First, one might want to rethink the phrasing of this question. The main area of cultural contact between Greece and Persia was Asia Minor, where Greek city states were in geographical proximity to Persia and on major trade routes. Persians visited these city states and Greeks visited Persia leading to substantial cultural interchanges, especially in science, religion, medicine, the arts, and technology. This degree of contact was enhanced by peace rather than conflict, and the Persian Wars really were an interruption to this. Also, you should note that Greece was not a unified empire, but a group of city-states with constantly changing allegiances. Athens, briefly, tried to transform the Delian League into something approximating an empire, but this was short-lived and generally unsuccessful.
A far greater impetus to cultural exchange was internal conflict in Greece, especially the Peloponnesian wars. During these, Greek factions would ally themselves to Persia in an attempt to gain support against other Greek city states. Many Greek soldiers served as mercenaries in Persian armies, leading to increased cultural interchange in the late fifth and fourth centuries. Xenophon provides us with an account of serving as a mercenary for Persia and is one of our best sources on such cultural interchanges.