From Quantum to Classical: the Quantum State Diffusion model

The vagueness of the quantum-classical border in physics makes some physicists uncertain about the foundation of their science. Others, by contrast, feel that the accuracy of the verified predictions makes these foundations especially solid. This longtime state of affairs is changing in an interesting way. Recent progress in mesoscopic physics, and the so-called “new science of nanotechnology,” open new domains to be explored by theoreticians and experimentalists alike. In this contribution we will exploit numerical and conceptual tools that were developed to explore this new domain of mesoscopic physics, and apply them to systems that are close to the quantum-classical border. Central to classical mechanics are the concepts of trajectories and the phase space structures associated with them. Hence, we would like to illustrate our results with figures showing these trajectories and structures as they appear in our models. These figures, and the fact that they can be computed without any supplementary parameters, constitute the main result of this contribution. As examples we could use several kinds of oscillators. The first, and the simplest, is the harmonic oscillator with its regular orbits; but this is boring! Next, systems that exhibit classical chaos, such as the Kicked Anharmonic OScillator (KAOS) [1, 2, 3, 4] and the double-well Duffing oscillator [5, 6]. In both cases the structure of the strange attractors emerges out of quantum cloudiness when the quantum-classical border is approached. Another example worth mentioning is a nonlinear oscillator exhibiting hysteresis [7], the hysteresis curve appearing when the system becomes more classical. The feature which characterizes all these systems is that they are open; they interact significantly with some kind of external environment. One cannot in general describe an open system by a single state vector; one requires a density matrix, essentially giving classical probabilities for the system to be in given quantum states. In the limit of Markovian systems, this density matrix evolves according to a linear master equation, as we shall see below.