Two chatbots - One teaches another I am seeking the information for this kind of chatbot architecture : There are two chatbots. One plays the role of teacher, and another is a student who is learning. The goal is to test the student's quality, and to improve the student's ability.

I didn't find much reference. There are :

Bottester: Testing Conversational Systems with Simulated Users

And the ParlAI, a python-based platform for enabling dialog AI research has the notion of "Teacher agent", which seems to be what I am looking for.

Of course, we also have deep reinforcement learning which might be related.

I prefer to have some classical references for this approach to chatbots.
Currently, reinforcement learning is not in my consideration.

Constructing two chatbots talking to each other, like what Facebook did, is not what I want. Because in this case, both of them are student agents.
The Hidden Agenda User Simulation Model (Schatzmann/Young) describes a chatbot training design in which a user simulator assembles and executes a conversational agenda, with the direct goal being to train the target chatbot. 

Perhaps you can add specificity to this design by casting the user simulator as the teacher, and creating an agenda in which it is communicating information to the (student) chatbot. The trained behavior expected is, perhaps, correct responses to topical questioning by the teacher.