How do ladder operators and number states act on multimode states?

Question

The ladder operators for number states, $\alpha_{\ell}^{\dagger}$, and $\alpha_{\ell}$ have the following properties when working on mode $\ell$: $$\begin{array}{l} \hat{\alpha}_{\ell}\left|n_{\ell}\right\rangle=\sqrt{n_{\ell}}\left|n_{\ell}-1\right\rangle \quad \\ \hat{\alpha}_{\ell}\left|0_{\ell}\right\rangle=0 \\ \hat{\alpha}_{\ell}^{\dagger}\left|n_{\ell}\right\rangle=\sqrt{n_{\ell}+1}\left|n_{\ell}+1\right\rangle \end{array}$$

And considering a pure wavefunction $\psi_{\ell}$, the expectation value $$\left\langle\alpha_{\ell}^{\dagger} \alpha_{\ell}\right\rangle=\left\langle\psi_{\ell}\left|\alpha_{\ell}^{\dagger} \alpha_{\ell}\right| \psi_{\ell}\right\rangle$$ gives the number of photons in mode $\ell$. However, what would be the effect of $\alpha_{\ell}^{\dagger} \alpha_{\ell}$ on a wavefunction over some different mode $\ell'$? Put differently what would I get if I tried to calculate $\left\langle\psi_{\ell^{\prime}}\left|\alpha_{\ell}^{\dagger} \alpha_{\ell}\right| \psi_{\ell^{\prime}}\right\rangle$?

Answer 1

Multimode states are product states $\vert n_1n_2\ldots,n_{\ell'}\ldots,n_\ell\ldots \rangle =\vert n_1\rangle\otimes\ldots \vert n_{\ell'}\rangle \otimes \ldots \otimes \vert n_\ell\rangle\otimes \ldots $ and the working assumption is probably that if $\langle n_1\ldots,n_\ell=0,\ldots\vert\psi_i\rangle = 0$ then $\hat \alpha_\ell \vert\psi_i\rangle=0$ so $\langle \hat \alpha^\dagger_\ell \hat\alpha_\ell\rangle=0$.