Quantum Brownian Motion in C-Numbers

The theory of Brownian motion as proposed by Einstein is now hundred years old. Over the span of a century the theory has grown in various directions to understand stochastic processes in physics, chemistry and biology. An important endeavour in this direction is the quantisation of Brownian motion. While the early development of quantum optics initiated in sixties was based on density operator, noise operator and master equation methods primarily within weak-coupling and Markov approximations, path integral approach to quantum Brownian motion attracted wide attention in early eighties. Although this development had widened the scope of condensed matter physics and chemical physics significantly so far as the large coupling limit and finite correlation time of the noise processes are concerned several problems still need to be addressed. These include, for example, search for quantum analogues of equations of motion for true probability distribution functions, treatment of rate processes in the deep tunnelling regimes where semi-classical approximations are untenable, development of simpler numerical schemes for calculation of rate of activated processes and others. Keeping in view of these aspects it is worthwhile to ask how to extend classical theory of Brownian motion to quantum domain for arbitrary friction and temperature down to vacuum limit. Based on a coherent state representation of noise operators and Wigner canonical thermal distribution for harmonic bath oscillators we have recently developed a scheme for quantum Brownian motion in terms of c-number generalised quantum Langevin equation. The approach allows us to use classical methods of non-Markovian dynamics to study various quantum stochastic processes.