Yes, there’s lots of evidence for that - probably much more evidence for medieval trade than for Chinese-Roman trade, actually. There was some contact with the ancient Romans under the Han dynasty up to the 3rd century AD, when Rome was called “Daqin” in Chinese. There was no direct contact with China though, because Persia was in the way (initially the Sassanid dynasty and then the Islamic empires ruling in Persia):

“The overland caravan route led from the western borders of China through Sogdiana (now Bokhara or Bukhara) to the Persian border, where the wares were transferred by Chinese merchants to the Persians, who transported them to the customhouses on the Byzantine border…Byzantine commerce with the East, therefore, depended very closely upon the relations between the Empire and Persia, and since wars with Persia were a regular occurrence in Byzantine life, trade relations with the East suffered constant interruptions and great harm. The main article of trade was Chinese silk, the production of which was guarded in deep secrecy by China. In view of the difficulties involved in its production, its prices and the prices of silk stuffs greatly in demand on Byzantine markets rose at times to unbelievable figures. Besides Chinese silk, China and India exported to the West perfumes, spices, cotton, precious stones, and other articles demanded primarily in the Byzantine Empire.” (Vasiliev, 163)

The Byzantines loved silk so much that they were willing to look for other routes to China to avoid having to go through Persia. When Turkic tribes arrived around the Black Sea and Caspian Sea, the Byzantines tried to use them as middlemen with China. Emperor Justinian was also interested in finding a sea route to China - they knew the Red Sea and Persian Gulf emptied into the Indian Ocean but they didn’t really know if they could get to China that way (of course they could! But they didn’t know that!). Justinian also found another way to get silk - the story goes that he sent two monks to China, and they smuggled back some silkworm larvae along with the knowledge needed to get them to make silk (they need to eat mulberry trees, for one thing). So after the 6th century, the Byzantines figured out how to make their own silk, and they didn’t depend on China as much.

Contact with China was easier when Chinese control extended almost all the way to the Caspian Sea, as it did during the Tang Dynasty. In the 10th century though, the Song Dynasty was smaller and did not extend as far west. China had less contact with Persia, so contact between Byzantium and China decreased as well. In the 12th and 13th centuries, when the Mongols controlled the trade routes through Central Asia, trade was easier, although by then the Byzantine Empire was much smaller - Anatolia was ruled by the Seljuk Turks, for a few decades western crusaders ruled Constantinople, then the Byzantines were slowly conquered by the Ottomans…but there was a lot of contact between China and European states further west. By the time Marco Polo was there, French and Italian travellers and ambassadors and missionaries had already been visiting the Mongols for decades.

What else did the Byzantines trade with China? There is a 13th-century Chinese work called Zhu Fan Zhi by Zhao Rukuo, which describes “Lu-mei” - possible Seljuk Anatolia, maybe Constantinople, maybe Rome…the Chinese typically did not travel that far west in person so they had no idea where anything really was. They were relying on third- or fourth-hand information from other merchants and travellers. In any case, the products of Lu-mei were:

“byssus [“sea silk”], gold-spangled yue-no cloth, brocades with alternating stripes of gold and silk, bezoar stones, wu-ming-i, rose-water, gardenia flowers, liquid storax, borax, and a superior quality of engraved opaque glassware. The people are fond of breeding camels, horses, and dogs.” (pg 141)

As for Chinese/Byzantine slaves, I don’t think there were any in the early periods, since they didn’t have any direct contact like that. Much later there were “Tartar” slaves in Europe, who were probably Mongols but maybe also Chinese. But they came through the Genoese colonies in the Black Sea, not Byzantium (which was largely confined to Constantinople at that point, and was no longer a big player in the trade with the east).

Sources:

- A.A. Vasiliev, History of the Byzantine Empire (University of Wisconsin Press, 1952)

- Angeliki E. Laiou, ed., The Economic History of Byzantium (Dumbarton Oaks, 2002)

- Angeliki E. Laiou and Cécile Morrisson, The Byzantine Economy (Cambridge University Press, 2007)

- Chau Ju-Kua: His Work on the Chinese and Arab Trade in the Twelfth and Thirteenth Centuries, entitled Chu-fan-chi, trans. Friedrich Hirth and W.W. Rockhill (St. Petersburg, 1911)

- Feng Chengjun, ed., Zhu Fan Zhi Jiao Zhu (Description of Various Countries) (Shanghai, 1956) - this is the Chinese edition of Zhu Fan Zhi, if that helps…I can't read it myself...

- The Christian Topography of Cosmas, trans. J.W. McCrindle (Cambridge University Press, 1897) - Cosmas “Indicopleustes” (essentially “the guy who travelled to India”) tried to find a sea route to China and wrote a lot about India and Sri Lanka, although not so much about trade exactly

- two articles by Jialing Xu, “Narratives of the Roman-Byzantine World in Ancient Chinese Sources” and Chen Zhi-Qiang, “Narrative Materials about the Byzantines in Chinese Sources”, in John Burke, ed., Byzantine Narrative: Papers in Honour of Roger Scot (Brill, 2006)

See also these previous answers about the Romans and the Silk Road:

Why didn't Romans take a ship to the far east? by u/Tiako

What led to the end of the Silk Road? by u/Sandafluffoid

How "Busy" was the Silk Road at its most used point in History? by u/anchaescastilla