Atomic Coherent States

An interactive visualization of the stretched representation for many coupled two-level systems.

What are atomic coherent states?

Consider many two-level quantum systems (e.g. atoms). The state of each individual system can be expressed with a two-component vector of probability amplitudes.

By treating these vectors as two-component spinors, we may define a pseudo-spin operator S_zⁱ for each atom. Then the Hamiltonian for each individual atom can be written as some multiple of S_zⁱ.

If there is no spatial inhomogeneity in the system, and negligible (or mean-field) interactions between the atoms, then the total Hamiltonian can be written as as a multiple of the total pseudo-spin operator S_z := Σ_i S_zⁱ. Moreover, the pseudo-spin of the atoms can be coupled in the usual way, and then decomposed into irreducible spin representations.

The top-level (highest-spin) irrep is known as the stretched representation. Since the magnitude of the spin vector is fixed in this representation, we can visualize its state as a quasi-probability distribution over a sphere. This is what is visualized in this app.

The ground state is the lowest-S_z state of the stretched representation, and the state with all atoms excited has the highest S_z. Absorbing one photon increments S_z, and vice-versa. Many common experimental techinques (e.g. Rabi/Ramsey sequences) will keep the system in the stretched representation if homogeneity is not broken, so this component of the spin decomposition is particularly useful.

See this paper for more details.