Abstract
Molecules in open systems may be modeled using so-called reduced descriptions that keep focus on the molecule while including the effects of the environment. Mathematically, the matrices governing the Markovian equations of motion for the reduced density matrix, such as the Lindblad and Redfield equations, belong to the family of non-normal matrices. Tools for predicting the behavior of normal matrices (e.g., eigenvalue decompositions) are inadequate for describing the qualitative dynamics of systems governed by non-normal matrices. For example, such a system may relax to equilibrium on timescales much longer than expected from the eigenvalues. In this paper we contrast normal and non-normal matrices, expose mathematical tools for analyzing non-normal matrices, and apply these tools to a representative example system. We show how these tools may be used to predict dissipation timescales at both intermediate and asymptotic times, and we compare these tools to the conventional eigenvalue analyses. Interactions between the molecule and the environment, while generally resulting in dissipation on long timescales, can directly induce transient or even amplified behavior on short and intermediate timescales.
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References
A.H. Al-Mohy, N.J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33(2), 488–511 (2011). https://doi.org/10.1137/100788860
A. Ben-Israel, Linear equations and inequalities on a finite dimensional, real or complex, vector space: a unified theory. J. Math. Anal. Appl. 27, 367–389 (1969). https://doi.org/10.1016/0022-247X(69)90054-7
K. Blum, Density Matrix Theory and Applications, 3rd edn. (Springer, Heldelberg, 2012)
H.P. Breuer, F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)
L. Chen, T. Hansen, I. Franco, Simple and accurate method for time-dependent transport along nanoscale junctions. J. Phys. Chem. C 118(34), 20009–20017 (2014)
E.S. Coakley, V. Rokhlin, A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices. Appl. Comput. Harmon. Anal. 34(3), 379–414 (2013). https://doi.org/10.1016/j.acha.2012.06.003
G.S. Engel, T.R. Calhoun, E.L. Read, T.K. Ahn, T. Mančal, Y.C. Cheng, R.E. Blankenship, G.R. Fleming, Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446, 782–786 (2007)
G.H. Golub, C.F. Van Loan, Matrix Computations, 4th edn. (The Johns Hopkins University Press, Baltimore, 2012)
V. Gorini, A. Kossakowski, E.C.G. Sudarshan, Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17(5), 821–825 (1976). https://doi.org/10.1063/1.522979
R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)
A. Iserles, S. MacNamara, Applications of magnus expansions and pseudospectra to markov processes. Eur. J. Appl. Math. 30(2), 400–425 (2019). https://doi.org/10.1017/S0956792518000177
J.M. Jean, R.A. Friesner, G.R. Fleming, Application of a multilevel redfield theory to electron transfer in condensed phases. J. Chem. Phys. 96(8), 5827–5842 (1992). https://doi.org/10.1063/1.462858
D. Krejčřík, P. Siegl, M. Tater, J. Viola, Pseudospectra in non-Hermitian quantum mechanics. J. Math. Phys. 56(10), 103513 (2015). https://doi.org/10.1063/1.4934378
G. Lindblad, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48(2), 119–130 (1976). https://doi.org/10.1007/BF01608499
K.G. Makris, L. Ge, H.E. Türeci, Anomalous transient amplification of waves in non-normal photonic media. Phys. Rev. X 4, 041044 (2014). https://doi.org/10.1103/PhysRevX.4.041044
C. Moler, C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix. Twenty-five years later. SIAM Rev. 45(1), 3–49 (2003). https://doi.org/10.1137/S00361445024180
A. Nitzan, Chemical Dynamics in Condensed Phases (Oxford University Press, Oxford, 2006)
M.B. Plenio, J. Almeida, S.F. Huelga, Origin of long-lived oscillations in 2d-spectra of a quantum vibronic model: electronic versus vibrational coherence. J. Chem. Phys. 139(23), 235102 (2013). https://doi.org/10.1063/1.4846275
M.G. Reuter, Associating the invariant subspaces of a non-normal matrix with transient effects in its matrix exponential or matrix powers. (2019). http://arxiv.org/abs/1909.05931
E. Thyrhaug, R. Tempelaar, M.J.P. Alcocer, K. Žídek, D. Bína, J. Knoester, T.L.C. Jansen, D. Zigmantas, Identification and characterization of diverse coherences in the Fenna–Matthews–Olson complex. Nat. Chem. 10, 780–786 (2018)
L.N. Trefethen, M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University Press, Princeton, 2005)
L.N. Trefethen, A.E. Trefethen, S.C. Reddy, T.A. Driscoll, Hydrodynamic stability without eigenvalues. Science 261(5121), 578–584 (1993). https://doi.org/10.1126/science.261.5121.578
T. Zelovich, L. Kronik, O. Hod, State representation approach for atomistic time-dependent transport calculations in molecular junctions. J. Chem. Theory Comput. 10(8), 2927–2941 (2014)
Acknowledgements
F.P. and T.H. thank the Lundbeck Foundation for generous financial support. M.G.R. was supported by startup funds from the Institute for Advanced Computational Science at Stony Brook University.
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Poulsen, F., Hansen, T. & Reuter, M.G. Predicting slow relaxation timescales in open quantum systems. J Math Chem 60, 1542–1554 (2022). https://doi.org/10.1007/s10910-022-01367-2
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DOI: https://doi.org/10.1007/s10910-022-01367-2