About how many years passed between the end of the Babylonian Empire and the beginning of the Persian Empire?

Answer 1

None, actually:
"In 536 B.C., after 70 years of supremacy, the Babylonian empire, the "head of gold" in Daniel's Statue, came to an end when it fell to the Persians (see Ancient Empires - Persia)."

"After the death of Assurbanipal, the last great Assyrian monarch, Nabopolassar, the ruler of Babylonia, established (625 B.C.) his independence. He allied himself with the Medes and Persians and helped to bring about the capture of Nineveh (612 B.C.) and the fall of the Assyrian Empire. He established what is generally known as the Chaldaean or New Babylonian Empire. Under his son, Nebuchadnezzar, the new empire reached its height (see Babylon). The recalcitrant Hebrews were defeated and punished with the Babylonian captivity. Egypt had already been defeated by Nebuchadnezzar in the great battle of Carchemish (605) while Nabopolassar was still alive. The empire seemed secure, but it was actually transitory. The steady growth of Persian power spelled the end of Babylonia, and in 538 B.C. the last of the Babylonian rulers surrendered to Cyrus the Great (see also Belshazzar). Babylonia became an important region of the Persian Empire."

Unless, that is, you're talking about the OLD Babylonian Empire:
"The armies of Babylonia were well-disciplined, and they conquered the city-states of Isin, Elam, and Uruk, and the strong Kingdom of Mari. The rule of Babylon was even obeyed as far as the shores of the Mediterranean. But Mesopotamia had no clear boundaries, making it vulnerable to attack. Trade and culture thrived for 150 years, until the fall of Babylon in 1595 BC."

In which case, 1057 years would have elapsed.

Answer 2

The Persian Empire conquered the Babylonian Empire, and thus (or to some extent,even before) it became an empire.