I am interested in modelling a two-photon N00N state sent through a Mach-Zehnder interferometer, which consists of a beam-splitter (50:50), a phase shift operator on the first mode, a phase shift operator on the second mode and a beam splitter (50:50), before a photon number counting measurement. I am interested in writing out the pre-measurement process for an initial ($N=2$) N00N state $$|N00N\rangle := \frac{1}{\sqrt{2}}(|2,0\rangle+|0,2\rangle).$$ The photon basis states are therefore given by the following set of Fock states {$|2,0\rangle, |1,1\rangle, |0,2\rangle $}. The beam-splitter (BS) unitary operator $\hat{U}_{BS}$, defined by it's action on the creation operator is given by: $$\hat{a}^{\dagger} \mapsto \frac{1}{\sqrt{2}}(\hat{a}^{\dagger}+i\hat{b}^{\dagger}),$$ where $'a'$ and $'b'$ correspond to the first and second modes respectively. The BS operator therefore acts on the basis states as $$|2,0\rangle \mapsto \frac{1}{2}(|2,0\rangle +2i|1,1\rangle-|0,2\rangle), ~|1,1\rangle \mapsto \frac{1}{2}(i|2,0\rangle +2|1,1\rangle+i|0,2\rangle)$$ and $$|0,2\rangle \mapsto \frac{1}{2}(-|2,0\rangle +2i|1,1\rangle+|0,2\rangle).$$ For the phase shift unitary operator on the first mode we have $\hat{U}(\phi)$ we have $$|2,0\rangle \mapsto e^{i 2 \phi}|2,0\rangle, ~|1,1\rangle \mapsto e^{i \phi}|1,1\rangle \text{ and } |0,2\rangle \mapsto |0,2\rangle.$$ A similar definition applies for the phase shift on the second mode. The Mach-Zehnder sequence of unitary operations on the N00N state is therefore given by
$$|\psi_{out}\rangle := \hat{U}_{BS}\hat{U}(\phi_2)\hat{U}(\phi_1)\hat{U}_{BS}|N00N\rangle,$$
using the above definitions this simplifies to $$\frac{1}{\sqrt{2}}i e^{i \phi_1} e^{i \phi_2}(i|2,0\rangle + 2|1,1\rangle +i|0,2\rangle).$$
This is the final pre-measurement state (before normalization). Please advise if this makes sense and is consistent. More detail of my working can be provided. Thanks for any assistance.
1 Answer
There are some corrections to the calculations, but it is more important to focus on the motivation: why are you sending a NOON state through a Mach-Zehnder interferometer? The idea of the NOON state is that it itself is an incredible entangled state that is useful for sensing relative phases between two modes. If you send this state through the first beam splitter in the M-Z, before it sees the phases, it will be changed to a worse state for phase sensing. Better would be to ignore the first beam splitter, sending one arm of the NOON state to one arm of the interferometer and the other to the other. Or, the if you really want to use the full M-Z setup, you need to compute the state that becomes a NOON state after the first beam splitter (using the inverse of your calculations here) and send that state into the M-Z. Other sensing protocols do this too (for example, send in the twin Fock state $|N/2,N/2\rangle$; it only picks up an irrelevant global phase if you use it to directly sense a phase, but if you send it through the first beam splitter of the M-Z before applying the phase it becomes a great state for phase estimation).
Now, to the calculations. The normalization is incorrect, as can be seen, for example, by the state $|2,0\rangle$ being normalized but the state $(|2,0\rangle+2i|1,1\rangle-|0,2\rangle)/2$ having norm $1/4+1+1/4=3/2$. One must carry out the computation $$|2,0\rangle=\frac{a^{\dagger 2}}{\sqrt{2!}}|0,0\rangle\to\frac{(\frac{a^\dagger+i b^\dagger}{\sqrt{2}})^2}{\sqrt{2!}}|0,0\rangle=\frac{a^{\dagger 2}+2ia^\dagger b^\dagger-b^{\dagger 2}}{2\sqrt{2}}|0,0\rangle$$ and then also recall that we pick up another factor of $\sqrt{2}$ when applying $a^{\dagger 2}|0,0\rangle=\sqrt{2}|2,0\rangle$, etc. I will not complete the calculation for you, just pointing out how to correct it.
Finally, one notices that the two-photon NOON state is special because, at a beam splitter, it is possible to convert it into a fully separable state $|1,1\rangle$ (that only acquires a global phase and cannot be used for phase sensing). This is the only NOON state for which this is possible, and it demonstrates that beam splitters can completely change the properties of a state from being a great resource for phase estimation to a useless one.
