How did the Persians defeat the Spartans?
The Spartans never really recovered from the huge loss of life
in 465 BC earthquake and subsequent helot revolt as well as wars
with Argives and Arcadians (except Mantineans) which came
immediately after epic war with Persia, and basically ended in the
eve of the Great earthquake.
Wounded Sparta then embarked on a very tough, civil war
equivalent - Peloponnesian war(s). But it ended it not as a former
military power, though still strong, but as more of a political
power, which at the end of an exausting war gave them a victory.
But at what cost? Spartan system crumbled, adn with citizen
population decimated, their army was no longer an elite, their
warriors weren't bred from childhood to be the best in Greece, but
most of the army were now either free semi trained periokoi (second
class citizens) or even freed helots (public slaves).
Finally, Thebans, which rose to power in Greece in 4th century
BC after almost 800 years since last heyday, defeated Spartan army
in the battle of Leuctra in 371 BC, officially ending the very slow
process of Spartan power fading, which started almost immediately
after Persian wars. That wasn't the first nor the last defeat
Spartans had suffered, but it was probably the most decisive, as it
never recovered its power.
But only few decades later, Thebans also fell to the power of
Alexander the Great and Hellenistic kingdoms, who again, after
Alexander's death soon fell to the Romans, together with entire
Greece.
Sparta was slowly reduced to a remote village, a Roman tourist
attraction, and in the early middle ages, after several barbarian
attacks and fall of Rome, it ceased to exist until refounded by
modern Greeks as Sparti in the end of 19th century where it still
stands.