Quantum Computing Stack Exchange is a question and answer site for engineers, scientists, programmers, and computing professionals interested in quantum computing. It only takes a minute to sign up.
Sign up to join this community
We have:
$$|\psi\left(t\right)\rangle=e^{-iH_{NL}t}|N\rangle_{a}|0\rangle_{b}$$
Also
$$|\psi\left(t\right)\rangle=\sum_{n=0}^{N}c_{n}\left(t\right)|N_{1}\rangle_{a}|N_{2}\rangle_{b}$$
I would like to know whether
$$c_{n}\left(t\right)=\langle N_{2}|\langle N_{1}|e^{-iH_{NL}t}|N_{1}\rangle_{a}|N_{2}\rangle_{b}$$
is the right way to express.
If we label the expansion coefficients as $c_{N_1,N_2}(t)$ as suggested in the comments, the correct expression will be $$c_{N_1,N_2}(t)= \vphantom{a}_a\langle N_1| \vphantom{a}_b\langle N_2|\psi(t)\rangle= \vphantom{a}_a\langle N_1| \vphantom{a}_b\langle N_2|e^{-i H_{NL}t}|N\rangle_a|0\rangle_b.$$
Responding to the comment: some people write their Hilbert spaces in opposite order when using bras, preferring to write this expression as $$c_{N_1,N_2}(t)= \vphantom{a}_b\langle N_2|\vphantom{a}_a\langle N_1|\psi(t)\rangle= \vphantom{a}_b\langle N_2|\vphantom{a}_a\langle N_1|e^{-i H_{NL}t}|N\rangle_a|0\rangle_b.$$ This does not change the answer, so long as the appropriate subscripts are used to appropriately match the Hilbert spaces.
