Nonlinear extension of the quantum dynamical semigroup

Jakub Rembieliński; Paweł Caban

doi:10.22331/q-2021-03-23-420

Nonlinear extension of the quantum dynamical semigroup

Jakub Rembieliński and Paweł Caban

Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Lodz, Pomorska 149/153, 90-236 Łódź, Poland

Published:	2021-03-23, volume 5, page 420
Eprint:	arXiv:2003.09170v3
Doi:	https://doi.org/10.22331/q-2021-03-23-420
Citation:	Quantum 5, 420 (2021).

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Abstract

In this paper we consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that if family of linear non-trace-preserving maps satisfies the semigroup property then the generated family of convex quasi-linear operations also possesses the semigroup property. Next we generalize the Gorini-Kossakowski-Sudarshan-Lindblad type equation for the considered evolution. As examples we discuss the general qubit evolution in our model as well as an extension of the Jaynes-Cummings model. We apply our formalism to spin density matrix of a charged particle moving in the electromagnetic field as well as to flavor evolution of solar neutrinos.

Featured image: Evolution of a relativistic qubit in the electromagnetic field

► BibTeX data

@article{Rembielinski2021nonlinearextension,
  doi = {10.22331/q-2021-03-23-420},
  url = {https://doi.org/10.22331/q-2021-03-23-420},
  title = {Nonlinear extension of the quantum dynamical semigroup},
  author = {Rembieli{\'{n}}ski, Jakub and Caban, Pawe{\l{}}},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {420},
  month = mar,
  year = {2021}
}

► References

[1] V. I. Arnol'd, V. S. Afrajmovich, Yu. S. Il'yashenko, and L. P. Shil'niov. Bifurcation Theory, volume 5. Springer-Verlag, Berlin-Heidelberg, 1994. 10.1007/978-3-642-57884-7.
https://doi.org/10.1007/978-3-642-57884-7

[2] V. Bargmann, L. Michel, and V. L. Telegdi. Precession of the polarization of particles moving in a homogeneous electromagnetic field. Phys. Rev. Lett., 2: 435–436, 1959. 10.1103/PhysRevLett.2.435.
https://doi.org/10.1103/PhysRevLett.2.435

[3] A. Bassi and K. Hejazi. No-faster-than-light-signaling implies linear evolution. A re-derivation. European J. Phys., 36: 055027, 2015. 10.1088/0143-0807/36/5/055027.
https://doi.org/10.1088/0143-0807/36/5/055027

[4] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht. Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys., 85: 471–527, 2013. 10.1103/RevModPhys.85.471.
https://doi.org/10.1103/RevModPhys.85.471

[5] C. M. Bender. Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys., 70: 947–1018, 2007. 10.1088/0034-4885/70/6/r03.
https://doi.org/10.1088/0034-4885/70/6/r03

[6] W. B. Berestetzki, E. M. Lifschitz, and L. P. Pitajewski. Relativistic Quantum Theory, volume 1. Nauka, Moscow, 1968.

[7] I. Białynicki-Birula and J. Mycielski. Nonlinear wave mechanics. Ann. Phys. (New York), 100: 62, 1976. 10.1016/0003-4916(76)90057-9.
https://doi.org/10.1016/0003-4916(76)90057-9

[8] N. N. Bogolubov, A. A. Logunov, and I. T. Todorov. Introduction to Axiomatic Quantum Field Theory. W. A. Benjamin, Reading, Mass., 1975.

[9] H.-J. Briegel, B.-G. Englert, N. Sterpi, and H. Walther. One-atom master: Statistics of detector clicks. Phys. Rev. A, 49: 2962–2985, 1994. 10.1103/PhysRevA.49.2962.
https://doi.org/10.1103/PhysRevA.49.2962

[10] D. C. Brody and E.-M. Graefe. Mixed-state evolution in the presence of gain and loss. Phys. Rev. Lett., 109: 230405, 2012. 10.1103/PhysRevLett.109.230405.
https://doi.org/10.1103/PhysRevLett.109.230405

[11] P. Caban and J. Rembieliński. Lorentz-covariant reduced spin density matrix and Einstein–Podolsky–Rosen–Bohm correlations. Phys. Rev. A, 72: 012103, 2005. 10.1103/PhysRevA.72.012103.
https://doi.org/10.1103/PhysRevA.72.012103

[12] C. Cohen-Tannoudji, B. Diu, and F. Laloë. Quantum mechanics. Wiley-VCH, 1991.

[13] M. Czachor. Mobility and non-separability. Found. Phys. Lett., 4: 351–361, 1991. 10.1007/BF00665894.
https://doi.org/10.1007/BF00665894

[14] M. Czachor and H.-D. Doebner. Correlation experiments in nonlinear quantum mechanics. Phys. Lett. A, 301: 139–152, 2002. 10.1016/S0375-9601(02)00959-3.
https://doi.org/10.1016/S0375-9601(02)00959-3

[15] G. C. Ghirardi, A. Rimini, and T. Weber. Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D, 34: 470–491, 1986. 10.1103/PhysRevD.34.470.
https://doi.org/10.1103/PhysRevD.34.470

[16] N. Gisin. A simple nonlinear dissipative quantum evolution equation. J. Phys. A: Math. Gen., 14: 2259–2267, 1981. 10.1088/0305-4470/14/9/021.
https://doi.org/10.1088/0305-4470/14/9/021

[17] N. Gisin. Irreversible quantum dynamics and the Hilbert space structure of quantum kinematics. J. Math. Phys., 24: 1779, 1983. 10.1063/1.525895.
https://doi.org/10.1063/1.525895

[18] N. Gisin. Weinberg's non-linear quantum mechanics and supraluminal communications. Phys. Lett. A, 143: 1–2, 1990. 10.1016/0375-9601(90)90786-N.
https://doi.org/10.1016/0375-9601(90)90786-N

[19] N. Gisin and M. Rigo. Relevant and irrelevant nonlinear Schrödinger equations. J. Phys. A: Math. Gen., 28: 7375–7390, 1995. 10.1088/0305-4470/28/24/030.
https://doi.org/10.1088/0305-4470/28/24/030

[20] J. Grabowski, M. Kuś, and G. Marmo. Symmetries, group actions, and entanglement. Open Systems and Information Dynamics, 13: 343–362, 2006. 10.1007/s11080-006-9013-3.
https://doi.org/10.1007/s11080-006-9013-3

[21] R. Grimaudo, A. S. M. de Castro, M. Kuś, and A. Messina. Exactly solvable time-dependent pseudo-Hermitian su(1,1) Hamiltonian models. Phys. Rev. A, 98: 033835, 2018. 10.1103/PhysRevA.98.033835.
https://doi.org/10.1103/PhysRevA.98.033835

[22] B. Helou and Y. Chen. Extensions of Born's rule to non-linear quantum mechanics, some of which do not imply superluminal communication. Journal of Physics: Conference Series, 880: 012021, 2017. 10.1088/1742-6596/880/1/012021.
https://doi.org/10.1088/1742-6596/880/1/012021

[23] John D. Jackson. Classical electrodynamics. John Wiley & Sons, New York, 1999.

[24] E. T. Jaynes and F. W. Cummings. Comparison of quantum and semiclassical radiation theories with application to the beam maser. Proc. IEEE, 51: 89, 1963. 10.1109/PROC.1963.1664.
https://doi.org/10.1109/PROC.1963.1664

[25] K. Kawabata, Y. Ashida, and M. Ueda. Information retrieval and criticality in parity-time-symmetric systems. Phys. Rev. Lett., 119: 190401, 2017. 10.1103/PhysRevLett.119.190401.
https://doi.org/10.1103/PhysRevLett.119.190401

[26] K. Kraus. States, Effects, and Operations. Springer, Berlin, Heidelberg, 1983. 10.1007/3-540-12732-1.
https://doi.org/10.1007/3-540-12732-1

[27] M. Maltoni and A. Yu. Smirnov. Solar neutrinos and neutrino physics. Eur. Phys. J. A, 52: 87, 2016. 10.1140/epja/i2016-16087-0.
https://doi.org/10.1140/epja/i2016-16087-0

[28] N. Moiseyev. Non-Hermitian quantum mechanics. Cambridge University Press, Cambridge, UK, 2011. 10.1017/CBO9780511976186.
https://doi.org/10.1017/CBO9780511976186

[29] J. Polchinski. Weinberg's nonlinear quantum mechanics and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett., 66: 397–400, 1991. 10.1103/PhysRevLett.66.397.
https://doi.org/10.1103/PhysRevLett.66.397

[30] J. Rembieliński and P. Caban. Nonlinear evolution and signaling. Phys. Rev. Research, 2: 012027, 2020. 10.1103/PhysRevResearch.2.012027.
https://doi.org/10.1103/PhysRevResearch.2.012027

[31] J. Rembieliński and J. Ciborowski. in preparation.

[32] A. Sergi and K. G. Zloshchastiev. Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments. Int. J. Mod. Phys. B, 27: 1350163, 2013. 10.1142/S0217979213501634.
https://doi.org/10.1142/S0217979213501634

[33] A. Sergi and K. G. Zloshchastiev. Time correlation functions for non-Hermitian quantum systems. Phys. Rev. A, 91: 062108, 2015. 10.1103/PhysRevA.91.062108.
https://doi.org/10.1103/PhysRevA.91.062108

[34] B. W. Shore and P. L. Knight. The Jaynes–Cummings model. J. Modern Optics, 40: 1195–1238, 1993. 10.1080/09500349314551321.
https://doi.org/10.1080/09500349314551321

[35] S. Weinberg. Testing quantum mechanics. Ann. Phys. (New York), 194: 336–386, 1989a. https://doi.org/10.1016/0003-4916(89)90276-5.
https://doi.org/10.1016/0003-4916(89)90276-5

[36] S. Weinberg. Precision tests of quantum mechanics. Phys. Rev. Lett., 62: 485–488, 1989b. 10.1103/PhysRevLett.62.485.
https://doi.org/10.1103/PhysRevLett.62.485

[37] K. G. Zloshchastiev. Non-Hermitian Hamiltonians and stability of pure states. Eur. Phys. J. D, 69: 253, 2015. 10.1140/epjd/e2015-60384-0.
https://doi.org/10.1140/epjd/e2015-60384-0

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