Disturbing quantum systems by changing either interactions or external potentials often leads to intriguing non-equilibrium quantum phenomena. Whereas some of these quantum non-equilibrium dynamics have been well understood and some others remain mysterious to physicists, in this talk, I will discuss an underlying hyperbolic geometry governing a wide range of quantum dynamics. On the one hand, this framework establishes a geometric interpretation for prototypical quantum dynamical phenomena, such as the dynamical instability or the parametric amplification, and offers natural solutions to some mysteries about breathers of Bose-Einstein condensates. On the other hand, it delivers a new tool to control quantum systems using SU(1,1) echoes, which could reverse quantum many-body dynamics and provide a new type of quantum-enhanced sensing.
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