# Two different expressions of Jaynes-Cummings Hamiltonian

## Main Question or Discussion Point

Hi, I have a question about two different expressions of Jaynes-Cummings Hamiltonian

$H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} + g (a^{\dagger}\sigma_{-} +a\sigma_{+} )$

and

$H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} +i g (a^{\dagger}\sigma_{-} -a\sigma_{+} )$.$(\hbar=1)$

I read them from different papers and books.

Why are they equal, and how to derive one from another?
How to choose the appropriate expression when utilizing the Jaynes-Cummings Hamiltonian?

Thanks!

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Suppose I took the first form and wrote $-i \tilde{a} = a$. How does the Hamiltonian look in terms of $\tilde{a}$?

Suppose I took the first form and wrote $-i \tilde{a} = a$. How does the Hamiltonian look in terms of $\tilde{a}$?
But in both expressions, $a$ and $a^{\dagger}$ are annihilation and creation operators, respectively, and they are real. Is the transformation still valid?

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