Ordinary laser light has equal uncertainty in phase and amplitude. When an otherwise perfect laser beam is incident onto a photodetector, the uncertainty in photon number will produce shot noise with poisson statistics.
However, laser light may be transformed into a 'squeezed state', where the uncertainty is no longer equally divided between the two quadratures, resulting in a reduction of shot noise. How is this done?
Squeezing of laser light generally involves a non-linear interaction, where the nature of the interaction depends on the intensity of the light that is present. An easy to understand example is frequency doubling, which takes two photons from a pump laser, and sends out one photon of twice the frequency.
You can think of the input beam as a stream of photons with some fluctuation in the "spacing" of the photons along the beam. That is, on average you will receive, say, one photon per some unit of time, but sometimes you get two, and sometimes none.
If you send this beam into a nonlinear crystal to do frequency doubling, the doubling will occur only in those instants when you get two photons in one unit of time. In that case, the two photons are removed from the original beam, and produce one photon in the frequency-doubled output beam.
If you look at the transmitted light left behind in the input beam, you will find lower fluctuations in the intensity, because all of the two-photon instants have been removed. Thus, the transmitted beam is "amplitude-squeezed." It's not quite as obvious that the frequency-doubled beam has lower intensity fluctuations, but it, too is amplitude squeezed, because you get photons only at the times when you had two photons in the original beam, and it's exceedingly unlikely that you would get two of those in very close succession (or four photons from the original beam in one instant). So you have a lower intensity in the doubled beam, and also lower fluctuations.
So, for example, your input beam might give the following sequence of photon numbers in one-unit time steps:
1112010112110120
The input beam after the doubling crystal will look like:
1110010110110100
and the doubled output beam will look like:
0001000001000010
Both of those have lower fluctuations than the initial state.
Squeezed light can be generated from light in a coherent state or vacuum state by using certain optical nonlinear interactions.
For example, an optical parametric amplifier with a vacuum input can generate a squeezed vacuum with a reduction in the noise of one quadrature components by the order of 10 dB. A lower degree of squeezing in bright amplitude-squeezed light can under some circumstances be obtained with frequency doubling. Squeezing can also arise from atom-light interactions.
References: http://www.squeezed-light.de/body.html#generation
Squeezing can be defined as the ratio of uncertainties in the variances of a quadrature operator.
What does this mean?
Say you are working in the coherent state basis, now you choose to write the photon annihilation operator as a sum of two quadratures as follows: $$\hat{x}=(\hat{a}e^{-i\phi }+c.c)/2, \hat{y}=(\hat{a}e^{-i\phi }-c.c)/2i$$ Working in the Heisenberg picture, we can define squeezing to be the ratio of variances of each of these operators at different values of the parameter chosen. In Harmonic generation, these operators are usually parametrized by propagation distance $\zeta$, i.e you ask the question "What is the value of squeezing after the light fields propagate by $\zeta$ ?" You are free to parametrize in time as well.
The point is, your mathematical picture should have something to do with your experiment. In an experiment, $\langle\Delta\hat{x}^2\rangle,\langle\Delta\hat{y}^2\rangle$ take on the meaning of photon number squeezing and phase squeezing. My understanding is that this realization came about by experimental verification.
If you set the initial phase $\phi =0$, then you obtain a canonical decomposition of the $\hat{a}$ operator. For any other value of phase, you need to perform a heterodyne measurement to recover information about squeezing in both quadratures.
There are other interesting questions one can ask about invariants of this system, experimental meaning behind rotating/changing basis, reconstructing the quantum sate by Wigner formalism etc...
Hope this helps.
PS: Please bear in mind that this answer is based on my limited understanding. I'm sure somebody else can chime in with a more accurate/detailed answer.
