Senior Honors Thesis

Thursday, June 2, 2016, Wilder 102, 2:00 PM

Oscar Friedman​, Department of Physics and Astronomy, Dartmouth College

Title:  The Wigner Flow Function for Open Quantum Systems

Abstract:  In classical mechanics, one can fully describe the dynamics of any one-dimensional system using a single trajectory in phase space. Since the rules of quantum mechanics beget an inherent uncertainty associated with a simultaneous measurement of the position and momentum observables, the quantum analog to the classical phase space trajectory requires a more nuanced description. The Wigner Function, first published by Eugene Wigner in 1932, provides a map from the density operator to a phase space quasiprobability distribution. As the name implies, since the Wigner Function admits negative values, it is not strictly a probability distribution, although it shares certain features with the classical phase space probability distribution. In order to describe the dynamics of a closed quantum system, we can recast the Time-Development Schroedinger Equation in terms of the Wigner Function forming the Wigner Evolution Function (WEF). For an open quantum system we follow a similar procedure using the Lindblad Master Equation in place of the Schroedinger Equation. From the WEF we obtain the Wigner Flow Function (WFF), analogous to classical particle flow, which provides a novel way to envision the dynamics of a quantum system, having implications for quantum tunneling, the quantum-classical crossover, and the mechanisms by which quantum effects appear for nonlinear systems. This thesis examines the properties of the WFF for the nonlinear Duffing Oscillator using a numerical simulation within the framework of the open-source Quantum Toolbox in Python (QuTiP).