Dissipative Quantum Systems

Abstract

Dissipation is a ubiquitous phenomenon in real physical systems. Its nature is made clear by considering the damped harmonic oscillator, a paradigm for dissipative systems in the classical as well as the quantum regime. After starting at a nonequilibrium position, the system will perform damped oscillations and end up in the equilibrium position. Looking closely, one will notice that even in equilibrium the oscillator coordinate fluctuates. This effect is related to the Brownian motion of a free particle. Damping and fluctuations are both caused by the coupling of the harmonic oscillator to other degrees of freedom. A pendulum's motion is damped because of collisions with molecules in the air during the oscillations. For the same reason the pendulum will fluctuate around its equilibrium position. The identical origin of both effects manifests itself in the fluctuation-dissipation theorem which will be discussed later in this chapter.Another system where the coupling to other degrees of freedom plays a prominent role, is the decay of a metastable state. At not too low temperatures a state in a potential minimum, which is separated by a barrier from an energetically lower region, may decay by thermal activation. Here, the other degrees of freedom, which are also collectively called heat bath or reservoir, provide the system with the necessary energy to surmount the barrier. At low enough temperatures, the decay of the metastable state will be dominated by quantum tunneling through the barrier, and one might ask how the other degrees of freedom influence the tunneling rate.In this chapter, we will specifically consider the damped harmonic oscillator, the damped free particle, and the decay of a metastable state to illustrate the techniques introduced to describe dissipation in quantum mechanics. Of course, there are other interesting problems like the damped motion in a periodic potential and the dissipative two-level system [1]. For a more complete discussion of many aspects of quantum dissipation, we refer the reader e.g. to the textbook by Weiss [2] which may also serve as a guide to the literature.A rather recent area of research activities is concerned with the interplay between quantum dissipation and chaos which is discussed in Chapter 6. In this context, techniques for the treatment of driven systems discussed in Chapter 5 are required.