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A Resonance Theory for Open Quantum Systems with Time-Dependent Dynamics

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Abstract

We develop a resonance theory to describe the evolution of open systems with time-dependent dynamics. Our approach is based on piecewise constant Hamiltonians: we represent the evolution on each constant bit using a recently developed dynamical resonance theory, and we piece them together to obtain the total evolution. The initial state corresponding to one time-interval with constant Hamiltonian is the final state of the system corresponding to the interval before. This results in a non-Markovian dynamics. We find a representation of the dynamics in terms of resonance energies and resonance states associated to the Hamiltonians, valid for all times t≥0 and for small (but fixed) interaction strengths. The representation has the form of a path integral over resonances. We present applications to a spin-fermion system, where the energy levels of the spin may undergo rather arbitrary crossings in the course of time. In particular, we find the probability for transition between ground- and excited state at all times.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada

    Marco Merkli

  2. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Shannon Starr

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  1. Marco Merkli
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  2. Shannon Starr
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Correspondence to Marco Merkli.

Additional information

Dedicated to Jürg Fröhlich and Tom Spencer with our respect and affection.

M. Merkli supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant 205247. S. Starr supported in part by a US National Science Foundation grant, DMS-0706927.

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Merkli, M., Starr, S. A Resonance Theory for Open Quantum Systems with Time-Dependent Dynamics. J Stat Phys 134, 871–898 (2009). https://doi.org/10.1007/s10955-008-9645-5

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  • DOI: https://doi.org/10.1007/s10955-008-9645-5

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