The Neo-Babylonian Empire, the last resurgence of Babylonian autonomy, actually ended in approximately 539 BCE. Cyrus the Great, the founder of the Achaemenid Empire (one of several imperial dynasties which might be called "the Persian Empire", preceded by the Median and reigning for two hundred years, until Alexander the also Great thoroughly conquered its entirety) overthrew the last Babylonian king, Nabonidus, and seized the capital, making it a province of his new imperial territory. The Achaemenid Empire dates to about 550 BCE - 330 BCE, and while it controlled the territory of the Babylonians and incorporated much of their culture, it was a distinctly Persian-ruled entity.

As for the Romans, the beginning of their Republic is usually dated to 509 BCE (with the city itself being allegedly founded in 753 BCE, but ruled by semi-mythical kings for about 250 years), and it lasted until the creation of the Roman Empire in 27 BCE.

So, the last self-governed Babylonian state ended about a generation before the rise of Roman political significance, approximately 50 years. The time between the end of the Neo-Babylonian empire and the formation of Imperial Rome is about 500 years.

Although the Babylonian Empire was dismantled slightly before the rise of Rome, Babylonia (the cultural region) and Babylon (the eponymous city proper) remained vital and flourishing provincial components of the Achaemenid, Macedonian, Parthian and Sassanid (Persian) empires for almost a millennia after the death of its last king, and Romans from every period could've potentially visited the area. The city began to decline and fall into disuse sometime after the Muslim conquest of the 7th century CE (A.D.), paralleling the gradual deterioration and marginalization of Aramaic/Babylonian culture under Islam.

I don’t quite know what question is driving your questions. I suppose some basic information will help set you straight here.

Presuming that we’re talking about the Neo-Babylonian Empire, this dates from the rebellion of Nabopolassar against Assyrian rule in 627, through to Nabonidus in 539, when Cyrus II of Persia conquered and subsumed the Babylonian empire into his Achaemenid Empire.

Rome is technically an Empire, in terms of ‘having an Emperor’ from 27BC onwards, which is well after the existence of the Neo-Babylonian empire. Although for a broader definition of ‘empire’, I think one could make a good case that from the end of the Pyrrhic War in 275 BC, the expansion of Rome into the Mediterranean and general ‘foreign policy’ constituted Empire just as much as, say, earlier Athenian dominance of the Greek states was a form of empire. However that still doesn’t get you close to the end of the Bablyonian empire, time-wise.

On those dates, you are looking at a gap of 539 to 275 BC, some 264 years.

So, in response to your first question, no - while the Babylonian Empire existed, there was not much to go and see in Rome. If the republic is dated back to 509 BC, as per Livy, then going back any further and all you have is the kingdom of Rome, not very impressive at all.

I'll keep these explanations to your questions as light as possible. The list below is in chronological order – oldest down to most recent.

• Babylon was already a powerful city that controlled almost all of Mesopotamia as early as c.1770 BCE under king Hammurabi. In effect it was already an empire by this time. (On a side note, c. simply means about. Therefore "c.1770 BCE" reads as "about 1770 BCE").

• But by c.1590 BCE, the Hittites raided and left Babylon for dead. The power vacuum that the Hittites left there became a priceless opportunity for the semi-nomadic Kassites to migrate into Babylonia and take control of the region. The Kassites were from the area of the Zagros Mountains (in present day Iran). But within a generation the Kassites became babylonized.

• Babylon didn't achieve imperial greatness again until 609 BCE, when Nabopolassar of Babylon defeated Pharaoh Necho of Egypt. Nebuchadnezzer (Nabopolassar's son) continued his father's conquest and successfully establish a large empire that stretched from the Sinai region to River Tigris. Even managed to conduct a raid into Egypt towards the end of his long reign.

• In 539 BCE, Cyrus the Great of Persia conquered Babylon, and by the time of his death, he controlled an empire that stretched from the Sinai region northward to the Aegean Sea and eastward past the Persian Gulf. Cyrus is the founder of the Persia Empire.

• In 525 BCE, the Persians conquered and added Egypt to their empire.

• In 509 BCE, the Roman Republic was established.

• In 480 BCE, Xerxes I of Persia attacked Greece but failed. This is the basis for the historical-fantasy film 300.

• In 330 BC, Alexander the Great matched into Persepolis, the capital of Persia and ended the empire. Alexander's new empire didn't last. Immediately after he died, his subordinates broke his empire apart and started fighting among themselves.

• In 27 BCE, the Roman Republic, which by this time already ruled a territory larger than the Babylonian empire, became an empire. It was the year Augustus became the first emperor of Rome. In fact, Rome was already an empire before this, just that it was led essentially by the Roman senate. By this time, Babylon was just a little town without any regional significance.

I hope this helps.