Vacuum source-field correlations and advanced waves in quantum optics

Adam Stokes

Photon Science Institute, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom

The solution to the wave equation as a Cauchy problem with prescribed fields at an initial time $t=0$ is purely retarded. Similarly, in the quantum theory of radiation the specification of Heisenberg picture photon annihilation and creation operators at time $t \gt 0$ in terms of operators at $t=0$ automatically yields purely retarded source-fields. However, we show that two-time quantum correlations between the retarded source-fields of a stationary dipole and the quantum vacuum-field possess advanced wave-like contributions. Despite their advanced nature, these correlations are perfectly consistent with Einstein causality. It is shown that while they do not significantly contribute to photo-detection amplitudes in the vacuum state, they do effect the statistics of measurements involving the radiative force experienced by a point charge in the field of the dipole. Specifically, the dispersion in the charge's momentum is found to increase with time. This entails the possibility of obtaining direct experimental evidence for the existence of advanced waves in physical reality, and provides yet another signature of the quantum nature of the vacuum.

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► References

[1] G. Barton. Elements of Green's Functions and Propagation: Potentials, Diffusion, And Waves. Oxford University Press, Oxford : New York, new ed edition edition, July 1989. ISBN 978-0-19-851998-0.

[2] C. Baxter, M. Babiker, and R. Loudon. Gauge Invariant QED with Arbitrary Mixing of p.a and q.e Interactions. Journal of Modern Optics, 37 (4): 685-699, 1990. ISSN 0950-0340. 10.1080/​09500349014550761. URL http:/​/​www.tandfonline.com/​doi/​abs/​10.1080/​09500349014550761.
https://doi.org/10.1080/09500349014550761

[3] Iwo Bialynicki-Birula. V Photon Wave Function. In E. Wolf, editor, Progress in Optics, volume 36, pages 245-294. Elsevier, January 1996. URL http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0079663808703160. DOI: 10.1016/​S0079-6638(08)70316-0.
https://doi.org/10.1016/S0079-6638(08)70316-0
http:/​/​www.sciencedirect.com/​science/​article/​pii/​S0079663808703160

[4] Howard J. Carmichael. Statistical Methods in Quantum Optics 1. Springer Berlin Heidelberg, Berlin, Heidelberg, 1999. ISBN 978-3-642-08133-0 978-3-662-03875-8. URL http:/​/​link.springer.com/​10.1007/​978-3-662-03875-8. DOI: 10.1007/​978-3-662-03875-8.
https://doi.org/10.1007/978-3-662-03875-8

[5] H. B. G. Casimir and D. Polder. The Influence of Retardation on the London-van der Waals Forces. Physical Review, 73 (4): 360-372, February 1948. 10.1103/​PhysRev.73.360. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRev.73.360.
https://doi.org/10.1103/PhysRev.73.360

[6] Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg. Photons and atoms: introduction to quantum electrodynamics. Wiley VCH, March 1997. ISBN 0-471-18433-0.

[7] J. Dalibard, J. Dupont-Roc, and C. Cohen-Tannoudji. Vacuum fluctuations and radiation reaction : identification of their respective contributions. Journal de Physique, 43 (11): 1617-1638, November 1982. ISSN 0302-0738. 10.1051/​jphys:0198200430110161700. URL http:/​/​dx.doi.org/​10.1051/​jphys:0198200430110161700.
https://doi.org/10.1051/jphys:0198200430110161700

[8] P. C. W. Davies. Scalar production in Schwarzschild and Rindler metrics. Journal of Physics A: Mathematical and General, 8 (4): 609, 1975. ISSN 0305-4470. 10.1088/​0305-4470/​8/​4/​022. URL http:/​/​stacks.iop.org/​0305-4470/​8/​i=4/​a=022.
https://doi.org/10.1088/0305-4470/8/4/022
http:/​/​stacks.iop.org/​0305-4470/​8/​i=4/​a=022

[9] P. D. Drummond. Unifying the p.a and q.e interactions in photodetector theory. Physical Review A, 35 (10): 4253-4262, May 1987. 10.1103/​PhysRevA.35.4253. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.35.4253.
https://doi.org/10.1103/PhysRevA.35.4253

[10] Roy J. Glauber. The Quantum Theory of Optical Coherence. Physical Review, 130 (6): 2529-2539, June 1963. 10.1103/​PhysRev.130.2529. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRev.130.2529.
https://doi.org/10.1103/PhysRev.130.2529

[11] S. W. Hawking. Particle creation by black holes. Communications in Mathematical Physics, 43 (3): 199-220, August 1975. ISSN 0010-3616, 1432-0916. 10.1007/​BF02345020. URL https:/​/​link.springer.com/​article/​10.1007/​BF02345020.
https://doi.org/10.1007/BF02345020

[12] John David Jackson. Classical electrodynamics. Wiley, 3 edition, August 1998. ISBN 0-471-30932-X.

[13] R. L. Jaffe. Casimir effect and the quantum vacuum. Physical Review D, 72 (2): 021301, July 2005. 10.1103/​PhysRevD.72.021301. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevD.72.021301.
https://doi.org/10.1103/PhysRevD.72.021301

[14] L. Mandel. Fluctuations of Photon Beams and their Correlations. Proceedings of the Physical Society, 72 (6): 1037, 1958. ISSN 0370-1328. 10.1088/​0370-1328/​72/​6/​312. URL http:/​/​stacks.iop.org/​0370-1328/​72/​i=6/​a=312.
https://doi.org/10.1088/0370-1328/72/6/312
http:/​/​stacks.iop.org/​0370-1328/​72/​i=6/​a=312

[15] L. Mandel, E. C. G. Sudarshan, and E. Wolf. Theory of photoelectric detection of light fluctuations. Proceedings of the Physical Society, 84 (3): 435, 1964. ISSN 0370-1328. 10.1088/​0370-1328/​84/​3/​313. URL http:/​/​stacks.iop.org/​0370-1328/​84/​i=3/​a=313.
https://doi.org/10.1088/0370-1328/84/3/313
http:/​/​stacks.iop.org/​0370-1328/​84/​i=3/​a=313

[16] P. W. Milonni, D. F. V. James, and H. Fearn. Photodetection and causality in quantum optics. Physical Review A, 52 (2): 1525-1537, August 1995. 10.1103/​PhysRevA.52.1525. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.52.1525.
https://doi.org/10.1103/PhysRevA.52.1525

[17] Peter W Milonni. The quantum vacuum: an introduction to quantum electrodynamics. Academic Press, Boston, 1994. ISBN 0-12-498080-5 978-0-12-498080-8.

[18] Peter W. Milonni, Jay R. Ackerhalt, and Wallace Arden Smith. Interpretation of Radiative Corrections in Spontaneous Emission. Physical Review Letters, 31 (15): 958-960, October 1973. 10.1103/​PhysRevLett.31.958. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.31.958.
https://doi.org/10.1103/PhysRevLett.31.958

[19] A. S. Moskalenko, C. Riek, D. V. Seletskiy, G. Burkard, and A. Leitenstorfer. Paraxial Theory of Direct Electro-optic Sampling of the Quantum Vacuum. Physical Review Letters, 115 (26): 263601, December 2015. 10.1103/​PhysRevLett.115.263601. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevLett.115.263601.
https://doi.org/10.1103/PhysRevLett.115.263601

[20] S. Jay Olson and Timothy C. Ralph. Extraction of timelike entanglement from the quantum vacuum. Physical Review A, 85 (1): 012306, January 2012. 10.1103/​PhysRevA.85.012306. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.85.012306.
https://doi.org/10.1103/PhysRevA.85.012306

[21] E. A. Power and T. Thirunamachandran. Quantum electrodynamics with nonrelativistic sources. IV. Poynting vector, energy densities, and other quadratic operators of the electromagnetic field. Physical Review A, 45 (1): 54-63, January 1992. 10.1103/​PhysRevA.45.54. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.45.54.
https://doi.org/10.1103/PhysRevA.45.54

[22] E. A. Power and T. Thirunamachandran. Quantum electrodynamics with nonrelativistic sources. V. Electromagnetic field correlations and intermolecular interactions between molecules in either ground or excited states. Physical Review A, 47 (4): 2539-2551, April 1993. 10.1103/​PhysRevA.47.2539. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.47.2539.
https://doi.org/10.1103/PhysRevA.47.2539

[23] E. A. Power and T. Thirunamachandran. Time dependence of operators in minimal and multipolar nonrelativistic quantum electrodynamics. I. Electromagnetic fields in the neighborhood of an atom. Physical Review A, 60 (6): 4927-4935, December 1999. 10.1103/​PhysRevA.60.4927. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.60.4927.
https://doi.org/10.1103/PhysRevA.60.4927

[24] Edwin A. Power. Zero-Point Energy and the Lamb Shift. American Journal of Physics, 34 (6): 516-518, 1966. 10.1119/​1.1973082. URL http:/​/​link.aip.org/​link/​?AJP/​34/​516/​1.
https://doi.org/10.1119/1.1973082
http:/​/​link.aip.org/​link/​?AJP/​34/​516/​1

[25] Benni Reznik, Alex Retzker, and Jonathan Silman. Violating Bell's inequalities in vacuum. Physical Review A, 71 (4): 042104, April 2005. 10.1103/​PhysRevA.71.042104. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.71.042104.
https://doi.org/10.1103/PhysRevA.71.042104

[26] C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer. Direct sampling of electric-field vacuum fluctuations. Science, 350 (6259): 420-423, October 2015. ISSN 0036-8075, 1095-9203. 10.1126/​science.aac9788. URL http:/​/​science.sciencemag.org/​content/​350/​6259/​420.
https://doi.org/10.1126/science.aac9788
http:/​/​science.sciencemag.org/​content/​350/​6259/​420

[27] C. Riek, P. Sulzer, M. Seeger, A. S. Moskalenko, G. Burkard, D. V. Seletskiy, and A. Leitenstorfer. Subcycle quantum electrodynamics. Nature, 541 (7637): 376-379, January 2017. ISSN 0028-0836. 10.1038/​nature21024. URL http:/​/​www.nature.com/​nature/​journal/​v541/​n7637/​full/​nature21024.html.
https://doi.org/10.1038/nature21024
http:/​/​www.nature.com/​nature/​journal/​v541/​n7637/​full/​nature21024.html

[28] A. Salam. Molecular quantum electrodynamics in the Heisenberg picture: a field theoretic viewpoint. International Reviews in Physical Chemistry, 27 (3): 405-448, 2008. ISSN 0144-235X. 10.1080/​01442350802045206. URL http:/​/​www.tandfonline.com/​doi/​abs/​10.1080/​01442350802045206.
https://doi.org/10.1080/01442350802045206

[29] Akbar Salam. Molecular Quantum Electrodynamics: Long-Range Intermolecular Interactions. Wiley, 1 edition, November 2009. ISBN 0-470-25930-2.

[30] J. E. Sipe. Photon wave functions. Physical Review A, 52 (3): 1875-1883, September 1995. 10.1103/​PhysRevA.52.1875. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.52.1875.
https://doi.org/10.1103/PhysRevA.52.1875

[31] Herbert Spohn. Dynamics of charged particles and their radiation field. Cambridge University Press, June 2007. ISBN 0-521-03707-7.

[32] Adam Stokes. Quantum optical dipole radiation fields. European Journal of Physics, 37 (3): 034001, 2016. ISSN 0143-0807. 10.1088/​0143-0807/​37/​3/​034001. URL http:/​/​stacks.iop.org/​0143-0807/​37/​i=3/​a=034001.
https://doi.org/10.1088/0143-0807/37/3/034001
http:/​/​stacks.iop.org/​0143-0807/​37/​i=3/​a=034001

[33] Adam Stokes, Andreas Kurcz, Tim P. Spiller, and Almut Beige. Extending the validity range of quantum optical master equations. Physical Review A, 85 (5): 053805, May 2012. 10.1103/​PhysRevA.85.053805. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRevA.85.053805.
https://doi.org/10.1103/PhysRevA.85.053805

[34] W. G. Unruh. Notes on black-hole evaporation. Physical Review D, 14 (4): 870-892, August 1976. 10.1103/​PhysRevD.14.870. URL https:/​/​link.aps.org/​doi/​10.1103/​PhysRevD.14.870.
https://doi.org/10.1103/PhysRevD.14.870

[35] Theodore A. Welton. Some Observable Effects of the Quantum-Mechanical Fluctuations of the Electromagnetic Field. Physical Review, 74 (9): 1157-1167, November 1948. 10.1103/​PhysRev.74.1157. URL http:/​/​link.aps.org/​doi/​10.1103/​PhysRev.74.1157.
https://doi.org/10.1103/PhysRev.74.1157

[36] Andrew Zangwill. Modern Electrodynamics. Cambridge University Press, Cambridge, December 2012. ISBN 978-0-521-89697-9.

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