The Rabi Problem: Solving with the Semiclassical Approach

[Vineesha]

What is the Rabi problem? How to solve it using the semiclassical approach? I am looking for content which is easy to understand.

 

[hilbert2]

It usually means a problem of a two-level system where the state vectors have only two complex number components, and the Hamiltonian is a $2\times 2$ matrix that has a periodic time dependence, as in

$H = \begin{bmatrix}a & b\sin \omega t \\ b\sin \omega t & c\end{bmatrix}$ .

Then you use time dependent perturbation theory or some other method to find the evolution of some initial state vector. The most appropriate way may depend on how large the frequency $\omega$ is.

EDIT: A practical application of this could be a system where an atom in ground state is subjected to an electromagnetic wave with low amplitude and frequency $\omega$ that is chosen in such a way that it's unlikely that the atom will be excited to any other but a single excited state. Then you can think of it as an effective two-level system even though any atom has an infinite number of energy eigenstates in principle.

 

[Vineesha]

It usually means a problem of a two-level system where the state vectors have only two complex number components, and the Hamiltonian is a $2\times 2$ matrix that has a periodic time dependence, as in

$H = \begin{bmatrix}a & b\sin \omega t \\ b\sin \omega t & c\end{bmatrix}$ .

Then you use time dependent perturbation theory or some other method to find the evolution of some initial state vector. The most appropriate way may depend on how large the frequency $\omega$ is.

EDIT: A practical application of this could be a system where an atom in ground state is subjected to an electromagnetic wave with low amplitude and frequency $\omega$ that is chosen in such a way that it's unlikely that the atom will be excited to any other but a single excited state. Then you can think of it as an effective two-level system even though any atom has an infinite number of energy eigenstates in principle.

Thank you very much. However, I wish to work out the exact mathematics of the problem. Can you suggest a good, easy book for this?

 

[hilbert2]

The Bransden & Joachain's "Quantum Mechanics 2nd ed." I have myself seems to contain this in Chapter 9.

The idea is to write the state of the system as a vector with two time-dependent components,

$\left|\right.\psi (t) \left.\right> = \begin{bmatrix}a(t) \\ b(t)\end{bmatrix}$

and then make the time dependent Schrödinger equation

$i\hbar\frac{\partial}{\partial t}\begin{bmatrix}a(t) \\ b(t)\end{bmatrix} = H\begin{bmatrix}a(t) \\ b(t)\end{bmatrix}$,

where $H$ is a matrix like that in my previous post. Now you have two coupled differential equations for the functions $a(t),b(t)$ and they can be solved with some kind of approximations.

 

[Vineesha]

The Bransden & Joachain's "Quantum Mechanics 2nd ed." I have myself seems to contain this in Chapter 9.

The idea is to write the state of the system as a vector with two time-dependent components,

$\left|\right.\psi (t) \left.\right> = \begin{bmatrix}a(t) \\ b(t)\end{bmatrix}$

and then make the time dependent Schrödinger equation

$i\hbar\frac{\partial}{\partial t}\begin{bmatrix}a(t) \\ b(t)\end{bmatrix} = H\begin{bmatrix}a(t) \\ b(t)\end{bmatrix}$,

where $H$ is a matrix like that in my previous post. Now you have two coupled differential equations for the functions $a(t),b(t)$ and they can be solved with some kind of approximations.

Thanks a lot!