Energy Conservation in Open Quantum Systems

Klerk, Albertus de (14. September 2018): Energy Conservation in Open Quantum Systems. Masterarbeit, Ludwig-Maximilians-Universität München

Abstract

The Lindblad theory of open quantum systems has been successfully applied in various fields ranging from quantum optics, condensed matter physics, and quantum thermodynamics to quantum chemistry. Nevertheless, there are situations where we can compare its predictions to those of exact methods or numerically exact results and find discrepancies between them. One example is the case where a single harmonic oscillator is coupled to a thermal bath of independent harmonic oscillators. In this case, for certain parameters, the exact master equation describes equilibration of the system to a thermal state with a temperature different from the one predicted by the Lindblad master equation. The Lindblad theory, based on a weak coupling between the system and environment as well as the Born-Markov approximation, predicts that the system will evolve to a thermal state whose temperature is equal to the initial temperature of the bath. This result is a consequence of the Born approximation, which supposes that throughout the evolution, the bath only fluctuates around its initial equilibrium state. In this master thesis we study the validity of the Born approximation as well as its consistency with other approximations.

By starting from a Hamiltonian for the total system we show that, if the the Born approximation is not invoked, the dynamics of the system is determined by a hierarchy of equations for matrices which can be used to reconstruct the reduced density matrix of the system. This hierarchy approach is fundamentally different from the standard approach, which describes the system dynamics with a weak-coupling master equation, because the hierarchy does not restrict the amount of entanglement possible between the system and the environment. Furthermore, we show how invoking the Born approximation reduces to the usual result and how a generalised ansatz, which we refer to as a generalised Born approximation, reduces the hierarchy to a master equation where the bath temperature is an additional degree of freedom. Finally, we analyse the conservation of the energy of the total system and its relationship to the Markov and secular approximations, and discuss whether imposing it as a condition can fix the time-dependent temperature.

• Dokumententyp: Hochschulschrift (Masterarbeit)
• Keywords: Open Quantum Systems; Energy Conservation; Born Approximation; Master Equation; Generalized Lindblad Form
• Fakultät: Physik > Studienabschlussarbeiten
• Themengebiete: 500 Naturwissenschaften und Mathematik > 530 Physik
• Sprache: Englisch

Literaturliste

1. De Vega, I. & Alonso, D. Dynamics of non-Markovian open quantum systems. Reviews of Modern Physics 89. issn: 0034-6861, 1539-0756. doi:10.1103/RevModPhys.89.015001 (2017).

2. Yang, C.-J., An, J.-H., Luo, H.-G., Li, Y. & Oh, C. H. Canonical versus noncanonical equilibration dynamics of open quantum systems. Phys. Rev. E 90. issn: 1539-3755, 1550-2376. doi:10.1103/PhysRevE.90.022122 (2014).

3. Feynman, R. P. & Vernon Jr, F. L. The theory of a general quantum system interacting with a linear dissipative system. Ann. Phys. 281, 547–607 (2000).

4. Carmichael, H. An open systems approach to quantum optics isbn: 978-0-387-56634-4 978-3-540-56634-2 (Springer-Verlag, Berlin; New York, 1993).

5. Scully, M. O. & Zubairy, M. S. Quantum Optics 1 edition. isbn: 978-0-524-23595-9 (Cambridge University Press, 1997).

6. Loudon, R. The Quantum Theory of Light 3 edition. isbn: 978-0-19-850176-3 (Oxford University Press, Oxford ; New York, 2000).

7. Kosloff, R. Quantum Thermodynamics: A Dynamical Viewpoint. Entropy 15, 2100–2128 (2013).

8. Breuer, H.-P. The Theory of Open Quantum Systems New Ed edition. isbn: 978-0-19-921390-0 (Oxford University Press, USA, Oxford, 2007).

9. Hall, M. J. W., Cresser, J. D., Li, L. & Andersson, E. Canonical form of master equations and characterization of non-Markovianity. Phys. Rev. A 89. issn: 1050-2947, 1094-1622. doi:10.1103/PhysRevA. 89.042120 (2014).

10. Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976).

11. Gorini, V. Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821. issn: 00222488 (1976).

12. Esposito, M. & Gaspard, P. Quantum master equation for a system influencing its environment. Physical Review E 68. issn: 1063-651X, 1095-3787. doi:10.1103/PhysRevE.68.066112 (2003).

13. Spohn, H. & Lebowitz, J. L. Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys 38, 109–142 (1978).

14. Breuer, H.-P. Non-Markovian generalization of the Lindblad theory of open quantum systems. Phys. Rev. A 75. issn: 1050-2947, 1094-1622. doi:10.1103/PhysRevA.75.022103 (2007).

15. Budini, A. A. Random Lindblad equations from complex environments. Phys. Rev. E 72. issn: 1539-3755, 1550-2376. doi:10. 1103/PhysRevE.72.056106 (2005).

16. Esposito, M. & Gaspard, P. Spin relaxation in a complex environment. Phys. Rev. E 68. issn: 1063-651X, 1095-3787. doi:10.1103/PhysRevE.68.066113 (2003).

17. Blasone, M., Jizba, P. & Vitiello, G. Quantum Field Theory And Its Macroscopic Manifestations: Boson Condensation, Ordered Patterns And Topological Defects Reprint edition. isbn: 978-1-911299-72-1 (Icp, 2011).

18. Chin, A. W., Rivas, A., Huelga, S. F. & Plenio, M. B. Exact mapping between system-reservoir quantum models and semi-infinite discrete chains using orthogonal polynomials. J. Math. Phys. 51, 092109. issn: 0022-2488 (2010).

19. Xiao, M.-w. Theory of transformation for the diagonalization of quadratic Hamiltonians. arXiv:0908.0787 (2009).