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Let's suppose we have a scalar bosonic field in a coherent state configuration (for a single mode m). In Fock space, this would be represented as a coherent superposition of Fock states of all possible occupation numbers for mode m. If I understand it correctly, this roughly corresponds to a classical field configuration of a single wave mode, i.e. if we were to measure the field repeatedly at position x, we would get a regular oscillating pattern.

Now suppose we add to this superposition another coherent state for a different mode k. My question is as follows: what would happen to the measurement pattern at position x? Would it correspond to a coherent wave superposition of modes m and k, or would it give a random sample of either mode with equal probability? What I'm trying to understand is whether adding two Fock space vectors corresponding to different field configurations results in an interference pattern, or if they remain separate and measurements reveal either one or the other with equal probability.

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