Conditions for the quantum-to-classical transition: Trajectories versus phase-space distributions

We contrast two sets of conditions that govern the transition in which classical dynamics emerges from the evolution of a quantum system. The first was derived by considering the trajectories seen by an observer (dubbed the “strong” transition) [Bhattacharya et al., Phys. Rev. Lett. 85, 4852 (2000)], and the second by considering phase-space densities (the “weak” transition) [Greenbaum et al., Chaos 15, 033302 (2005)]. On the face of it these conditions appear rather different. We show, however, that in the semiclassical regime, in which the action of the system is large compared to $\hslash$, and the measurement noise is small, they both offer an essentially equivalent local picture. Within this regime, the weak conditions dominate while in the opposite regime where the action is not much larger than $\hslash$, the strong conditions dominate.

Figure 1

(Color online) (a) A typical Wigner function for the Duffing oscillator when $\hslash =0.1$ and the measurement strength $k=1$. In this plot luminosity denotes the absolute value of the real part of the Wigner function (thus black corresponds to zero). (b) The associated probability density for the position of the oscillator. The wave function is spread over a significant region of the phase-space.

Figure 2

(Color online) The mean position of the observed Duffing oscillator when $\hslash =0.1$ and $k=1$. The position uncertainty of the quantum state is manifest in the noise that is visible on this trajectory.