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# **$g^{(2)}$**

* gee-two* refers to a central quantity in quantum optics, introduced (chiefly) by Roy Glauber for which he basically got the Nobel Prize. It is a normalized intensity-intensity correlation function. So if your intensity as function of time $t$ is $I(t)$, then

$$g^{(2)}(t_1,t_2)\equiv{\langle I(t_1)I(t_2)\rangle\over\langle I(t_1)\rangle\langle I(t_2)\rangle}\,.$$

This is intrinsically a two-times function, although in many cases of interest, the signal is stationary, and **$g^{(2)}$** becomes a one-time, but still two-body function for the time delay $\tau\equiv t_2-t_1$:

$$g^{(2)}(\tau)={\langle I(0)I(\tau)\rangle\over\langle I(0)\rangle^2}$$

where the denominator gets squared because now $\langle I(t)\rangle$ is a constant of $t$.

The meaning of **$g^{(2)}$** is as a calibrated density probability of two-photons. Every word is important. We write "calibrated" and not normalized because a density probability should be normalized. The calibration means that the two-photon density probability itself, let us call it $G(\tau)$, gets divided by the average intensity. As a probability density, it is not a probability, and can take values arbitrarily large (this is called bunching) although it has to be positive. The two photons do not have to be successive, this is important and one of the main value of this quantity, that it is resilient to losses: intermediate photons can be lost without changing the function (only its noise). The Mathematical transcription of this definition reads (in a steady state):

$$g^{(2)}(\tau)={G(\tau)\over\langle I(0)\rangle}$$

In terms of a Bose annihilation operator $a$, i.e., for a quantized field, this becomes:

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