As Biblical events go, there is a staggering amount of archaeological/non-Biblical written evidence for the Babylonian deportation. But before I get into that, I want to clarify something about the Judean community in exile. They were not enslaved. 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 cases, another deported population would then be sent to take their place.

As for evidence, it's actually difficult to do a survey of the old city, the so-called "City of David," in Jerusalem and not identify the layer of arrow heads, ash, and rubble from the Babylonian capture in 587 BC. The University of Michigan's excavations in the the 1970s-80s are particularly noteworthy, but additional material evidence is found regularly. Those links should provide a broad sampling of different finds in the old city. The Bible Unearthed by Israel Finkelstein and Neil Asher Silberman provides a good overview of Biblical archaeology in general.

In terms of written evidence, we do not actually have obvious Babylonian records for the Second Siege of Jerusalem in 587, which led to the largest deportation. That's more because the relevant chronicle has never been found rather than evidence against the event. There just are not detailed chronographic records for the latter half of Nebuchadnezzar's reign. However, we do have a chronicle (sometimes even called the "Jerusalem Chronicle") from the first decade of Nebuchadnezzar's rule, which references the first rebellion and siege of Jerusalem in 598-7 BC:

In the seventh year, the month of Kislîmu, the king of Akkad mustered his troops, marched to the Hatti-land, and besieged the city of Judah and on the second day of the month of Addaru he seized the city and captured the king.

This event is described in more detail in 2 Kings 24, as the Bible is significantly more concerned with events in Judah than the Babylonians. However, its consequences are quite evident in the Babylonian records. There are four known tablets referencing rations for "Jehoiachin, king of the land of Judah" or his sons provided by the Babylonian government, confirming the Biblical record of his captivity after 597. Extra-biblical writing about the second siege and resulting sack of Jerusalem is rarer, with the most prominent example probably coming from the Lachish Letters, which record communications from a Judean officer in the city of Lachish and one of his outlying subordinates in the run up to the Babylonian invasion of their territory.

The Jerusalem Chronicle also helps contextualize when Jerusalem first fell into the Babylonian orbit (they had to rule Judah already for the siege in 598 to a rebellion after all). 2 Kings 24 opens with a statement that Jehoiachim served Nebuchadnezzar for 3 years before rebelling, which reflects the other campaigns described in the chronicle. Nebuchadnezzar gradually pried Syria and the Levant ("the Hatti-land") from Egyptian control and forced local powers into submission from the later years of his father's reign until 601 BC, which supports the Biblical claim of when he would have taken control of Judah.

There is also ample evidence of the Judean presence in Babylonia during and after the Exile in the form of simple financial and legal records in various government and mercantile archives. Hebrew names appear regularly, with a much larger number of vaguely "West Semitic" or Canaanite names that could identify Jews or people from other Levantine cultures. They also reveal that many Judeans adopted or gave their children Babylonian names (something also represented in the Bible with Zerubabel, the Davidic leader of the first returnees). So there are almost certainly unidentified Judeans in the records with less distinct names.

There is also evidence in Egypt for the fallout from the Judean revolts and exiles. Both 2 Kings and Jeremiah describe a failed third attempt to rebel against Babylon that opened with the assassination of Gedeliah, the new provincial governor. This last round of rebels (and an unwilling prophet Jeremiah) fled to Egypt, where they were resettled by Pharaoh Apries. The most prominent Jewish settlement in self-imposed Egyptian exile was the military colony on the southern island of Elephantine, which continued and thrived for almost 200 years, even constructing a new temple of YHWH. Their history is largely documented through letters and receipts included in the Elephantine Papyri.