Quantum linear Boltzmann equation

Physics Reports

Volumn 478, Issue 4-6, 2009, Pages 71-120

  Vacchini, Bassano ^a,b   Hornberger, Klaus ^c  

^a UNIVERSITY OF MILAN   (Italy)

^b INFN SEZIONE DI MILANO   (Italy)

^c UNIVERSITY OF MUNICH   (Germany)

Author keywords

Atomic index of refraction; Collisional decoherence; Dynamic structure factor; Quantum Boltzmann equation; Quantum Brownian motion

Indexed keywords

EID: 67650881946     PISSN: 03701573     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physrep.2009.06.001     Document Type: Review

References (198)

1. L. Boltzmann, Vorlesungen über Gastheorie, Barth, Leipzig, 1898
2. Cercignani C. Theory and Application of the Boltzmann Equation (1975), Scottisch Academic Press, Edinburgh
3. Spohn H. Large Scale Dynamics of Interacting Particles (1991), Springer, Berlin
4. Harris S. An Introduction to the Theory of the Boltzmann Equation (2004), Courier Dover Publications
5. Balian R. From Microphysics to Macrophysics: Methods and Applications of Statistical Physics. Vol. II (2007), Springer, Berlin
6. Davison B., and Sykes B. Neutron Transport Theory (1957), Clarendon Press, Oxford
7. Williams M.M.R. The Slowing Down and Thermalization of Neutrons (1966), North-Holland, Amsterdam
8. Nordheim L.W. On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. Roy. Soc. 119 (1928) 689-698
9. Uehling E.A., and Uhlenbeck G.E. Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev. 43 (1933) 552-561
10. Ross J., and Kirkwood J.G. The statistical-mechanical theory of transport processes. VIII. Quantum theory of transport in gases. J. Chem. Phys. 22 (1954) 1094-1103
11. Gardiner C.W., and Zoller P. Quantum kinetic theory: A quantum kinetic master equation for condensation of a weakly interacting Bose gas without a trapping potential. Phys. Rev. A 55 (1997) 2902-2921
12. Jaksch D., Gardiner C.W., and Zoller P. Quantum kinetic theory. II. Simulation of the quantum Boltzmann master equation. Phys. Rev. A 56 (1997) 575-586
13. Gardiner C.W., Zoller P., Ballagh R.J., and Davis M.J. Kinetics of Bose-Einstein condensation in a trap. Phys. Rev. Lett. 79 (1997) 1793-1796
14. Gardiner C.W., and Zoller P. Quantum kinetic theory. III. Quantum kinetic master equation for strongly condensed trapped systems. Phys. Rev. A 58 (1998) 536-556
15. Jaksch D., Gardiner C.W., Gheri K.M., and Zoller P. Quantum kinetic theory. IV. Intensity and amplitude fluctuations of a Bose-Einstein condensate at finite temperature including trap loss. Phys. Rev. A 58 (1998) 1450-1464
16. Gardiner C.W., Lee M.D., Ballagh R.J., Davis M.J., and Zoller P. Quantum kinetic theory of condensate growth: Comparison of experiment and theory. Phys. Rev. Lett. 81 (1998) 5266-5269
17. Gardiner C.W., and Zoller P. Quantum kinetic theory. V. Quantum kinetic master equation for mutual interaction of condensate and noncondensate. Phys. Rev. A 61 (2000) 033601
18. Spohn H. Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Modern Phys. 52 (1980) 569-615
19. Snider R.F. Relaxation and transport of molecular systems in the gas phase. Int. Rev. Phys. Chem. 17 (1998) 185-225
20. Erds L., Salmhofer M., and Yau H. On the quantum Boltzmann equation. J. Stat. Phys. 116 (2004) 367-380
21. Benedetto D., Castella F., Esposito R., and Pulvirenti M. A short review on the derivation of the nonlinear quantum Boltzmann equations. Commun. Math. Sci. (2007) 55-71
22. Lukkarinen J., and Spohn H. Not to normal order-notes on the kinetic limit for weakly interacting quantum fluids. J. Stat. Phys. 134 (2009) 1133-1172
23. H. Spohn, Kinetic equations for quantum many-particle systems, in: Modern Encyclopedia of Mathematical Physics, Springer (in press) preprint arXiv:0706.0807
24. Castella F. From the von-Neumann equation to the quantum Boltzmann equation in a deterministic framework. J. Stat. Phys. 104 (2001) 387-447
25. Castella F. From the von Neumann equation to the quantum Boltzmann equation II: Identifying the Born series. J. Stat. Phys. 106 (2002) 1197-1220
26. Eng D., and Erds L. The linear Boltzmann equation as the low density limit of a random Schrödinger equation. Rev. Math. Phys. 17 (2005) 669-743
27. Alicki R., and Lendi K. Quantum Dynamical Semigroups and Applications. 2nd edition. Lecture Notes in Physics vol. 717 (2007), Springer, Berlin
28. Holevo A.S. Statistical Structure of Quantum Theory. Lecture Notes in Physics vol. 67 (2001), Springer, Berlin
29. Breuer H.-P., and Petruccione F. The Theory of Open Quantum Systems (2007), Oxford University Press, Oxford
30. Gorini V., Kossakowski A., and Sudarshan E.C.G. Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17 (1976) 821-825
31. Lindblad G. On the generators of quantum dynamical semigroups. Comm. Math. Phys. 48 (1976) 119-130
32. Hornberger K., and Vacchini B. Monitoring derivation of the quantum linear Boltzmann equation. Phys. Rev. A 77 (2008) 022112
33. Holevo A.S. Covariant quantum dynamical semigroups: Unbounded generators. In: Böhm A., Doebner H.D., and Kielanowski P. (Eds). Irreversibility and Causality. Lecture Notes in Physics vol. 504 (1998), Springer, Berlin 67-81
34. Arndt M., Hornberger K., and Zeilinger A. Probing the limits of the quantum world. Phys. World 18 (2005) 35-40
35. A. Cronin, J. Schmiedmayer, D. Pritchard, Optics and interferometry with atoms and molecules, Rev. Modern. Phys. (2009) (in press) eprint arXiv: 0712.3703
36. Gerlich S., Hackermüller L., Hornberger K., Stibor A., Ulbricht H., Gring M., Goldfarb F., Savas T., Müri M., Mayor M., and Arndt M. A Kapitza-Dirac-Talbot-Lau interferometer for highly polarizable molecules. Nature Phys. 3 (2007) 711-715
37. Schmiedmayer J., Chapman M.S., Ekstrom C.R., Hammond T.D., Wehinger S., and Pritchard D.E. Index of refraction of various gases for sodium matter waves. Phys. Rev. Lett. 74 (1995) 1043-1047
38. Jacquey M., Buchner M., Trenec G., and Vigue J. First measurements of the index of refraction of gases for lithium atomic waves. Phys. Rev. Lett. 98 (2007) 240405
39. Hornberger K., Uttenthaler S., Brezger B., Hackermüller L., Arndt M., and Zeilinger A. Collisional decoherence observed in matter wave interferometry. Phys. Rev. Lett. 90 (2003) 160401
40. Joos E., and Zeh H.D. The emergence of classical properties through interaction with the environment. Z. Phys. B: Condens. Matter 59 (1985) 223-243
41. Hornberger K. Master equation for a quantum particle in a gas. Phys. Rev. Lett. 97 (2006) 060601
42. Vacchini B. Test particle in a quantum gas. Phys. Rev. E 63 (2001) 066115
43. Gallis M.R., and Fleming G.N. Environmental and spontaneous localization. Phys. Rev. A 42 (1990) 38-48
44. Diósi L. Quantum master equation of a particle in a gas environment. Europhys. Lett. 30 (1995) 63-68
45. Lanz L., and Vacchini B. Incoherent dynamics in neutron-matter interaction. Phys. Rev. A 56 (1997) 4826-4838
46. Vacchini B. Completely positive quantum dissipation. Phys. Rev. Lett. 84 (2000) 1374-1377
47. Vacchini B. Translation-covariant Markovian master equation for a test particle in a quantum fluid. J. Math. Phys. 42 (2001) 4291-4312
48. Alicki R. Search for a border between classical and quantum worlds. Phys. Rev. A 65 (2002) 034104
49. Dodd P.J., and Halliwell J.J. Decoherence and records for the case of a scattering environment. Phys. Rev. D 67 (2003) 105018
50. Clark J. Decoherence rates for Galilean covariant dynamics. J. Math. Phys. 49 (2008) 052103
51. Clark J. An infinite-temperature limit for a quantum scattering process. Rep. Math. Phys. 63 (2009) 131-152
52. Clark J. The reduced effect of a single scattering with a low-mass particle via a point interaction. J. Funct. Anal. 256 (2009) 2894-2916
53. J. Clark, W. De Roeck, C. Maes, Diffusive behavior from a quantum master equation, reprint arXiv:0812.2858v1
54. Hornberger K., and Sipe J.E. Collisional decoherence reexamined. Phys. Rev. A 68 (2003) 012105
55. Hackermüller L., Hornberger K., Brezger B., Zeilinger A., and Arndt M. Decoherence in a Talbot Lau interferometer: The influence of molecular scattering. Appl. Phys. B 77 (2003) 781-787
56. Petruccione F., and Vacchini B. Quantum description of Einstein's Brownian motion. Phys. Rev. E 71 (2005) 046134
57. Lanz L., and Vacchini B. Dynamical semigroup description of coherent and incoherent particle-matter interaction. Int. J. Theor. Phys. 36 (1997) 67-88
58. Altenmüller T.P., Müller R., and Schenzle A. Atom-interferomeric study of Bose-Einstein condensation. Phys. Rev. A 56 (1997) 2959-2971
59. Teta A. On a rigorous proof of the Joos-Zeh formula for decoherence in a two-body problem. Multiscale Methods in Quantum Mechanics. Trends Math. (2004), Birkhäuser, Boston 197-205
60. Dürr D., Figari R., and Teta A. Decoherence in a two-particle model. J. Math. Phys. 45 (2004) 1291-1309
61. Adami R., Figari R., Finco D., and Teta A. On the asymptotic behaviour of a quantum two-body system in the small mass ratio limit. J. Phys. A 37 (2004) 7567-7580
62. Cacciapuoti C., Carlone R., and Figari R. Decoherence induced by scattering: A three-dimensional model. J. Phys. A 38 (2005) 4933-4946
63. Adami R., Figari R., Finco D., and Teta A. On the asymptotic dynamics of a quantum system composed by heavy and light particles. J. Phys. A 268 (2006) 819-852
64. Stenholm S. Occurrences, observations and measurements in quantum mechanics. Phys. Scripta 47 (1993) 724-731
65. Hornberger K. Monitoring approach to open quantum dynamics using scattering theory. EPL 77 (2007) 50007
66. Hornberger K. Open quantum dynamics via environmental monitoring. J. Phys.: Conf. Ser. 67 (2007) 012002
67. Jacobs K., and Steck D.A. A straightforward introduction to continuous quantum measurement. Contemp. Phys. 47 (2006) 279-303
68. Barchielli A., and Gregoratti M. Quantum Trajectories and Measurements in Continuous Time. Lecture Notes in Physics vol. 782 (2009), Springer, Berlin
69. Dümcke R. The low density limit for an N-level system interacting with a free Bose or Fermi gas. Comm. Math. Phys. 97 (1985) 331-359
70. Raffelt G., Sigl G., and Stodolsky L. Non-Abelian Boltzmann equation for mixing and decoherence. Phys. Rev. Lett. 70 (1993) 2363-2366
71. Tsonchev S., and Pechukas P. Binary collision model for quantum Brownian motion. Phys. Rev. E 61 (2000) 6171-6182
72. Kleckner M., and Ron A. Decoherence of a pointer by a gas reservoir. Phys. Rev. A 63 (2001) 022110
73. Hellmich M. Alicki's model of scattering-induced decoherence derived from Hamiltonian dynamics. J. Phys. A 37 (2004) 8711-8719
74. Pechen A.N. Quantum stochastic equation for a test particle interacting with a dilute Bose gas. J. Math. Phys. 45 (2004) 400-417
75. Pechen A.N. White noise approach to the low density limit of a quantum particle in a gas. In: Schuermann M., and Franz U. (Eds). QP-PQ: Quantum Probability and White Noise Analysis vol. 18 (2005), World Scientific, Singapore 428
76. Barnett S.M., and Cresser J.D. Quantum theory of friction. Phys. Rev. A 72 (2005) 022107
77. Adler S.L. Normalization of collisional decoherence: Squaring the delta function, and an independent cross-check. J. Phys. A 39 (2006) 14067-14074
78. Halliwell J.J. Two derivations of the master equation of quantum Brownian motion. J. Phys. A: Math. Theor. 40 (2007) 3067-3080
79. Dominguez-Clarimon A. A particle across a medium: How decoherence relates to the index of refraction. Ann. Phys. 322 (2007) 2085-2103
80. F. Mintert, E.J. Heller, Simulation of open quantum systems, arXiv:0803.3883
81. Benedetto D., Castella F., Esposito R., and Pulvirenti M. Some considerations on the derivation of the nonlinear quantum Boltzmann equation. J. Stat. Phys. 116 (2004) 381-410
82. Benedetto D., Castella F., Esposito R., and Pulvirenti M. Some considerations on the derivation of the nonlinear quantum Boltzmann equation II: The low density regime. J. Stat. Phys. 124 (2006) 951-996
83. Benedetto D., Castella F., Esposito R., and Pulvirenti M. From the N-body Schrödinger equation to the quantum Boltzmann equation: A term-by-term convergence result in the weak coupling regime. Comm. Math. Phys. 277 (2008) 1-44
84. Steinigeweg R., Breuer H.-P., and Gemmer J. Transition from diffusive to ballistic dynamics for a class of finite quantum models. Phys. Rev. Lett. 99 (2007) 150601
85. Kadiroglu M., and Gemmer J. Boltzmann-equation approach to transport in finite modular quantum systems. Phys. Rev. B 76 (2007) 024306
86. Kossakowski A. On quantum statistical mechanics of non-Hamiltonian systems. Rep. Math. Phys. 3 (1972) 247-274
87. Manita A.D. Properties of translation-invariant quantum-dynamical semigroups. Theoret. Math. Phys. 89 (1991) 1271-1281
88. Botvich D.D., Malyshev V.A., and Manita A.D. Translation invariant quantum master equation. Helv. Phys. Acta 64 (1991) 1072-1092
89. Holevo A.S. A note on covariant dynamical semigroups. Rep. Math. Phys. 32 (1993) 211-216
90. Holevo A.S. On conservativity of covariant dynamical semigroups. Rep. Math. Phys. 33 (1993) 95-110
91. Holevo A.S. On translation-covariant quantum Markov equations. Izv. Math. 59 (1995) 427-443
92. Holevo A.S. Covariant quantum Markovian evolutions. J. Math. Phys. 37 (1996) 1812-1832
93. B. Vacchini, Covariant mappings for the description of measurement, dissipation and decoherence in quantum mechanics, in: E. Brüning, F. Petruccione (Eds), Theoretical Foundations of Quantum Information Processing and Communication, in: Lecture Notes in Physics, vol. 787, Springer, Berlin, 2010. eprint arXiv:quant-ph/0707.0603
94. Huang K. Statistical Mechanics (1987), John Wiley & Sons, New York
95. Spohn H., and Lebowitz J.L. Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38 (1978) 109-142
96. Lindblad G. Non-Equilibrium Entropy and Irreversibility (1983), Reidel Publishing Company, Dordrecht
97. Alicki R., and Fannes M. Quantum Dynamical Systems (2001), Oxford University Press, Oxford
98. Wehrl A. General properties of entropy. Rev. Modern Phys. 50 (1978) 221-260
99. Ingarden R.S., Kossakowski A., and Ohya M. Information Dynamics and Open Systems (1997), Kluwer Academic Publishers, Dordrecht
100. Vacchini B., and Hornberger K. Relaxation dynamics of a quantum Brownian particle in an ideal gas. Eur. Phys. J. ST 151 (2007) 59-72
101. Breuer H.-P., and Vacchini B. Three-dimensional Monte Carlo simulations of the quantum linear Boltzmann equation. Phys. Rev. E 76 (2007) 036706
102. Diósi L. Stochastic pure state representation for open quantum systems. Phys. Lett. 114A (1986) 451-454
103. Gardiner C.W., Parkins A.S., and Zoller P. Wave-function quantum stochastic differential equations and quantum-jump simulation methods. Phys. Rev. A 46 (1992) 4363-4381
104. Carmichael H. An Open Systems Approach to Quantum Optics (1993), Springer, Berlin
105. Molmer K., Castin Y., and Dalibard J. Monte Carlo wave-function method in quantum optics. J. Opt. Soc. Am. B 10 (1993) 524-538
106. Vacchini B. Quantum and classical features in the explanation of collisional decoherence. J. Mod. Opt. 51 (2004) 1025-1029
107. Gillespie D.T. Markov Processes (1992), Academic Press, Boston
108. Grabert H., Schramm P., and Ingold G. Quantum Brownian motion: The functional integral approach. Phys. Rep. 168 (1988) 115-207
109. Ingold G. Path integrals and their application to dissipative quantum systems. In: Buchleitner A., and Hornberger K. (Eds). Coherent Evolution in Noisy Environments. Lecture Notes in Physics vol. 611 (2002), Springer, Berlin 1-53
110. Hänggi P., and Ingold G. Fundamental aspects of quantum Brownian motion. Chaos 15 (2005) 026105
111. Weiss U. Quantum Dissipative Systems. 3rd edition (2008), World Scientific, Singapore
112. Rayleigh J.W.S. Dynamical problems in illustration of the theory of gases. Phil. Mag. 32 (1891) 424-445
113. Green M.S. Brownian motion in a gas of noninteracting molecules. J. Chem. Phys. 19 (1951) 1036-1046
114. Chang C.S.W., and Uhlenbeck G.E. The Kinetic Theory of Gases. In: Boer J.D. (Ed). Studies in statistical mechanics vol. 5 (1970), North-Holland, Amsterdam Chapter V.
115. Ferrari L. An improved differential form of the Boltzmann collision operator for a Rayleigh gas (or Brownian particles). Physica A 142 (1987) 441-466
116. Ambegaokar V. Quantum Brownian motion and its classical limit. Ber. Bunsenges. Phys. Chem. 95 (1991) 400-404
117. Pechukas P. Quantum Brownian Motion. In: Gans W., Blumen A., and Amann A. (Eds). Large Scale Molecular Systems. NATO ASI Series vol. 258 (1991), Plenum Press, New York 123
118. Bassi A., Ippoliti E., and Vacchini B. On the energy increase in space-collapse models. J. Phys. A 38 (2005) 8017-8038
119. Caldeira A.O., and Leggett A.J. Path integral approach to quantum Brownian motion. Physica A 121 (1983) 587-616
120. Caldeira A.O., and Leggett A.J. Quantum tunnelling in a dissipative system. Ann. Phys. NY 149 (1983) 374-456
121. Leggett A.J., Chakravarty S., Dorsey A.T., Fisher M.P.A., Garg A., and Zwerger W. Dynamics of the dissipative two-state system. Rev. Modern Phys. 59 (1987) 1-85
122. Barchielli A. Continual measurements for quantum open systems. Nuovo Cimento B 74 (1983) 113-138
123. Sandulescu A., and Scutaru H. Open quantum systems and the damping of collective modes in deep inelastic collisions. Ann. Physics 173 (1987) 277-317
124. Isar A. Uncertainty, entropy and decoherence of the damped harmonic oscillator in the Lindblad theory of open quantum systems. Fortschr. Phys. 47 (1999) 855-879
125. Vacchini B. Quantum optical versus quantum Brownian motion master equation in terms of covariance and equilibrium properties. J. Math. Phys. 43 (2002) 5446-5458
126. Isar A., Sandulescu A., Scutaru H., Stefanescu E., and Scheid W. Open quantum systems. Int. J. Mod. Phys. E 3 (1994) 635-714
127. Jacobs K. A Monte Carlo method for modeling thermal damping: Beyond the Brownian motion master equation. EPL 85 (2009) 40002
128. Diósi L. On high temperature Markovian equation for quantum Brownian motion. Europhys. Lett. 22 (1993) 1-3
129. Diósi L. Caldeira-Leggett master equation and medium temperatures. Physica A 199 (1993) 517-526
130. Vacchini B. Non-abelian linear Boltzmann equation and quantum correction to Kramers and Smoluchowski equation. Phys. Rev. E 66 (2002) 027107
131. Sears V.F. Neutron Optics (1989), Oxford University Press, Oxford
132. Werner S.A., and Rauch H. Neutron Interferometry (2000), Oxford University Press, Oxford
133. Adams C.S., Siegel M., and Mlynek J. Atom optics. Phys. Rep. 240 (1994) 143-210
134. Miffre A., Jacquey M., Büchner M., Trénec G., and Vigué J. Atom interferometry. Phys. Scripta 74 (2006) C15-C23
135. M. Arndt, K. Hornberger, Quantum interferometry with complex molecules, in: Quantum Coherence in Solid State Systems, in: Proceedings of the International School of Physics Enrico Fermi, Course CLXXI, Società Italiana di Fisica, 2009 (in press), arXiv:0903.1614
136. Vigue J. Index of refraction of dilute matter in atomic interferometry. Phys. Rev. A 52 (1995) 3973-3975
137. Rauch H., Treimer W., and Bonse U. Test of a single crystal neutron interferometer. Phys. Lett. A 47 (1974) 369-371
138. Taylor J.R. Scattering Theory (1972), John Wiley & Sons, New York
139. C. Champenois, Interférométrie atomique avec l'atome de lithium: analyse théorique et construction d'un interféromètre, applications., Ph.D. Thesis, Université Paul Sabatier, Toulouse. Available at: http://tel.archives-ouvertes.fr/tel-00003602, 1999
140. Champenois C., Jacquey M., Lepoutre S., Buchner M., Trenec G., and Vigue J. Index of refraction of gases for matter waves: Effect of the motion of the gas particles on the calculation of the index. Phys. Rev. A 77 (2008) 013621
141. Forrey R.C., You L., Kharchenko V., and Dalgarno A. Index of refraction of noble gases for sodium matter waves. Phys. Rev. A 54 (1996) 2180-2184
142. Kharchenko V., and Dalgarno A. Refractive index for matter waves in ultracold gases. Phys. Rev. A 63 (2001) 023615
143. Roberts T.D., Cronin A.D., Kokorowski D.A., and Pritchard D.E. Glory oscillations in the index of refraction for matter waves. Phys. Rev. Lett. 89 (2002) 200406
144. Sanders S.N., Mintert F., and Heller E.J. Coherent scattering from a free gas. Phys. Rev. A 79 (2009) 023610
145. Zurek W.H. Decoherence and the transition from quantum to classical. Phys. Today 44 (1991) 36-44
146. Joos E., Zeh H.D., Kiefer C., Giulini D., Kupsch J., and Stamatescu I.-O. Decoherence and the Appearance of a Classical World in Quantum Theory. 2nd edition (2003), Springer, Berlin
147. Zurek W.H. Decoherence, einselection, and the quantum origins of the classical. Rev. Modern Phys. 75 (2003) 715-775
148. Schlosshauer M. Decoherence and the Quantum-to-Classical Transition (2007), Springer-Verlag, Berlin
149. Hornberger K. Introduction to decoherence theory. In: Buchleitner A., Viviescas C., and Tiersch M. (Eds). Entanglement and Decoherence. Lecture Notes in Physics vol. 768 (2009), Springer, Berlin 221-276
150. Vacchini B. Master-equations for the study of decoherence. Int. J. Theor. Phys. 44 (2005) 1011-1021
151. Hornberger K., Sipe J.E., and Arndt M. Theory of decoherence in a matter mave Talbot-Lau interferometer. Phys. Rev. A 70 (2004) 053608
152. Feller W. An Introduction to Probability Theory and its Applications. Vol. II (1971), John Wiley & Sons Inc., New York
153. Lukacs L. Characteristic Functions (1966), Griffin, London
154. Vacchini B. Theory of decoherence due to scattering events and Lévy processes. Phys. Rev. Lett. 95 (2005) 230402
155. Uys H., Perreault J.D., and Cronin A.D. Matter-wave decoherence due to a gas environment in an atom interferometer. Phys. Rev. Lett. 95 (2005) 150403
156. 60 collisions. Chem. Phys. Lett. 270 (1997) 121-128
157. Maitland G.C., Rigby M., Smith E.B., and Wakeham W.A. Intermolecular Forces-Their Origin and Determination (1981), Clarendon Press, Oxford
158. Cheng C.-C., and Raymer M.G. Long-range saturation of spatial decoherence in wave-field transport in random multiple-scattering media. Phys. Rev. Lett. 82 (1999) 4807-4810
159. Pfau T., Spälter S., Kurtsiefer C., Ekstrom C., and Mlynek J. Loss of spatial coherence by a single spontaneous emission. Phys. Rev. Lett. 73 (1994) 1223-1226
160. Chapman M.S., Hammond T.D., Lenef A., Schmiedmayer J., Rubenstein R.A., Smith E., and Pritchard D.E. Photon scattering from atoms in an atom interferometer: Coherence lost and regained. Phys. Rev. Lett. 75 (1995) 3783-3787
161. Kokorowski D.A., Cronin A.D., Roberts T.D., and Pritchard D.E. From single- to multiple-photon decoherence in an atom interferometer. Phys. Rev. Lett. 86 (2001) 2191-2195
162. Hackermüller L., Hornberger K., Brezger B., Zeilinger A., and Arndt M. Decoherence of matter waves by thermal emission of radiation. Nature 427 (2004) 711-714
163. Hornberger K., Gerlich S., Ulbricht H., Hackermüller L., Nimmrichter S., Goldt I.V., Boltalina O., and Arndt M. Theory and experimental verification of Kapitza-Dirac Talbot-Lau interferometry. New J. Phys. 11 (2009) 043032
164. Facchi P. Thermal decoherence in mesoscopic interference. J. Mod. Opt. 51 (2004) 1049-1055
165. Hornberger K., Hackermüller L., and Arndt M. Influence of molecular temperature on the coherence of fullerenes in a near-field interferometer. Phys. Rev. A 71 (2005) 023601
166. Hornberger K. Thermal limitation of far-field matter-wave interference. Phys. Rev. A 73 (2006) 052102
167. Nimmrichter S., Hornberger K., Ulbricht H., and Arndt M. Absolute absorption spectroscopy based on molecule interferometry. Phys. Rev. A 78 (2008) 063607
168. Viale A., Vicari M., and Zanghi N. Analysis of the loss of coherence in interferometry with macromolecules. Phys. Rev. A 68 (2003) 063610
169. Lombardo F.C., Mazzitelli F.D., and Villar P.I. Decoherence induced by a fluctuating Aharonov-Casher phase. Phys. Rev. A 72 (2005) 042111
170. Villar P., and Lombardo F. Visibility fringe reduction due to noise-induced effects: Microscopic approach to interference experiments. Int. J. Mod. Phys. B 21 (2007) 4659-4676
171. Qureshi T., and Venugopalan A. Decoherence and matter wave interferometry. Int. J. Mod. Phys. B 22 (2008) 981-990
172. Rubenstein R., Kokorowski D.A., Dhirani A., Roberts T., Gupta S., Lehner J., Smith W., Smith E., Bernstein H., and Pritchard D. Measurement of the density matrix of a longitudinally modulated atomic beam. Phys. Rev. Lett. 83 (1999) 2285-2288
173. Rubenstein R., Dhirani A., Kokorowski D., Roberts T., Smith E., Smith W.W., Bernstein H., Lehner J., Gupta S., and Pritchard D. Search for off-diagonal density matrix elements for atoms in a supersonic beam. Phys. Rev. Lett. 82 (1999) 2018-2021
174. Anglin J., Paz J., and Zurek W. Deconstructing decoherence. Phys. Rev. A 55 (1997) 4041-4053
175. Kusnezov D., Bulgac A., and Dang G.D. Quantum Lévy processes and fractional kinetics. Phys. Rev. Lett. 82 (1999) 1136-1139
176. Lutz E. Anomalous Lévy decoherence. Phys. Lett. A 293 (2002) 123-128
177. Schomerus H., and Lutz E. Nonexponential decoherence and momentum subdiffusion in a quantum Lévy kicked rotator. Phys. Rev. Lett. 98 (2007) 260401
178. Bellomo B., Barnett S.M., and Jeffers J. Frictional quantum decoherence. J. Phys. A: Math. Theor. 40 (2007) 9437-9453
179. F.M. Ramazanoglu, Approach to thermal equilibrium in the Caldeira-Leggett model, eprint arXiv:0812:2520v1
180. Wigner E.P. Review of quantum mechanical measurement problem. In: Meystre P., and Scully M.O. (Eds). Quantum Optics, Experimental Gravity and Measurement Theory (1983), Plenum Press, New York 43
181. Einstein A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. (Leipzig) 322 (1905) 549-560
182. Gocke C., and Röpke G. Localization of coherent wave packets in plasmas due to decoherence. Contrib. Plasma Phys. 47 (2007) 291-296
183. Gocke C., and Röpke G. Master equation of the reduced statistical operator of an atom in a plasma. Theoret. Math. Phys. 154 (2008) 26-51
184. Lovesey S. Theory of Neutron Scattering from Condensed Matter. Nuclear Scattering vol.1 (1984), Clarendon Press, Oxford, UK
185. Schwabl F. Advanced Quantum Mechanics (2003), Springer, New York
186. Pitaevskii L., and Stringari S. Bose-Einstein Condensation (2003), Oxford University Press, Oxford
187. Lanz L., and Vacchini B. Subdynamics of relevant observables: A field theoretical approach. Internat. J. Modern Phys. A 17 (2002) 435-463
188. Brenig W. Statistical Theory of Heat (1989), Springer-Verlag, Berlin
189. Van Hove L. Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys. Rev. 95 (1954) 249-262
190. Alicki R., and Kryszewski S. Completely positive Bloch-Boltzmann equations. Phys. Rev. A 68 (2003) 013809
191. Kryszewski S., and Czechowska J. Positivity of Bloch-Boltzmann equations. Phys. Rev. A 74 (2006) 022719
192. Rautian S.G., and Shalagin A.M. Kinetic Problems of Non-Linear Spectroscopy (1991), North-Holland, Amsterdam
193. Vacchini B. Non-Markovian dynamics for bipartite systems. Phys. Rev. A 78 (2008) 022112
194. Budini A.A. Random Lindblad equations from complex environments. Phys. Rev. E 72 (2005) 056106
195. Budini A.A. Lindblad rate equations. Phys. Rev. A 74 (2006) 053815
196. Breuer H.-P., Gemmer J., and Michel M. Non-Markovian quantum dynamics: Correlated projection superoperators and Hilbert space averaging. Phys. Rev. E 73 (2006) 016139
197. Breuer H.-P. Non-Markovian generalization of the Lindblad theory of open quantum systems. Phys. Rev. A 75 (2007) 022103
198. Abramowitz M., and Stegun I. Handbook of Mathematical Functions (1965), Dover Publications, New York