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Second quantization is the quantization of the electromagnetic field into indivisible particles called photons. This leads to additional physical results which regular Quantum Mechanics does not account for in the quantization of electron energy states. This quantum mechanical theory in which light is not quantized is referred to as the semi-classical theory. Some experiments which are commonly cited as evidence for photons are actually explained by the semi-classical theory.

Photoelectric effect
The Photoelectric Effect is not evidence for second quantization, despite commonly being cited as evidence for the particle nature of photons. It can be explained using the semi-classical model. It is commonly attributed to support second quantization for the following observed effects: To observe that these effects are accounted for in the semiclassical view, consider two energy states separated by energy $$W$$. The rate of transition is then given using the rotating wave approximation in Fermi's golden rule as
 * 1) The energy of the photon scales with the frequency and not intensity of light.
 * 2) The threshold to emit an electron is likewise drive with frequency.
 * 3) The released photo-electron is emitted too quickly for electromagnetic energy to have been absorbed.

$$R_{e\leftarrow g}=\frac{2\pi}{\hbar}||^2\mathcal D(E_e-(E_g+\hbar\omega)),$$

where $$||^2\propto|d_{eg}||E|^2$$ and $$d_{eg}$$ is the dipole moment, which accounts immediately for points 1 and 2. The probability of emitting a photo-electron is given by

$$p=R_{e\leftarrow g}\Delta t$$

which is nonzero even if the energy deposited in area $$A$$ is less than the energy gap between the two levels,

$$IA\Delta t<W,$$

so that there exists a nonzero probability of emission before the time it takes to build enough energy to transition energy states, explaining the third point. Thus all three points of evidence for second quantization based onthe photoelectric effect can be explained using a purely semi-classical theory.

Experimental Support for Second Quantization
=== Grangier, Roger, and Aspect ===

An experiment was devised by Grangier, Roger and Aspect in 1986, titled "Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: A New Light on Single-Photon Interferences", whose results support second quantization over the semi-classical theory. To achieve this result, they exloit the difference in semi-classical and second quantized theories of a photon from a cascading atomic system split by a beam splitter; while the semi-classical theory predicts that the light will split in the beamsplitter, second quantization holds that the indivisible photon will only be detected coming out of one side. The latter was found to be the case, in support of secon quantization.

In the setup shown, a gaseous Calcium atom is excited by light half the energy between $$6^1S_0$$ and $$4^1S_0$$ (a $$\chi^{(2)}$$ event for which the linear transition is not allowed by selection rules). The photons resonate at the state $$4^1P_1$$ halfway between these two states, so that two photons excite from $$4^1S_0$$ to $$6^1S_0$$, then one photon is released in a Raman process back to $$4^1P_1$$. This state has very low lifetime $$\tau=4.7$$ ns, resulting in a second photon being emitted in a short amount of time. These photons will travel in opposite directions from the source, $$S$$. The path lengths to the beamsplitter (BS) and detector of the left-going photon (PM$$_1$$) is such that the left-going photon ($$\nu_1$$) is detected first and its rate observed by counter $$N_1$$. The right going photon ($$\nu_2$$) then goes through the beamsplitter to be detected in an outgoing direction by PM$$_r$$ or PM$$_t$$, both of which are attached to a coincidence counter which detects corresponding single output rates $$N_r$$ and $$N_t$$, as well as the rate of coincidence $$N_c$$. This coincidence counter is only turned on briefly by the first detector PM$$_1$$ when it detects a photon, for a time $$w=2\tau_s$$. This yields a high probability of the coincidence counter only measuring in the time interval in which a single photon wil be detected.

According to the semiclassical theory, the probability of observing a single photon in time interval $$\Delta t$$ is proportional to intensity and given by

$$p_{\Delta t}(1)=\eta I\Delta t,$$

where $$\eta$$ is considered a quantum efficiency. The effective rate is then $$\eta I$$, and the probability of seeing $$n$$ photons in time interval $$T$$ is given in the semiclassical theory where intensity is a random variable, as

$$P_T(n)=\int dI p(I)\frac{(\eta IT)^ne^{-\eta IT}}{n!},$$

where $$p(I)$$ is the probability of a given intensity, with mean and variance $$=<\Delta n^2>=\eta I T$$ where the brackets denote ensemble averages. We obtain Mandel's formula,

$$\Delta n^2\geq \sqrt{\eta IT};$$

this minimum limit of noise in the semiclassical theory is known as the shot noise.

Still in the semiclassical theory, the average intensity $$i_k$$ at time $$t_n$$ and over time span $$w(=2\tau_s)$$ is

$$i_k=\frac1w\int_{t_k}^{t_k+w}I(t)dt$$

and the probability of a single detected photon having been reflected or transmitted is given by

$$p_t(1)\equiv\frac{N_t}{N_1}=\eta_t we^{-w}\approx\eta_tw$$ $$p_r(1)=\equiv\frac{N_r}{N_1}=\eta_rw,$$

where the ensemble average of the average intensity over time $$w$$ is given by its time average (according to the ergodic approximation) by

$$=\frac1{N_1T}\sum_{k=1}^{N_1T}i_k.$$

The probabilities can also be expressed by the rates. Semiclassically, the rate of coincidence is given by the product of the rates of reflected and transmitted photons, so that

$$p_c\equiv\frac{N_c}{N_1}=\frac{N_t}{N_1}\frac{N_r}{N_1}=p_tp_r=\eta_t\eta_rw^2.$$

With

$$=\frac1{NT}\sum_{k=1}^{N_1T}i_k^2.$$

the semiclassical theory then predicts

$$\frac{p_c}{p_tp_r}=\frac{}{^2}\geq1;$$

this follows from the Cauchy-Schwarz inequality and the equations for $$$$ and $$$$.

So, while the second quantized theory with indivisible photons expects no probability of coincidence if the coincidence counter is kept on for a sufficiently small window, the semiclassic theory expects $$p_c\geq p_tp_r$$. As shown in the graph, the latter is observed, with the probability increasing as the coincidence counter time span $$w$$ increases (allowing a higher probability of detecting multiple photons). This is one of the simplest demonstrations of the quantized nature of the electromagnetic field.