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Tunable Spin-orbit Coupling and Quantum Phase Transition in a Trapped Bose-Einstein Condensate

Spin-orbit coupling (SOC), the intrinsic interaction between a particle spin and its motion, is responsible for various important phenomena, ranging from atomic fine structure to topological condensed matter physics. The recent experimental breakthrough on the realization of SOC for ultra-cold atoms provides a completely new platform for exploring spin-orbit coupled superfluid physics. However, the SOC strength in the experiment, determined by the applied laser wavelengths, is not tunable. In this Letter, we propose a scheme for tuning the SOC strength through a fast and coherent modulation of the laser intensities. We show that the many-body interaction between atoms, together with the tunable SOC, can drive a \textit{quantum phase transition} (QPT) from spin-balanced to spin-polarized ground states in a harmonic trapped Bose-Einstein condensate (BEC). This transition realizes the long-sought QPT in the quantum Dicke model, and may have important applications in quantum optics and quantum information. We characterize the QPT using the periods of collective oscillations (center of mass motion and scissors mode) of the BEC, which show pronounced peaks and damping around the quantum critical point.