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Dynamics of Shuttle Devices

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3.1. COHERENT DYNAMICS OF SMALL OPEN SYSTEMS 3.1 Coherent dynamics of small open systems The master equation is usually derived for models in which a“small” system with few degrees of freedom is in interaction with a “large” bath with effec- tively an infinite number of degrees of freedom. The Liouville von-Neumann equation of motion for the total density matrix is very complicated to solve and actually contains too much information since it also takes into account coherencies of the bath. It is useful to average it over bath variables and obtain an equation of motion for the density matrix of the system (the re- duced density matrix ). With no further simplification this equation is called Generalized Master Equation (GME) since it involves not only the pop- ulations but also the coherencies of the small subsystem. The derivation of the GME from the equation of Liouville-von Neumann is far from trivial and also non-universal: it involves a series of approximations justified by the physical properties of the model at hand. Despite the apparent similar- ities, the two equations are deeply different: the equation of Liouville-von Neumann describes the reversible dynamics of a closed system; the GME, instead, describes the irreversible dynamics of an open system that continu- ously exchange energy with the bath3. Shuttle devices are small systems coupled to different baths (leads, ther- mal bath) but they maintain a high degree of correlation between electrical and mechanical degrees of freedom captured by the coherencies of the reduced density matrix. The GME seems to be a good candidate for the description of their dynamics. In the next two sections we will derive two GMEs using two different approaches. They are both necessary for the description of the shuttling devices since they correspond to the different coupling of the system to the mechanical and electrical baths. 3.2 Quantum optical derivation The harmonic oscillator weakly coupled to a bosonic bath is a typical problem analyzed in quantum optics. This model well describes in shuttling devices the interaction of the mechanical degree of freedom of the NEMS with its environment. Following section 5.1 of the book “Quantum Noise” by C. W. Gardiner and P. Zoller [27] we start considering a small system S coupled to a large bath B described by the generic Hamiltonian: 3How can irreversibility be derived from reversibility? The solution of this dilemma lies in time scales: system+bath recurrence time is “infinite” on the time scale of the system. The GME holds on the time scales of the system. 27

Anteprima della Tesi di Andrea Donarini

Anteprima della tesi: Dynamics of Shuttle Devices, Pagina 15

Tesi di Dottorato

Dipartimento: Department if Micro and Nanotechnologies

Autore: Andrea Donarini Contatta »

Composta da 158 pagine.

 

Questa tesi ha raggiunto 319 click dal 01/02/2005.

Disponibile in PDF, la consultazione è esclusivamente in formato digitale.