Do Photons Exhibit Poisson Fluctuations?

Holly Maness

Photons are not classical objects.  Their behavior is governed by quantum statistics.  At short wavelengths, Poisson counting is a good approximation so that when \(n \) photons from a thermal source are detected, the fluctuations are given by \( \sqrt{n} \).  At infrared and radio wavelengths, the Poisson approximation breaks down.  At these long wavelengths, photons do not arrive independently and the fluctuations are of order \(n \), rather than \( \sqrt{n} \).

Consider the number of photons, \( n \) of thermal radiation, in a cavity at temperature, \( T \), and frequency, \( \nu \).  As photons are emitted and absorbed by the walls, the number of photons in the cavity will change. Boltzmann’s law gives the probability that a state with energy, \( E = n h \nu \), occurs, where \( n \) may take on any value \( 0, 1, 2, 3, \dots \): \[P(n) = \frac{1}{U} \exp \left(-n h \nu / k T \right), \] where \( h \) is Planck’s constant, and \( k \) is Boltzmann’s constant. The normalization, \( U \), is found from the condition that the probability must sum to unity: \[ \begin{eqnarray} U & = & \sum_{n=0}^{\infty} \exp \left( -n h \nu / k T \right) \\ &=& \frac{ \exp \left( h \nu / k T \right) }{ \exp \left( h \nu / k T\right) -1 }.\end{eqnarray}\] The quantity \( U \) is sometimes called the partition function.

The average energy at frequency, \( \nu \), is \[\begin{eqnarray} \langle E \rangle &=& \sum_{n=0}^{\infty} E (n) P(n) \\ &=& \frac{1}{U} \sum_{n=0}^{\infty} n h \nu \exp \left( -n h \nu / k T \right) \\ &=& \frac{h \nu}{\exp(h \nu / kT) -1}. \end{eqnarray}\] We can use this result to compute the energy density of photons. Photons are spin zero particles, and therefore, unlike half-integer spin particles like electrons, can congregate in a cell of phase space. The number of phase cells per unit volume for photons with momenta \( p \) to \( p + dp \) is \[ dn = 2 \frac{4 \pi p^2}{h^3} dp = \frac{8 \pi \nu^2}{c^3} d \nu, \] where the factor of two accounts for the two polarization states. Thus, the energy density of radiation at frequency \(\nu \) to \( \nu + d \nu \) is \[\begin{eqnarray} \rho_\nu d \nu & = & \langle E \rangle dn \\ & = & \frac{8 \pi \nu^2}{c^3} \frac{h \nu}{\exp(h \nu / kT) -1} d\nu, \end{eqnarray}\] which is the famous Planck formula for blackbody radiation.

The mean number of photons is \[ \langle n \rangle = \sum_{n=0}^{\infty} n P(n) = \frac{1}{\exp(h \nu / kT) -1}, \] and the average value of \( n^2 \) is \[ \langle n^2 \rangle = \sum_{n=0}^{\infty} n^2 P(n) = \frac{\exp(h \nu / kT) +1}{\left(\exp(h \nu / kT) -1\right)^2}. \] Hence, the variance, which is \( \sigma^2 = \langle n^2 \rangle - \langle n \rangle^2 \), is \[ \sigma^2 = \frac{\exp(h \nu / kT)}{\left(\exp(h \nu / kT) -1\right)^2}, \] or by substituting for \( \langle n \rangle \), we see that the variance is proportional to \( \langle n \rangle \) times a correction factor: \[\begin{eqnarray} \sigma^2 &=& \langle n \rangle \frac{1}{1 - \exp(h \nu / kT)} \\ &=& \langle n \rangle (1 + \langle n \rangle ).\end{eqnarray}\] Often the correction factor is very close to unity. The figure below shows the correction factor for an incandescent lamp (3000 K) and room temperature (300 K) at wavelengths between 100 nm and 100 \( \mu\)m.

Poisson correction factor as a function of wavelength.

In the ultraviolet and visible, we have to a very good approximation: \[ \sigma^2 = \langle n \rangle, \] i.e., the fluctuations are Poisson. This feature of photon statistics is used in our measurement of the gain of a CCD.

However, at infrared and radio wavelengths the Poisson approximation for thermal radiation becomes increasingly less accurate. When \(h \nu / kT \gg 1 \), the expression for \( \langle n \rangle \) shows that \( \langle n \rangle \ll 1\), and therefore the expression for the variance becomes \( \sigma^2 = \langle n \rangle \). For the detection of such light, individual photon events are uncorrelated. However, when \(h \nu / kT \ll 1 \), \( \langle n \rangle \gg 1\) and thus \( \sigma^2 = \langle n \rangle ^2 \). The increased fluctuations in this limit are a consequence of "photon bunching." Photons are correlated in this case in such a way as to increase the noise in the detected signal above that given by Poisson noise.

Further Reading

Lewis, W. B. (1947). Fluctuations in streams of thermal radiation. Proceedings of the Physical Society, 59(1), 34–40. doi:10.1088/0959-5309/59/1/307 

* The image at the top of the post shows a two-dimensional image of Poisson counts with a mean value of unity, displayed using a false color map.

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