Higher order coherence

[1] It is used to differentiate between optics experiments that require a quantum mechanical description from those for which classical fields suffice.

[4] Orders of coherence can be measured using classical correlation functions or by using the quantum analogue of those functions, which take quantum mechanical description of electric field operators as input.

The intensity of resulting field is measured as a function of the time delay.

Young's double slit experiment demonstrates the dependence of interference on coherence, specifically on the first-order correlation.

is Light field has highest degree of coherence when the corresponding interference pattern has the maximum contrast on the screen.

Correspondingly, in the quantum description the electric field operators are similarly related,

This implies The intensity fluctuates as a function of position i.e. the quantum mechanical treatment also predicts interference fringes.

If the system is observed to determine the path of the photon, then on average the interference of amplitudes will vanish.

[3] The normalised second order correlation function is written as:[7] Note that this is not a generalization of the first-order coherence If the electric fields are considered classical, we can reorder them to express

For classical fields, one can apply the Cauchy–Schwarz inequality to the intensities in the above expression (since they are real numbers) to show that

(calculated from averages) can be reduced down to zero with a proper discriminating trigger level applied to the signal (within the range of coherence).

Note the Hanbury Brown and Twiss effect uses this fact to find

, does not matter in the classical case, as they are merely numbers and hence commute, the ordering is vital in the quantum analogue of these correlation functions.

[4] The first order correlation function, measured at the same time and position gives us the intensity i.e.

In quantum mechanics, the positive and negative frequency components of the electric field are replaced by the operators

Subsequently, the n-th order normalized correlation function is defined as:

here measures the probability of coincidence of two photons being detected with a time difference

[4] For all varieties of chaotic light, the following relationship between the first order and second-order coherences holds:

This model fits the observation that was done by Hanbury Brown and Twiss using stellar light as demonstrated in figure 3.

[10] Thermal light can be modeled to be a Lorentzian power spectrum centered around frequency

Classically, we can think of a light beam as having a probability distribution as a function of mode amplitudes,

Classically, the light beams arrives as an electromagnetic wave and interferes owing to the superposition principle.

HBT's experiment allows for a fundamentally distinction in the way in which photons are emitted from a laser compared to a natural light source.

Higher order coherences are measured in photon-coincidence counting experiments.

Correlation interferometry uses coherences of fourth-order and higher to perform stellar measurements.

A field is called mth order coherent if there exists a function

[3] When dealing with classical optics, physicists often employ the assumption that the statistics of the system are stationary.

This means that while the observations might fluctuate, the underlying statistics of the system remains constant as time progresses.

The quantum analogue of stationary statistics is to require that the density operator, which contains the information about the wavefunction, commutes with the Hamiltonian.

, in the Heisenberg picture, giving us This means that under the assumption that the underlying statistics of the system are stationary, the nth order correlation functions do not change when every time argument is translated by the same amount.

This is a plot of the absolute value of g (1) as a function of the delay normalized to the coherence length τ/τ c . The blue curve is for a coherent state (an ideal laser or a single frequency). The red curve is for Lorentzian chaotic light (e.g. collision broadened). The green curve is for Gaussian chaotic light (e.g. Doppler broadened).
Figure 1. Schematic diagram for the setup of the Young's Double Slit Experiment.
Plot of g (2) as a function of the delay normalized to the coherence length τ/τ c . The blue curve is for a coherent state (an ideal laser or a single frequency). The red curve is for Lorentzian chaotic light (e.g. collision broadened). The green curve is for Gaussian chaotic light (e.g. Doppler broadened). The chaotic light is super-Poissonian and bunched.
Figure 2. A schematic diagram for the setup for Hanbury Brown and Twiss's original experiment.
Figure 3. The second order coherence for stellar light as measured in the Hanbury Brown and Twiss experiment as a function of the time delay introduced between the signals , where is the coherence length.
Figure 4. The second order coherence for thermal, stellar and coherent light as a function of time delay. is the coherence length of the light beam.