This is a good question. Wave properties of matter particles was actually seen long ago (1927) in Davisson-Germer experiment. Electron beam was targeted at a Nickel crystal whose inter-molecular spacing was of the order of the deBroglie wavelength of the electron, and hence the crystal acted as a grating there. The interference pattern could be seen in a movable detector.

In an atomic interferometer two beams of matter particles which have originated from the same source, are interfering with each other. They are usually split at source using a beam splitter. The two beams therefore have comparable wavelength and so they can interfere. Experimental apparatus indeed has a macroscopic size, but all that matters is that the two beams should have comparable wavelengths.

Dear Biswajoy, thank you for your clear answer. Your answer is very helpful. I also find some reports of interference of bigger and heavier molecule. See http://www.nature.com/articles/ncomms1263. This really subverts my past knowledge.

I am thinking that if there is an evident boundary between quantum and classical world or not. For a state of photons in continuous variable regime, coherent state seems like the boundary. But for atoms, is there a boundary state?

I not so sure that will be helpful,but in the most case of atom interferometer, the interference is due to a semi-classical behaviour and it is atom by atom that give you the interference, then you are right the interference will happen only on the range of the br

I would correct Biswajoy a little bit. To interfere, waves of any kind (electromagnetic or matter waves, in particular) must have not only comparable but exactly equal wavelengths and be coherent. Otherwise, the phase difference between them at a given point changes with time with the characteristic time constant of the order of the inverse frequency difference. As a result, the interference picture is smeared out.

More interestingly, not only two atoms with the same wavelengths can interfere, but a single atom (or electron, or any other particle) can interfere with itself in a double-slit experiment described above by Biswajoy. 

In an atom interferometer, (thermal source, i. e, not a BEC) the interference is a single atom with itself that generate the interference signal, just to make it clear. 

Let's get back to the original question. An atom beam can be seen wavelike, but it can't be dealt with quantum mechanics according to the description of Griffiths' book (that its de Broglie wavelength is smaller than its characteristic length). That's weird!  In fact most of the time, in quantum mechanics, we deal with atoms because the electrons‘ energy level is what we consider and electrons can be seen by quantum. I don't know how to deal with just atoms  without considering electrons in quantum mechanics.

Dear Alexander, "To interfere, waves of any kind must have" the same particle and nothing more. Another matter, we cannot work with particles which have essentially different velocities because we are not clever enough.

I have to agree with Marat, yet with a remark. Actually we can deal with particles with different velocities. This is the case when interference of two released Bose-Einstein condensates is observed where we take a snap-shot of the interference picture with a sufficiently short exposure time. In this experiment, however, we deal with many atoms. They are not the same particle but, of course, they are identical and originally in exactly the same quantum-mechanical state. In this case, velocities of two particles coming to the same point from different condensates at a particular time are essentially different (for two particles from the same condensate they are exactly equal), but they are low, the relevant frequencies are also low and hence the interference picture changes so slowly that we can take a snap-shot. My previous comment referred to a stable interference picture.

IMHO, the keyword is "temperature". At room temperatures, matter-wave interference is a strictly microscopic effect, unobservable in real life. At nano-Kelvins (Bose-condensates), matter-wave interference extends to macroscopic scales.

L. I. Plimak, you are right that temperature is important, but you don't need to get to BEC (nano-Kelvin temperatures), at micro Kelvin (simply laser cooled atoms) you can see interferences of an atom with himself as Carlos as mentioned. And on the other hand, you can also have interference in a beam (atomic, molecular, neutrons, electrons) if the transversal temperature is low enough.

I do not know the answer to this question.

Interference between two independent clouds of atoms was first done in 1996. In this way it was demonstrated that two Bose-condensates can interfere with each other.

Ref. http://cua.mit.edu/ketterle_group/Projects_1997/Pubs_97/andr97-Science_int.pdf