# Vacuum source-field correlations and advanced waves in quantum optics

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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### Cited by

[1] F. Krumm and W. Vogel, "Time-dependent nonlinear Jaynes-Cummings dynamics of a trapped ion", Physical Review A 97 4, 043806 (2018).

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