# Two different expressions of Jaynes-Cummings Hamiltonian

1. Aug 20, 2012

### 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!

Last edited: Aug 20, 2012
2. Aug 20, 2012

### Physics Monkey

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

3. Aug 20, 2012

### ZhangBUPT

But in both expressions, $a$ and $a^{\dagger}$ are annihilation and creation operators, respectively, and they are real. Is the transformation still valid?

Last edited: Aug 21, 2012
4. Aug 21, 2012

### 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?

5. Aug 21, 2012

### ZhangBUPT

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