As you have noted each state in the USA has different requirements. The general requirements are as follows:

 - Obtain a Bachelor's in Engineering (BSE) from an ABET accredited engineering program. The disciplines that I am familiar are Electrical, Civil, and Mechanical. I suggest you review the references below for more detailed information. 
 - Successfully complete the Fundamentals in Engineering exam to obtain Engineering in Training (EIT) status.
 - Gain professional experience (typically four years minimum).
 - Successfully complete the Principles and Practice of Engineering (PE) exam.

Considering that you have bachelor's and graduate degrees in physics, you have a few options:

 - Enroll in a ABET accredited engineering program to obtain a Bachelor's in Engineering. With your physics background you are most likely to get a significant amount of credit. It is likely that you will be able to obtain a Bachelor's in Engineering in about 2 year depending on the University.
 - Submit your credential to the licensing authority; hopefully they will accept your credentials and allow you sit for the EIT exam. You will have to do more research on this topic. I suggest that you review the [**Washington Accord**][1].
 - If you are already a licensed, practicing engineer, submit those credentials to the appropriate licensing authority. They may accept your current credentials. I have submitted my USA credentials to [Engineers Australia][2] and obtained registration in Australia for an Australia-based engineering project. 


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## References

 - [Washington Accord][3]
 - [Regulation and licensure in engineering][4]
 - [What is the sequence of exams to become a Professional Engineer in California?][5]
 - [How To Get Licensed (National society of Professional Engineers)][6]


  [1]: http://www.ieagreements.org/Washington-Accord/Washington-Accord-Overview.pdf
  [2]: https://www.engineersaustralia.org.au/national-engineering-register
  [3]: https://en.wikipedia.org/wiki/Washington_Accord
  [4]: https://en.wikipedia.org/wiki/Regulation_and_licensure_in_engineering
  [5]: https://engineering.stackexchange.com/a/5985/110
  [6]: http://www.nspe.org/resources/licensure/how-get-licensed