Dynamics of a Quantum Phase Transition in a Ferromagnetic Bose-Einstein Condensate
by Bogdan Damski and Wojciech H. Zurek
Journal club talk by Edward Taylor
Summary by Ganesh Ramachandran
Ed chose an interesting paper studying dynamics through a quantum phase transition in a spinor condensate. The discussion brought out some general ideas/features of the dynamics of quantum phase transitions.
Ed gave a lightning introduction to spinor condensates and wrote down the energy functional given in the paper. To put it in context, he described previous experimental study of the ferromagnetic-polar transition by Sadler et al.The phase diagram is known to have ferromagnetic, polar and normal(non-condensed) phases. Experiments have probed the dynamics as well as the detailed domain structure formed. The phase diagram is known to have ferromagnetic, polar and normal(non-condensed) phases.
The Hamiltonian possesses three Bogoliubov modes. In the polar phase, one is gapless, corresponding to the broken U(1) symmetry of the BEC. In the ferromagnet, broken U(1) and rotational symmetries give 2 gapless modes. The only energy scale then, is $\Delta$, the excitation gap of the third mode. As we move away from the critical point where even this mode is gapless(?), $\Delta$ rises from zero and eventually saturates. This gives us two time scales which should determine the dynamics - the relaxation time $1/\Delta$ and the transition time taken until saturation.
Tuning the rate of increase of magnetic field, we can explore an impulse regime and an adiabatic regime depending on which time scale dominates. The crossover between regimes occurs where the time scales are equal, which should scale as the one-third power of the 'quench time'. The numerical calculations in the paper do give this precise scaling.
Ed drew a typical plot of the z-magnetization as a function of changing magnetic field or time. The plot showed that the order parameter moved away from zero toward the expected equilibrium value, only after a delay. This was identified as the crossover between impulse and adiabatic regimes. The delay thus read off, showed the expected 1/3 scaling except for values close to zero.
Ed wound up with a neat quick summary, only to make way for a brisk discussion. Igor's question prompted a discussion on domain formation. Drawing a parallel to the early universe where the size of structures was limited by the speed of light, Ed brought out that the size of domains was given by the speed of sound. With two soundlike modes, for some reason, it is the slower mode velocity that plays a role. Kibble-Zurek theory predicts a 1/3 scaling for domain size, which has been observed in this paper.
There was a question from Michael which brought out that the above considerations only hold for intermediate time scales, where the transition is non-adiabatic, but slow enough. Michael also pointed out that it would be interesting to explore the normal region just above the critical point.
