TY - JOUR

T1 - The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation for two-dimensional systems

AU - Andrianov, Alexander A.

AU - Ioffe, Mikhail V.

AU - Izotova, Ekaterina A.

AU - Novikov , Oleg O.

N1 - Andrianov, A.A.; Ioffe, M.V.; Izotova, E.A.; Novikov, O.O. The Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) Equation for Two-Dimensional Systems. Symmetry 2022, 14, 754. https://doi.org/10.3390/sym14040754

PY - 2022/4/6

Y1 - 2022/4/6

N2 - Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to 1, and find a moment of time starting from which the FGKLS equation can be used—the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.

AB - Open quantum systems are, in general, described by a density matrix that is evolving under transformations belonging to a dynamical semigroup. They can obey the Franke–Gorini–Kossakowski–Lindblad–Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space of dimension 2. First, we find final fixed states (called pointers) of an evolution of an open system, and we then obtain a general solution to the FGKLS equation and confirm that it converges to a pointer. After this, we check that the solution has physical meaning, i.e., it is Hermitian, positive and has trace equal to 1, and find a moment of time starting from which the FGKLS equation can be used—the range of applicability of the semigroup symmetry. Next, we study the behavior of a solution for a weak interaction with an environment and make a distinction between interacting and non-interacting cases. Finally, we prove that there cannot exist oscillating solutions to the FGKLS equation, which would resemble the behavior of a closed quantum system.

KW - density matrix

KW - Franke–Gorini–Kossakowski–Lindblad–Sudarshan equation

KW - pointers

KW - decoherence

KW - time evolution

UR - https://www.mdpi.com/2073-8994/14/4/754

UR - http://www.scopus.com/inward/record.url?scp=85128791072&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/a7a554d6-3928-30f1-b383-d002827d27fd/

U2 - 10.3390/sym14040754

DO - 10.3390/sym14040754

M3 - Article

VL - 14

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 4

M1 - 754

ER -