Quantum Optics in Random Media

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A random medium in optics is a medium in which the dielectric function varies randomly with position. Frosted glass is a familiar example. In a laboratory setting, one can suspend particles in a fluid to achieve strong multiple scattering with little absorption. On a much larger scale, interstellar clouds and stellar atmospheres provide a random medium for the radiation propagating through them.

For many applications, a classical description of the electromagnetic field is sufficient. The theory of propagation of classical waves through random media has been developped extensively in the last two decades. Quantum effects such as vacuum fluctuations and spontaneous emission of radiation are not taken into account in these studies. These are the subject of quantum optics, which however rarely goes beyond one-dimensional scattering.

The essentially three-dimensional, chaotic scattering present in random media has so far received little attention in the context of quantum optics. This deficiency is felt particularly strongly for amplifying media (so called "random lasers"), where the interplay of spontaneous and stimulated emission with chaotic scattering plays a central role. My work tried to make a first step towards a bridging of the gap between random media and quantum optics.

Introduction

My thesis is available in PDF format.

Methods

Classically a linear sample can be described by a finite-dimensional scattering matrix S - assuming that the sample was put into a waveguide so that only a finite number of propagating modes exist. For a random medium, S is a random quantity, and one has to consider an ensemble of samples with small variations in the positions of the scatterers. Random-matrix theory gives information about the distribution of S.

Interestingly, the scattering matrix is all we need to know for a quantum-optical description of the sample. The quantum-mechanical operators describing the radiation leaving the sample are related to the operators describing the incident radiation by the this matrix. Additionally we need to include vacuum fluctuations and spontaneous emission but their magnitude are known from the fluctuation-dissipation theorem - which in turn only depends only on S. This formalism of calculating the state of the outgoing radiation field is called "input-output relations".

Introduction

Effects on coherent radiation

Classically a signal can be amplified by an arbitrary factor without a deterioration of the signal-to-noise ratio. Quantum-mechanically amplification introduces additional noise into the system. The "input-output relations" allow to compute the outgoing state (and thus also its noise) for an arbitrary incident state but since there are infinitely many possible incident states, so there are infinitely many answers to this question.

It therefore is much more sensible to restrict one to a particular class of ingoing radiation - at least at the beginning. The most obvious choice is coherent radiation: A coherent state is the closest quantum-optical equivalent to a classical light beam, and it is characterised by a single parameter only (its intensity). It will be seen that in a random media more noise will be added to the signal than in a comparible medium without scattering.

Excess noise for coherent radiation propagating through amplifying random media

Effects on squeezed radiation

After having examined the propagation of "classical" coherent radiation, the logical next step is an extension to states of the radiation field that are nonclassical by nature. Coherent radiation is the "most classical" radiation but also radiation with more noise (e.g. thermal radiation) can be described classically - at least to a certain degree. The prototype of nonclassical radiation is squeezed radiation where the intensity fluctuations can be lower than for coherent radiation (at the expense of stronger phase fluctuations).

The nonclassical properties of squeezed radiation are destroyed if too much noise is added to the signal. Similar to case of coherent radiation, where more noise is added if random scattering is present, squeezing is destroyed faster in a random medium than in a sample without scattering.

Propagation of squeezed radiation through amplifying or absorbing random media

Correlations

So far, the radiation leaving the sample was measured by a single photodetector (depending on the size of the detector as little as a single mode or as much as all radiation might have been detected). However, one can also ask about correlations between the radiation leaving the sample at different positions on its surface.

In a linear medium without scattering, radiation in one specific mode can propagate through the sample without being affected by the radiation in the other modes. Therefore, the radiation at different points on the surface is uncorrelated. In contrast in a random medium part of the radiation will be scattered into other modes, with the rest staying in the original mode. This way one can expect the build-up of correlations between the radiation in different modes. Actually, it is not even necessary that radiation is incident onto the sample: A single photon created by spontaneous emission inside the medium can leave the sample at different positions - due to scattering - causing spatial correlations.

Long-range correlation of thermal radiation

Decay rates

An essential "ingredient" for the scattering-matrix approach is the distribution of decay rates for the disordered medium. The decay rates tell how long it takes a photon to escape from the system. (Due to multiple scattering this time is much longer than the linear dimensions of the sample, divided by the speed of light.) This distribution is known analytically only for the case of a chaotic cavity with a small hole - a geometry that does not really resemble experiments but due to the lack of better formulas still had to be used for many of the results presented on this page. By using numerical computations we have determined the decay rate distribution of a disordered slab, i.e., the most common experimental geometry.

Decay rate distributions of disordered slabs and application to random lasers

Random laser

A laser consists of two important ingredients: amplification and feedback. In a "conventional" laser the latter is due to mirrors at both ends. However, feedback is also possible by scattering inside the laser. Such a laser is called a random laser.

Description of a laser might seem to be impossible within the input-output-relation approach but actually a decription is possible - except near the lasing threshold where nonlinearities become important. Far above the threshold - where one probably would want to operate such a device - one can compute the most important characteristic of a laser: its linewidth.

Since a random medium is described by an ensemble of sample, the computed linewidth is a random quantity, too. Lasing action will select a particular mode of the system (the one with the lowest losses) so that the statistics differ from the statistics as observed below the threshold (where all modes participate equally). If the outcoupling of the laser light is weak, the problem is relatively simple.

Quantum-limited linewidth of a chaotic laser cavity

This no longer is the case if the outcoupling becomes stronger.

Large Petermann factor in chaotic cavities with many scattering channels

A comparison and a more detailed derivation:

Quantum limit of the laser linewidth in chaotic cavities and statistics of residues of scattering matrix poles

Random laser with nonlinearities

For a proper description of the behaviour of the laser near threshold, nonlinearities (for example saturation of the medium) need to be included. While the average emitted intensity is easily computed, the noise properties are much more difficult. Since the characteristic of laser light lies in its noise properties, namely its coherence, these properties are of extreme importance. We were able to compute them using a Langevin approach.

Theory for the photon statistics of random lasers

Input-output relations vs. diffusion approach

The method of input-output relation is very powerful in that it allows to treat samples of arbitrary cross-sectional area. In the limit of a very wide sample and detector (or equivalently the limit of a very short wavelength of the radiation) a different method may be computationally easier: Photons are assumed to be "just particles" that are diffusing through the medium, and computing the radiation properties reduces to solving a diffusion equation.

For certain geometries (e.g. a sphere) the solution of the diffusion equation is known whereas the statistics of scattering matrices are not known yet, so only the diffusion approach can be used. On the other hand, the limit of a very wide detector does not allow the description of single-mode measurements or spatial correlations. It is shown that both methods do agree for the cases where both methods can be applied (and the resulting equations be solved).

Frequency dependence of the photonic noise spectrum in an absorbing or amplifying diffusive medium

Localised regime

The calculations done to solve the problems described so far were always done in the "diffusive regime". If the length of the samples is increased, a transition to the so called "localised regime" is observed, and the intensity of the transmitted radiation becomes exponentially small. (This transition is known as Anderson localisation in semiconductor physics.)

Since a medium which transmitts almost no radiation is of only limited relevance to most "real life" applications, the localised case was "saved" until now. Interestingly, if the sample is sufficiently far in the localised regime, universal behaviour appears.

General description of the different regimes in which a random medium can be, and a comparison with the corresponding regimes in semiconductors.

Photon shot noise

Deep in the localised regimes both the transmitted intensity and the noise of the signal vary widely but its ratio (the noise figure) approaches an universal value.

Photonic excess noise and wave localization