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Master equation and initial factorization

Master equation and initial factorization. Paolo Facchi Università di Bari, Italy in collaboration with S. Pascazio, K. Yuasa (Bari) H. Nakazato, I. Ohba, S. Tasaki (Tokyo) G. Kimura (Sendai). Introduction. Closed quantum system unitary evolution (reversible) Schrödinger equation

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Master equation and initial factorization

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  1. Master equation and initial factorization Paolo Facchi Università di Bari, Italy in collaboration with S. Pascazio, K. Yuasa (Bari) H. Nakazato, I. Ohba, S. Tasaki (Tokyo) G. Kimura (Sendai)

  2. Introduction • Closed quantum system • unitary evolution (reversible) • Schrödinger equation • small systems / discrete spectra • Open Quantum Systems • dissipation/decoherence (irreversible) • system S + reservoir B • derivation of master equation for the reduced density ops. infinitely extended systems System S • (Thermal) Quantum Field Theory Reservoir B

  3. Total System S+B Reduced Density Op. Master Eq. for S (Markov approximation) Master equation

  4. Derivation of master equation • System S + Reservoir B • B is infinitely large • Weak coupling • Trace over B • Nakajima-Zwanzig’s Projection Method • Weak-Coupling + Markov Approximation • van Hove’s Limit • No Initial Correlation between S and B

  5. interaction picture Assumptions of Factorization factorized initial state factorization at later times D.F. Walls and G.J. Milburn, Quantum Optics (1994); M.O. Scully and M.S. Zubairy, Quantum Optics (1997); C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (1998); H.J. Carmichael, Statistical Methods in Quantum Optics 1 (1999); H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (2002); C.W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (2004); …

  6. For a factorized initial state For a correlated initial state choice of the reference state ? Reference State

  7. Notation • Liouvillians • Projections • Properties

  8. Projection by and Integrating out Projection Method non-Markovian initial correlation F. Haake, in Quantum Statistics in Optics and Solid-State Physics, Vol. 66 of Springer Tracts in Modern Physics, edited by G. Höhler (Springer, Berlin, 1973), pp. 98-168; R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II, 2nd ed. (Springer, Berlin, 1995); L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (1995); …

  9. Requirements • {0}= point spectrum of + (absolutely) continuous spectrum • is a bounded perturbation of • Projection Tasaki et al., Ann. Phys. 322, 631-656 (2007)

  10. S D. Ruelle, J. Stat. Phys. 98, 57 (2000); W. Aschbacher, V. Jakšić, Y. Pautrat, and C.-A. Pillet, mp_arc 05-207 (2005); S. Tasaki and J. Takahashi, cond-mat/0606259 (2006). Non-Equilibrium Steady State (NESS) a steady current flowing between two reservoirs … a mixing state B for bounded ops. Mixing reservoir for any bounded ops. • is mixing (with respect to ). • e.g.)

  11. … factorization! Mixing reservoir is mixing (with respect to ). for any bounded ops. For the total system S+B mixing

  12. Open system in a mixing reservoir perturbations Perturbations/correlation propagate away to infinity. There remains the mixing state and the system is factorized. Correlation through interaction. → Mixing quickly in t. System S free evolution int correlation … S+B looks factorized in Reservoir B

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