When did Ezekiel experience captivity?

The prophet Ezekiel did not experience captivity in the sense that we often think of it—that is, being kept in prison—but was taken into exile when the Babylonians attacked and defeated Judah (the southern kingdom) in 597 BCE. It was standard practice for the ancient Babylonians: after they had conquered another nation, they gathered up the wealthy and influential citizens and brought them back to Babylon to work. Since Ezekiel was part of a priestly family, he was considered important enough to take into exile. This was before he delivered any of his prophetic oracles, since he received his call to become a prophet only once he was in Babylon:

In the thirtieth year, in the fourth month, on the fifth day of the month, as I was among the exiles by the river Chebar, the heavens were opened, and I saw visions of God. (Ezekiel 1:1)

It appears that Ezekiel, like many other exiles, was not imprisoned but allowed to live and work with some degree of freedom. Thus Ezekiel lived in his own house (Ezekiel 3:1) and was married (Ezekiel 24:15-18).

A decade after the first defeat of Judah, in 587/586 BCE, the Babylonians invaded again. This time they destroyed the kingdom more completely, including Jerusalem and its temple, and brought even more prisoners back to Babylon with them. It was not until 538 BCE, when Cyrus the Great conquered Babylon, that the people of Judah were allowed to return and rebuild the city of Jerusalem and their temple.