Advertisement

Journal of Statistical Physics

, Volume 148, Issue 5, pp 800–823 | Cite as

Adiabatic Response for Lindblad Dynamics

  • J. E. Avron
  • M. FraasEmail author
  • G. M. Graf
Article

Abstract

We study the adiabatic response of open systems governed by Lindblad evolutions. In such systems, there is an ambiguity in the assignment of observables to fluxes (rates) such as velocities and currents. For the appropriate notion of flux, the formulas for the transport coefficients are simple and explicit and are governed by the parallel transport on the manifold of instantaneous stationary states. Among our results we show that the response coefficients of open systems, whose stationary states are projections, is given by the adiabatic curvature.

Keywords

Adiabatic response theory Open quantum system Lindblad dynamics Fluxes Currents Principle of virtual work Adiabatic curvature 

Notes

Acknowledgements

J.E.A. is supported by the ISF, the NSF under Grant No. PHY11-25915 and the fund for promotion of research at the Technion. M.F. was supported by UNESCO and ISF. We thank M. Porta for useful discussions.

References

  1. 1.
    Aizenman, M., Graf, G.M.: Localization bounds for an electron gas. J. Phys. A, Math. Gen. 31(32), 6783–6806 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Aschbacher, W., Jaksic, V., Pautrat, Y., Pillet, C.-A.: Topics in non-equilibrium quantum statistical mechanics. In: Attal, S., Joye, A., Pillet, C.-A. (eds.) Open Quantum Systems III. Lecture Notes in Mathematics, vol. 1882, pp. 1–66. Springer, Berlin/Heidelberg (2006). CrossRefGoogle Scholar
  3. 3.
    Attal, S., Joye, A., Pillet, C.-A.: Open Quantum Systems: The Markovian Approach. Springer, Berlin (2006) zbMATHGoogle Scholar
  4. 4.
    Avron, J.E., Seiler, R., Yaffe, L.G.: Adiabatic theorems and applications to the quantum Hall effect. Commun. Math. Phys. 110, 33–49 (1987) MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Avron, J.E., Fraas, M., Graf, G.M., Grech, P.: Adiabatic theorems for generators of contracting evolutions. arXiv e-prints (June 2011) Google Scholar
  6. 6.
    Avron, J.E., Fraas, M., Graf, G.M., Kenneth, O.: Quantum response of dephasing open systems. New J. Phys. 13, 053042 (2011). arXiv:1008.4079 MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Avron, J.E., Seiler, R., Simon, B.: Quantum Hall effect and the relative index for projections. Phys. Rev. Lett. 65(17), 2185–2188 (1990) ADSCrossRefGoogle Scholar
  8. 8.
    Bellissard, J.: Coherent and dissipative transport in aperiodic solids: An overview. In: Lecture Notes in Physics, pp. 413–485. Springer, Berlin/Heidelberg (2002) Google Scholar
  9. 9.
    Bellissard, J., van Elst, A., Schulz-Baldes, H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10), 5373–5451 (1994) MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Berry, M.V., Robbins, J.M.: Chaotic classical and half-classical adiabatic reactions: geometric magnetism and deterministic friction. Proc. R. Soc. Lond. A 442, 659–672 (1993) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Bloch, F.: Flux quantization and dimensionality. Phys. Rev. 166, 415–423 (1968) ADSCrossRefGoogle Scholar
  12. 12.
    Bohm, D.: Note on a theorem of Bloch concerning possible causes of superconductivity. Phys. Rev. 75(3), 502–504 (1949) ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, London (2007) zbMATHCrossRefGoogle Scholar
  14. 14.
    Davies, E.B.: Quantum Theory of Open Systems. Academic Press/Harcourt Brace Jovanovich, London (1976) zbMATHGoogle Scholar
  15. 15.
    Davies, E.B., Spohn, H.: Open quantum systems with time-dependent Hamiltonians and their linear response. J. Stat. Phys. 19, 511–523 (1978) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Ericsson, M., Sjöqvist, E., Brännlund, J., Oi, D.K.L., Pati, A.K.: Generalization of the geometric phase to completely positive maps. Phys. Rev. A 67(Feb.), 020101 (2003) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Gebauer, R., Car, R.: Current in open quantum systems. Phys. Rev. Lett. 93(16), 160404 (2004) ADSCrossRefGoogle Scholar
  18. 18.
    Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17(5), 821–825 (1976) MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn. 5, 435–439 (1950) ADSCrossRefGoogle Scholar
  20. 20.
    Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976) MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Pekola, J.P., Brosco, V., Möttönen, M., Solinas, P., Shnirman, A.: Decoherence in adiabatic quantum evolution: Application to Cooper pair pumping. Phys. Rev. Lett. 105(Jul.), 030401 (2010) ADSCrossRefGoogle Scholar
  22. 22.
    Read, N., Rezayi, E.H.: Hall viscosity, orbital spin, and geometry: paired superfluids and quantum Hall systems. arXiv e-prints (August 2010) Google Scholar
  23. 23.
    Sarandy, M.S., Lidar, D.A.: Adiabatic approximation in open quantum systems. Phys. Rev. A 71(Jan.), 012331 (2005) ADSCrossRefGoogle Scholar
  24. 24.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Texts and Monographs in Physics. Springer, Berlin (1991) zbMATHCrossRefGoogle Scholar
  25. 25.
    Spohn, H., Lebowitz, J.L.: Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs. Adv. Chem. Phys. 38, 109–142 (1978) CrossRefGoogle Scholar
  26. 26.
    Thouless, D.J.: Quantization of particle transport. Phys. Rev. B 27(10), 6083–6087 (1983) MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Thouless, D.J., Kohmoto, M., Nightingale, M.P., den Nijs, M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49(6), 405–408 (1982) ADSCrossRefGoogle Scholar
  28. 28.
    Whitney, R.S., Makhlin, Y., Shnirman, A., Gefen, Y.: Geometric nature of the environment-induced berry phase and geometric dephasing. Phys. Rev. Lett. 94, 070407 (2005) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of PhysicsTechnionHaifaIsrael
  2. 2.Theoretische PhysikETH ZurichZurichSwitzerland

Personalised recommendations