Abstract
Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.
Similar content being viewed by others
References
N. N. Bogoliubov and N. M. Krylov, “On the Fokker-Planck equations generated in perturbation theory by a method based on the spectral properties of a perturbed Hamiltonian [in Russian],” Zap. Kafedry Matem. Fiz. AN USSR, 4, 5–80 (1939)
N. N. Bogoliubov, Collection of Scientific Works in Twelve Volumes [in Russian], Vol. 5, Nonequilibrium Statistical Mechanics: 1939–1980, Nauka, Moscow (2006).
N. N. Bogoliubov, On Some Statistical Methods in Mathematical Physics [in Russian], Akad. Nauk USSR (1945)
N. N. Bogolyubov, Selected Works in Three Volumes [in Russian], Vol. 2, Naukova Dumka, Kiev (1970).
N. N. Bogoliubov, Problems of Dynamical Theory in Statistical Physics [in Russian], Gostekhizdat, Moscow (1946); Selected Works in Three Volumes [in Russian], Vol. 2, Naukova Dumka, Kiev (1970), pp. 99–196.
D. N. Zubarev, V. G. Morozov, and G. Röpke, Statistical Mechanics of Nonequilibrium Processes, Vols. 1 and 2, Akademie Verlag, Berlin (1996).
H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford Univ. Press, Oxford (2002).
E. Fick, G. Sauermann, The Quantum Statistics of Dynamic Processes (Springer Ser. Solid-State Sci., Vol. 86), Springer, Berlin (1990).
F. T. Hioe and J. H. Eberly, “N-level coherence vector and higher conservation laws in quantum optics and quantum mechanics,” Phys. Rev. Lett., 47, 838–841 (1981).
R. R. Puri, “SU(m, n) coherent states in the bosonic representation and their generation in optical parametric processes,” Phys. Rev. A, 50, 5309–5316 (1994).
R. R. Puri, Mathematical Methods of Quantum Optics (Springer Ser. Optical Sci., Vol. 79), Springer, Berlin (2001).
R. Zwanzig, “Ensemble method in the theory of irreversibility,” J. Chem. Phys., 33, 1338–1341 (1960).
F. Shibata, Y. Takahashi, and N. Hashitsume, “A generalized stochastic Liouville equation: Non-Markovian versus memoryless master equations,” J. Statist. Phys., 17, 171–187 (1977).
H.-P. Breuer, B. Kappler, and F. Petruccione, “The time-convolutionless projection operator technique in the quantum theory of dissipation and decoherence,” Ann. Phys., 291, 36–70 (2001).
A. G. Redfield, “On the theory of relaxation processes,” IBM J. Res. Dev., 1, 19–31 (1957).
M. Suzuki, “Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems,” Comm. Math. Phys., 51, 183–190 (1976).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 194, No. 2, pp. 259–276, February, 2018.
Rights and permissions
About this article
Cite this article
Bogolyubov, N.N., Soldatov, A.V. Algebraic Aspects of the Dynamics of Quantum Multilevel Systems in the Projection Operator Technique. Theor Math Phys 194, 220–235 (2018). https://doi.org/10.1134/S0040577918020034
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577918020034