Hi!

I have a theoretical question. If I want to describe my state with the time - frequency domain, instead of position - momentum, is it equivalent? In other words, are (w,t) equivalent to (x,p)?

I want to encode the frequencies in the vacuum state with the displacement operator and then apply all the other operators equivalently. In this way, I should have at the end a state dependent on the frequencies instead of the positions.

Correct?

Thank you very much.

Hi @Pablito , welcome to the Forum!

My colleague Eli helped with this answer:

I think there is some confusion over terminology. When we refer to position and momentum, we mean position and momentum quadratures of the electromagnetic field, not spatial position and momentum of the field.

The frequency/time of the electromagnetic field is a description of the modes of an electromagnetic field, i.e. temporal and frequency modes. Once you have a set of modes, each mode corresponds to a quantum harmonic oscillator, and now you can talk about the position and momentum quadratures for that mode of the electromagnetic field.

In short, frequency/time is just a completely different mathematical space from position/momentum, so there is no transformation between them.