# Quantum non-demolition

Experimental aspects of general relativity

First Online:

## Keywords

Gravity Wave Quantum Limit Annihilation Operator Gravitational Radiation Thermal Bath## Preview

Unable to display preview. Download preview PDF.

## Notes and References

- 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 - 1a.Braginsky V.B., and Vorontsov, Y.I.,
*Usp.Fiz.Nauk*. 114, 41, (1974) [*Sov.Phys. Usp.*17, 644 (1975)]Google Scholar - 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 - 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
- 3.As the collapse time for a solar mass black hole is about 10
^{−5}to 10^{−4}sec., 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 - 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 - 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 - 6.
- 7.
- 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 - 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 - 10.See reference 9.Google Scholar
- 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
- 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 - 13.
- 14.Paper in preparation.Google Scholar
- 15.
- 16.Helstrom C.W.,
*Quantum Detection and Estimation Theory*Acad. Press (N.Y.) 1976Google Scholar - 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
- 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 - 19.This analysis was actually derived by Thorne K., et.al. in ref 21 and Unruh W. in ref 23 before Hollenhorst's works.Google Scholar
- 20.See J. Lipa lectures in this volume.Google Scholar
- 21.Thorne K., Drever, R.W.P., Caves C.M., Zimmerman, M., and Sandberg V.D.,
*Phys. Rev.Lett.*40, 667 (1978)Google Scholar - 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 - 23.
- 24.See for example the discussion in pp. 331f in Louisell W.H.,
*Quantum Statistical Properties of Radiation*Wiley, (N.Y.) 1973.Google Scholar - 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 - 26.See Misner, Thorne, Wheeler (ref 5) on p. 1031f.Google Scholar
- 27.
- 28.See Misner, Thorne, Wheeler (ref 5) 1975 on p. 946f.Google Scholar
- 29.The geodesic equations for the spatial components of the position d
^{2}X^{i}/dλ^{2}+ Γ_{μv}^{i}(dX^{μ}/dλ)(dX^{v})/dλ) = 0 will maintain Xl constant if dX^{i}/dλ is initially zero for all i since Γ_{∞}^{i}depends only on h_{ot}.Google Scholar - 30.μ and λ are the usual Lame coefficients for an isotropic medium.Google Scholar

## Copyright information

© Springer-Verlag 1980