Basic properties of nonlinear stochastic Schrödinger equations driven by Brownian motions

Annals of Applied Probability

Volumn 18, Issue 2, 2008, Pages 591-619

  Mora, Carlos M ^a   Rebolledo, Rolando ^b  

^a UNIVERSITY OF CONCEPCIÓN   (Chile)

^b PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE   (Chile)

Author keywords

Existence and uniqueness of solutions; Nonlinear stochastic Schr dinger equations; Quantum mechanics; Regular invariant measures; Stochastic evolution equations

Indexed keywords

EID: 52949083967     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051607000000311     Document Type: Article

References (62)

1. ARNOLD, L. (1974). Stochastic Differential Equations: Theory and Applications. Wiley, New York. MR0443083
2. ACCARDI, L., FAGNOLA, F. and HACHICHA, S. (2006). Generic q-Markov semigroups and speed of convergence of q-algorithms. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 567-594. MR2282720
3. ALICKI, R. and FANNES, M. (1987). Dilations of quantum dynamical semigroups with classical Brownian motion. Comm. Math. Phys. 108 353-361. MR0874898
4. BARCHIELLI, A. and BELAVKIN, V. P. (199.1). Measurements continuous in time and a posteriori states in quantum mechanics. J. Phys. A Math. Gen. 24 1495-1514. MR1121822
5. BARCHIELLI, A. and HOLEVO, A. S. (1995). Constructing quantum measurement processes via classical stochastic calculus. Stochastic Process. Appl. 58 293-317. MR1348380
6. BARCHIELLI, A., PAGANONI, A. M. and ZUCCA, F. (1998). On stochastic differential equa-tions and semigroups of probability operators in quantum probability. Stochastic Process. Appl. 73 69-86. MR1603834
7. BARCHIELLI, A. and PAGANONI, A. M. (2003). On the asymptotic behaviour of some stochastic differential equations for quantum states. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 223-243. MR1991493
8. BELAVKIN, V. P. (1989). Nondemolition measurements, nonlinear filtering and dynamic programming of quantum stochastic processes. Lecture Notes in Control and Inform. Sci. 121 245-265. Springer, Berlin. MR1231124
9. BELAVKIN, V. P. and KOLOKOLTSOV, V. N. (1991). Quasiclassical asymptotics of quantum stochastic equations. Teoret. Mat. Fiz. 89 163-177. [Translation in Theoret. Math. Phys. 89 (1992) 1127-1138.] MR1151379
10. BELAVKIN, V. P. (1992). Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42 171-201. MR1183841.
11. BELAVKIN, V. P. (1993, 1994). Quantum diffusion, measurement and filtering. I, II. Theor. Probab. Appl. 38, 39 573-585, 363-378. MR1317995, MR1347181
12. BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. MR0233396
13. CARMICHAEL, H. (1993). An Open Systems Approach to Quantum Optics. Springer, Berlin.
14. CHEBOTAREV, A. M. and FAGNOLA, F. (1998). Sufficient conditions for conservativity of minimal quantum dynamical semigroups. J. Funct. Anal. 153 382-404. MR1614586
15. CHEBOTAREV, A. M. (2000). Lectures on Quantum Probability. Sociedad Matemática Mexicana, Mexico. MR1925129
16. COHEN-TANNOUDJI, C., DIU, B. and LALOË, F. (1977). Quantum Mechanics 1. Hermann, Paris.
17. DALIBARD, J. and CASTIN, Y. (1992). Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett. 68 580-583.
18. DELLACHERIE, C. and MEYER, P.-A. (1978). Probabilities and Potential. North-Holland, Amsterdam. MR0521810
19. DIÓSI, L. (1988). Continuous quantum measurement and Itô formalism. Phys. Lett. A 129 419-423. MR095251.9
20. DUDLEY, R. M. (1989). Real Analysis and Probability. Wadsworth, Pacific Grove, CA. MR0982264
21. ENGLERT, B.-G. and MORIGI, G. (2002). Five lectures on dissipative master equations. In Coherent Evolution in Noisy Environments. Lecture Notes in Phys. 611 55-106. Springer, Berlin. MR2012175
22. ETHIER, S. N. and KURTZ, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York. MR0838085
23. FAGNOLA, F. (1999). Quantum Markov semigroups and quantum flows. Proyecciones 18 1-144, MR1814506
24. FAGNOLA, F. and REBOLLEDO, R. (2001). On the existence of stationary states for quantum dynamical semigroups. J. Math. Phys. 42 1296-1308. MR1814690
25. FAGNOLA, F. and REBOLLEDO, R. (2002). Subharmonic projections for a quantum Markov semigroup. J. Math. Phys. 43 1074-1082. MR1878987
26. FAGNOLA, F. and REBOLLEDO, R. (2003). Transience and recurrence of quantum Markov semigroups. Probab. Theory Related Fields 126 289-306. MR1990058
27. FAGNOLA, F. and QUEZADA, R. (2005). Two-photon absorption and emission process. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 573-591. MR2184084
28. GARDINER, C. W. and ZOLLER, P. (2000). Quantum Noise. Springer, Berlin. MR1736115
29. GATAREK, D. and GISIN, N. (1991). Continuous quantum jumps and infinite-dimensional stochastic equations. J. Math. Phys. 32 2152-2157. MR1123608
30. GHOSE, S., ALSING, P., SANDERS, B. C. and DEUTSCH, I. (2005). Entanglement and the quantum-to-classical transition. Phys. Rev. A 72 014102. MR2168054
31. G.I.SIN, N. and PERCIVAL, I. C. (1992). The quantum-state diffusion model applied to open systems. J. Phys. A Math. Gen. 25 5677-5691. MR1192024
32. GOUGH, J. and SOBOLEV, A. (2004). Continuous measurement of canonical observables and limit stochastic Schrodinger equations. Phys. Rev. A 69 032107. MR2062685
33. HUDSON, R. L. and PARTHASARATHY, K. R. (1984). Quantum Itô's formula and stochastic evolutions. Comm. Math. Phys. 93 301-323. MR0745686
34. HOLEVO, A. S. (1996). On dissipative stochastic equations in a Hubert space. Probab. Theory Related Fields 104 483-500. MR1384042
35. HOLEVO, A. S. (1996). Stochastic differential equations in Hilbert space and quantum Markovian evolutions. In Probability Theory and Mathematical Statistics (Tokyo, 1995) 122-131. World Science, River Edge, NJ. MR1467932
36. IKEDA, N. and WATANABE, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam. MR1011252
37. 2+. Phys. Rev. Lett. 7 229-231.
38. KHAS'MINSKII, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff and Nordhoff, Alphen aan den Rijn. MR0600653
39. KOLOKOLTSOV, V. N. (1998). Long time behavior of continuously observed and controlled quantum systems (a study of the Belavkin quantum filtering equation). Quantum Probab. Comm. 10 229-243. MR1689486
40. KOLOKOLTSOV, V. N. (1998). Localization and analytic properties of the solutions of the sim-plest quantum filtering equation. Rev. Math. Phys. 10 801-828. MR1643961
41. KOLOKOLTSOV, V. N. (2000). Semiclassical Analysis for Diffusions and Stochastic Processes. Springer, Berlin. MR1755149
42. MATTINGLY, J. C., STUART, A. M. and HIGHAM, D. J. (2002). Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101 185-232. MR1931266
43. MEYER, P. A. (1993). Quantum Probability for P robabilists. Springer, Berlin. MR1222649
44. MIKULEVICIUS, R. and ROZOVSKII, B. L. (1999). Martingale problems for stochastic PDE's. In Stochastic Partial Differential Equations: Six Perspectives 243-325. Amer. Math. Soc., Providence, RI. MR1661767
45. MORA, C. M. (2004). Numerical simulation of stochastic evolution equations associated to quantum Markov semigroups. Math. Comp. 73 1393-1415. MR2047093
46. MORA, C. M. (2005). Numerical solution of conservative finite-dimensional stochastic Schrodinger equations. Ann. Appl. Probab. 15 2.144-2.171. MR2152256
47. MORA, C. M. and REBOLLEDO, R. (2007). Regularity of solutions to linear stochastic Schrodinger equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 237-259. MR2337521
48. ONDREJAT, M. (2004). Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. 426 63. MR2067962
49. PARTHASARATHY, K. R. (1992). An Introduction to Quantum Stochastic Calculus. Birkhäuser, Basel. MR1164866
50. PROTTER, P. (2004). Stochastic Integration and Differential Equations, 2nd ed. Springer, Berlin. MR2020294
51. PERCIVAL, I. C. (1998). Quantum State Diffusion. Cambridge Univ. Press. MR1666826
52. REBOLLEDO, R. (2005). A view on decoherence via master equations. Open Syst. Inf. Dyn. 12 37-54. MR2135004
53. REVUZ, D. and YORK, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin. MR1725357
54. ROA, L. (1994). Phase squeezing in a three-level atom micromaser. Phys. Rev. A 50 R1995- R1998.
55. RUDIN, W. (1973). Functional Analysis. McGraw-Hill, New York. MR0365062
56. SAKURAI, J. J. (1994). Modern Quantum Mechanics, rev. ed. Addison-Wesley, Reading, MA.
57. SCOTT, A. J. and MILBURN, G. J. (2001). Quantum nonlinear dynamics of continuously measured systems. Phys. Rev. A 63 42101. MR1812792
58. SCHACK, R., BRUN, T. A. and PERCIVAL, I. C. (1995). Quantum state diffusion, localization and computation. J. Phys. A 28 5401-5413. MR1364146
59. SPEHNER, D. and ORSZAG, M. (2002). Temperature enhanced squeezing in cavity QED. J. Opt. B 4 326-335.
60. UMANITA, V. (2006). Classification and decomposition of quantum Markov semigroup. Probab. Theory Related Fields 134 603-623. MR2214906
61. WISEMAN, H. M. (1996). Quantum trajectories and quantum measurement theory. Quantum Semiclass. Optics 8 205-222. MR1374517
62. ZEIDLER, E. (1990). Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators. Springer, New York. MR1033497