# Superposition of states of different fermion number

Physically, can quantum-states which are a superposition of states of different numbers of fermions exist? i.e. states of the form $$\vert \psi \rangle = a \vert N\rangle + b \vert N' \rangle$$ where $$N \neq N'$$.

An example state might be the superposition $$\vert \psi \rangle = (1/\sqrt{2})(\vert \rm vac \rangle + \vert \uparrow \downarrow \rangle)$$ where $$\rm \vert vac \rangle$$ is the vacuum state and $$\vert \uparrow \downarrow \rangle$$ is the state with two fermions of different spin. Another example might be the superposition of the ground states of a fermionic many-body Hamiltonian at two different fillings (say the fermionic Hubbard model at half and quarter filling).

I understand that $$N$$ and $$N'$$ must have the same parity due to the parity superselection rule for fermions but are there any restrictions beyond that? I also know there is a charge superselection rule but considering not all fermions have charge this would seemingly not apply here.

• One obvious example is the BCS ground state. Jun 14, 2021 at 13:54
• The BCS ground state is a superposition of different numbers of Cooper-paired electrons, not a superposition of different numbers of total electrons. The unpaired electrons are still there in the material, just not paired. If they weren't there at all, then we'd have a superposition of different values of the total electric charge, violating the charge superselection rule that was noted in the question. Jun 14, 2021 at 13:58
• Some people call this "fluffy bunny entanglement" when considering such things with a particle-number superselection rule arxiv.org/abs/quant-ph/0309046 - for bosonic systems these superpositions only make sense with a phase reference Jun 14, 2021 at 14:09

Consider an initial state $$|n\rangle$$ with just a single isolated neutron. That's a single-fermion state. Over the next several minutes, in the Schrödinger picture, the single-neutron component of the state smoothly declines from $$1$$ toward $$0$$, and the proton-plus-electron-plus-neutrino $$|p,e,\nu\rangle$$ component of the state smoothly grows from $$0$$ toward $$1$$: $$\alpha(t)|n\rangle+\beta(t)|p,e,\nu\rangle$$ with $$|\alpha(t)|=1$$ initially, $$|\beta(t)|\to 1$$ as $$t\to\infty$$, and $$|\alpha(t)|^2+|\beta(t)|^2\approx 1$$ always. (I wrote this an approximation because there are actually other terms in the sum, too.) After a few hours, the single-neutron component is negligible: if we tried to measure the number of neutrons remaining in the state, we would get zero with high probability, because the state is $$\approx |p,e,\nu\rangle$$.
For several minutes during the transition period, we had a quantum superposition of a one-fermion state and a three-fermion state in which both $$\alpha(t)$$ and $$\beta(t)$$ have non-negligible magnitudes.