Two different expressions of Jaynes-Cummings Hamiltonian

  • Thread starter ZhangBUPT
  • Start date
  • #1
3
0

Main Question or Discussion Point

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

[itex]H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} +
g (a^{\dagger}\sigma_{-} +a\sigma_{+} )[/itex]

and

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

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:

Answers and Replies

  • #2
Physics Monkey
Science Advisor
Homework Helper
1,363
34
Suppose I took the first form and wrote [itex] -i \tilde{a} = a [/itex]. How does the Hamiltonian look in terms of [itex] \tilde{a} [/itex]?
 
  • #3
3
0
Suppose I took the first form and wrote [itex] -i \tilde{a} = a [/itex]. How does the Hamiltonian look in terms of [itex] \tilde{a} [/itex]?
Thanks for your answer. I can derive the second equantion using your transformation.
But in both expressions, [itex] a[/itex] and [itex] a^{\dagger}[/itex] are annihilation and creation operators, respectively, and they are real. Is the transformation still valid?
 
Last edited:
  • #4
Physics Monkey
Science Advisor
Homework Helper
1,363
34
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
3
0
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.
 

Related Threads for: Two different expressions of Jaynes-Cummings Hamiltonian

  • Last Post
Replies
14
Views
3K
Replies
4
Views
2K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
5
Views
2K
Replies
1
Views
1K
  • Last Post
Replies
0
Views
2K
Top