department stockroom has a cavity that is roughly one foot long, which means a coherence time of the time required for light to travel one foot, which is one nanosecond.
So, you would want to be able to pick up the pattern in less than a nanosecond, which means your detector needs to be able to handle count rates of at least one gigahertz. That's not too difficult to manage-- you can get photodiodes with a bandwidth of 50-60GHz pretty easily. In order to see a spatial pattern, though, you really want a linear array of these at the very least, and a CCD camera would be even better. You also need to be able to pick up multiple photons in that span, in order to be able to see the pattern, so you can clearly resolve the difference between bright and dark fringes-- I would say you probably want at least 100 photons/pixel in the bright fringes to be able to get decent fringe contrast. This is doable with fairly basic lasers-- a few milliwatts in the red range of wavelengths is about 10^15 photons/s, or about 10^6 photons/ns, so you can spread that out over a few thousand pixels and still be safe. And, of course, you need a low level of "dark counts" in your detector, so it can readily tell the difference between a bright spot with 100 photons and a dark spot with none.
That's a pretty challenging set of detector requirements. You want, essentially, a CCD detector with single-photon sensitivity and a bandwidth of 100 GHz. I don't think you're going to find that just lying around. Even the fast CCD detectors people use for real-time monitoring of BEC experiments and the like have a frame readout time of a millisecond or so, and those aren't single-photon sensitive. Single-photon detectors tend to be single avalanche photodiodes, which are easily damaged by photon count rates of a few tens of kilohertz. You might be able to construct a detector with these specs, but it would be an extremely difficult problem, and not something that would be worth doing just to see a spatial interference pattern between two independent lasers.
If you can bump the coherent time of the lasers up by a factor of 1000 (which is not a trivial undertaking) then it becomes a little easier to do, but it's still very difficult-- you're looking for a CCD with a frame readout time of a microsecond or so, able to handle photon counts in the megahertz range. Which is still a very hard problem.
Which is probably why it hasn't been done, at least not that I'm aware of.
EDIT: The closest to the sort of thing you're talking about is the Pfleegor-Mandel experiment, which used a statistical technique to show that there were spatial fringes in the interference pattern of two independent lasers. It's a long way from a direct observation of the pattern, though.