The Jaynes-Cummings Hamiltonian

  • Thread starter kof9595995
  • Start date
  • Tags
    Hamiltonian
In summary: In that case the equations are defining properties of the dipole transition operator, which is a mathematical object that describes the transition between two energy levels.
  • #1
kof9595995
679
2
This Hamiltonian popped up when I was reading an article, as a reference(wikipedia): http://en.wikipedia.org/wiki/Jaynes–Cummings_model#cite_note-1
I don't understand why the Hamiltonian [itex]\hat H_{atom}[/itex] and [itex]\hat H_{int}[/itex] look the way they are. Usually we we just take a classical Hamiltonian and "operatorize" it, but I fail to see the classical counterpart for [itex]\hat H_{atom}[/itex] and [itex]\hat H_{int}[/itex]
 
Physics news on Phys.org
  • #2
kof9595995 said:
This Hamiltonian popped up when I was reading an article, as a reference(wikipedia): http://en.wikipedia.org/wiki/Jaynes–Cummings_model#cite_note-1
I don't understand why the Hamiltonian [itex]\hat H_{atom}[/itex] and [itex]\hat H_{int}[/itex] look the way they are. Usually we we just take a classical Hamiltonian and "operatorize" it, but I fail to see the classical counterpart for [itex]\hat H_{atom}[/itex] and [itex]\hat H_{int}[/itex]

[itex]\hat H_{atom}[/itex] is just a generic two-level system. Therefore in the easiest case you get just two energy levels at [itex]E_{1/2}=\pm\hbar \omega[/itex]. The interaction Hamiltonian just describes the interaction between the bosonic light field and this two-level system. The energy-conserving terms of [itex]\hat E \hat S[/itex] describe the destruction of a photon combined with the simultaneous excitation of the two-level system and the creation of a photon combined with the simultaneous transition of the two-level system from the excited to the ground state.
 
  • #3
Cthugha said:
The energy-conserving terms of [itex]\hat E \hat S[/itex] describe the destruction of a photon combined with the simultaneous excitation of the two-level system and the creation of a photon combined with the simultaneous transition of the two-level system from the excited to the ground state.
I can't see the physics you said form [itex]\hat E \hat S[/itex]. Actually I did find something more elaborate on this: http://uncw.edu/phy/documents/Shafer499Talk.pdf
But I don't understand the content on page 27, i.e. why are those four equations the defining properties of a dipole operator, the author gave a handwaving reason "The dipole operator is responsible for “moving” the atom between energy levels.", but I don't really see why it has to be the way it is.
EDIT: I think I understand now, the author is probably referring to dipole transition.
 
Last edited:

1. What is the Jaynes-Cummings Hamiltonian?

The Jaynes-Cummings Hamiltonian is a mathematical model used in quantum optics to describe the interaction between a single atom and a quantized electromagnetic field. It was developed by E. T. Jaynes and F. W. Cummings in 1963.

2. How does the Jaynes-Cummings Hamiltonian work?

The Jaynes-Cummings Hamiltonian describes the system of a two-level atom interacting with a single mode of the quantized electromagnetic field. It takes into account both the energy levels of the atom and the energy levels of the field, and how they interact with each other.

3. What are the applications of the Jaynes-Cummings Hamiltonian?

The Jaynes-Cummings Hamiltonian has applications in various areas of physics, such as quantum optics, cavity quantum electrodynamics, and quantum information processing. It is used to study phenomena such as photon emission, photon absorption, and the behavior of quantum systems.

4. What is the significance of the Jaynes-Cummings Hamiltonian in quantum mechanics?

The Jaynes-Cummings Hamiltonian is significant in quantum mechanics because it provides a theoretical framework for understanding the interaction between light and matter at the quantum level. It has been used to make predictions and explain experimental observations in various quantum systems.

5. Are there any limitations to the Jaynes-Cummings Hamiltonian?

Like any mathematical model, the Jaynes-Cummings Hamiltonian has its limitations. It assumes a simple two-level atom and a single mode of the electromagnetic field, which may not accurately describe more complex systems. Additionally, it does not take into account the effects of decoherence or environmental noise on the system.

Similar threads

Replies
4
Views
958
Replies
1
Views
2K
Replies
16
Views
1K
Replies
3
Views
789
  • Quantum Physics
Replies
2
Views
2K
  • Quantum Physics
Replies
3
Views
743
Replies
3
Views
1K
  • Atomic and Condensed Matter
5
Replies
156
Views
7K
  • Quantum Physics
Replies
11
Views
2K
  • Special and General Relativity
Replies
1
Views
616
Back
Top