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Journal of Chemical Physics / Volume 129 / Issue 19 / ARTICLES / Theoretical Methods and Algorithms
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J. Chem. Phys. 129, 194104 (2008); http://dx.doi.org/10.1063/1.3010370 (5 pages)

Coarse grained open system quantum dynamics

Ioannis Thanopulos1, Paul Brumer2, and Moshe Shapiro1,3

1Department of Chemistry, The University of British Columbia, Vancouver V6T 1Z3, Canada
2Chemical Physics Theory Group, Department of Chemistry, and Center of Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario M5S 3H6, Canada
3Department of Chemical Physics, The Weizmann Institute, Rehovot 76100, Israel

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(Received 4 September 2008; accepted 10 October 2008; published online 17 November 2008)

We show that the quantum dynamics of a system comprised of a subspace Q coupled to a larger subspace P can be recast as a reduced set of “coarse grained” ordinary differential equations with constant coefficients. These equations can be solved by a single diagonalization of a general complex matrix. The method makes no assumptions about the strength of the couplings between the Q and the P subspaces, nor is there any limitation on the initial population in P. The utility of the method is demonstrated via computations in three following areas: molecular compounds, photonic materials, and condensed phases.

© 2008 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. THEORY
  3. APPLICATIONS
  4. CONCLUSION

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KEYWORDS and PACS

Keywords

differential equations, matrix algebra, quantum theory, Schrodinger equation

PACS

  • 03.65.Yz

    Decoherence; open systems; quantum statistical methods

  • 02.10.Yn

    Matrix theory

  • 03.65.Ge

    Solutions of wave equations: bound states

  • 02.30.Hq

    Ordinary differential equations

  • 03.65.Vf

    Phases: geometric; dynamic or topological

  • 03.65.Fd

    Algebraic methods

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3010370

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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Figures (click on thumbnails to view enlargements)

FIG.1
Coarse-grained population evolution of |1〉 (black lines), |2〉 (red lines), and |3〉 (blue lines) ∊ Q interacting with p = 50 states ∊ P, when P is initially empty (left panel) and half-populated (right panel). Results for M = 10 (dashed lines), M = 30 (dash-dotted lines), and M = 50 (solid lines) are shown. The solution of the corresponding Schrödinger equation is also presented (solid circles). The M = 10 (dashed) and M = 30 (dash-dotted) state |3〉 curves (blue) coincide in both panels.

FIG.1 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.2
The dynamics of the |16〉|0〉P Q-state when P is initially empty. Exact evolution (blue solid line, both panels), evolution of this state in the Wλk,ℓ coarse-graining procedure, with M = 50 (green dashed line), and M = 100 (red dot-dashed line) [panel (a)]. Panel (b): dynamics under the Vλ,ℓ coarse-graining procedure with M = 1000 (green dashed line).

FIG.2 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.3
The Jaynes–Cummings model with detuning on resonance (blue dashed line) and off resonance (red dot-dashed line), as well as non-Markovian spontaneous decay into a photonic band gap (black solid line). The solid circles indicate corresponding results by previous works (Refs. 3 , 11).

FIG.3 Download High Resolution Image (.zip file) | Export Figure to PowerPoint

FIG.4
Top panel: Re[C(t)] (blue solid line) and Im[C(t)] (green dashed line) of the exact correlation function C(t) for the Ohmic Gohm(E) at zero temperature, and the real part (red) and imaginary part (black) of the correlation function obtained by fitting Gohm(E) to Eq. ( 7 ) with M = 12. Bottom: σ(t) ≡ (|c2(t)|2−|c1(t)|2) for two interacting states in Q coupled to an Ohmic, initially empty, P space, using the fitted spectral density with M = 12.

FIG.4 Download High Resolution Image (.zip file) | Export Figure to PowerPoint


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