Abstract
We consider the weak coupling limit for a quantum system consisting of a small subsystem and reservoirs. It is known rigorously since [10] that the Heisenberg evolution restricted to the small system converges in an appropriate sense to a Markovian semigroup. In the nineties, Accardi, Frigerio and Lu [1] initiated an investigation of the convergence of the unreduced unitary evolution to a singular unitary evolution generated by a Langevin-Schrödinger equation. We present a version of this convergence which is both simpler and stronger than the formulations which we know. Our main result says that in an appropriately understood weak coupling limit the interaction of the small system with environment can be expressed in terms of the so-called quantum white noise.
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Accardi L., Frigerio A. and Lu Y.G. (1990). Weak coupling limit as a quantum functional central limit theorem. Commun. Math. Phys. 131: 537–570
Accardi L., Gough J. and Lu Y.G. (1995). On the stochastic limit of quantum field theory. Rep. Math. Phys. 36: 155–187
Accardi L., Lu Y.G. and Volovich I.V. (2002). Quantum Theory and Its Stochastic Limit. Springer, New York
Accardi L. and Lu Y.G. (1991). The low-density limit of quantum systems. J. Phys. A: Math. Gen. 24: 3483–3512
Accardi L., Pechen A.N. and Volovich I.V. (2002). Quantum stochastic equation for the low density limit. J. Phys. A:Math. Gen. 35: 4889–4902
Attal, S.: Quantum noises. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Quantum Open Systems II: The Markovian approach, Lecture Notes in Mathematics 1881, Berlin Heidelberg-New York: Springer, 2006
Barchielli, A.: Continual measurements in quantum mechanics. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Open Quantum Systems III. Recent developments, Lecture Notes in Mathematics 1882, Berlin Heidelberg-New York: Springer, 2006
Bouten L., Maassen H. and Kümmerer B. (2003). Constructing the davies process of resonance fluorescence with quantum stochastic calculus. Optics and Spectroscopy 94: 911–919
Chebotarev A.M. (1996). Symmetric form of the hudson-parthasarathy stochastic equation. Mat. Zametki [Math. Notes] 60(5): 726–750
Davies E.B. (1974). Markovian master equations. Commun. Math. Phys. 39: 91–110
Dereziński, J.: Introduction to representations of canonical commutation and anticommutation relations. In: Derezinski, J., Siedentop, H. (eds.) Large Coulomb Systems, Lecture Notes in Physics 695, Berlin Heidelberg-New York: Springer, 2006
Dereziński J. and De Roeck W. (2007). Extended weak coupling limit for friedrichs hamiltonians. J. Math. Phys. 48: 012103
Dereziński, J., De Roeck, W., Maes, C.: Fluctuations of quantum currents and unravelings of master equations. http://arxiv.org/list/cond-mat/0703594, 2007
Dümcke R. (1983). Convergence of multitime correlation functions in the weak and singular coupling limits. J. Math. Phys. 24(2): 311–315
Dümcke R. (1985). The low density limit for an n-level system interacting with a free bose or fermi gas. Commun. Math. Phys. 97: 331–359
Frigerio A. and Gorini V. (1984). Diffusion processes, quantum dynamical semigroups and the classical kms condition. J. Math. Phys. 25(4): 1050–1065
Gardiner C.W. and Collet M.J. (1985). Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation. Phys. Rev. A 31: 3761–3774
Gardiner C.W. and Zoller P. (2004). Quantum Noise: A Handbook of Markovian and Non-Markovian Quantum Stochastic Methods With Applications to Quantum Optics. Springer Verlag, Berlin Heidelberg-New York
Gough J. (1999). Asymptotic stochastic transformations for non-linear quantum dynamical systems. Rep. Math. Phys. 44(3): 313–338
Gough J. (2005). Quantum flows as markovian limit of emission, absorption and scattering interactions. Commun. Math. Phys. 254: 489–512
Gregoratti M. (2001). The hamiltonian operator associated with some quantum stochastic evolutions. Commun. Math. Phys. 222: 181–200
Van Hove L. (1955). Quantum-mechanical perturbations giving rise to a statistical transport equation. Physica 21: 517–540
Hudson R.L. and Parathasaraty K.R. (1984). Quantum ito’s formula and stochastic evolutions. Commun. Math. Phys. 93(3): 301–323
Lindblad G. (1975). Completely positive maps and entropy inequalities. Commun. Math. Phys. 40: 147–151
Maassen, H.: Quantum markov processes on fock spaces described by integral kernels. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications II, Volume 1136 of Lecture Notes in Mathematics. Berlin: Springer, 1984, pp. 361–374
Meyer, P.-A.: Quantum probability for probabilists. Volume 1538 of Lecture Notes in Mathematics. Berlin: Springer, 1995, pp. 384–411
De Roeck W. and Maes C. (2006). Fluctuations of the dissipated heat in a quantum stochastic model. Rev. Math. Phys. 18: 619–653
Rudnicki S., Alicki R. and Sadowski S. (1992). The low-density limit in terms of collective squeezed vectors. J. Math. Phys. 33(7): 2607–2617
Spohn H. (1980). Kinetic equations from hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52: 569–616
von Waldenfels, W.: Ito solution of the linear quantum stochastic differential equation describing light emission and absorption. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications IV, Volume 1055 of Lecture Notes in Mathematics, Berlin: Springer, 1986, pp. 384–411
Waldenfels W. (2005). Symmetric differentiation and hamiltonian of a quantum stochastic process. Inf. Dim. Anal. & Quantum Prob. 8(1): 73–116
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Communicated by H.-T. Yau
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Dereziński, J., De Roeck, W. Extended Weak Coupling Limit for Pauli-Fierz Operators. Commun. Math. Phys. 279, 1–30 (2008). https://doi.org/10.1007/s00220-008-0419-3
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DOI: https://doi.org/10.1007/s00220-008-0419-3