Chaos-non chaos transition caused by the quantum continuous measurement

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The derivation of classical Lyapunov exponents based only on quantum mechanics has till recently been one of the important open questions of quantum-classical correspondence. Recently Bhattacharya and other authors made great progress by showing that the trajectories of wave packets under continuous measurement of the Caves-Milburn type have the same Lyapunov exponents as the classical one in the classical limit (where "measurement strength" k is large, and Plack constant is small enough).Recalling that unitary evolution (k=0) gives zero Lyapunov exponent, their answer suggests the existence of "non-chaos to chaos" transition due to the continuous measurement. In this research project, we investigated the nature of this transition by computing the Lyapunov exponents, and constructing the stroboscopic maps, of the Duffing system with various values of k, especially small values of k (too small to be in the classical limit). We found that even when the wave packet is extended, its motion can be characterized by a finite (>0) Lyapunov exponent. Remarkably, our result implies the existence of chaos even in a quantum dynamical regime.