From repeated to continuous quantum interactions

Annales Henri Poincare

Volumn 7, Issue 1, 2006, Pages 59-104

  Attal, Stéphane ^a   Pautrat, Yan ^b  

^a UNIVERSITÉ LYON 1   (France)

^b UNIVERSITÉ PARIS SUD   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33644882745     PISSN: 14240637     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00023-005-0242-8     Document Type: Review

References (35)

1. [AFL1] L. Accardi, A. Frigerio, Y.G. Lu, Quantum Langevin equation in the weak coupling limit, Quantum probability and applications, V (Heidelberg, 1988), 1-16, Lecture Notes in Math., 1442, Springer, Berlin, 1990.
2. [AGL] L. Accardi, J. Gough, Y.G. Lu, On the stochastic limit for quantum theory, Proceedings of the XXVII Symposium on Mathematical Physics (Toruń, 1994), Rep. Math. Phys. 36, no. 2-3, 155-187 (1995).
3. [ALV] L. Accardi, Y.G. Lu, I. Volovich, Quantum theory and its stochastic limit, Springer-Verlag, Berlin, 2002.
4. [A-L1] L. Accardi, U.G. Lu, The low density limit and the quantum Poisson process, Probability theory and mathematical statistics, Vol. I (Vilnius, 1989), 1-27, "Mokslas", Vilnius, 1990.
5. [APV] L. Accardi, A.N. Pechen, I.V. Volovich, A stochastic golden rule and quantum Langevin equation for low density limit, preprint.
6. [AFL2] L. Accardi, A. Frigerio, Y.G. Lu, The weak coupling limit as a quantum functional central limit, Comm. Math. Phys. 131, no. 3, 537-570 (1990).
7. [At1] S. Attal, Extensions of the quantum stochastic calculus, Quantum Probability Communications XI, 1-38, World Scientific (2003).
8. [At2] S. Attal, Approximating the Fock space with the toy Fock space, Séminaire de Probabilités XXXVI, 477-491, Lecture Notes in Math., 1801, Springer, Berlin (2003).
9. [At3] S. Attal, Quantum Noise Theory, book in preparation, Springer Verlag.
10. [A-L2] S. Attal, J.M. Lindsay, Quantum stochastic calculus with maximal operator domains, Ann. Probab. 32, no 1A, 488-529 (2004).
11. [AJ1] S. Attal, A. Joye, Weak coupling and continuous limits for repeated quantum interactions, preprint.
12. [AJ2] S. Attal, A. Joye, The Langevin equation for a quantum heat bath, preprint.
13. [A-M] S. Attal, P.-A. Meyer, "Interprétation probabiliste et extension des intégrales stochastiques non commutatives", Séminaire de Probabilités XXVII, 312-327, Lecture Notes in Math., 1557, Springer, Berlin (1993).
14. [A-P] S. Attal, Y. Pautrat, From (n + 1)-level atom chains to n dimensional noises, Annales de l'Institut Henri Poincaré (B) Probability and Statistics 41, Issue 3, 255-630 (2005).
15. [Ba1] A. Barchielli, Measurement theory and stochastic differential equations in quantum mechanics, Phys. Rev. A (3)34, no. 3, 1642-1649 (1986).
16. [Ba2] A. Barchielli, Some stochastic differential equations in quantum optics and measurement theory: the case of counting processes, Stochastic evolution of quantum states in open systems and in measurement processes (Budapest, 1993), 1-14, World Sci. Publishing, River Edge, NJ, 1994.
17. [B-B] A. Barchielli, V.P. Belavkin, Measurements continuous in time and a posteriori states in quantum mechanics, J. Phys. A 24, no. 7, 1495-1514 (1991).
18. [BRSW] J. Bellissard, R. Rebolledo, D. Spehner, W. von Waldenfels, The Quantum Flow of Electronic transport, preprint.
19. [Coq] A. Coquio, Why are there only three quantum noises?, Probab. Theory Related Fields 118, no. 3, 349-364 (2000).
20. [Dav] E.B. Davies, Markovian master equations, Comm. Math. Phys. 39, 91-110 (1974).
21. [D-J] J. Derezinski, V. Jakšić, On the nature of Fermi Golden Rule for open quantum systems, J. Stat. Phys. 116, No 1-4, 411-423 (2004).
22. [F-R] F. Fagnola, R. Rebolledo, A view on stochastic differential equations derived from quantum optics, Stochastic models (Guanajuato, 1998), 193-214, Aportaciones Mat. Investig., 14, Soc. Mat. Mexicana, Mexico, 1998.
23. [FRS] F. Fagnola, R. Rebolledo, C. Saavedra, Quantum flows associated to master equations in quantum optics, J. Math. Phys. 35, no. 1, 1-12 (1994).
24. [FKM] G.W. Ford, M. Kac, P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Mathematical Phys. 6, 504-515 (1965).
25. [FLO] G.W. Ford, J.T. Lewis, R.F. O'Connell, Quantum Langevin equation, Phys. Rev. A (3) 37, no. 11, 4419-4428 (1988).
26. [G-Z] C.W. Gardiner, P. Zoller, Quantum noise. A handbook of Markovian and non-Markovian quantum stochastic methods with applications to quantum optics., Second edition. Springer Series in Synergetics. Springer-Verlag, Berlin, 2000.
27. [GSI] H. Grabert, P. Schramm,G.-L. Ingold Quantum Brownian motion: the functional integral approach, Phys. Rep. 168, no. 3, 115-207 (1988).
28. [Gui] A. Guichardet, Symmetric Hilbert spaces and related topics., Lecture Notes in Mathematics, Vol. 261. Springer-Verlag, Berlin-New York, 1972.
29. [H-P] R.L. Hudson, K.R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys. 93, no. 3, 301-323 (1984).
30. [L-M] J.M. Lindsay, H. Maassen, Stochastic calculus for quantum Brownian motion of nonminimal variance - an approach using integral-sum kernel operators, Mark Kac Seminar on Probability and Physics Syllabus 1987-1992 (Amsterdam, 1987-1992), 97-167, CWI Syllabi, 32, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1992.
31. [M-R] H. Maassen, P. Robinson, Quantum stochastic calculus and the dynamical Stark effect, Rep. Math. Phys. 30, no. 2, 185-203 (1992) (1991).
32. [Par] K.R. Parthasarathy, An introduction to quantum stochastic calculus, Monographs in Mathematics 85, Birkhäuser (1992).
33. [Pa1] Y. Pautrat, From Pauli matrices to quantum Ito formula, Mathematical Physics, Analysis and Geometry, to appear.
34. [Pa2] Y. Pautrat, Kernel and integral representations of operators on infinite dimensional toy Fock space, Séminaire de Probabilités, to appear.
35. [Pa3] Y. Pautrat, Des matrices de Pauli aux bruits quantiques, Thèse de doctorat de l'Université Joseph Fourier, (2003).