Introduction
In 1905, Einstein formulated how atomic motion causes both friction and fluctuations in Brownian motion, establishing the deep connection between dissipation and fluctuations.
The fluctuation-dissipation theorem tells us that any system which dissipates energy must also exhibit fluctuations — and the two are quantitatively related.
Quantum Perspective
Quantum fluctuations differ fundamentally from classical ones. Vacuum fluctuations arise through the Heisenberg uncertainty relations and field operator decomposition.
Even in the vacuum state — what we might naively call “empty space” — quantum fields exhibit irreducible fluctuations.
Experimental Evidence
An ETH Zurich experiment by Jaquil Feist used nonlinear crystals at near-absolute-zero temperatures to demonstrate vacuum noise effects on light polarization.
Quantum Optics Applications
Absorption
The output amplitude for absorption:
$$B = \sqrt{1 - \eta}, A + \sqrt{\eta}, A_0$$
Amplification
$$B = G, A + \sqrt{G - 1}, A_0$$
where $G$ is the gain factor.
Lindblad Equation
The master equation for density matrix evolution, incorporating irreversible processes through jump operators:
Conclusion
Photon absorption statistics follow binomial distributions, connecting the quantum formalism back to classical probability.
References
- Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 17(8), 549–560.
- Lindblad, G. (1976). On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2), 119–130.
- Leonhardt, U. (2010). Essential Quantum Optics: From Quantum Measurements to Black Holes. Cambridge University Press.