Advertisement

Controlling the Stability of Steady States in Continuous Variable Quantum Systems

  • Philipp StrasbergEmail author
  • Gernot Schaller
  • Tobias Brandes
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

For the paradigmatic case of the damped quantum harmonic oscillator we present two measurement-based feedback schemes to control the stability of its fixed point. The first scheme feeds back a Pyragas-like time-delayed reference signal and the second uses a predetermined instead of time-delayed reference signal. We show that both schemes can reverse the effect of the damping by turning the stable fixed point into an unstable one. Finally, by taking the classical limit \(\hbar \rightarrow 0\) we explicitly distinguish between inherent quantum effects and effects, which would be also present in a classical noisy feedback loop. In particular, we point out that the correct description of a classical particle conditioned on a noisy measurement record is given by a non-linear stochastic Fokker-Planck equation and not a Langevin equation, which has observable consequences on average as soon as feedback is considered.

Keywords

Harmonic Oscillator Master Equation Classical Limit Wigner Function Feedback Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

PS wishes to thank Philipp Hövel, Lina Jaurigue and Wassilij Kopylov for helpful discussions about time-delayed feedback control. Financial support by the DFG (SCHA 1646/3-1, SFB 910, and GRK 1558) is gratefully acknowledged.

References

  1. 1.
    M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997)CrossRefzbMATHGoogle Scholar
  2. 2.
    S.L. Braunstein, P. van Loock, Rev. Mod. Phys. 77, 513 (2005)ADSCrossRefGoogle Scholar
  3. 3.
    C. Weedbrook, S. Pirandola, R. García-Patrón, N.J. Cerf, T.C. Ralph, J.H. Shapiro, S. Lloyd, Rev. Mod. Phys. 84, 621 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    M. Aspelmeyer, T.J. Kippenberg, F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    L.D. Faddeev, O.A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students, Student Mathematical Library, vol. 47. (American Mathematical Society, 2009)Google Scholar
  6. 6.
    H.M. Wiseman, G.J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, 2010)zbMATHGoogle Scholar
  7. 7.
    W.P. Smith, J.E. Reiner, L.A. Orozco, S. Kuhr, H.M. Wiseman, Phys. Rev. Lett. 89, 133601 (2002)ADSCrossRefGoogle Scholar
  8. 8.
    P. Bushev, D. Rotter, A. Wilson, F. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, P. Zoller, Phys. Rev. Lett. 96, 043003 (2006)ADSCrossRefGoogle Scholar
  9. 9.
    G.G. Gillett, R.B. Dalton, B.P. Lanyon, M.P. Almeida, M. Barbieri, G.J. Pryde, J.L. OBrien, K.J. Resch, S.D. Bartlett, A.G. White. Phys. Rev. Lett. 104, 080503 (2010)Google Scholar
  10. 10.
    C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Rybarczyk, S. Gleyzes, P. Rouchon, M. Mirrahimi, H. Amini, M. Brune, J.M. Raimond, S. Haroche, Nature (London) 477, 73 (2011)ADSCrossRefGoogle Scholar
  11. 11.
    R. Vijay, C. Macklin, D.H. Slichter, S.J. Weber, K.W. Murch, R. Naik, A.N. Korotkov, I. Siddiqi, Nature (London) 490, 77 (2012)ADSCrossRefGoogle Scholar
  12. 12.
    D. Riste, C.C. Bultink, K.W. Lehnert, L. DiCarlo, Phys. Rev. Lett. 109, 240502 (2012)ADSCrossRefGoogle Scholar
  13. 13.
    H.J. Carmichael, An Open Systems Approach to Quantum Optics (Lecture Notes, Springer, Berlin, 1993)zbMATHGoogle Scholar
  14. 14.
    G. Lindblad, Commun. Math. Phys. 48, 119 (1976)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    V. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17, 821 (1976)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    K. Jacobs, D.A. Steck, Contemp. Phys. 47, 279 (2006)ADSCrossRefGoogle Scholar
  17. 17.
    A. Barchielli, L. Lanz, G.M. Prosperi, Nuovo Cimento B 72, 79 (1982)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    A. Barchielli, L. Lanz, G.M. Prosperi, Found. Phys. 13, 779 (1983)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    C.M. Caves, G.J. Milburn, Phys. Rev. A 36, 5543 (1987)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    H.M. Wiseman, G.J. Milburn, Phys. Rev. A 47, 642 (1993)ADSCrossRefGoogle Scholar
  21. 21.
    A.C. Doherty, K. Jacobs, Phys. Rev. A 60, 2700 (1999)ADSCrossRefGoogle Scholar
  22. 22.
    C. Gardiner, P. Zoller, Quantum Noise (Springer-Verlag, Berlin Heidelberg, 2004)zbMATHGoogle Scholar
  23. 23.
    H.M. Wiseman, G.J. Milburn, Phys. Rev. Lett. 70, 548 (1993)ADSCrossRefGoogle Scholar
  24. 24.
    H.M. Wiseman, G.J. Milburn, Phys. Rev. A 49, 1350 (1994)ADSCrossRefGoogle Scholar
  25. 25.
    H.M. Wiseman, Phys. Rev. A 49, 2133 (1994)ADSCrossRefGoogle Scholar
  26. 26.
    M. Hillery, R.F. O’Connell, M.O. Scully, E. Wigner, Phys. Rep. 106, 121 (1984)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    H.J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, New York, 1999)CrossRefzbMATHGoogle Scholar
  28. 28.
    C. Zachos, D. Fairlie, T. Curtright (eds.). Quantum mechanics in phase space: an overview with selected papers (World Scientific, vol. 34, 2005)Google Scholar
  29. 29.
    H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin Heidelberg, 1984)CrossRefzbMATHGoogle Scholar
  30. 30.
    P. Hövel, E. Schöll, Phys. Rev. E 72, 046203 (2005)CrossRefGoogle Scholar
  31. 31.
    E. Arthurs, J.L. Kelly, Bell Syst. Tech. J. 44, 725 (1965)CrossRefGoogle Scholar
  32. 32.
    A.J. Scott, G.J. Milburn, Phys. Rev. A 63, 042101 (2001)ADSCrossRefGoogle Scholar
  33. 33.
    K. Pyragas, Phys. Lett. A 170, 421 (1992)ADSCrossRefGoogle Scholar
  34. 34.
    E. Schöll, H.G. Schuster (eds.), Handbook of Chaos Control (Wiley-VCH, Weinheim, 2007)zbMATHGoogle Scholar
  35. 35.
    V. Flunkert, I. Fischer, E. Schöll (eds.). Dynamics, control and information in delay-coupled systems: an overview. Phil. Trans. R. Soc. A, 371, 1999 (2013)Google Scholar
  36. 36.
    S.J. Whalen, M.J. Collett, A.S. Parkins, H.J. Carmichael, Quantum Electronics Conference & Lasers and Electro-Optics (CLEO/IQEC/PACIFIC RIM), IEEE pp. 1756–1757 (2011)Google Scholar
  37. 37.
    A. Carmele, J. Kabuss, F. Schulze, S. Reitzenstein, A. Knorr, Phys. Rev. Lett. 110, 013601 (2013)ADSCrossRefGoogle Scholar
  38. 38.
    F. Schulze, B. Lingnau, S.M. Hein, A. Carmele, E. Schöll, K. Lüdge, A. Knorr, Phys. Rev. A 89, 041801(R) (2014)ADSCrossRefGoogle Scholar
  39. 39.
    S.M. Hein, F. Schulze, A. Carmele, A. Knorr, Phys. Rev. Lett. 113, 027401 (2014)ADSCrossRefGoogle Scholar
  40. 40.
    A.L. Grimsmo, A.S. Parkins, B.S. Skagerstam, New J. Phys. 16, 065004 (2014)ADSCrossRefGoogle Scholar
  41. 41.
    W. Kopylov, C. Emary, E. Schöll, T. Brandes, New J. Phys. 17, 013040 (2015)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    A.L. Grimsmo, Phys. Rev. Lett. 115, 060402 (2015)ADSCrossRefGoogle Scholar
  43. 43.
    J. Kabuss, D.O. Krimer, S. Rotter, K. Stannigel, A. Knorr, A. Carmele, arXiv:1503.05722
  44. 44.
    W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70, 1187 (1993)ADSCrossRefGoogle Scholar
  45. 45.
    G.J. Milburn, Quantum Semiclass. Opt. 8, 269 (1996)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Philipp Strasberg
    • 1
    Email author
  • Gernot Schaller
    • 1
  • Tobias Brandes
    • 1
  1. 1.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

Personalised recommendations