Quantum Theory: A Two-Time Success Story pp 213-246 | Cite as

# A Gravitational Aharonov-Bohm Effect, and Its Connection to Parametric Oscillators and Gravitational Radiation

- 1.6k Downloads

## Abstract

A thought experiment is proposed to demonstrate the existence of a gravitational, vector Aharonov-Bohm effect. We begin the analysis starting from four Maxwell-like equations for weak gravitational fields interacting with slowly moving matter. A connection is made between the gravitational, vector Aharonov-Bohm effect and the principle of local gauge invariance for nonrelativistic quantum matter interacting with weak gravitational fields. The compensating vector fields that are necessitated by this local gauge principle are shown to be incorporated by the DeWitt minimal coupling rule. The nonrelativistic Hamiltonian for weak, time-independent fields interacting with quantum matter is then extended to time-dependent fields, and applied to the problem of the interaction of radiation with macroscopically coherent quantum systems, including the problem of gravitational radiation interacting with superconductors. But first we examine the interaction of EM radiation with superconductors in a parametric oscillator consisting of a superconducting wire placed at the center of a high *Q* superconducting cavity driven by pump microwaves. Some room-temperature data will be presented demonstrating the splitting of a single microwave cavity resonance into a spectral doublet due to the insertion of a central wire. This would represent an *unseparated* kind of parametric oscillator, in which the signal and idler waves would occupy the same volume of space. We then propose a *separated* parametric oscillator experiment, in which the signal and idler waves are generated in two disjoint regions of space, which are separated from each other by means of an impermeable superconducting membrane. We find that the threshold for parametric oscillation for EM microwave generation is much lower for the separated configuration than the unseparated one, which then leads to an observable dynamical Casimir effect. We speculate that a separated parametric oscillator for generating coherent GR microwaves could also be built. [*Editor’s note*: for a video of the talk given by Prof. Chiao at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-20.]

## Keywords

Parametric Oscillator Pump Wave Stokes Wave Transverse Magnetic Mode Idle Wave## Notes

### Acknowledgements

D.A.S. acknowledges the support of a 2012–2013 Fulbright Senior Scholar Grant. We thank Jay Sharping for his help in our experiments.

## References

- 1.M.A. Hohensee, B. Estey, P. Hamilton, A. Zeilinger, H. Müller, Force-free gravitational redshift: proposed gravitational Aharonov-Bohm experiment. Phys. Rev. Lett.
**108**, 230404 (2012). Here we consider a thought experiment to see the*vector*gravitational AB effect instead of the*scalar*gravitational AB effect. ADSCrossRefGoogle Scholar - 2.Y. Aharonov, G. Carmi, Quantum aspects of the equivalence principle. Found. Phys.
**3**, 493 (1973) ADSCrossRefGoogle Scholar - 3.E.G. Harris, The gravitational Aharonov-Bohm effect with photons. Am. J. Phys.
**64**, 378 (1996). Here we consider the AB effect with electrons rather than photons. The AB effect with photons could be understood entirely classically in terms of classical EM waves diffracting around a “solenoid.” However, no such classical explanation would exist for the electron interference experiment described in Fig. 14.1 ADSCrossRefGoogle Scholar - 4.J.M. Cohen, B. Mashhoon, Standard clocks, interferometry, and gravitomagnetism. Phys. Lett. A
**181**, 353 (1993) ADSCrossRefGoogle Scholar - 5.A. Tartaglia, Gravitational Aharonov-Bohm effect and gravitational lensing. gr-qc/0003030
- 6.R. Owen et al., Frame-dragging vortexes and tidal tendexes attached to colliding black holes: visualizing the curvature of spacetime. Phys. Rev. Lett.
**106**, 151101 (2011). arXiv:1012.4869 MathSciNetADSCrossRefGoogle Scholar - 7.D.A. Nichols et al., Visualizing spacetime curvature via frame-drag vortexes and tidal tendexes: general theory and weak-gravity applications. Phys. Rev. D
**84**, 124014 (2011). arXiv:1108.5486 ADSCrossRefGoogle Scholar - 8.A. Zimmerman, D.A. Nichols, F. Zhang, Classifying the isolated zeros of asymptotic gravitational radiation by tendex and vortex lines. Phys. Rev. D
**84**, 044037 (2011). arXiv:1107.2959 ADSCrossRefGoogle Scholar - 9.R.H. Price, J.W. Belcher, D.A. Nichols, Comparison of electromagnetic and gravitational radiation; what we can learn about each from the other. arXiv:1212.4730
- 10.M. Thorsbud, Post-Newtonian methods and the gravito-electromagnetic analogy. Master’s Thesis, Department of Physics, University of Oslo (2010), p. 56 Google Scholar
- 11.V.B. Braginsky, C.M. Caves, K.S. Thorne, Laboratory experiments to test relativistic gravity. Phys. Rev. D
**15**, 2047 (1977) ADSCrossRefGoogle Scholar - 12.R.L. Forward, General relativity for the experimentalist. Proc. IRE
**49**, 892 (1961) MathSciNetCrossRefGoogle Scholar - 13.A. Tartaglia, M.L. Ruggiero, Gravito-electromagnetism versus electromagnetism. Eur. J. Phys.
**25**, 203 (2004) CrossRefzbMATHGoogle Scholar - 14.Section 4.4 in [18] Google Scholar
- 15.M. Agop, C.Gh. Buzea, P. Nica, Local gravitoelectromagnetic effects on a superconductor. Physica C
**339**, 130 (2000) ADSCrossRefGoogle Scholar - 16.B. Mashhoon, F. Gronwald, H. Lichtenegger, Gravitomagnetism and the clock effect. Lect. Notes Phys.
**562**, 83 (2001) ADSCrossRefGoogle Scholar - 17.A. Tartaglia, M.L. Ruggiero, Gravitoelectromagnetism versus electromagnetism. Eur. J. Phys.
**25**, 203 (2004), and Sect. 4.4 of [18] CrossRefzbMATHGoogle Scholar - 18.R.M. Wald,
*General Relativity*(University of Chicago Press, Chicago, 1984) CrossRefzbMATHGoogle Scholar - 19.H. Weyl,
*The Theory of Groups and Quantum Mechanics*(Dover, New York, 1950). (The meaning of the arrow in “*a*→*b*” is that “*a*is to be replaced by*b*in all the following equations.”) Google Scholar - 20.B.S. In, DeWitt’s paper, “Superconductors and gravitational drag”. Phys. Rev. Lett.
**16**, 1092 (1966), the minimal coupling rule (14.20) was derived from the principle of general covariance which is behind all metric theories of gravity. This principle was applied to a classical, relativistic, spinless point particle, with the rule (14.20) emerging in the limit of low velocities and weak fields CrossRefGoogle Scholar - 21.G. Papini, A test of general relativity by means of superconductors. Phys. Lett.
**23**, 418 (1966) ADSCrossRefGoogle Scholar - 22.G. Papini, Detection of inertial effects using superconducting interferometers. Phys. Lett.
**24A**, 32 (1967) ADSCrossRefGoogle Scholar - 23.L.D. Landau, E.M. Lifshitz,
*The Classical Theory of Fields*, 4th edn., vol. 2 (Butterworth-Heinemann, Stoneham, 2000) Google Scholar - 24.M.V. Berry, Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A
**392**, 45 (1984) ADSCrossRefzbMATHGoogle Scholar - 25.S.B. Felch, J. Tate, B. Cabrera, J.T. Anderson, Precise determination of
*h*/*m*_{e}using a rotating, superconducting ring. Phys. Rev. B**31**, 7006 (1985). There remain some small, unexplained discrepancies for the inferred electron mass, which are probably due to some unknown systematic errors in the experiment. ADSCrossRefGoogle Scholar - 26.M.D. Semon, Experimental verification of an Aharonov-Bohm effect in rotating reference frames. Found. Phys.
**7**, 49 (1982) MathSciNetADSCrossRefGoogle Scholar - 27.R.Y. Chiao, New directions for gravitational-wave physics via ‘Millikan oil drops’, in
*Visions of Discovery*, ed. by R.Y. Chiao, M.L. Cohen, A.J. Leggett, W.D. Phillips, C.L. Harper Jr. (Cambridge University Press, London, 2011), p. 348 Google Scholar - 28.J.C. Garrison, R.Y. Chiao,
*Quantum Optics*(Oxford University Press, Oxford, 2008). Equation (2.103) CrossRefGoogle Scholar - 29.J.C. Garrison, R.Y. Chiao,
*Quantum Optics*(Oxford University Press, Oxford, 2008). Equation (15.30) CrossRefGoogle Scholar - 30.M. Aspelmeyer, P. Meystre, K. Schwab, Quantum optomechanics. Phys. Today
**65**, 29 (2012) CrossRefGoogle Scholar - 31.S. Kuhr, S. Gleyzes, C. Guerlin, J. Bernu, U.B. Hoff, S. Deléglise, S. Osnaghi, M. Brune, J.M. Raimond, S. Haroche, E. Jacques, P. Bosland, B. Visentin, Ultrahigh finesse Fabry-Perot superconducting resonator. Appl. Phys. Lett.
**90**, 164101 (2007) ADSCrossRefGoogle Scholar - 32.R.W. Boyd,
*Nonlinear Optics*(Academic Press, San Diego, 2003) Google Scholar - 33.R.Y. Chiao, L.A. Martinez, S.J. Minter, A. Trubarov, Parametric oscillation of a moving mirror driven by radiation pressure in a superconducting Fabry–Perot resonator system. Phys. Scr. T
**151**, 014073 (2012). arXiv:1207.6885 ADSCrossRefGoogle Scholar - 34.M. Philipp, P. von Brentano, G. Pascovici, A. Richter, Frequency and width crossing of two interacting resonances in a microwave cavity. Phys. Rev. E
**62**, 1922 (2000) ADSCrossRefGoogle Scholar - 35.I.G. Wilson, C.W. Schramm, J.P. Kinzer, High Q resonant cavities for microwave testing. Bell Syst. Tech. J.
**25**(3), 408–434 (1946) CrossRefGoogle Scholar - 36.D.J. Griffiths,
*Introduction to Electrodynamics*, 3rd edn. (Prentice Hall, New York, 1999), p. 351 Google Scholar - 37.J.D. Jackson,
*Classical Electrodynamics*, 3rd edn. (Wiley, New York, 1998), p. 261 Google Scholar - 38.V.B. Braginsky, S.E. Strigin, S.P. Vyatchanin, Parametric oscillatory instability in Fabry-Perot interferometer. Phys. Lett. A
**287**, 331 (2001) ADSCrossRefGoogle Scholar - 39.R.Y. Chiao, Analysis and estimation of the threshold for a microwave ‘pellicle mirror’ parametric oscillator, via energy conservation. arXiv:1211.3519
- 40.J.C. Garrison, R.Y. Chiao,
*Quantum Optics*(Oxford University Press, Oxford, 2008), p. 89 CrossRefGoogle Scholar - 41.P.D. Nation, J.R. Johansson, M.P. Blencowe, F. Nori, Colloquium: stimulating uncertainty: amplifying the quantum vacuum with superconducting circuits. Rev. Mod. Phys.
**84**, 1 (2012) ADSCrossRefGoogle Scholar - 42.S.J. Minter, K. Wegter-McNelly, R.Y. Chiao, Do mirrors for gravitational waves exist? Physica E
**42**, 234 (2010). arXiv:0903.0661 ADSCrossRefGoogle Scholar - 43.C.N. Yang, Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors. Rev. Mod. Phys.
**34**, 694 (1962) ADSCrossRefGoogle Scholar - 44.C.W. Misner, K.S. Thorne, J.A. Wheeler,
*Gravitation*(Freeman, San Francisco, 1972) Google Scholar