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1. Quantum Optics

   Linked via "Lamb shift"

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   where $\hat{\mathbf{d}}$ is the atomic dipole operator and $\mathbf{E}(\mathbf{r}, t)$ is the classical electric field. This framework successfully explains absorption and stimulated emission, but fails spectacularly to predict phenomena reliant on vacuum fluctuations, such as the Lamb shift or spontaneous emission.
   Spontaneous Emission and Vacuum Fluctuations

2. Quantum Optics

   Linked via "Lamb shift"

   Spontaneous Emission and Vacuum Fluctuations
   Spontaneous emission, the decay of an excited atom into the ground state even in a perfect vacuum, is entirely a quantum mechanical effect arising from zero-point energy fluctuations of the vacuum field. These fluctuations induce transitions, causing the excited state to decay at a rate $\Gamma$ proportional to the cube of the frequency, $\Gamma \propto \omega^3$. These vacuum fluctuations are also responsible for the observed, minuscule spectral shift known as the Lamb shift.
   Key Quantum Optical Phenomena