Abstract
We consider a general class of models consisting of a small quantum system \(S\)interacting with a reservoir \(R\). We compare three applications of 2nd order perturbation theory (the Fermi Golden Rule) to the study of such models: (1) the van Hove (weak coupling) limit for the dynamics reduced to \(S\); (2) the Fermi Golden Rule applied to the Liouvillean—an argument that was used in recent papers on the return to equilibrium; (3) the Fermi Golden Rule applied to the so-called C-Liouvillean. These three applications lead to three Level Shift Operators. As our main result, we prove that if the reservoir \(R\)is thermal (if it satisfies the KMS condition), then the Level Shift Operator obtained in (1) (often called the Davies generator) and the Level Shift Operator constructed in (2) are connected by a similarity transformation. We also show that the Davies generator coincides with the Level Shift Operator obtained in (3) for a general \(R\).
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REFERENCES
O. Brattelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volume 1, 2nd edn. (Springer-Verlag, Berlin, 1987).
O. Brattelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volume 2, 2nd edn. (Springer-Verlag, Berlin, 1996).
V. Bach, J. Fröhlich, and I. Sigal, Return to equilibrium, J. Math. Phys. 41:3985 (2000).
E. B. Davies, Markovian master equations, Comm. Math. Phys. 39:91 (1974).
E. B. Davies, Markovian master equations II, Math. Ann. 219:147 (1976).
6. E. B. Davies, One Parameter Semigroups(Academic Press, 1980).
J. Derezinski and V. Jakšic, Spectral theory of Pauli-Fierz operators, J. Func. Anal. 180:241 (2001).
J. Derezinski and V. Jakšic, Return to equilibrium for Pauli-Fierz systems, Ann. Henri Poincaré 4:739–793 (2003).
J. Derezinski and V. Jakšic, in preparation.
J. Derezinski, V. Jakšic, and C. A. Pillet, Perturbation theory of W*-dynamics, Liouvilleans and KMS-states, to appear in Rev. Math. Phys.
E. Fermi, Nuclear Physics,Notes compiled by J. Orear, A. H. Rosenfeld, and R. A. Schluter (The University of Chicago Press, Chicago, 1950).
V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, and E. C. G. Sudarshan, Properties of quantum markovian master equations, Rep. Math. Phys. 13:149 (1978).
R. Haag, Local Quantum Physics(Springer-Verlag, Berlin, 1993).
F. Haake, Statistical treatment of open systems by generalized master equation, in Springer Tracts in Modern Physics, Vol. 66 (Springer-Verlag, Berlin, 1973).
V. Jakšic and C.-A. Pillet, On a model for quantum friction III. Ergodic properties of the spin-boson system, Comm. Math. Phys. 178:627 (1996).
V. Jakšic and C.-A. Pillet, Spectral theory of thermal relaxation, J. Math. Phys. 38:1757 (1997).
V. Jakšic and C.-A. Pillet, From resonances to master equations, Ann. Inst. Henri Poincaré 67:425 (1997).
V. Jakšic and C.-A. Pillet, Non-equilibrium steady states for finite quantum systems coupled to thermal reservoirs, Comm. Math. Phys. 226:131 (2002).
V. Jakšic and C.-A. Pillet, Mathematical theory of non-equilibrium quantum statistical mechanics, J. Stat. Phys. 108:787 (2002).
R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II. Nonequilibrium Statistical Mechanics(Springer-Verlag, Berlin, 1985).
M. Merkli, Positive commutators in non-equilibrium quantum statistical mechanics, Comm. Math. Phys. 223:327 (2001).
J. Lebowitz and H. Spohn, Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs, Adv. Chem. Phys. 39:109 (1978).
W. Pauli, Fesrshrift zum 60. Gerburstage A. Sommerfeld(Hirzel, Leipzig, 1928), p. 30.
W. Pauli, Pauli Lectures on Physics: Volume 4. Statistical Mechanics,C. P. Enz, ed. (MIT Press, Cambridge, 1973).
C.-A. Pillet, Private communication.
D. Ruelle, Natural nonequilibrium states in quantum statistical mechanics, J. Stat. Phys. 98:57 (2000).
S. Stratila, Modular Theory in Operator Algebras(Abacus Press, Turnbridge Wells, 1981).
S. Stratila and L. Zsido, Lectures on von Neumann Algebras(Abacus Press, Turnbridge Wells, 1979).
L. Van Hove, Master equation and approach to equilibrium for quantum systems, in Fundamental Problems in Statistical Mechanics, combined by E. G. D. Cohen, (North-Holand, Amsterdam, 1962).
V. Weisskopf and E. P. Wigner, Berechnung der natürlichen Linienbreite auf Grund der Diracschen Lichttheorie, Z. Phys. 63:54 (1930).
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Dereziński, J., Jakšić, V. On the Nature of Fermi Golden Rule for Open Quantum Systems. Journal of Statistical Physics 116, 411–423 (2004). https://doi.org/10.1023/B:JOSS.0000037208.99352.0a
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DOI: https://doi.org/10.1023/B:JOSS.0000037208.99352.0a