Nonlinear dynamics of open quantum systems


The evolution of a composite closed system using the integral wave equation with the kernel in the form of path integral is considered. It is supposed that a quantum particle is a subsystem of this system. The evolution of the reduced density matrix of the subsystem is described on the basis of the integral wave equation for a composite closed system. The equation for the density matrix for such a system is derived. This equation is nonlinear and depends on the history of the processes in the closed system. It is shown that, in general, the reduced density matrix trace does not conserve in the evolution processes progressing in open systems and the procedure of the trace normalization is necessary as the mathematical image of a real nonlocal physical process. The wave function collapse and EPR correlation are described using this approach.

Full Text

1. Introduction. The principle differences between the dynamics of an open system and the evolution of a closed one can not solely restricted by the irreversibility of the former. In general, there exist nonlinear transformations of the reduced density matrices of open systems. Depending on the specific properties of the system, these nonlinear processes can take the form of the wave function collapse in process of the measurement, the decoherence phenomenon, etc. Any open system can be considered as a subsystem of a large closed system obeying the linear evolution law. The impossibility to describe the nonlinear state transformation of an open system under the measurement using the Schrödinger equation 214 Nonlinear dynamics of open quantum systems led to the necessity to formulate a particular reduction postulate [1] (the quantum jump notion [2]). The peculiarity of the problem is that there is no cause, expressed in precise physical terms, determining the form of the transformation of the quantum state [3]. Except for nonlinearity the open system dynamics, in general, has one more specific property - the dependence on the evolution history. This property already emerges in the correlation of the uncertainty of the measured value of the stationary state energy with the duration of the measurement process and becomes apparent when considering the EPR paradox [4]1 . We assume that the Schrödinger equation is absolutely accurate when describing the evolution of closed quantum systems for infinitesimal time intervals2. The unique strict generalization of Schrödinger’s equation on finite time intervals is the integral wave equation with the kernel in the form of path integral [6, 7]. The action functionals entering into the integral evolution operator generates the dependence of the quantum system state on the evolution history. Besides, the mathematical form of this law supposes the existence of a subsystem nonlinear evolution [8, 9]. Open system quantum states can be described by reduced density matrices. A corresponding evolution equation is usually derived by considering a large closed system including this open system

About the authors

Alexey Yu Samarin

Samara State Technical University

244, Molodogvardeyskaya st., Samara, 443100, Russian Federation
Cand. Phys. & Math. Sci.; Associate Professor; Dept. of General Physics and Physics of Oil and Gas Production


  1. von Neumann J. Mathematische Grundlagen der Quantenmechanik. Berlin, Heidelberg, Springer, 1932, ix+262 pp. doi: 10.1007/978-3-642-61409-5.
  2. Dirac P. A. M. The Principles of Quantum Mechanics, The International Series of Monographs on Physics. London, Clarendon Press, 1958, xii+312 pp.
  3. Bell J. S. Against ‘measurement’, In: Speakable and unspeakable in quantum mechanics. Collected Papers on Quantum Philosophy. Cambridge, Cambridge Univ. Press, 2004, pp. 213-231. doi: 10.1017/CBO9780511815676.025.
  4. Einstein A., Podolsky B., Rosen N. Can quantum-mechanics description of physical reality be considered complete?, Physical Review, 1935, vol. 47, pp. 777-780. doi: 10.1103/PhysRev.47.777.
  5. Bassi A., Ghirardi G. C. Dynamical reduction models, Phys. Rept., 2003, vol. 379, no. 5-6, pp. 257-426, arXiv: quant-ph/0302164 [quant-ph]. doi: 10.1016/S0370-1573(03)00103-0.
  6. Feynman R. P. Space-Time Approach to Non-Relativistic Quantum Mechanics, Rev. of Mod. Phys., 1948, vol. 20, no. 2, pp. 367-387. doi: 10.1103/RevModPhys.20.367.
  7. Feynman R. P., Hibbs A. R. Quantum Mechanics and Path Integrals, International Earth & Planetary Sciences. New York, McGraw-Hill Co., 1965.
  8. Samarin A. Yu. Nonlocal transformation of the internal quantum particle structure, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, vol. 20, no. 3, pp. 423-456 (In Russian). doi: 10.14498/vsgtu1484.
  9. Samarin A. Yu. Non-mechanical nature of the wave function collapse, 2015, arXiv: 1507.07202 [quant-ph].
  10. Lindblad G. On the generators of quantum dynamical semigroups, Comm. Math. Phys., 1976, vol. 48, no. 2, pp. 119-130. doi: 10.1007/BF01608499.
  11. Carmichael H. An Open Systems Approach to Quantum Optics, Lecture Notes in Physics Monographs, vol. 18. Berlin, Heidelberg, Springer-Verlag, 1991, x+182 pp. doi: 10.1007/978-3-540-47620-7.
  12. Zinn-Justin J. Path Integrals in Quantum Mechanics. Oxford, Oxford Press, 2004. doi: 10. 1093/acprof:oso/9780198566748.001.0001.
  13. Samarin A. Yu. Space localization of the quantum particle, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2013, no. 1(30), pp. 387-397 (In Russian). doi: 10.14498/vsgtu1138.
  14. Samarin A. Yu. Macroscopic Body Motion in Terms of Quantum Evolution, 2014, arXiv: 1408.0340 [quant-ph].
  15. Meleshko N. V., Samarin A. Yu. Complex time transformation pecularities for wave function collapse description using quntum path integrals, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, no. 4(37), pp. 170-177 (In Russian). doi: 10.14498/vsgtu1352.
  16. Ghirardi G. C., Weber T. Quantum mechanics and faster-than-light communication: Methodological considerations, Nuov. Cim. B, 1983, vol. 78, no. 1, pp. 9-20. doi: 10.1007/BF02721378.
  17. Gisin N. Stochastic quantum dynamics and relativity, Helvetica Physica Acta, 1989, vol. 62, pp. 363-371, Retrieved from (November 24, 2017).
  18. Maudlin T. What Bell did, J. Phys. A: Math. Theor., 2014, vol. 47, no. 42, 424010. doi: 10.1088/1751-8113/47/42/424010.
  19. Werner R. F. Comment on ‘What Bell did’, J. Phys. A: Math. Theor., 2014, vol. 47, no. 42, 424011. doi: 10.1088/1751-8113/47/42/424011.
  20. Samarin A. Yu. Quantum particle motion in physical space, Adv. Studies Theor. Phys., 2014, vol. 8, no. 1, pp. 27-34, arXiv: 1407.3559 [quant-ph]. doi: 10.12988/astp.2014.311136.
  21. Eberhard P. H. Bell’s theorem and the different concepts of locality, Nuov. Cim. B, 1978, vol. 46, no. 2, pp. 392-419. doi: 10.1007/BF02728628.



Abstract - 23

PDF (Russian) - 6





  • There are currently no refbacks.

Copyright (c) 2018 Samara State Technical University

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies