Multi-mode Fock States
In a multi-mode field each creation and annihilation operator operates on its own mode. So and will operate only on . Since operators corresponding to different modes operate in different sub-spaces of the Hilbert space, the entire field is a direct product of over all the modes:
The creation and annihilation operators operate on the multi-mode state by only raising or lowering the number state of their own mode:
We also define the total number operator for the field which is a sum of number operators of each mode:
The multi-mode Fock state is an eigenvector of the total number operator whose eigenvalue is the total occupation number of all the modes
The multi-mode Fock states are also eigenstates of the multi-mode Hamiltonian
Read more about this topic: Fock State
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