Quantum trajectories in random environment: the statistical model for a heat bath
Confluentes Mathematici, Volume 1 (2009) no. 2, pp. 249-289.

In this paper, we derive the stochastic master equations corresponding to the statistical model of a heat bath. These stochastic differential equations are obtained as continuous time limits of discrete models of quantum repeated measurements. Physically, they describe the evolution of a small system in contact with a heat bath undergoing continuous measurement. The equations obtained in the present work are qualitatively different from the ones derived in [6], where the Gibbs model of heat bath has been studied. It is shown that the statistical model of a heat bath has a clear physical interpretation in terms of emissions and absorptions of photons. Our approach yields models of random environment and unravelings of stochastic master equations. The equations are rigorously obtained as solutions of martingale problems using the convergence of Markov generators.

Published online:
DOI: 10.1142/S1793744209000109
Ion Nechita 1; Clément Pellegrini 1

1
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Ion Nechita; Clément Pellegrini. Quantum trajectories in random environment: the statistical model for a heat bath. Confluentes Mathematici, Volume 1 (2009) no. 2, pp. 249-289. doi : 10.1142/S1793744209000109. https://cml.centre-mersenne.org/articles/10.1142/S1793744209000109/

[1] S. Attal, Open Quantum Systems II, Lecture Notes in Math. 1881 (Springer, 2006) pp. 79-147.

[2] S. Attal, Quantum Noises, book in preparation .

[3] S. Attal and A. Joye, J. Funct. Anal. 247, 253 (2007), DOI: 10.1016/j.jfa.2006.09.019 .

[4] S. Attal and Y. Pautrat, Ann. Henri Poincaré 7, 59 (2006), DOI: 10.1007/s00023-005-0242-8 .

[5] S. Attal and Y. Pautrat, Ann. Inst. H. Poincaré Probab. Statist. 41, 391 (2005), DOI: 10.1016/j.anihpb.2004.10.003 .

[6] S. Attal and C. Pellegrini, Stochastic master equations for a heat bath, preprint, 2007 .

[7] A. Barchielli, Quantum Opt. 2, 423 (1990), DOI: 10.1088/0954-8998/2/6/002 .

[8] A. Barchielli and M. Gregoratti , Quantum Trajectories and Measurements in Continuous Time The Diffusive Case , Lecture Notes in Physics 782 ( Springer , 2009 ) .

[9] A. Barchielli, Open Quantum Systems III, Lecture Notes in Math. 1882 (Springer, 2006) pp. 207-292.

[10] A. Barchielli, Classical and Quantum Systems (Goslar, 1991) (World Scientific, 1993) pp. 488-491.

[11] A. Barchielli and G. Lupieri, Quantum Probability, Banach Center Publ. 73 (Polish Acad. Sci., 2006) pp. 65-80.

[12] A. Barchielli and G. Lupieri, Quantum Inform. Comput. 4, 437 (2004).

[13] A. Barchielli and F. Zucca, Rend. Sem. Mat. Fis. Milano 66, 355 (1996), DOI: 10.1007/BF02925365 .

[14] A. Barchielli, A. M. Paganoni and F. Zucca, Stoch. Process. Appl. 73, 69 (1998), DOI: 10.1016/S0304-4149(97)00093-8 .

[15] A. Barchielli and A. S. Holevo, Stoch. Process. Appl. 58, 293 (1995), DOI: 10.1016/0304-4149(95)00011-U .

[16] V. P. Belavkin, J. Multivariate Anal. 42, 171 (1992), DOI: 10.1016/0047-259X(92)90042-E .

[17] L. Bouten, M. Guta and H. Maassen, J. Phys. A 37, 3189 (2004), DOI: 10.1088/0305-4470/37/9/010 .

[18] L. Bouten, R. van Handel and M. James, A discrete invitation to quantum filtering and feedback control, to appear in SIAM Rev , arXiv:math.PR/0606118 .

[19] L. Bouten, R. van Handel and M. James, SIAM J. Control Optim. 46, 2199 (2007), DOI: 10.1137/060651239 .

[20] F. Fagnola, Open Quantum Systems II, Lecture Notes in Math. 1881 (Springer, 2006) pp. 183-220.

[21] C. W. Gardiner and P. Zoller , A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods with Applications to Quantum Optics , 3rd edn. ( Springer-Verlag , 2004 ) .

[22] C. M. Mora and R. Rebolledo, Ann. Appl. Probab. 18, 591 (2008), DOI: 10.1214/105051607000000311 .

[23] I. Nechita and C. Pellegrini, Random repeated quantum interactions and random invariant states , arXiv:0902.2634 .

[24] K. R. Parthasarathy , An Introduction to Quantum Stochastic Calculus , Monographs in Mathematics 85 ( Birkhäuser , 1992 ) .

[25] C. Pellegrini, Ann. Probab. 36, 2332 (2008), DOI: 10.1214/08-AOP391 .

[26] C. Pellegrini, Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case , arXiv:0709.3713 .

[27] C. Pellegrini, Markov chains approximation of jump-diffusion quantum trajectories, preprint, 2008 .

[28] H.-P. Breuer and F. Petruccione , The Theory of Open Quantum Systems ( Oxford Univ. Press , 2002 ) .

[29] P. Billingsley , Convergence of Probability Measures , 2nd edn. ( Wiley-Interscience , 1999 ) .

[30] S. N. Ethier and T. G. Kurtz , Markov Processes ( John Wiley and Sons , 1986 ) .

[31] J. Jacod , Calcul Stochastique et Problèmes de Martingales , Lecture Notes in Mathematics 714 ( Springer , 1979 ) .

[32] J. Jacod and P. Protter, Seminar on Probability, XVI, Lecture Notes in Math. 920 (Springer, 1982) pp. 447-458.

[33] J. Jacod and A. N. Shiryaev , Limit Theorems for Stochastic Processes , 2nd edn. , Fundamental Principles of Mathematical Sciences 288 ( Springer-Verlag , 2003 ) .

[34] P. E. Protter , Stochastic Integration and Differential Equations , 2nd edn. , Applications of Mathematics 21 ( Springer-Verlag , 2004 ) .

[35] T. G. Kurtz and P. Protter, Ann. Probab. 19, 1035 (1991), DOI: 10.1214/aop/1176990334 .

[36] T. G. Kurtz and P. Protter, Stochastic Analysis (Academic Press, 1991) pp. 331-346.

[37] P. Brémaud , Point Processes and Queues ( Springer-Verlag , 1981 ) .

[38] T. C. Brown, Ann. Probab. 11, 726 (1983), DOI: 10.1214/aop/1176993517 .

[39] H. M. Wiseman and G. J. Milburn, Phys. Rev. A 47, 1652 (1993), DOI: 10.1103/PhysRevA.47.1652 .

[40] H. M. Wiseman, Quantum trajectories and feedback, Ph.D Thesis 1994 .

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