Nonlinear extension of the quantum dynamical semigroup
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). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
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
► 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
Cited by
On Crossref's cited-by service no data on citing works was found (last attempt 2021-04-22 02:59:06). On SAO/NASA ADS no data on citing works was found (last attempt 2021-04-22 02:59:06).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.