Spin-1 microcondensates paper out
09/03/11 22:29
Just out in PRA today is my paper on spin-1 Bose condensates in what I’ve dubbed the microcondensate regime, meaning that there is no variation in the spin state across the system. In some ways this is analogous to studying a single magnetic domain of a ferromagnet, but considerably richer because of the larger phase space afforded by spin-1.
This project turned into a terrific excuse to learn some fascinating things about classical integrable systems -- notably the idea of Hamiltonian monodromy, which relates to the global existence of action-angle variables -- semiclassical quantization, hyperbolic geometry, and the Bethe ansatz.
This system can be reduced to two coupled hyperbolic (or SU(1,1)) spins and the resulting phase space analyzed in some neat ways:
Here the black line is a classical trajectory of one of the hyperbolic spins, and the red lines are projections onto a hemisphere below the hyperbola, and then from the hemisphere to a half plane. The latter is the celebrated Poincaré half-plane model of hyperbolic geometry.
This project turned into a terrific excuse to learn some fascinating things about classical integrable systems -- notably the idea of Hamiltonian monodromy, which relates to the global existence of action-angle variables -- semiclassical quantization, hyperbolic geometry, and the Bethe ansatz.
This system can be reduced to two coupled hyperbolic (or SU(1,1)) spins and the resulting phase space analyzed in some neat ways:
Here the black line is a classical trajectory of one of the hyperbolic spins, and the red lines are projections onto a hemisphere below the hyperbola, and then from the hemisphere to a half plane. The latter is the celebrated Poincaré half-plane model of hyperbolic geometry.