Quantum Dynamics of Cold Atomic Gas with $SU(1,1)$ Symmetry

Abstract

Motivated by recent advances in quantum dynamics, we investigate the dynamics of the system with SU (1, 1) symmetry. Instead of performing the time-ordered integral for the evolution operator of the time-dependent Hamiltonian, we show that the time evolution operator can be expressed as an SU (1, 1) group element. Since the SU (1, 1) group describes the "rotation" on a hyperbolic surface, the dynamics can be visualized on a Poincaré disk, a stereographic projection of the upper hyperboloid. As an example, we present the trajectory of the revival of Bose-Einstein condensation and that of the scale-invariant Fermi gas on the Poincaré disk. Further considering the quantum gas in the oscillating lattice, we also study the dynamics of the system with time-dependent singleparticle dispersion. Our results are hopefully to be checked in current experiments.

Emergent spacetimes from Hermitian and non-Hermitian quantum dynamics

“…which are fundamental ingredients in holographic tensor networks, determining how quantum entanglement is developed with changing the length scales [6,7]. They also produce Hamiltonians of a wide range of quantum systems with SU (1,1) symmetry, i.e., H(τ ) = i ξ i (τ )K i , where ξ i are real [16][17][18][19][20]. We define the cost function as…”

“…The mass M is fixed for unitary fermions in 3D harmonic traps, but can, in general, be tuned as a function of time. For instance, the effective mass at band bottom in an optical lattice could be tuned through a time-dependent tunnelling strength [20]. Since ξ 0 = ω(δ −1 + δ), ξ 1 = ω(δ −1 − δ), when ω(t) = 0, ξ 0 = ξ 1 , and ξ = 0.…”