# Rabi oscillations and spin 1/2 systems.

1. Mar 23, 2008

### noospace

Hi all,

Can anybody please explain to me the connection between Rabi oscillations and spin-1/2 systems?

I believe the connection lies in the bloch sphere and the ability to represent the spin-1/2 system by a superposition of Pauli matrices but I'm just not getting it.

Thanks

2. Mar 23, 2008

### Mentz114

The Bloch sphere can represent any 2-state system. It happens that it can be used to describe spin-states and 2-level atoms. The basic Rabi model describes a 2 level atom, so that's where one can use the BS. I don't think there's a direct connection between the Rabi model and spin 1/2 systems.

But, I'm no expert and there might be a connection I don't know about.

3. Mar 23, 2008

### f95toli

Mentz114 is correct.
However, maybe it would be worth adding that the main "connection" nowadays is that the notation with spin matrices etc that were originally developed for spin-1/2 systems (which when placed in a magnetic field have the two states "spin up" and "spin down") is now used for virtually all 2-level systems (e.g. qubits) regardless if they have anything to do with spin or not. Spin-1/2 systems are just archetypal 2-level systems.
Hence, as far as I know the connection is mainly historical.

4. Mar 23, 2008

### noospace

Any two-level system can be written in the form $e^{-i\phi/2}\cos(\theta/2) | 0 \rangle + \sin\theta(\theta/2) e^{i\phi/2}|1\rangle$ justifying the Bloch sphere interpretation.

The density operator of the two-level system can be expanded in the basis of Pauli matrices $\{1,\sigma_x,\sigma_y,\sigma_z\}$ giving

$\sigma = \frac{1}{2}(\mathbf{1} + \hat{n} \cdot \vec{\sigma})$

where $\hat{n} = (\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ as expected.

For a spin-1/2 system, the vector $\hat{n}$ characterizes the polarization of the spin.

What does it correspond to for two-level atom undergoing Rabi oscillations subject to sinusoidal electric field?