Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials

2010/06/30 by Alex W. Chin, Ángel Rivas, Susana F. Huelga +1 · 290 citations
Computer Science · Mathematics · Physics and Astronomy · #Chain (unit) #Density matrix renormalization group #Hamiltonian (control theory) #Orthogonal polynomials #Quantum #Quantum Computing Algorithms and Architecture #Quantum many-body systems #Quantum system #Recurrence relation #Spectroscopy and Quantum Chemical Studies #Unitary state #Unitary transformation #cond-mat.stat-mech #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1063/1.3490188

published in Journal of Mathematical Physics 51(9) (American Institute of Physics) · 24 pages, 1 figure, final version

openalex publication_date 2010/09/01 · arxiv created 2010/09/30 · arxiv updated 2010/10/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.

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