Understanding the role of respresentation in controlled quantum-dynamical mechanism analysis

Abhra Mitra, Ignacio R. Sola, Herschel Albert Rabitz

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Abstract

Hamiltonian Encoding (HE) has been proposed as a technique for analyzing the mechanism of controlled quantum dynamics, where mechanism is understood in terms of the set of amplitudes of the dominant pathways connecting the initial and final states of the system. The choice of representation for the system wave function is often motivated by seeking simplicity for the structure of the Hamiltonian and not necessarily for the generated dynamics. However, the mechanism revealed by HE is strongly dependent on the basis in which the wave function is represented. The degree of mechanistic complexity is indicated by the relevant orders of the Dyson series contributing to the dynamics. An appropriate choice of representation can yield a simpler view of the dynamical mechanism by shifting some of the complexity into the representation itself. In this work the choice of representation is set up as the solution to a variational optimization problem. For unconstrained basis transformations, the optimization of the representation is formally equivalent to solving the time-dependent Schrödinger equation; different constrained basis transformations provide distinct dynamical perspectives. Specific constrained variational Ansätze are compared and analyzed by performing HE on several simple Hamiltonians with an observation of the extent to which the mechanism assessment varies with representation. The general variational formulation for determining representation can flexibly admit other Ansätze with the ultimate aim of balancing the ease of determining and understanding the representation with the reduction in mechanistic complexity.

Original languageEnglish (US)
Article number043415
JournalPhysical Review A - Atomic, Molecular, and Optical Physics
Volume77
Issue number4
DOIs
StatePublished - Apr 22 2008

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics

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