7. TRAJECTORIES BASED ON QUANTUM DISTRIBUTIONS

** **

Applications of the Wigner function for quantum electronic transport appeared in the 1980s [67-74], mostly of a formal nature. The first computation for a realistic time-dependent physical semiconductor system was made in [75]. Quantum Monte Carlo for device simulation has been extensively investigated since then [76-79]. In many cases the non-positive definiteness of the Wigner distribution was overcome by using damping theoretic methods and by averaging procedures. The equations of motion for carriers have been obtained by moment expansions of the non-local Wigner integral equation of motion and solved using Ensemble Monte Carlo simulation by trajectory tracking [78]. Recently a powerful method [79] has emerged for handling the non-positive definite Wigner function by particle tracking but it is highly compute-intensive. Unfortunately even this method suffers from second problem that follows because the Wigner function does not have *compact support *in phase space [17]. This problem derives from the geometric *centre-of-mass* construction. Thus a wave-packet incident on a simple 1D tunnel barrier may split into two well-defined reflected and transmitted packets; but in the Wigner representation this situation leads to well-defined exit packets in phase space plus a wildly oscillating structure midway between the exiting packets in a region where the position distribution and momentum distributions are essentially zero. This problem has been overcome by our introduction of a new unique quantum distribution [80, 81] which we have called a *C*-distribution that has manifest compact support in phase space (*C* for compact support and complex valued). The *C*-distributions may be derived from the density matrix using a mixed real space-momentum representation and the approach generalises to double-time double-space non-equilibrium Green’s functions. The equations of motion and possible Monte Carlo trajectory computational schemes are discussed within exactly soluble models [81] that illustrate the formalism and its interpretation. The near-classical limit is easily obtained and lends itself to path-variable iterative methods including Monte Carlo trajectory schemes. The formalism has well-defined phase space trajectories for stationary states, time-dependent states and open systems.

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