Geometrizing quantum dynamics of Bose-Einstein condensates

Zhao-Xian Yu, Fuping Liu, Zhi-Yong Jiao

Benjamin Schlein

Zhao-Xian Yu, Jiao Zhi-Yong, Shuo Jin +1 more

By using of the invariant theory, we study a two energy-level Bose–Einstein condensate interacting with a time-dependent laser field, the dynamical and geometric phases are given respectively. The Aharonov–Anandan phase is also obtained under the cyclical evolution.

Wu‐Ming Liu

Chang-Yan Wang, Yan He

The method of geometrization arises as an important tool in understanding the entanglement of quantum fields and the behavior of the many-body system. The symplectic structure of the boson operators provide a natural way to geometrize the quantum dynamics of the bosonic systems of quadratic Hamiltonians, by recognizing that the time evolution operator corresponds to a real symplectic matrix in $Sp(4,R)$ group. We apply this geometrization scheme to study the quantum dynamics of the spinor Bose-Einstein condensate systems, demonstrating that the quantum dynamics of this system can be represented by trajectories in a six dimensional manifold. It is found that the trajectory is quasi-periodic for coupled bosons. The expectation value of the observables can also be naturally calculated through this approach.

Changyuan Lyu, Chenwei Lv, Qi Zhou

We show that quantum dynamics of Bose-Einstein condensates in the weakly interacting regime can be geometrized by a Poincaré disk. Each point on such a disk represents a thermofield double state, the overlap between which equals the metric of this hyperbolic space. This approach leads to a unique geometric interpretation of stable and unstable modes as closed and open trajectories on the Poincaré disk, respectively. The resonant modes that follow geodesics naturally equate fundamental quantities including the time, the length, and the temperature. Our work suggests a new geometric framework to coherently control quantum systems and reverse their dynamics using SU(1,1) echoes. In the presence of perturbations breaking the SU(1,1) symmetry, SU(1,1) echoes deliver a new means to measure these perturbations such as the interactions between excited particles.