Abstract
For the most general quantum master equations, also called Nakajima–Zwanzig or non-Markovian equations, we define suitable boundedness conditions on integral kernels and inhomogeneity terms in order to derive with mathematical rigor an upper bound on solutions, as required by the von Neumann conditions. Such equations are of importance for quantum dynamics of open systems with arbitrary couplings to environment and arbitrary entangled initial states. The derivation is based on an equivalent coherence-vector representation in finite dimension n leading to coupled Volterra integro-differential equations of second kind and convolution type in an (n 2−1)-dimensional real vector space. As examples, analytical and numerical model solutions are worked out for 2-level systems in order to test suitable trial functions for input quantities. All this is motivated by the fact that exact solutions can hardly be found but appropriate trial functions may provide a reasonable semiphenomenological description of complicated quantum dynamics.
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Reference
V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. 17:821(1976).
G. Lindblad, Commun. Math. Phys. 48:119(1976).
H. Spohn and J. L. Lebowitz, in Advances in Chemical Physics, Vol. 38, S. A. Rice, ed. (Wiley, New York, 1978).
R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Lecture Notes in Physics, Vol. 286 (Springer, Berlin, 1987).
F. Farhadmotamed, Entropy Production in Open Quantum Systems, Ph.D. thesis (University of Zürich, Zürich, 1998).
F. Farhadmotamed, A. J. van Wonderen, and K. Lendi, J. Phys. A: Math. Gen. 31:3395(1998).
S. Stenholm, Phys. Rep. 6:1(1973).
J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. 44:1323(1980).
A. J. van Wonderen, Phys. Rev. A 56:3116(1997).
R. Zwanzig, in Boulder Lectures in Theoretical Physics, Vol. III (Interscience, 1960), p. 106
R. Zwanzig, J. Chem. Phys. 33:1338(1960).
J. Kupsch, Open quantum systems, in Decoherence and the Appearance of a Classical World in Quantum Theory, D. Giulini et al., eds. (Berlin, Springer, 1996).
A. Carrington and A. D. McLauchlan, Introduction to Magnetic Resonance (Harper & Row, New York, 1967).
A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1986).
K. Lendi, J. Phys. A: Math. Gen. 20:15(1987).
A. Aissani, Sur la détion d'éue et sur l'étude d'une équation intégro-différentielle,Ph.D. thesis (University of Metz,Metz2000)
T. A. Burton, Volterra Integral and Differential Equations(Academic, New York, 1983).
P. H. M. Wolkenfelt, IMA J. Numer. Anal. 2:131(1982).
Ch. Lubich, IMA J. Numer. Anal. 3:439(1983).
L. F. Shampine, Math. Comp. 46:135(1986).
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Aissani, A., Lendi, K. Conditions for Bounded Solutions of Non-Markovian Quantum Master Equations. Journal of Statistical Physics 111, 1353–1362 (2003). https://doi.org/10.1023/A:1023064518885
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DOI: https://doi.org/10.1023/A:1023064518885