Read in more detail in our recent publication:
- Sagi, Y., Brook, M., Almog, I. & Davidson, N. Observation of Anomalous Diffusion and Fractional Self-similarity in One Dimension. Phys. Rev. Lett. 108, 093002 (2012). See also arXiv:1109.1503.
What is anomalous diffusion?
We imagine an ensemble of atoms which initially are all starting at the same point in space (i.e. the origin). We also assume that each atom has a different velocity, and as time goes on it is moving accordingly. If the velocity of each atom is constant in time, the cloud of atoms expand ballistically. All characteristic sizes of the cloud, such as the width, scale linearly with time. We shall call this process diffusion when the velocity of each atom is not constant. In regular diffusion the characteristic lengths scale as a square root of time. To see how this happens, note that accumulated distance of each atom can be regarded as a random variables which is a sum of small distances in which the velocity of the atom was approximately constant. The central limit therm state that such a sum of random variables distribute to a normal distribution, as long as the distribution of each of the random variables has a finite variance. For simplicity, lets assume that the time such an atom stays in a constant velocity is well distributed (has a finite variance). This is the case, for example, for a process in which the atoms change their velocity in elastic collisions, since the time between collisions has a Poisson distribution. Out conclusion is that for regular diffusion to happen, the distribution of velocities need to have a finite second moment.
Anomalous diffusion is a diffusion process for which the accumulated distances does not follow the central limit theorem. As a result, the scaling with time not anymore as a square root. It is natural to seek for a generalization of the central limit theorem for sums of random variable, each of which has a distribution with a diverging second moment. Such a generalization was achieved by the famous French mathematician Paul Pierre Levy, and therefore the limit distributions of these sums are called Levy stable distributions. Each of these distributions has a characteristic exponent which defines its normalization scaling with the number of summons. A normal distribution is also a Levy stable distribution with a characteristic exponent of 2. In anomalous diffusion, the Levy stable distributions play a central role; they are the kernel of the fractional diffusion equation which describe this process. In other words, if in a physical system which exhibit anomalous diffusion the initial condition is a delta function, then at later times the resulting atomic cloud is given by a Levy stable function. Each anomalous diffusion process is characterized by the same characteristic exponent of the corresponding stable distribution. Moreover, the scaling with time of the characteristic lengths is a power law with an exponent inversely proportional to the characteristic exponent of the process.
In order to observe anomalous diffusion it is necessary that the steady-state velocity distribution will have heavy tails with a diverging second moment. In the following section we explain one way this can be done in an experiment.
How to create a non-Gaussian velocity distribution?
Distribution with heavy tails means that its asymptotic decay at large values goes as a power law. The first thing one should do to produces such velocity distribution is not to be in thermal equilibrium. In thermal equilibrium the velocity distribution is following the Maxwell-Boltzmann distribution which has an exponentially decaying tails. The experimental goal, therefore, is to reach a steady state with some external field which is by itself not thermal. This can be done, for example, by letting the atoms scatter photons from a laser field.
The specific configuration of the lasers is that of the well-known Sisyphus cooling scheme. In one dimension, the cooling scheme works in a lattice created by two counter propagating lasers at the same frequency but orthogonal polarizations. This results in a polarization lattice, i.e. the average intensity is the same in all places, but the polarization is changing periodically. The important parameters are the lasers intensity and detuning from the atomic resonance. As long as the detuning is large compared to the natural linewidth of the atomic transition, these parameters are combined to a single parameter – the lattice depth. The atomic motion in these lattices is quite complicated and was investigated in numerous theoretical studies. The photon scattering rate depends on position; the average rate in higher for positions with higher potential energy. This means that the cooling works by continuously pumping the atoms from high potential energy to lower ones, thus the atoms loose dissipate kinetic energy to photons.
In usual laser cooling experiments the lattice parameters are chosen such that at the end of the cooling sequence the energy of the atom in much less than the lattice depth and the atoms are trapped in some potential well. In contrast, when going to shallow lattices the atoms have high probability to scatter enough photons so they are no longer trapped in the potential. In this case, the atoms move a large distance before scattering enough photon to be trapped in the lattice again. The distance covered by the atoms at these periods were found to follow the Levy distributions, and accordingly they are called Levy flights. The steady state velocity distribution in a Sisyphus lattice was found to be a power law (Tsallis distribution) with an exponent that depends on the lattice depth. This was also confirmed in experiments. In conclusion, letting the atoms equilibrate in a polarization lattice give rise to a power law velocity distribution, and as a result the diffusion process is expected to be anomalous.
The experimental setup
A sketch of the experiment setup is shown below:
We start by trapping and cooling the atoms in a Magneto-Optical Trap (MOT). We follow with cooling in a Sisyphus lattice (this is not the same lattice of the main experiment,but a 3D lattice created by the MOT beams) and a Raman sideband cooling scheme. The atoms are then loaded in a dipole trap made by two crossing Ytterbium laser beam. We sometimes perform evaporation by reducing the laser intensity.
To simplify things we want to confine the dynamics to one dimension. This is done, in part, to increase the signal to noise and to eliminate the effect of the gravitation force from the experiment. We superimpose another far-off resonance laser with a waist large enough so it can be considered as an effective “tube” in which the atoms move. The lattice itself is also one dimensional, running at the same direction as the tube trap. The detection is done by absorption imaging in an axis orthogonal to both gravitation and the tube trap axes.
When we first switch on the lattice beams and the tube trap, we keep the crossed trap beams on so the diffusion process does not start immediately but there is some time for initial equilibration. An important point is that in the radial directions there is no lattice (or cooling) and therefore atoms can eventually escape after scattering enough photons. The t=0 of the experiment is defined as the instance when the crossed trap is witched off.
Results
In the image below there is an example of a measurement of the density profiles after different waiting times:
We integrate over the y-axis of this image to obtain the axial density distribution:
From this data we can study many aspect of anomalous diffusion. For example, we extract the full width at half the maximum (FWHM) and plot it as a function of time. We expect the width to scale as a power of time, with an exponent that depends on the lattice depth. In the following graph you can see the width as a function of time:
We fit these lines with a power law and extract the characteristic exponent. This exponent is than plotted as a function of the lattice depth, and a typical result is given below:
In this graph, the y-axis is actually twice the diffusion exponent, which means that 1 correspond to normal diffusion, 2 to ballistic expansion, and all the range in between is anomalous diffusion. We can clearly see anomalous diffusion for low enough lattices. This is an on-going work, and these days we are working to better characterize the exact form of the distribution.