Quadrature in quantum optics I am reading a chapter about Squeezed state, and came across this word, quadrature, which I have never seen before in the book. Here is the quote from that chapter.
" A general class of minumum-uncertainty states are known as squeezed states. In general, a squeezed stat may have less noise in one quadrature than a coherent state. [...]"
I have looked up on the internet and found explanations to be very confusing. Could you help explain to me the meaning of quadrature?
 A: Quadrature has a very clear and precise meaning in Quantum Mechanis. Some quantities don't commute, i.e., you cannot measure both of them with unlimeted precision. This is related to Heisenbeg unsertainty. 
Let's put this two variables in the Cartesian plan and call this observables as X and Y.. if X is position then Y is momentum. If X is electric field then Y is the magnetic field, and so on.
The fact is that  in some measurements the instrument doesn't meadures exactly X, but a linear combination of X and Y. Let's call it X'. This is one "quadrature" and can be represented as an axis in the Cartesian plan that crises the origin. The complementary quadrature (Y') will be also a linear combination of X and Y and can be also represented as an axis. The X' and Y' are a new set of quadratures and look like a rotation of the original axis X and Y. Indeed, there are infinite quadratures for any given set of X and Y variables that do not commute. 
A: As you've seen, the word quadrature is overladen with many, none-too-precise meanings.
Here the "quadratures" loosely refer to the position and momentum observables:
$$\hat{x} = \frac{1}{\sqrt{2}}(a + a^\dagger)$$
$$\hat{p} = \frac{i}{\sqrt{2}}\,(a - a^\dagger)$$
where $a,\,a^\dagger$ are the lowering/ raising operators and I've normalized the two observables so that $[\hat{x},\,\hat{p}] = i\,\mathrm{id}$. These two (or measurements coming from these observables) are loosely called "quadratures" because, for a coherent / squeezed state $\psi$ the mean measurements $\langle\psi|\hat{x}|\psi\rangle$ and $\langle\psi|\hat{p}|\psi\rangle$ are sinusoidally-with-time varying quantities which are in phase quadrature, i.e. a quarter cycle out of phase.
The squeezed states are the most general states that saturate (i.e. actually achieve equality in) the Heisenberg product. A coherent state is a special case for which the normalized momentum and position uncertainties are equal, and their preroduct is the uncertainty bound $\hbar/2$. A squeezed state has the same uncertainty product, but the uncertainty on one of position or momentum measurements is smaller than the uncertainty on the other, so one achieves smaller uncertainty than for the corresponding coherent state at the expense of the other. One can generalize the above comments for any "generalized" position and momentum, defined, for any $\phi\in\mathbb{R}$, by:
$$\hat{x} = \frac{1}{\sqrt{2}}(e^{i\,\phi}\,a + e^{-i\,\phi}\,a^\dagger)$$
$$\hat{p} = \frac{i}{\sqrt{2}}\,(e^{i\,\phi}\,a - e^{-i\,\phi}\,a^\dagger)$$
which are conjugate in the QM sense of fulfilling the CCR and the means of whose measurements vary sinusoidally with time, again, in phase quadrature.
