Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes

The Gross-Pitaevskii (GP) or nonlinear Schr\"odinger equation relevant to ultracold atomic gaseous Bose-Einstein condensates possess a modified Schr\"odinger symmetry in two spatial dimensions, in the presence of a harmonic trapping potential, an (artificial) constant magnetic field (or rotation) and an electric field of a quadratic electrostatic potential. We find that a variance and a center of a trapped gas with or without a vorticity can be regarded as massive Nambu-Goldstone (NG) modes associated with spontaneous breaking of the modified Schr\"odinger symmetry. We show that the Noether theorem for the modified Schr\"odinger symmetry gives universal equations of motion which describe exact time-evolutions of the trapped gases such as a harmonic oscillation, a cyclotron motion and a breathing oscillation with frequencies determined by the symmetry independently of the details of the system. We further construct an exact effective action for all the NG modes.