While you remember correctly that large numbers of Jews, or more accurately the Judahites (the people of the kingdom of Judah) were conquered and deported by Babylon, and that they were allowed to return by the Persians, there is one major misconception: The Jews were not enslaved by the Babylonians.

The Babylonians had slavery, but it was never the sort of mass chattel slavery more familiar modern colonial history or the Roman Empire. The bulk of labor in the ancient Near East was performed by nominally free peasants. Instead, the Biblical history of the Babylonian Captivity is related to a process of mass deportation, but not enslavement. This was a system pioneered by the Assyrians in which a large swath of people from a particularly obstinate or rebellious city or small territory would be forcibly moved somewhere else, under the basic premise that they would be productive but not rebellious elsewhere in the Empire. In some concept, another deported population would then be sent to take their place, but it's unclear how exactly this worked, and whether or not it ever happened to Judah is likewise up for debate.

The Assyrian version of the same practice also appears in the Bible. When Assyria conquered the northern kingdom of Israel (as it is known in the Bible) or Samaria (as it is known in contemporary sources), they too deported a large swath of the Hebrew population. This is the origin of the so-called "Ten Lost Tribes of Israel," referring to the ten Israelite tribes that the Bible describes inhabiting the northern kingdom. They are lost because neither the Bible nor the Assyrian records help us identify where they were resettled and they were never re-identified by later ancient Jews.

The Achaemenids mostly seem to have abandoned this practice, though examples of it still appear to have occurred in the Persian period. For example, the population of Miletus, the Greek city which lead the Ionian Revolt, was deported to a town in southwestern Iran, and Greeks from Cyrenaica were sent to Bactria after the Achaemenid satrap of Egypt conquered Lybia.

More importantly, Cyrus the Great intentionally portrayed himself as beneficent and divinely ordained during his conquest of Babylon. The city of Babylon itself also apparently faced minimal repercussions for its resistance, and the Cyrus Cylinder implies that the city fell peacefully, though other sources cast that claim into doubt. Depending on how you interpret the Cyrus Cylinder, it may be taken as evidence that Cyrus ended the exile of all peoples deported by the Babylonians and allowed them to go home as a way of fostering support among his new subjects. Sending people to their parents' and grandparents' homelands would also, ironically, have the same effect as the original deportations.

Achaemenid attitudes toward slavery are also often misrepresented to imply that Cyrus was somehow an abolitionist. This is not true. The Persians both practiced and permitted slavery in their territory, which I discuss more in this thread.