Two different expressions of Jaynes-Cummings Hamiltonian

[ZhangBUPT]

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!

 

[Physics Monkey]

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

 

[ZhangBUPT]

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

Thanks for your answer. I can derive the second equantion using your transformation.
But in both expressions, $a$ and $a^{\dagger}$ are annihilation and creation operators, respectively, and they are real. Is the transformation still valid?

 

[Physics Monkey]

Why do you think "they are real"? It seems to me you need to go back and get some basics down first. For example, how does the creation operator evolve in time? How is it related to x and p variables in a harmonic oscillator or equivalently to the physical electromagnetic field?

 

[ZhangBUPT]

Why do you think "they are real"? It seems to me you need to go back and get some basics down first. For example, how does the creation operator evolve in time? How is it related to x and p variables in a harmonic oscillator or equivalently to the physical electromagnetic field?

Thank you very much! And I'll figure it out.