I was reading this response on the importance of the Soviet Union in Japan's surrender, discussion of the why Japan surrendered aside (I dont really want my question bogged by that) it struck me that one of the arguments that is often floated around is that:
But... Nazi Germany while not developing one was aware of the implications of Einsteins theory and had a program to develop its own, which run into a lot of difficulties, so did the Soviet Union, it seems reasonable to believe that Japan had scientists that realized the implications as well even if unlike the other powers those implications werent refined into a project.
It also seems weird to me that the Empire of Japan being a modern and technologically advanced country would be unaware of how difficult it was to produce an atomic bomb, and that they would take a decision out of uncertainty, surely they had a scientist somewhere that would inform them that it wasnt logically or logistically possible for the US to just pump out a dozen more bombs by the end of the week.
You have to remember the United States kept their bomb effort secret to a massive degree. Even if Japan was up-to-date on nuclear technology, they would have no way of knowing how far along the US was.
According to a memo documenting a conversation between General John E. Hull and Colonel L.E. Seeman, there was a third bomb ready for shipment to Tinian, which would be ready for use by August 19th, with three more ready to go for October (https://nsarchive2.gwu.edu//NSAEBB/NSAEBB162/72.pdf)
So it does seem the US was ready to begin a full-scale nuclear bombardment of Japan if necessary. Japan's fears, it would seem, were well founded.
The Japanese knew about the theory behind atomic bombs as well, and did have their own small research programs.
That does not, by itself, tell you what the US production capacity was — how many bombs they could produce. The US Manhattan Project could produce 3-4 atomic bombs per month by mid-1945. In a sense, the US did have more: they had the capacity to use another one by August 17-18, a little over a week after Nagasaki. And more could be deployed beyond that. The idea that the atomic bombs were a "bluff" — in that the US had only two — is a misconception. The US had created an atomic bomb production system that, with time, could produce many more weapons, and they were planning to do so.
In any case, the issue of future atomic bombing attacks played very little role in any Japanese deliberation about surrender. If they did play a role, it was more abstract than concrete. It is not clear that the Japanese were worried that there would be more by the end of the week.
Hasegawa's Racing the Enemy is the best source on Japanese perceptions of the bomb and its relative role in their surrender.