User:Robbyk42/Photon Antibunching

= Photon Antibunching = Photon antibunching is a quantum-mechanical effect observed in resonance fluorescence, predicted in 1976 and observed in 1977 by H. Jeff. Kimble and Mandel. Specifically, it describes a photon distribution where individual photons are not likely to be observed immediately before or after each other, but arrived evenly-spaced in time. This is in contrast to photon bunching, where photons tend to be observed in groups. Antibunching is important because it provides definitive evidence for nonclassical light, i.e. the quantization of the electromagnetic field.

It is a common misconception that the first definitive evidence for the existence of photons was given by Albert Einstein's explanation of the photoelectric effect in 1905. However, while Einstein's prediction of the particle-like nature of light would prove true, there is a perfectly suitable explanation of the photoelectric effect that uses a semi-classical field interacting with quantized matter. In fact, it is impossible to definitively demonstrate the quantized nature of the electromagnetic field using single-photon observables. Thus, in order to observe the particulate nature of light, we must consider higher-order photon correlations.

The Two-Photon Correlation Function
The second-order correlation between two photodetectors is a measurement of the intensity of the field correlated with itself some time later. The typical device used to measure such correlations is a Hanbury Brown and Twiss interferometer, which sends incoming light through a 50-50 beam splitter into one of two photo detectors. The signals are compared and the coincidence of the two detectors contributes to the measured correlation. Putting a signal delay of time $$\tau$$ onto one of the detectors allows one to compare the field to itself at different times.

Mathematically, the second-order correlation as a function of delay time $$\tau$$ can be written as:

$$\hat{g}^{(2)}(\tau) = \left \langle \hat{I}(t) \hat{I}(t+\tau) \right \rangle$$.

Writing this in normal order with respect to the electric field operators gives:

$$\hat{g}^{(2)}(\tau) = \frac{\langle \hat{E}^{(-)}(t)\hat{E}^{(-)}(t+\tau)\hat{E}^{(+)}(t+\tau)\hat{E}^{(+)}(\tau) \rangle} {\langle\hat{E}^{(-)}(t) \hat{E}^{(+)}(t)\rangle^2}$$.

for Classical Light
Writing a general state using the Glauber-Sudarshan P representation:

$$\rho = \int P(\alpha) |\alpha \rangle \langle \alpha| d^2 \alpha $$

it is straightforward to calculate the $g^{(2)}(0) $ correlation for with no delay ($\tau = 0 $ ):

$$\hat{g}^{(2)}(0)[\hat\rho(t)] = \frac{\langle \hat E^{(-)2}(t) \hat E^{(+)2}(t) \rangle_{\rho}} {\langle \hat E^{(-)}(t) \hat E^{(+)}(t) \rangle^2_{\rho}} = 1 + \frac{\int P(\alpha)(|\alpha|^2-\langle|\alpha|^2\rangle^2) d^2\alpha}{\langle|\alpha|^2\rangle^2}   $$

using $$\hat{E}^{(+)}|\alpha\rangle = \alpha |\alpha \rangle $$. If the distribution describes a classical distribution of coherent states, because every quantity in the final term is positive, we conclude $g^{(2)}(0) \geq 1  $  for classical light. Furthermore, a condition for classical light is $$g^{(2)}(t + \tau) \leq g^{(2)}(t)$$, which implies that the correlation is maximized at $$g^{(2)}(0)$$. This means that a positive rate of change in the correlation function also implies photon antibunching.

for A Single Two-Level Atom
Consider a single atom with ground and excited states denoted $|g\rangle        $, $|e\rangle         $ , respectively. Furthermore, assume we drive the atom at coherent laser light for long enough such that the atom is in some steady state $$\rho_{ss} $$

Considering the interaction between the atom and the field, replacing $$\hat{E}^{(+)} \propto \hat\sigma_-         $$, $$\hat{E}^{(-)}\propto \hat\sigma_+          $$, where $$\sigma_+ = |e\rangle\langle g| = \sigma_-^\dagger$$:

$$\hat{g}^{(2)}(\tau) \propto \langle \hat{\sigma}_+(0)\hat{\sigma}_+(\tau)\hat{\sigma}_-(\tau)\hat{\sigma}_-(0) \rangle =Tr[\rho(|e\rangle\langle g|)_0(|e\rangle\langle g|)_\tau(|g\rangle\langle e|)_\tau(|g\rangle\langle e|)_0] = P_e(\rho_{ss})[P_e(\tau)]_g  $$

where $$P_e(\rho_{ss}) $$ is the steady-state excitation population, and $$[P_e(\tau)]_g  $$ is the probability to find the atom in the excited state at time $t=\tau   $  given that the atom was in the ground state at $t=0   $. In the limit where $$\tau \rightarrow 0  $$, the only logical result is $$\hat{g}^{(2)}(\tau) = 0   $$, which means the light from the photon will be antibunched.

Of course, this analysis is limited because it assumes an oversimplified picture of the atom-laser interaction. In a real physical system, a laser has a finite linewidth and non-zero interaction time with the atom, so the situation becomes more complicated. A more complete analysis is given by Carmichael, which still finds a vanishing correlation function in a physically realizable system.

Experiments
Early experiments (late 1970's-1980's) were focused on improving the precision of optical measurements to simply observe the predicted antibunching effect. More modern experiments attempt to observe antibunching from more contrived sources and environmental regimes.
 * Antibunching was first observed by Kimble and Mandel in 1977 by exciting a beam of sodium atoms. They found the correlation function to have a positive slope for small delay times. However, they found the correlation to be higher at zero delay than in the infinite time limit. This was due to fluctuations of the atomic beam and the finite interaction time between the atoms and the laser.
 * Observations of a Sub-Poissonian photon distribution was first reported by R. Short and L. Mandel in 1983. This is important because a Sub-Poissonian distribution can be an indicator of antibunching. However, it is important to note that there exist Sub-Poissonian distributions that generate bunching, rather than antibunching.
 * Nearly 10 years after the first observation of antibunching, Diedrich and Walther used a radio-frequency trap to measure the resonance of individual magnesium atoms. This minimizes the issues with atom statistics and interaction times, so they were able to find $$g^{(2)}(0) = 0$$ for a single atom.
 * In 2012, Nothaft et. al. demonstrated photon antibunching at room temperature using organic molecules with low excitation energies.
 * In 2020, Hanscheke et. al. show that filtering the photons emitted by a source to sub-natural linewidth removes the antibunching effect from the system. This demonstrates that antibunching is a non-linear optical effect, since it requires the contribution of photons of different frequency to take place.