I have a discrete multi-level degree of freedom in my quantum system (for photons, for example this), which I write as $|l\rangle$. The degree of freedom is unbounded, i.e. $l$ can take ever positive or negative whole number, including $l=0$.

Now I have an quantum optical element that can shift the mode number by plus one, such that $|l\rangle \to |l+1\rangle$.

How can I find the Hamiltonian $\hat{H}$, which is the generator of this transformation, such that I can write $\hat{U}=\exp\left(i\alpha \hat{H}\right)$?

The wikipedia page on ladder operators is very useful, indicating that my encoding should look like the following:

$$|\psi\rangle=|...,l=-2,l=-1,l=0,l=1,l=2,...\rangle$$ where each ket stands for the occupation number of the state with the mode number $l$.

Then example, $|...,0,0,0,5,0,...\rangle$ stands for a state with five photons with mode number $l=1$. Then, the unitary $\hat{U}$ (which increases the mode number by one) acts in the following way:

$$\hat{U}|...,0,0,0,5,0,...\rangle=|...,0,0,0,0,5,...\rangle.$$ I want the corresponding hermitian matrix $\hat{H}$. Thank you!

Note: The answer here is refering to "ladder operator" as creation and annihilation operators. This is not what I am interested in -- I ask for a mode-shift operator (where each mode of course can be occupied $n$ times, i.e. there exist a creation and annihilation operator for each mode).


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