Survival of quantum effects for observables after decoherence

When a quantum nonlinear system is linearly coupled to an infinite bath of harmonic oscillators, quantum coherence of the system is lost on a decoherence time-scale $\tau_D$. Nevertheless, quantum effects for observables may still survive environment-induced decoherence, and be observed for times much larger than the decoherence time-scale. In particular, we show that the Ehrenfest time, which characterizes a departure of quantum dynamics for observables from the corresponding classical dynamics, can be observed for a quasi-classical nonlinear oscillator for times $\tau \gg\tau_D$. We discuss this observation in relation to recent experiments on quantum nonlinear systems in the quasi-classical region of parameters.