Two different expressions of Jaynes-Cummings Hamiltonian

In summary, the conversation discusses two different expressions of the Jaynes-Cummings Hamiltonian and the transformation between them. The first expression includes a term for the interaction between a cavity mode and a two-level atom, while the second expression includes an additional imaginary term. The participant is seeking to understand why the two expressions are equal and how to derive one from the other. They also inquire about the validity of the transformation between the two expressions and the nature of the annihilation and creation operators in the Hamiltonian.
  • #1
ZhangBUPT
3
0
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!
 
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  • #2
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
Physics Monkey said:
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?
 
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  • #4
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
Physics Monkey said:
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.
 

1. What is the Jaynes-Cummings Hamiltonian?

The Jaynes-Cummings Hamiltonian is a mathematical model used in quantum mechanics to describe the interaction between a two-level atom (such as a qubit) and a quantized electromagnetic field. It was first proposed by E. T. Jaynes and F. W. Cummings in 1963.

2. What are the two different expressions of the Jaynes-Cummings Hamiltonian?

The two different expressions of the Jaynes-Cummings Hamiltonian are the rotating wave approximation (RWA) and the full Hamiltonian. The RWA neglects certain terms in the full Hamiltonian that are considered to be small compared to others, simplifying the calculations.

3. How do the two expressions of the Jaynes-Cummings Hamiltonian differ?

The RWA and full Hamiltonian differ in the way they account for the energy levels of the atom and the electromagnetic field. The RWA only considers transitions between the ground and excited states of the atom, while the full Hamiltonian takes into account all energy levels of both the atom and the field.

4. Which expression of the Jaynes-Cummings Hamiltonian is more accurate?

The full Hamiltonian is considered to be more accurate than the RWA, as it takes into account all energy levels and interactions between the atom and the field. However, the RWA is often used in practical calculations due to its simplicity and close approximation to the full Hamiltonian.

5. How is the Jaynes-Cummings Hamiltonian used in research?

The Jaynes-Cummings Hamiltonian is used in various areas of research, such as quantum computing, quantum optics, and quantum information processing. It has also been used to study phenomena like quantum entanglement and the quantum Zeno effect. Researchers use the Hamiltonian to make predictions and analyze the behavior of quantum systems in these fields.

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