# Rabi oscillations and spin 1/2 systems.

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

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.

f95toli
Gold Member
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.

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?