/u/farquier is absolutely correct. In fact, I wrote this comment about Achaemenid slavery about a year ago:

Regarding the Achaemenid Empire, there is a great deal of textual evidence for the existence of slaves during their empire. Just as we have several sale contracts of slaves from Neo-Assyrian, Neo-Babylonian, and Seleucid Empires from Uruk (in Babylonia), we also have some from the Persian Period. One such document (Text 143, from Strassmaier, 1890, Inschriften von Kambyses) talks about the "branding" of the hand of a slave in both Akkadian and Aramaic. Similarly, two Aramaic documents from Achaemenid Egypt cite a similar practice of slave-branding (Texts 22 & 41 from Grelot, 1972, Documents araméens d'Égypte). Perhaps surprisingly, the same practice exists in Quintus Curtius Rufus' Alexandrian history. At V.5.5-6, he describes some Greek prisoners of war in the hands of the Persians, and they also have the same branding with "Persian letters." Now, they are not directly named "slaves," but the connection is not difficult to imagine.

Now, none of these texts describe Achaemenid attitudes toward slavery, nor describe the situation in Persia proper. Nonetheless, slavery did exist in at least some parts of the Achaemenid Empire. For further reading (and the sources of most of my information), see Briant's work that you reference elsewhere, From Cyrus to Alexander, pp. 456-9, and Dandamaev & Lukonin, 1989, The culture and social institutions of ancient Iran, 152-77.

Although Cyrus the Great did ostensibly abolish slavery, it either didn't stick or was simply a nominal proclamation, because there is a lot of evidence which supports that widespread slavery did exist in the Achaemenid Empire. Wars provided a substantial source of slaves, as captives were often sold into slavery as "booty of the bow". Even the royal estate kept a large retinue of slaves, and state-owned slaves worked in mines (which was evidently extremely grueling work).

One form of slavery which did die out in the Achaemenid period was debt slavery, where debtors could sell themselves into slavery. This was popular in Babylon but was stamped out under Persian rule.

Sources:
The Culture and Social Institutions of Ancient Iran
History of the Persian Empire

Just to be clear, you are referring to the Achaemenid empire, correct?