Self-Similar Modes of Coherent Diffusion

Abstract

Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase surface, based on the elegant Gaussian modes of optical diffraction. In an experiment of light storage in a gas of diffusing atoms, a complex initial condition is imprinted, and its diffusion dynamics is monitored.The self-similarity of both the amplitude and the phase pattern is demonstrated, and an algebraic decay associated with the mode order is measured. Notably, as opposed to a regular diffusion spreading, a self-similar contraction of a special subset of the solutions is predicted and observed.Typeset by REVT E X 1

Shape-preserving diffusion of a high-order mode

“…The diffusion coefficient obtained from fitting these results to the theoretical expectation is D 12 cm 2 ∕s. This value is consistent with previous results [9] but is here affirmed for significantly longer storage durations, due to improved uniformity of the magnetic field.…”

“…We verify that the standard modes are only shape preserving under free-space propagation, whereas the elegant ones are only shape preserving during storage. Such a difference between diffusion and diffraction did not exist in [9], where only the low-order common modes were studied. The distinction between diffusion and diffraction and, in particular, verification of the diffusion approximation during storage, is the main contribution of the current experiment.…”

“…Utilizing either electromagnetically induced transparency (EIT) [9] or gradient echo memory [10], the complex electric field of a probe light pulse was spatially mapped onto the ground-state coherence (dipoles) of diffusing atoms in a vapor cell with a buffer gas. During storage, the field of atomic coherence ψ undergoes coherent diffusion _ ψ D∇ 2 ψ − γψ, where D is the diffusion coefficient of the atoms and ψ an atomic dumping rate [11,12].…”

Spatial-mode storage in a gradient-echo memory

“…The Hermite-Gauss modes are not in general self-similar under diffusion [48], and the changing spatial intensity distribution provides additional evidence that spin wave diffusion is the dominant source of decoherence during storage. The TEM 20 mode, for example, has a small central lobe that is out of phase with the outer peaks.…”

Diffusion effects in gradient echo memory

“…For a plane wave, only axial diffusion is important, but transverse diffusion becomes significant when a realistic beam profile is included. There has been recent interest in the effects of diffusion in the EIT [23][24][25].…”