Numerical simulation of stochastic evolution equations associated to quantum markov semigroups

Mathematics of Computation

Volumn 73, Issue 247, 2004, Pages 1393-1415

  Mora, Carlos M ^a  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

Author keywords

Euler scheme; Galerkin method; Numerical solution; Quantum dynamical semigroup; Quantum master equation; Rate of convergence; Stochastic evolution equation

Indexed keywords

EID: 2942635940     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01595-3     Document Type: Article

References (26)

1. R. Balescu, Equilibrium and nonequilibrium statistical mechanics. Krieger Publishing Company (1991) (original edition 1975). MR 53:12405
2. A. Barchielli, A.M. Paganoni, F. Zucca, On stochastic differential equations and semigroups of probability operators in quantum probability. Stochastic Proc. and their Appl. 73 (1998), 69-86. MR 99k:81138
3. H. Carmichael, An open systems approach to quantum optics. Lecture Notes in Physics. Springer-Verlag (1993).
4. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum mechanics. Volume 1. Hermann (1977).
5. A.M. Chebotarev, F. Fagnola, Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153 (1998), 382-404. MR 99d:81064
6. F. Fagnola, R. Rebolledo, C. Saavedra, Quantum flows associated to master equations in quantum optics. J. Math. Phys. 35 (1994), 1-12. MR 94k:81353
7. F. Fagnola, Quantum markov semigroups and quantum flows. Proyecciones 18 (1999), 1-144. MR 2002f:81054
8. F. Fagnola, S. J. Wills, Solving quantum stochastic differential equations with unbounded coefficients. J. Funct. Anal. 198 (2003), 279-310.
9. N. Gisin, I.C. Percival, The quantum-state diffusion model applied to open systems. J. Phys. A 25 (1992), 5677-5691. MR 94b:82028
10. A.S. Holevo, On dissipative stochastic equations in a Hilbert space. Probab. Theory Related Fields 104 (1996), 483-500. MR 98c:60068
11. P.T. Jorgensen, Approximately reducing subspaces for unbounded linear operators. J. Funct. Anal. 23 (1976), 392-414. MR 98c:60068
12. T. Kato, Perturbation theory for linear operators. Corrected printing of the second edition. Springer-Verlag (1980). MR 96a:47025
13. P.E. Kloeden, E. Platen, Numerical solution of stochastic differential equations. Springer, Berlin (1992). MR 94b:60069
14. Tarso B.L. Kist, M. Orszag, T.A. Brun, L. Davidovich, Stochastic Schrödinger equations in cavity QDE: physical interpretation and localization. J. Opt. B: Quantum Semiclass. Opt. 1 (1999), 251-263.
15. R. Dautray, J.-L. Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Evolution Problems I, Springer-Verlag (1992). MR 92k:00006
16. G.N. Milstein, The numerical integration of stochastic differentials equations. Ural University Press (1988).
17. C. Mora, Solución numérica de ecuaciones diferenciales estocásticas mediante métodos exponenciales. Ph.D. Thesis, Pontificia Universidad Católica de Chile (2002).
18. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer-Verlag (1975). MR 85g:47061
19. I.C. Percival, Quantum state diffusion. Cambridge University Press (1998). MR 2000f:81023
20. P. Protter, Stochastic integration and differential equations. Springer (1990). MR 91i:60148
21. M.O. Scully, M.S. Zubairy, Quantum optics. Cambridge University Press (1997).
22. W. T. Strunz, L. Diósi, N. Gisin, T. Yu, Quantum trajectories for Brownian motion. Phys. Rev. Lett. 83 (1999), 4909-4913. MR 2000h:82042
23. D. Talay, Efficient numerical schemes for the approximation of expectations of functional of the solution of an SDE and applications. Springer Lecture Notes in Control and Inform. Sc. 61 (1984), 294-313. MR 88b:60140
24. D. Talay, Discrétisation d 'une E.D.S. et calcul approaché d'espérances de fonctionnelles de la solution. Math. Mod. Numer. Anal. 20 (1986), 141-179. MR 87k:60153
25. D. Talay, L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 (1990), 483-509. MR 92e:60124
26. D. Talay, Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Processes. Related Fields 8 (2002), 163-198. MR 2003e:60129