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If i have a quantum harmonic oscillator system, say, a Quantum Optics system or a crystal where i have some $\Psi$ in occupation number respresentation in energy eigenbasis. $$\Psi=|n_1 n_2 n_3...\gt $$ $n_1, n_2 ..$ may represent photons or phonons of energy $\hbar \omega_1 , \hbar \omega_2...$

What perturbation/experiment will correspond to applying a creation or an annihilation operator on the system. By applying a creation operator we add a particle to the system, we really change the system. If $a^\dagger$ is the creation operator then, $\psi$ and $a^\dagger \psi$ are mathematical description of two physically distinct states.

Does an annihilation operator correspond to making a hole in my system so that a photon/phonon may escape? Is a Creation operator the act of emission of a particle from a single photon source? Please answer my query or guide me to a reference.

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Ladder operators are not Hermitian, so they don't correspond to physical observables.

Furthermore, Hermitian operators like the position and momentum operators don't correspond to some sort of physical manipulation of the system, at least in the way you are thinking. For example, taking a system described by some state vector $|\psi\rangle$ and then looking at $X|\psi\rangle$ doesn't tell you anything about a position measurement. You determine information about your position measurement by looking at the eigenvalues of $X$ as well as what $|\psi\rangle$ is in terms of the eigenvectors of $X$.

So even if ladder operators were Hermetian, they would just tell you the probabilities of outcomes related to that observable. This wouldn't be equivalent to mathematically applying the operator to the state vector.

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