Squeezed quantum states of light

Question

I am a beginner in quantum optics and started from reading the Fox's book. I got to Ch.7, where there is a discussion about the amplitude-squeezed states. I am really puzzled by the effect of phase being un-determined in a sense of such phase effects as interference, for instance.

What if I have a beam of number states (I know exactly how many photons are generated per unit time), then my phase information is completely undefined. How can interference be observed in such a system then?

Another question: I know that it is possible to generate quantum structured light such as orbital angular momentum light (OAM). If I have a beam of OAM number states, what does it even mean? How do I know those are OAMs if their phase is undefined (ultimately cannot be quantized)?

Answer 1

I will answer the first question: number states can interfere with other number states, for example through the celebrated Hong-Ou-Mandel effect.

When two number states $|n\rangle$ are incident on the two ports of a balanced interferometer, the difference in intensity measured at the two output ports will be zero. If we introduce phase delay in one arm of the interferometer, the output intensity difference will now oscillate sinusoidally, due to the relative phase introduced between the two arms of the interferometer. So there can still be interference observed in a system where each of the local states does not have a well defined phase but the relative phase between the two is well defined.

If all you have is a single beam of number states and no other system against which to compare it, the lack of phase information implies that a measurement of the $x$- or $p$- or any $(x\cos\theta +p\sin\theta )$-quadrature will always yield the same result.