After you file

Only the state tax agency will know what all they are doing with your return.  Some states are quicker than others; some are quite slow. And any of them can decide to examine a return more closely for various reasons.

If you efiled, you can double-check with this tool to be sure the state return was efiled successfully and accepted:
https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp

The following FAQ tells how to track a state refund.   There may also be a FAQ on your state tax agency's website with into on about how long it takes to get a refund, and how long to wait before phoning them.

FAQ: How do I track my state refund?

https://ttlc.intuit.com/replies/3300211