Geometrizing Quantum Dynamics of a Bose-Einstein Condensate

Scale Invariance of a Spherical Unitary Fermi Gas

A unitary Fermi gas in an isotropic harmonic trap is predicted to show scale and conformal symmetry that have important consequences in its thermodynamic and dynamical properties. By experimentally realizing a unitary Fermi gas in an isotropic harmonic trap, we demonstrate its universal expansion dynamics along each direction and at different temperatures. We show that as a consequence of SO(2,1) symmetry, the measured release energy is equal to that of the trapping energy. We further observe the breathing mode with an oscillation frequency twice the trapping frequency and a small damping rate, providing the evidence of SO(2,1) symmetry. In addition, away from resonance when scale invariance is broken, we determine the effective exponent γ that relates the chemical potential and average density along the BEC-BCS crossover, which qualitatively agrees with the mean field predictions. This Letter opens the possibility of studying nonequilibrium dynamics in a conformal invariant system in the future.

The quantum dynamics of two-component Bose-Einstein condensate: anSp(4,R)symmetry approach

The compact groups such asSU(n) andSO(n) groups have been heavily studied and applied in the study of quantum many body systems. However, the non-compact groups such as the real symplectic groups are less touched. In this paper, it is revealed that the quantum dynamics of two-component Bose-Einstein condensate can be described by a non-compact real symplectic groupSp(4,R). With this group, an explicit form of the wavefunction in any time of the evolution can be given, meanwhile, this whole time evolution can be shown to correspond to a trajectory in a six-dimensional manifold. By introducing a polar coordinate, we can visualize this six-dimensional manifold in 2d unit disk and reveal the relation between the behavior of the trajectory in this manifold and the eigenenergies of the Hamiltonian. Furthermore, the time evolution of expectation value of a physical observable such as number operator is proven closely related to the behavior of the trajectory in this manifold.

SU(1,1) Echoes for Breathers in Quantum Gases

Though the celebrated spin echoes have been widely used to reverse quantum dynamics, they are not applicable to systems whose constituents are beyond the control of the su(2) algebra. Here, we design echoes to reverse quantum dynamics of breathers in three-dimensional unitary fermions and two-dimensional bosons and fermions with contact interactions, which are governed by an underlying su(1,1) algebra. Geometrically, SU(1,1) echoes produce closed trajectories on a single or multiple Poincaré disks and thus could recover any initial states without changing the sign of the Hamiltonian. In particular, the initial shape of a breather determines the superposition of trajectories on multiple Poincaré disks and whether the revival time has period multiplication. Our work provides physicists with a recipe to tailor collective excitations of interacting many-body systems.