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Fractional Dynamics of Open Quantum Systems

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Fractional Dynamics

Part of the book series: Nonlinear Physical Science ((NPS,volume 0))

Abstract

We can describe an open quantum system starting from a closed Hamiltonian system if the open system is a part of the closed system (Weiss, 1993). However situations can arise where it is difficult or impossible to find a Hamiltonian system comprising the given quantum system. As a result, the theory of open and non-Hamiltonian quantum systems can be considered as a fundamental generalization (Kossakowski, 1972; Davies, 1976; Ingarden and Kossakowski, 1975; Tarasov, 2005, 2008b) of the quantum Hamiltonian mechanics. The quantum operations that describe dynamics of open systems can be considered as real completely positive trace-preserving superoperators on the operator space. These superoperators form a completely positive semigroup. The infinitesimal generator of this semigroup is completely dissipative (Kossakowski, 1972; Davies, 1976; Ingarden and Kossakowski, 1975; Tarasov, 2008b). Fractional power of operators (Balakrishnan, 1960; Komatsu, 1966; Berens et al., 1968; Yosida, 1995; Martinez and Sanz, 2000) and superoperators (Tarasov, 2008b, 2009a) can be used as a possible approach to describe fractional dynamics of open quantum systems. We consider superoperators that are fractional powers of completely dissipative superoperators (Tarasov, 2009a). We prove that the suggested superoperators are infinitesimal generators of completely positive semigroups for fractional quantum dynamics. The quantum Markovian equation, which includes an explicit form of completely dissipative superoperator, is the most general type of Markovian master equation describing non-unitary evolution of the density operator that is trace-preserving and completely positive for any initial condition.

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References

  • R. Alicki, K. Lendi, 1987, Quantum Dynamical Semigroups and Applications, Springer, Berlin.

    MATH  Google Scholar 

  • C. Anastopoulous, J.J. Halliwell, 1995, Generalized uncertainty relations and longtime limits for quantum Brownian motion models, Physical Review D, 51, 6870–6885.

    Article  MathSciNet  ADS  Google Scholar 

  • V. Balakrishnan, 1960, Fractional power of closed operator and the semigroup generated by them, Pacific Journal of Mathematics, 10, 419–437.

    MathSciNet  MATH  Google Scholar 

  • H. Barnum, C.M. Caves, C.A. Fuchs, R. Jozsa, B. Schumacher, 1996, Noncommuting mixed states cannot be broadcast, Physical Review Letters, 76, 2818–2821.

    Article  ADS  Google Scholar 

  • S. Bochner, 1949, Diffusion equations and stochastic processes, Proceedings of the National Academy of Sciences USA, 35, 369–370.

    Article  MathSciNet  ADS  Google Scholar 

  • H. Berens, P.L. Butzer, U. Westphal, 1968, Representation of fractional powers of infinitesimal generators of semigroups, Bulletin of the American Mathematical Society, 74, 191–196.

    Article  MathSciNet  MATH  Google Scholar 

  • V. Buzek, M. Hillery, 1996, Quantum copying: Beyond the no-cloning theorem, Physical Review A, 54, 1844–1852.

    Article  MathSciNet  ADS  Google Scholar 

  • T.R. Cech, 1986, A model for the RNA-catalyzed replication of RNA, Proceedings of the National Academy of Sciences USA, 83, 4360–4363.

    Article  ADS  Google Scholar 

  • V. Daftardar-Gejji, A. Babakhani, 2004, Analysis of a system of fractional differential equations, Journal of Mathematical Analysis and Applications, 293, 511–522.

    Article  MathSciNet  MATH  Google Scholar 

  • E.P. Davies, 1970, Quantum stochastic processes II, Communication in Mathematical Physics, 19, 83–105.

    Article  ADS  MATH  Google Scholar 

  • E.B. Davies, 1976, Quantum Theory of Open Systems, Academic Press, London, New York, San Francisco.

    MATH  Google Scholar 

  • E.B. Davies, 1977, Quantum dynamical semigroups and neutron diffusion equation, Reports in Mathematical Physics, 11, 169–188.

    Article  ADS  MATH  Google Scholar 

  • E.B. Davies, 1981, Symmetry breaking for molecular open systems, Annales de l’Institut Henri Poincaré, Section A, 35, 149–171.

    MATH  Google Scholar 

  • K. Dietz, 2002, Asymptotic solutions of Lindblad equations, Journal of Physics A, 35, 10573–10590.

    Article  MathSciNet  MATH  Google Scholar 

  • L.M. Duan, G.C. Guo, 1998, A probabilistic cloning machine for replicating two non-orthogonal states, Physical Letters A, 243, 261–264.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • R.A. Freitas Jr., R.C. Merkle, 2004, Kinematic Self-Replicating Machines, Landes Bioscience, see also http://www.molecularassembler.com/KSRM.htm

    Google Scholar 

  • C.W. Gardiner, 1985, Handbook of Stochastic Methods for Physics, Chemistry and Natural Sciences, 2nd ed., Springer, Berlin.

    Google Scholar 

  • V. Gorini, A. Kossakowski, E.C.G. Sudarshan, 1976, Completely positive dynamical semigroups of N-level systems, Journal of Mathematical Physics, 17, 821–825.

    Article  MathSciNet  ADS  Google Scholar 

  • V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, E.C.G. Sudarshan, 1978, Properties of quantum markovian master equations, Reports in Mathematical Physics, 13, 149–173.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • K.E. Hellwing, K. Kraus, 1969, Pure operations and measurements, Communication in Mathematical Physics, 11, 214–220.

    Article  ADS  Google Scholar 

  • K.E. Hellwing, K. Kraus, 1970, Operations and measurements II, Communication in Mathematical Physics, 16, 142–147.

    Article  ADS  Google Scholar 

  • E. Hille, R.S. Phillips, 1957, Functional Analysis and Semigroups, American Mathematical Society, Providence.

    MATH  Google Scholar 

  • R.S. Ingarden, A. Kossakowski, 1975, On the connection of nonequilibrium information thermodynamics with non-Hamiltonian quantum mechanics of open systems, Annals of Physics, 89, 451–485.

    Article  MathSciNet  ADS  Google Scholar 

  • A. Isar, A. Sandulescu, H. Scutaru, E. Stefanescu, W. Scheid, 1994, Open quantum systems, International Journal of Modern Physics E, 3, 635–714; and E-print: quant-ph/0411189.

    Article  MathSciNet  ADS  Google Scholar 

  • A. Isar, A. Sandulescu, W. Scheid, 1996, Phase space representation for open quantum systems with the Lindblad theory, International Journal of Modern Physics B, 10, 2767–2779; and E-print: quant-ph/9605041.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.

    MATH  Google Scholar 

  • H. Komatsu, 1966, Fractional powers of operators, Pacific Journal of Mathematics, 19, 285–346.

    MathSciNet  MATH  Google Scholar 

  • A. Kossakowski, 1972, On quantum statistical mechanics of non-Hamiltonian systems, Reports in Mathematical Physics, 3, 247–274.

    Article  MathSciNet  ADS  Google Scholar 

  • K. Kraus, 1971, General state changes in quantum theory, Annals of Physics, 64, 311–335.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • K. Kraus, 1983, States, Effects and Operations. Fundamental Notions of Quantum Theory, Springer, Berlin.

    Book  MATH  Google Scholar 

  • S.G. Krein, 1971, Linear Differential Equations in Banach Space, Translations of Mathematical Monographs, Vol.29, American Mathematical Society, Translated from Russian: Nauka, Moscow, 1967.

    Google Scholar 

  • D.A. Lidar, Z. Bihary, K.B. Whaley, 2001, From completely positive maps to the quantum Markovian semigroup master equation, Chemical Physics, 268, 35–53; and E-print: cond-mat/0011204.

    Article  ADS  Google Scholar 

  • G. Lindblad, 1976a, On the generators of quantum dynamical semigroups, Communication in Mathematical Physics, 48, 119–130.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • G. Lindblad, 1976b, Brownian motion of a quantum harmonic oscillator, Reports in Mathematical Physics, 10, 393–406.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • C. Martinez, M. Sanz, 2000, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies. Vol.187, Elsevier, Amsterdam.

    Google Scholar 

  • H. Nakazato, Y. Hida, K. Yuasa, B. Militello, A. Napoli, A. Messina, 2006, Solution of the Lindblad equation in the Kraus representation, Physical Review A, 74, 062113; and E-print: quant-ph/0606193.

    Article  MathSciNet  ADS  Google Scholar 

  • J. von Neumann, 1966, Theory of Self-Reproducing Automata Source, University of Illinois.

    Google Scholar 

  • K.B. Oldham, J. Spanier, 1974, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York.

    MATH  Google Scholar 

  • A. Oza, A. Pechen, J. Dominy, V. Beltrani, K. Moore, H. Rabitz, 2009, Optimization search effort over the control landscapes for open quantum systems with Krausmap evolution, Journal of Physics A, 42, 205305.

    Article  MathSciNet  Google Scholar 

  • R.S. Phillips, 1952, On the generation of semigroups of linear operators, Pacific Journal of Mathematics, 2, 343–369.

    MathSciNet  MATH  Google Scholar 

  • A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, 1986, Integrals and Series, Voll: Elementary Functions, Gordon and Breach, New York.

    Google Scholar 

  • S.G. Samko, A.A. Kubas, O.I. Marichev, 1993, Integrals and Derivatives of Fractional Order and Applications, Nauka i Tehnika, Minsk, 1987, in Russian; and Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.

    Google Scholar 

  • A. Sandulescu, H. Scutaru, 1987, Open quantum systems and the damping of collective models in deep inelastic collisions, Annals of Physics, 173, 277–317.

    Article  MathSciNet  ADS  Google Scholar 

  • V. Scarani, S. Iblisdir, N. Gisin, 2005, Quantum cloning, Review of Modern Physics, 11, 1225–1256.

    MathSciNet  ADS  Google Scholar 

  • B. Schumacher, 1996, Sending entanglement through noisy quantum channels, Physical Review A, 54, 2614–2628.

    Article  ADS  Google Scholar 

  • H. Spohn, 1976, Approach to equilibrium for completely positive dynamical semigroups of N-level systems, Reports in Mathematical Physics, 10, 189–194.

    Article  MathSciNet  ADS  Google Scholar 

  • H. Spohn, 1977, An algebraic condition for the approach to equilibrium of an open N-level system, Letters in Mathematical Physics, 2, 33–38.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2002a, Pure stationary states of open quantum systems, Physical Review E, 66, 056116.

    Article  MathSciNet  ADS  Google Scholar 

  • V.E. Tarasov, 2002b, Quantum computer with mixed states and four-valued logic, Journal of Physics A, 35, 5207–5235.

    Article  MathSciNet  MATH  Google Scholar 

  • V.E. Tarasov, 2002c, Stationary states of dissipative quantum systems, Physics Letters A, 299, 173–178.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2004, Path integral for quantum operations, Journal of Physics A, 37, 3241–3257.

    Article  MathSciNet  MATH  Google Scholar 

  • V.E. Tarasov, 2005, Quantum Mechanics: Lectures on Foundations of the Theory, 2nd ed., Vuzovskaya Kniga, Moscow. In Russian.

    Google Scholar 

  • V.E. Tarasov, 2008a, Fractional Heisenberg equation, Physics Letters A, 372, 2984–2988.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2008b, Quantum Mechanics of Non-Hamiltonian and Dissipative Systems, Elsevier, Amsterdam.

    MATH  Google Scholar 

  • V.E. Tarasov, 2009a, Fractional generalization of the quantum Markovian master equation, Theoretical and Mathematical Physics, 158, 179–195.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • V.E. Tarasov, 2009b, Quantum Nanotechnology, International Journal of Nanoscience, 8, 337–344.

    Article  Google Scholar 

  • J.D. Watson, N.H. Hopkins, J.W. Roberts, J.A. Steitz, A.M. Weiner, 1987, Molecular Biology of the Gene, Vol.2, 3th ed., Benjamin/Cumming, California, 1103–1124.

    Google Scholar 

  • U. Weiss, 1993, Quantum Dissipative Systems, World Scientific Publishing, Singapore.

    MATH  Google Scholar 

  • E.P. Wigner, 1961, The probability of the existence of the self-reproducing unit, in The Logic of Personal Knowledge. Essays presented to Michael Polanyi, Routledge and Paul, London, 231–238.

    Google Scholar 

  • W.K. Wootters, W.H. Zurek, 1982, A single quantum cannot be cloned, Nature (London), 299, 802–803.

    Article  ADS  Google Scholar 

  • R. Wu, A. Pechen, C. Brif, H. Rabitz, 2007, Controllability of open quantum systems with Kraus-map dynamics, Journal of Physics A, 40, 5681–5693.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Yosida, 1995, Functional Analysis, 6th ed., Springer, Berlin.

    Google Scholar 

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  1. Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119992, Moscow, Russia

    Vasily E. Tarasov

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  1. Vasily E. Tarasov
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© 2010 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Tarasov, V.E. (2010). Fractional Dynamics of Open Quantum Systems. In: Fractional Dynamics. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7_20

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