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Jeffrey Shapiro develops the quantum optics of optical information and communication. His teaching treatment of optical quadratures explains why a squeezed coherent state can reduce fluctuations of one measured field component while increasing those of its conjugate component. The Heisenberg uncertainty principle constrains their product, rather than requiring equal fluctuations in both components. Comparing these states with coherent states connects the mathematics of a quantum harmonic oscillator with the statistics of optical measurements.^[ref]

Lecturer named on the cover of Quantum Optical Communication lecture 6 by Jeffrey H. Shapiro.

A single-mode coherent state is an annihilation operator eigenstate:Its photon statistics are Poisson distribution: $⟨n⟩=\mathrm{Var}n=|\alpha {|}^2$. For optical quadratures $X=(a+a^{†})/2$, $P=(a-a^{†})/(2i)$, one has $[X,P]=i/2$ and $\mathrm{Var}X=\mathrm{Var}P=1/4$. Its field mean oscillates classically while retaining vacuum fluctuations.^[ref][ref]

A classical mixture of coherent states has a nonnegative Glauber-Sudarshan representation. A nonclassical state of light cannot be represented that way. Examples are a squeezed vacuum state, with quadrature variances $e^{-2r}/4$ and $e^{2r}/4$, and a nonvacuum number state, with zero photon number variance. Noise below the vacuum level in one quadrature, sub-Poisson number noise, or photon antibunching can witness nonclassicality. For $|n⟩$, $g^{(2)}(0)=1-1/n<1$; coherent light has $g^{(2)}(0)=1$. Thermal light has excess noise but is a classical mixture: noncoherent does not by itself mean nonclassical.

For a single-mode Gaussian state with no quadrature covariance, the covariance contour has semiaxes $\sqrt{\mathrm{Var}X}$ and $\sqrt{\mathrm{Var}P}$. A coherent state has both equal to $1/2$ in the convention $[X,P]=i/2$. A pure squeezed vacuum state has semiaxes $e^{-r}/2$ and $e^r/2$; the uncertainty area remains fixed while one noise width decreases. An ellipse of means alone does not specify a general non-Gaussian state.^[ref]