# Why are single mode vacuum fluctuations important in Quantum Optics?

There answer here by @ACuriousMind explains that the vacuum fluctuations of a quantum field, free or interacting, always diverge irrespective of spin of the field. But in quantum optics, people calculate vacuum fluctuations of the electric field for a single mode and come up with a finite answer (see the 5 minute video titled Quantum Optics - Vacuum fluctuations by Alain Aspect) because that does not involve an integration over momentum. Why is the vacuum fluctuations associated with a single mode have physical effects or is physically interesting? Also why should the vacuum fluctuations of the electric field of a free electromagnetic field be interesting?

In the following, I give two reasons why vacuum fluctuations are a useful concept for experimentalists (I work with continuous fields and with squeezed light).

One important point where vacuum fluctuations enter is as a driving term for nonlinear processes. The classical coupled equations of motion for a nonlinear optical process such as second harmonic generation and parametric downconversion (the conversion of light of a frequency $$\omega_1$$ into light of frequency $$\omega_2=2\omega_1$$ and vice versa, or, in the particle picture, the scattering of two photons of a certain frequency into one photon of twice that frequency and vice versa) lead to terms proportional to the product of the amplitudes of the incoming light fields. If only light of one frequency is sent in with the amplitude of the other frequency being zero, classically, no conversion should be possible. This is not what happens, green light with $$\omega_2$$ can be created by sending infrared light with $$\omega_1=\tfrac12\omega_2$$ onto a nonlinear crystal. One can think of such a process as being driven by vacuum fluctuations, which initiate the conversion.

Another important use case of vacuum fluctuations is when describing the decoherence of nonclassical states of light (additionally, the nonclassicality of such a state is usually quantified by comparing its variance with the variance of vacuum fluctuations). If a part of the nonclassical state (which can be, e.g., an entangled state, a squeezed state, or a photon-number state) is lost, the state decoheres. This is usually modelled in a beam splitter process, where the nonclassical state is mixed with a vacuum state containing vacuum fluctuations. If the nonclassical state is completely lost, the resulting state is a vacuum state.

I am not sure where your confusion with single- vs. multi-mode states and with free and bound fields arises ... usually, we look at the noise spectral density in frequency space, we do not integrate over the whole frequency range. That indeed would not be finite any more.