Practical quantum synchronization for the Schrödinger-Lohe system

We present a practical synchronization method for the Schrödinger-Lohe (S-L) system distinct potentials. The S-L model describes the spatial-temporal evolution of the wave functions of quantum Lohe oscillators on a network with Lohe couplings. When the potential effects are ignored, complete wave function synchronization (CWFS) can emerge in the sense that the L ²-distance between wave functions exponentially approaches zero for a class of initial wave functions. In contrast, when the Lohe oscillators are under the effect of potential forces, CWFS cannot occur. In this study, we employ a weaker concept of quantum synchronization for discussing the asymptotic collective behavior of the S-L model. This concept leads to 'practical synchronization' of the S-L model. In practical synchronization, the L ²-distance between wave functions can be upper bounded by the inverse power of the square root of the coupling strength; as the coupling strength increases. Thus, the L ²-discrepancy between wave functions arbitrarily decreases as the coupling strength increases. We present a sufficient analytical framework for this practical synchronization, which is a generalization of the earlier result in (Choi S-H and Ha S-Y 2014 J. Phys. A: Math. Theor. 47 355104.