Quantum non-demolition

  • W. G. Unruh
Experimental aspects of general relativity
Part of the Lecture Notes in Physics book series (LNP, volume 124)


Gravity Wave Quantum Limit Annihilation Operator Gravitational Radiation Thermal Bath 
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Notes and References

  1. 1.
    The first person to seriously worry about this quantum limit and think about techniques for avoiding this limit was V. Braginsky. See Braginsky, V.B., and Manukin, A.B., in Measurement of Weak Forces in Physics Experiments, University of Chicago Press, Chicago, 1977 ed. Douglas, D.H. See alsoGoogle Scholar
  2. 1a.
    Braginsky V.B., and Vorontsov, Y.I., Usp.Fiz.Nauk. 114, 41, (1974) [Sov.Phys. Usp. 17, 644 (1975)]Google Scholar
  3. 1b.
    Braginsky, V.B., Vorontsov, Y.I., Krivchewkov, V.D., Zh.Exsp.Tesp.Teor.Fiz. 68,55, (1975) [J.E.T.P. 41,28,(1975)].Google Scholar
  4. 2.
    One would expect the strongest sources to be highly nonlinear in the source region, but to give a deviation from flatness as seen at infinity of less than unity by the time one arrived at the radiation zone, i.e. ∼ one wavelength from the source. Actual sources are probably much weaker than this.Google Scholar
  5. 3.
    As the collapse time for a solar mass black hole is about 10−5 to 10−4sec., the spectrum of gravity waves should extend to 10 Khz, with a reasonable strength of wave emitted for an asymmetric, rapidly rotating final collapse stage.Google Scholar
  6. 4.
    See for exaple Tamman, G.A., “Statistic of Supernovae in External Galaxies” in Eighth Texas Symposium on Relativistic Astrophysics ed. Papagiannis M.D., New York Acad. Sc.,(N.Y.)1977who derives a figure of about 1 per 10 years for our galaxy.Google Scholar
  7. 5.
    This uses the classical estimate of the energy in a gravity wave given, for example, in Misner, C., Thorne, K., and Wheeler J., Gravitation Freedman (N.Y.) 1975 p.955f.Google Scholar
  8. 6.
    Heffner, H., Proc. I.R.E. 50, 1604 (1962)Google Scholar
  9. 7.
    Haus, H.A., and Mullen, J.A., Phys.Rev. 128, 2407.Google Scholar
  10. 8.
    The (1962) “well known” commutation relation between number and phase is not exact and not derivable from quantum theory because of the non existence of an operator corresponding to phase conjugate to N. See for example Carruthers, Nieto, M.M., Rev.Mod.Phys. 40, 411 (1968) for a discussion of some of these problems.Google Scholar
  11. 9.
    These field normal modes are the C-number solutions of the wave equations for Ψ and Φ under the assumption of no coupling between the fields. See for example Bjorken, J., Drell, S., Relativistic Quantum Fields McGraw Hill (N.Y.) 1964.Google Scholar
  12. 10.
    See reference 9.Google Scholar
  13. 11.
    Even in the case of non linear interactions, the commutation relations place strong restrictions on the form of the S-matrix which maps the ingoing states to the outgoing states.Google Scholar
  14. 12.
    Von Neuman J., in Mathematical Foundations of Quantum Mechanics (Tr. Beyer,R.T.) Princeton University Press (1955) discusses the problem of breaking the chain of analysis in any quantum measurement process.Google Scholar
  15. 13.
    Einstein A., Phys.Zeits. 18, 121 (1917)Google Scholar
  16. 14.
    Paper in preparation.Google Scholar
  17. 15.
    Hollenhorst, J.N., Phys.Rev.D. 19, 1669 (1979)Google Scholar
  18. 16.
    Helstrom C.W., Quantum Detection and Estimation Theory Acad. Press (N.Y.) 1976Google Scholar
  19. 17.
    This is of course the property which sets quantum mechanics off from classical mechanics, that different states can have some probability of being indistinguishable.Google Scholar
  20. 18.
    The coherent states were introduced by Schroedinger E., Z.Physik.,14, 664 (1926), and are minimum uncertainty (δpδq = h/2) states. They are essentially eigenstates of the annihilation operator. See also Glauber, R.J., Phys.Rev. 131, 2766 (1963)Google Scholar
  21. 19.
    This analysis was actually derived by Thorne K., in ref 21 and Unruh W. in ref 23 before Hollenhorst's works.Google Scholar
  22. 20.
    See J. Lipa lectures in this volume.Google Scholar
  23. 21.
    Thorne K., Drever, R.W.P., Caves C.M., Zimmerman, M., and Sandberg V.D., Phys. Rev.Lett. 40, 667 (1978)Google Scholar
  24. 22.
    Caves, C.M., Thorne, K.S., Drever R.W.P., Sandberg V.D., and Zimmerman, M., “On the Measurement of a Weak Classical Force Coupled to a Quantum Mechanical Oscillator I. Issues of Principle” cal. Tech. preprint Apr. 1979.Google Scholar
  25. 23.
    Unruh W., Phys.Rev.D. 19, 2888 (1979)Google Scholar
  26. 24.
    See for example the discussion in pp. 331f in Louisell W.H., Quantum Statistical Properties of RadiationWiley, (N.Y.) 1973.Google Scholar
  27. 25.
    See for example the discussion in Messiah A., Quantum Mechanics Wiley, (N.Y.) 1966 on pp. 139–149. The argument presented in this paper demonstrates how the quantum uncertainties in the readout system preserve the uncertainties of any variables being measured.Google Scholar
  28. 26.
    See Misner, Thorne, Wheeler (ref 5) on p. 1031f.Google Scholar
  29. 27.
    Carter B., Quintana H., Phys.Rev.D. 16, 2928 (1977) Dyson F.,Ap. J. 156, 529 (1969).Google Scholar
  30. 28.
    See Misner, Thorne, Wheeler (ref 5) 1975 on p. 946f.Google Scholar
  31. 29.
    The geodesic equations for the spatial components of the position d2Xi/dλ2 + Γμvi(dXμ/dλ)(dXv)/dλ) = 0 will maintain Xl constant if dXi/dλ is initially zero for all i since Γi depends only on hot.Google Scholar
  32. 30.
    μ and λ are the usual Lame coefficients for an isotropic medium.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • W. G. Unruh
    • 1
  1. 1.Department of PhysicsUniversity of British CoZumbiaVancouver, B.C.Canada

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