Dynamical versus spectral localization in dissipative systems

A short summary of this paper which was published in Nature Photonics last week.

Waves in random or disordered media can exhibit Anderson localization. In Anderson localization, interference between different wave scattering paths (and in particular, constructive interference of backscattering) results in a complete suppression of wave propagation; waves remain localized around their sources indefinitely, with an amplitude (or intensity) decaying exponentially with the distance from the source.

Anderson localization is a universal phenomenon. It was originally predicted in a 1958 publication analyzing electrons in disordered crystalline materials, and has since been observed for a wide variety of waves including matter waves (Bose-Einstein condenstates), optics, and acoustics.

Historically, a thorny issue complicating the observation of Anderson localization in classical wave systems such as optics and acoustics has been the question of how to distinguish Anderson localization from absorption-induced localization; both lead to a similar exponential decay of the wave amplitude.

The present work concerns the generalization of Anderson localization to disordered dissipative optical wave systems with random distributions of gain and/or loss, and the subtle distinction between spectral localization and dynamical localization.

Spectral localization refers to localization of the modes of the medium. Each mode has a specific energy (frequency). The mode may be excited by placing a source (e.g. a speaker in the case of acoustic waves) in the medium tuned to that frequency, in which case the amplitude profile of the generated wave will match the profile of the correspond mode.

Dynamical localization refers to the time evolution behaviour of wavepackets comprising a range of frequencies, excited by switching the source on for a short time.

In wave systems that are Hermitian (conservative, i.e. no gain or loss of energy), spectral and dynamical localization coincide because the time evolution of any wavepacket can be obtained by expanding it as a sum of the medium's modes. The paper shows that this is not the case for wave systems with dissipative disorder; dynamical delocalization can occur despite spectral localization. In other words, waves generated by a monochromatic source will have an exponentially localized amplitude profile, whereas waves emitted by a broadband source will spread to distant parts of the system.

To demonstrate this dynamical delocalization, the authors had to carefully distinguish between energy transport and energy loss due to absorption. In particular, for lossy media the wave amplitudes are always decaying exponentially in time. At any moment in time we can consider the shape of the wave's amplitude distribution (e.g. by increasing the sensitivity of the camera or microphone used to detect the waves) and how it decays with separation from the source. 

In dissipative wave media the modal expansion is still valid, but different modes will have different loss rates. Therefore, the relative amplitudes of the different modes in the expansion will change in time, leading to large changes and in particular spreading in the normalized wave amplitude profile - dynamical delocalization. This is the main result of the study, which observes the phenomenon using a cleverly-designed system of coupled optical fibres.