Understanding Spin-Orbit Coupling in Hamiltonian

[Tianyou]

Hello,
I see a Hamiltonian of spin-orbit coupling, it is like this
H=$\frac{1}{2m} \sum_{\alpha} \left( \left( -\partial_{\alpha}^2 - 2i\kappa_{\alpha} \sigma_{\alpha} \partial_{\alpha} \right) + \kappa_{\alpha}^2\right)$

Here $\hbar = 1. \kappa_{\alpha}$ is the strength of spin-orbit coupling, and $\sigma_{\alpha} (\alpha = x, y, z)$ are Pauli matrices.
Can someone explain to me where this hamiltonian comes from? What is the meaning of each term?
Thank you.

 

[DrDu]

Where did you see it?

 

[Tianyou]

I saw this hamiltonian in an article in arXiv titled "Enhancement of condensate depletion due to spin-orbit coupling".

 

[DrDu]

Next time, it would be kind of you to give either a link or at least a full reference for an article.
The hamiltonian in that article does not coincide completely with the one of your first post.
However, I don't know where it comes from. Did you check the articles they are citing?
To me, it looks rather dubious. Spin 1/2 for bosons? Certainly not as general as they claim it to be.

 

[Tianyou]

I am confused for the Spin 1/2 for the bosons before. I checked it and found that in this system there is a pseudo Spin 1/2 states for the bosons.