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Quantum Nondemolition Filtering

  • J. W. Clark
  • T. J. Tarn
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 294)

Abstract

A continuous-time filter is formulated for a physical system modeled as an infinite-dimensional bilinear system. Efforts focus on a quantum system with Hamiltonian operator of the form H 0 + u(t) H 1 where H 0 is the Hamiltonian of the undisturbed system, H 1 couples the system to an external classical field, and u(t) represents the time-varying signal carried by this field. An important problem is to determine when and how the signal u(t) can be extracted from the time development of the measured value of a suitable system observable C. There exist certain observables, called quantum nondemolition observables (QNDO), which have the property that their expected and measured values coincide. The invertibility problem has been posed and solved for these quasiclassical observables. Since one is addressing an infinite-dimensional bilinear system, the domain issue for the operators H 0, H 1 and C becomes nontrivial. An additional complication is that the input observable C is in general time dependent. Having derived conditions for invertibility, necessary and sufficient conditions are developed for an observable to qualify as a QNDO. If an observable meets both sets of criteria, it is said to constitute a quantum nondemolition filter (QNDF). By construction, the associated filtering algorithm separates cleanly into the choice of output observable (a QNDO) and the choice of procedure for processing measurement outcomes. This approach has the advantage over previous schemes that no optimization is necessary. QNDF’s may see practical application in the demodulation of optical signals and the detection and monitoring of gravitational waves.

Keywords

Gravitational Wave Analytic Domain Bilinear System Inverse System Heisenberg Picture 
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.

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References

  1. [1]
    Ong, C. K., G. M. Huang, T. J. Tarn and J. W. Clark: Invertibility of quantum-mechanical control systems, Mathematical Systems Theory, 17 (1984), 335–350.CrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    Messiah, A.: Quantum Mechanics, Vol. 1, John Wiley and Sons, New York 1962.Google Scholar
  3. [3]
    Braginsky, V. B., Y. I. Vorontsov and K. S. Thorne: Quantum nondemolition measurements: Science, 209 (1980), 547–557.ADSCrossRefGoogle Scholar
  4. [4]
    Tarn, T. J., J. W. Clark, C. K. Ong and G. M. Huang: Continuous-time quantum mechanical filter, in: Proceedings of the Joint Workshop on Feedback and Synthesis of Linear and Nonlinear Systems, Bielefeld and Rome (Ed. D. Hinrichsen and A. Isidori), Springer-Verlag, Berlin 1982.Google Scholar
  5. [5]
    Clark, J. W., C. K. Ong, T. J. Tarn and G. M. Huang: Quantum nondemolition filters, Mathematical Systems Theory, 18 (1985), 33–55.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    Nelson, E.: Analytic vectors, Annals of Mathematics 70 (1959), 572–615.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    Caves, C. M., K. S. Thorne, R. W. P. Drever, V. D. Sandberg and M. Zimmermann: On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle, Reviews of Modern Physics, 52 (1980), 341–392.ADSCrossRefGoogle Scholar
  8. [8]
    Braginsky, V. B.: The prospects for high sensitivity gravitational antennae, in: Gravitational Radiation and Gravitational Collapse (Ed. C. DeWitt-Morette), Reidel, Dordrecht 1974, 28–34.Google Scholar
  9. [9]
    Thome, K. S., R. W. P. Drever, C. M. Caves, M. Zimmermann and V. D. Sandberg: Quantum nondemolition measurements of harmonic oscillators, Physical Review Letters, 40 (1978), 667–671.ADSCrossRefGoogle Scholar
  10. [10]
    Caves, C. M.: Quantum nondemolition measurements, in: Quantum Optics, Experimental Gravitation and Measurement Theory (Ed. P. Meystre and M. O. Scully), Plenum Press, New York 1982.Google Scholar
  11. [11]
    Unruh, W. G.: Quantum nondemolition and gravity-wave detection, Physical Review D 19 (1979), 2888–2896.ADSCrossRefGoogle Scholar
  12. [12]
    Barut, A. O. and R. Raczka: Theory of Group Representations and Applications, Polish Scientific Publishers, Warsaw 1977.Google Scholar
  13. [13]
    Huang, G. M., T. J. Tarn and J. W. Clark: On the controllability of quantum-mechanical systems, Journal of Mathematical Physics, 24 (1983), 2608–2618.ADSCrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    Wei, J. and E. Norman: Lie algebraic solution of linear differential equations, Journal of Mathematical Physics, 4 (1963), 575–581.ADSCrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    Wei, J. and E. Norman: On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Am. Math. Soc., 15 (1964), 327–334.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    Hirschorn, R. M.: Invertibility of nonlinear control systems, SIAM Journal of Control and Optimization, 17 (1979), 289–297.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    Baras, J.: Continuous quantum filtering, in: Proceedings of the 15th Allerton Conference (1977), 68–77.Google Scholar
  18. [18]
    Baras, J. S., R. O. Hargar, Y. H. Park: Quantum-mechanical linear filtering of random signal sequences, IEEE Trans. on Information Theory, IT-22 (1976), 59–64.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1987

Authors and Affiliations

  • J. W. Clark
    • 1
  • T. J. Tarn
    • 2
  1. 1.McDonnell Center for the Space Sciences and Department of PhysicsWashington UniversitySt. LouisUSA
  2. 2.Department of Systems Science and MathematicsWashington UniversitySt. LouisUSA

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