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.