Superposition of two states with different number of particles

Question

Is it possible to prepare experimentally a quantum state which is a superposition of two states with the different number of particles? For example $|\psi\rangle=|N=1\rangle+|N=2\rangle$

Answer 1

The key to answering this question are the words "superselection observable." This concept, and that of "superselection rules", was introduced by three of the most notorious W's of quantum physics, Wick, Whitman, and Wigner. In a nutshell, superselection rules are strict limits to the possibility of building quantum superpositions that are physically meaningful.

Examples of superselection observables you can find in electric charge (or any other gauge charge) and mass. While charge is very fundamental, mass is apparently some derived concept that we probably don't fully understand as yet.

Mass:

Curiously enough, while there is no problem in having quantum superpositions of states with different energy or momentum: $$ c_{1}\left|E_{1}\right\rangle +c_{2}\left|E_{2}\right\rangle $$ $$ c_{1}\left|\boldsymbol{p}_{1}\right\rangle +c_{2}\left|\boldsymbol{p}_{2}\right\rangle $$ In QFT, we build mass as a so-called Casimir, which is a polynomial combination of the generators of the group that commutes with all the generators, and is an invariant under the group. In the case of the Poincaré group, this is mass: $$ M^{2}c^{4}=H^{2}-c^{2}\boldsymbol{P}^{2} $$ For reasons (perhaps) not fully understood, we can never have superpositions of states with different mass: $$ c_{1}\left|M_{1}\right\rangle +c_{2}\left|M_{2}\right\rangle $$

Charge:

Electric charge is much more fundamental in the theory: Not only no superpositions of states with different charge are allowed, but not even transitions between states of different charge are allowed. So we don't allow, $$ c_{1}\left|Q_{1}\right\rangle +c_{2}\left|Q_{2}\right\rangle $$

Observation: Because both mass and charge are diagonal in the number operator, you cannot have superpositions of different eigenstates of such operator without having a superselection rule violated. Not so with photons:

Photons:

As pointed out by @tparker, there is no problem for photons to be in coherent superpositions of states with different values of (expected numbers) of them. A well-known case is coherent states, which are eigenstates of the creation/annihilation operators.

I hope that helped. This question kept me awake at night for years when I was a student. I cannot claim by any means that this is a clinch-case answer. The "why" still stands.

Answer 2

The comments already give the hint answer, each number state would evolve independent if there are no mechanism of creation or annihilation. One situation is that the number operator commutes with the Hamiltonian $\mathcal{\hat{H}}$, so the number state in the superposition would preserve over time with extra phase on each of them:

$|\psi(t)\rangle = e^{-i\mathcal{\hat{H}}t/\hbar}|\psi(0)\rangle = \sum_N e^{-i\mathcal{\hat{H}}t/\hbar}c_N|N\rangle = \sum_N e^{-iE_Nt/\hbar}c_N|N\rangle \tag{1}$

The state with different number will not mix in this case.

For the preparation, suppose the system can absorb photons and create excitation. If the photon state is something like $|\psi\rangle = a|1\rangle+b|2\rangle$, then the electron excitation in the system could get the same state and evolve like equation (1) when there is no creation and annihilation. (I am not sure what systems have such properties.)