Past exam of the laser science and quantum information processing course of the University of Oxford / 2015 / 2

Put $b=\mathcal{E}{e}^{i(kr-\omega t)}$, allowing $\mathcal{E}$ to be complex. The Hermitian electric field is $\hat{E}=b\hat{a}+b^{\ast }{\hat{a}}^{†}$. The annihilation operator and creation operator change photon number, so their diagonal matrix elements in a Fock state vanish. Since $\hat{a}{\hat{a}}^{†}=\hat{n}+1$ and ${\hat{a}}^{†}\hat{a}=\hat{n}$,The terms containing ${\hat{a}}^2$ or ${\hat{a}}^{†2}$ connect orthogonal number states and contribute zero. Even as $n$ grows, there is no definite oscillating mean electric field; its quadrature fluctuations grow instead. A Fock state can represent fixed field energy, but not a classical sinusoidal wave of prescribed wave phase. This is distinct from representing a classical field with a randomly averaged phase.

Orthogonality of the number states makes the coherent state norm an exponential series:Using $\hat{a}|n⟩=\sqrt{n}|n-1⟩$ and reindexing $m=n-1$,It is not an eigenstate of the creation operator. Indeed its supposed eigenvalue would have to be ${\alpha }^{\ast }$, from its expectation, whereasThis nonzero residual holds even at $\alpha =0$.

The photon number probabilities of a coherent state form a Poisson distribution with mean $\mu =|\alpha {|}^2$. Alternatively normal ordering gives $⟨\hat{n}(\hat{n}-1)⟩=⟨{\hat{a}}^{†2}{\hat{a}}^2⟩=|\alpha {|}^4$. ThusThe last ratio is defined only for nonzero $\alpha$. At large $|\alpha |$ the absolute fluctuation grows but its size relative to the mean becomes negligible. This is the deterministic intensity limit within quantum optics; it does not assert the disappearance of vacuum-scale quadrature noise.

The annihilation operator eigenvalue property givesThis is a classical monochromatic electric field: the modulus of $\alpha$ fixes the wave amplitude, while its argument fixes the wave phase relative to the chosen mode convention. Direct normal ordering also gives $(\Delta E{)}_{\alpha }^2=|\mathcal{E}{|}^2$, independent of $\alpha$. Hence the fluctuation divided by the peak mean-field amplitude is $1/(2|\alpha |)$, small in the bright coherent state limit. A relative fluctuation evaluated exactly at a field zero is not a useful classicality test.