# How to represent a laser pulse in quantum optics

Every quantum optics textbook that I've found says something like "a coherent state represents the output of a laser", but a coherent state is a static thing (aside from the oscillating phase of the complex parameter); how do you represent a laser pulse propagating in space and time? I'm guessing you need to take some kind of sum over k-modes (as a single k-mode is presumably completely delocalised?)...

Also, any references you can suggest which discuss this in detail would be greatly appreciated!

• the propagation of a wave-packet follows shrodiger equation, with a suitable potential, which after a while (depending on application) the wave-packet becomes scattered or de-coherent – Nikos M. Nov 6 '14 at 13:06
• Can you write position/momentum-space wavefunctions for states of the EM field? I had assumed that, since a general state has an non-definite particle number, the wavefunctions are ill-defined. What does $|\psi(x)|^2$ represent then? I get the general idea of what you're saying, but I don't know how to start trying to actually tackle the problem... (thanks, by the way) – Max Lock Nov 6 '14 at 13:39
• i have a book "Lasers", O. Svelto, will take a look and update – Nikos M. Nov 6 '14 at 14:33
• Try this: "The Quantum State of a Propagating Laser Field", by S. J. van Enk and Christopher A. Fuchs, in the arXiv quant-ph, arxiv.org/abs/quant-ph/0111157. – Sofia Nov 11 '14 at 20:26

A coherent state is actually a mathematical idealization of a monochromatic laser. Strictly speaking, any continuous wave laser in the laboratory would be a statistical mixture of phase-randomized coherent states. Furthermore, it would also have a finite linewidth, and different frequency components in that linewidth would have no definite phase relationship between them. Such a field is termed as a stationary field, since it's physical characteristics do not depend on the origin of time. Let's understand this mathematically.

The positive analytic scalar electric field has the following representation in the frequency basis,

$E^{(+)}(t)=\int_{0}^{\infty}\frac{1}{2\sqrt{\omega}} \hat{a}(\omega) e^{-i\omega t} d\omega$

For a stationary field, the time correlation function $\langle E^{(+)}(t)E^{(-)}(t+\tau)\rangle$ which measures the correlation between the field at time $t$ and the field at time $t+\tau$ depends only on the time difference $\tau$. Incidentally, this also means that the mean intensity, which is just the correlation function evaluated for $\tau=0$ is constant, i.e, independent of $t$. Such a field may have a finite bandwidth, but the different frequency components must be completely uncorrelated. Mathematically this means that,

$\langle \hat{a}^{\dagger}(\omega)\hat{a}(\omega')\rangle = S(\omega)\delta (\omega-\omega')$

where $S(\omega)$ is defined as the spectral density of the field, and is the quantity that one measures on the spectrometer.

On the other hand, a pulsed field is not a stationary field. The physical properties of the field indeed do depend on the origin of time. For such a field, different frequency components would have to have a definite phase relationship in order to superpose and give a pulse. A single gaussian pulse, for instance, is well described by the Gaussian-Schell model which essentially assumes a gaussian linewidth of frequencies and that the correlations between different frequency components have a gaussian spread.

Also, if you have a train of pulses then it is easy to see that the corresponding frequency spectrum would be discrete, but again the different frequency components would have to be correlated.

I recommend the following references which provide a holistic description of these concepts:

1. Optical Coherence and Quantum Optics - L. Mandel and E. Wolf
2. Statistical Optics - Joseph. W. Goodman

A coherent state is a monochromatic sinusoidal field. The electric field pulse is inherently not monochromatic, but instead has a spectrum of frequencies which are superimposed on top of one another so as to all add constructively once every pulse repetition time. Therefore, to represent the field pulse in quantum optics you actually need to bring in more modes: the full state is a tensor product of the state of each mode, each being itself in a coherent state.