Quantum correlation functions of the electric field

Question

The first order correlation function for one mode (up to normalization) is definied as:

$$g^{(1)}=\langle a^\dagger(t) a(t+\tau) \rangle$$

The second order:

$$g^{(1)}= \langle a^\dagger(t)a^\dagger(t+\tau) a(t+\tau)a(t) \rangle$$

My questions are:

1. What is an intuitive explanation for these definitions? In the first order function, I destroy a photon at a time $t+\tau$ and create one at $t$? How does one interpret the second order function?

2. It is always written that the higher orders are a generalization of the first order. The second order is somehow symmetric in the time argument $(t,t+\tau,t+\tau,t)$, the first order is not. How can this be explained?

Answer 1

The first order correlation function given in the OP is the essential part of the electric field correlation function $$\langle E(\mathbf{r}_1,t_1)^*E(\mathbf{r}_2,t_2)\rangle,$$ taking into account the representation of the field in terms of the creation and annihilation operators $$E(\mathbf{r},t)\sim \lambda_\mathbf{k}a_\mathbf{k}e^{i\mathbf{k}\mathbf{x}}+\lambda_\mathbf{k}^*a^\dagger_\mathbf{k}e^{-i\mathbf{k}\mathbf{x}},$$ while neglecting the terms with two creation and two annihilation operators, which are zero, if the Hamiltonian conserves the photon number.

The second order correlation function originates from the correlation of the field intensities, $$I(\mathbf{r},t)=E(\mathbf{r},t)^*E(\mathbf{r},t)$$

The relevant Wikipedia article is called Degree of coherence, it gives more general definitions of the correlation functions. Useful reading on the subject is also the Glauber's Nobel prize lecture.

Answer 2

Correlation functions (in your case auto-correlation function) means the following: You take an initial field $e(t)$ in your case at a single point in space $x$. As a result of the surrounding medium (polarization etc.) of its vicinity, after a time $\tau$, it will be $e(t+\tau)$. $e(t+\tau)$ is of course related to $e(t)$ via the time evolution of the immediate local surroundings. So $e(t)e(t+\tau)$ will be $e(t)e(t)[1+r\tau +O(\tau^2)+...]$ where $r$ is a linear response etc. and you get $e(t)^2[1+r\tau+O(\tau^2)+...]$ If you now do the same at a second, different spatial point (you can also do this with different times due to the ergodic hypothesis), you will also get a similar expression. However, the coefficients, e.g. $r$ for linear response, will be different, because it sees a different local environment which affects $e(t+\tau)$; in some cases $e(t+\tau)$ will increase and in some it will decrease. When you average over many such points in space for $\tau \rightarrow \inf$, $e(t+\tau)$ will be sometimes large and sometimes smaller than $e(t)$ and as a result the average (if you use $e(t)-e(t_0)$) will vanish. How fast will it vanish depends on the medium; hence the correlation function measures how fast memory is lost; A particularly clear picture on a different setting is to be found in

S.Alexiou, Overview of Plasma Line Broadening, High Energy Density Physics Volume 5, Issue 4, December 2009, Pages 225–233;

It has a nice picture of what happens with time evolution although it discusses atomic dipole-dipole correlation functions. But to answer your question, correlation functions measure how fast initial memory is lost as a result of interactions with a random medium.