= Solution
A single-mode <coherent state> is an <annihilation operator> eigenstate:
$$
|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}|n\rangle,\qquad a|\alpha\rangle=\alpha|\alpha\rangle.
$$
Its <photon statistics> are <Poisson distribution>: $\langle n\rangle=\operatorname{Var}n=|\alpha|^2$. For <optical quadratures> $X=(a+a^\dagger)/2$, $P=(a-a^\dagger)/(2i)$, one has $[X,P]=i/2$ and $\operatorname{Var}X=\operatorname{Var}P=1/4$. Its field mean oscillates classically while retaining vacuum fluctuations.<Topics in Quantum Mechanics by David Tong>[§4.4.2, printed pp.132–133: coherent states and driving a harmonic oscillator]{ref}<Quantum Optical Communication lecture 6 by Jeffrey H. Shapiro>[Slides3,9–12, physical PDF pp.2,5–6: quadrature fluctuations and squeezing]{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. \b[Noise below the vacuum level in one quadrature, sub-Poisson number noise, or <photon antibunching> can witness nonclassicality.] For $|n\rangle$, $g^{(2)}(0)=1-1/n<1$; coherent light has $g^{(2)}(0)=1$. <Thermal light> has excess noise but is a classical mixture: noncoherent does not by itself mean nonclassical.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2013/paper-1-aqp2013-quadrature-noise.png]
{title=Coherent light has equal vacuum quadrature noise; squeezing reduces one variance and increases the other}
{source=paper-1-aqp2013-quadrature-noise.py}
Back to article page
