This is the first experiment built in our group, with the aim to observe coherent backscattering (CBS) of light by a laser cooled sample of Rubidium atoms. It is now more generally devoted to the study of multiple scattering of light in an optically thick cloud of cold atoms.
Topics :
1) The trap
Our magneto optical trap (MOT) uses a six beam configuration to load Rb85 atoms from a thermal vapor. Large trapping beams (Ø ~ 4 cm) are employed to trap large numbers of atoms (up to 1010). The rather large required laser power ( ~ 200mW) is produced by a MOPA (single pass) seeded by a DBR master laser (5 mW, ~ 2 MHz line width). Most of our experiments involve a pulsed operation of the MOT, where the trap (trapping laser, magnetic gradient, repumping laser) is switched off, the measurements are performed, then the trap is turned back on before the cold atoms leave the trap area.
In our experiments, it is crucial to know accurately the parameters of the atomic cloud (number of atoms, size of the cloud, temperature). We use several techniques to characterize the atomic cloud.
After the trap is switched off, a probe laser is shone to the cold atoms. The laser is detuned by several natural width to ensure homogeneous illumination of the cloud. The fluorescence can be imaged on a CCD from two orthogonal directions giving access to the cloud density profile in 3D. We typically obtain clouds with a more or less gaussian shape and large sizes (3-6 mm FWHM).
Time of flight : In this standard technique, we measure via the absorption of a "light sheet" the arrival time distribution of the cold atoms dropped from a height of 31 cm. The large height reduces the effect of the initial spatial distribution in the cloud.
The most important parameter is the optical thickness (b = -ln(T), where T is the cloud transmission) which measures the degree of multiple scattering in the sample. A weak probe laser of small diameter is sent through the center of the cloud. The transmitted intensity is detected as the laser frequency is scanned around resonance. The width of the transmission curve gives a reliable estimate of the optical thickness. The maximum optical thickness in our atomic cloud is about 40.
Coherent backscattering (CBS) of light has been widely studied in the past 15 years, with samples ranging from milk to semiconductor powders. Our experiment on Rubidium allowed the first observation of CBS with an atomic vapor. We can study new regimes owing to the very peculiar properties of cold atoms.
CBS setup
CBS with
cold Rb atoms : the role of internal structure
We show below an example of the
far field distribution of the backscattered intensity in the four standard
polarization channels :
The most striking observation is the very small enhancement factor in the h//h channel. The anisotropies observed in the linear channels are due to the dipole-like radiation pattern of the atoms. For a more quantitative analysis, we show below the cones cross sections (angular average).
It can be seen that the enhancement factor is much smaller than 2 in all channels, even in the parallel helicity channel where it is only 1.05 (the theory predicts 2 for "classical scatterers") ! We have shown that this is due to the atomic internal structure (Zeeman substates), an additional degree of freedom in the atomic scatterer. The theory developed in our group together with the team of D. Delande at LKB (Paris) gives predictions in very good agreement with the experiment (the solid curves in the figure above are MonteCarlo simulations without adjustable parameters). Recently, the second experiment in our group yielded a CBS cone from a cold Strontium gas with an enhancement factor close to 2 in the // helicity channel, confirming this interpretation.
Our publications on the subject :
experiments : PRL1999, J.Opt.B.2000(Rb experiment), PRL2002b(Sr experiment), JOSAB2004.
theory : PRL2000,PRA2001b,PR2002a,PR2002b.
comparison with MonteCarlo : EuroPhys2002.
Role of sample geometry
The theory developed in our group allows the analytic calculation of the CBS enhancement factor up to any scattering order for any atomic transition and a semi infinite medium. However, the fine comparison with experimental data requires a MonteCarlo simulation to take into account the peculiar geometry of our sample (ellipsoidal shape, quasi-gaussian density profile). The graph below shows the results of an experiment where the CBS cone is recorded in the orthogonal helicity channel as a function of the probe laser detuning, for a given atomic cloud (fixed geometry). We assume that varying the detuning only modifies the atomic scattering cross section, and thus the mean free path and the cloud's optical thickness. We plot the normalized cone width Dqkl (where k is the wave number and l the mean free path) and the enhancement factor as a function of the optical thickness.
The most striking feature is the
linear (in Log-Log scale !) and continuous decrease of the normalized cone
width as a function of the mean free path (inversely proportional to the
optical thickness). This means that the cone width is almost independent
of the mean free path at the center of the cloud, while it would scale
as 1/l in a homogenous sample. This is a signature of the inhomogenous (quasi-gaussian)
density profile in the atomic cloud. A careful analysis shows that the CBS
width is determined by the mean free path at unity optical depth.
For an isotropic gaussian cloud, it is of the order of the size of the cloud.
Thus, in our sample, the CBS width is determined by the size of
the cloud.
The behavior observed for the enhancement
factor is explained by the competition between different scattering orders
: for small b's single scattering dominates decreasing the enhancement factor
; for large b's the relative proportion of large (>>2) scattering
orders increases, but these yield a smaller contribution to the CBS interference
than double scattering, and thus the enhancement is also reduced.
These results and the comparison with a full MonteCarlo simulation including the internal structure and the sample geometry are reported in PRA2002.
Large Faraday rotation
Because the atomic polarizability is highly resonant and the electronic energy levels very sensitive to external magnetic fields, magneto optical effects e.g. the Faraday rotation of light polarization are potentially very strong in cold atomic vapors. The typical Verdet constant is 20°.G-1.mm-1, some 5 orders of magnitude above usual Faraday glasses. By analyzing the polarization state of a probe laser beam sent through the cold atoms, we measured the Faraday rotation in our sample.
In principle, the Faraday rotation is proportional to the optical thickness. However, for low probe transmitted power (i.e. high b's and small B's), the detected signal is biased by off resonant components in the laser spectrum which results in a depolarized transmitted light. This is the origin of the non linear behavior observed on the figure above (right, blue circles). Our measurement yield a Verdet constant of 10°.G-1.mm-1 and a maximum rotation of about 150° for the polarization.
For more details see our PRA2001a paper.
Observation of the Hanle effect in CBS
The shape of the CBS peak is given by the Fourier transform of the diffuse light intensity distribution on the sample's surface. Thus, any anisotropy in the way light propagates inside the sample will result in an anisotropy of the CBS signal. An applied magnetic field induces such an anisotropy, for instance via the Faraday and Cotton-Mouton effects which modify the polarization of light as it propagates inside the sample. The impact of these effects (CBS peak shifting or splitting, other anisotropies, ...) on CBS was experimentally studied in Faraday glass based samples by the group of G. Maret.When we look at the CBS peak shape in the presence of an applied magnetic field (collinear to the light incident wave vector), we observe some new B field induced anisotropies :
We plot here the 2D angular distribution of the backscattered light (red corresponds to maximum intensity). The incident light wave vector and the applied magnetic field are both orthogonal to the plane of the figure. The incident light polarization is linear and oriented along the vertical of the figure. The left column corresponds to the CBS peaks for zero B field in the parallel (top) and orthogonal (bottom) channels. The observed anisotropies are due to the dipole component of the atomic radiation pattern. The column at center corresponds to CBS signals in the orthogonal channel for an applied B field of ±8 G (resonant light). When the magnetic field is applied, the induced dipole rotates (Hanle effect) yielding a 45° flip of the CBS cone whose sign depends on the orientation of B. This is the first observation of CBS anisotropies due to a B-induced modification of the radiation pattern. When the laser is detuned (right column), the induced dipole can become circular yielding an isotropic CBS cone.
These observations are reported in our PRL2002a paper.
Coherence lengthrestoration with a magnetic field
The situation of CBS for atoms in a magnetic field is highly complicated by the presence of the internal structure. This gives rise to behaviors very different from the "classical" situation, as can be seen from the experimental result below where the enhancement factor in the parallel helicity channel increases with the magnetic field.
This experiment is performed by maintaining the laser frequency in resonance with the Zeeman-shifted mF = 3 -> m'F = 4 transition. Very crudely, this allows effectively to remove the internal structure and to recover a full interference contrast. However, the CBS enhancement does not reach 2 because the single scattering contribution can not be filtered out using polarization selection. The group of D. Delande has developed a MonteCarlo simulation to describe this very complex situation. As can be seen in the figure above (solid line), the agreement is remarkable.
Thus, in our situation, adding a magnetic field actually restores the coherence length of the light in the cloud, which was initially degraded by the internal structure of the atomic ground state. This technique may thus prove an important tool to study mesoscopic physics in cold atoms with an internal structure.
In atomic gases, several mechanisms are likely to reduce the mesoscopic coherence length. We have already mentioned the existence of an internal structure. Other possible mechanisms are inelastic scattering (see next chapter on saturation) and the motion of the atoms. Indeed, the frequency redistribution due to the Doppler effect induces a phase difference between the "reciprocal" amplitudes interfering to give rise to the CBS peak. This effect, quite analogous to the Sagnac effect in closed loop cavities, is here enhanced by the very narrow atomic resonance yielding a reduced CBS enhancement even for very small residual velocities.
We have experimentally demonstrated this effect by recording CBS peaks from initially ultra cold atomic clouds which were heated by applying an intense optical molasses :
These observations and MonteCarlo simulations by D. Delande (solid line in the figure above) indicate that extremely cold samples will be required to fully reach the mesoscopic regime in optically thick atomic clouds.
Frequency redistribution may also originate from inelastic scattering, which becomes dominant if the saturation parameter s , set by the light intensity inside the cloud, exceeds unity. Such a situation may play an important role in the case of random lasers or in the presence of localized states. As in the case of the atomic motion, CBS can be used to probe the coherence lenght of the ight in the sample.
An important effort is under way in our group and that of D. Delande to develop a consistent theory for this difficult problem. A simplified model as already given some insights on the physics at work (Wellens2004). The CBS reduction due to an increasing saturation has first been reported in our Strontium experiment (PhysRevE2004). An extensive experimental investigation of the effect of saturation has also been conducted on the Rubidium experiment, where the situation is more complex due to the internal structure of the ground state.
4) Radiation trapping
Coherent backscattering measurements
give access to static quantities, such as the scattering mean free
path. Complementary informations can be obtained from dynamic quantities,
such as the diffusion constant. Time resolved measurement have enabled
in the recent years the development of the rich field of Diffusive Wave Spectroscopy.
In a first series of experiments, we investigated the so-called phenomenon
of Radiation Trapping (RT) : due to multiple scattering, the resonant light
remains "trapped" for a certain time before leaving the sample. This situation
has been abundantly studied since the beginning of last century, in hot
vapors where strong frequency redistribution occurs due to various
mechanisms (Doppler effect, collisions, ...). In this regime, the transport
is essentially incoherent (see for instance the section Dynamical breakdown of CBS).
On the other hand, in cold atoms
the frequency redistribution is very small and the transport is essentially
coherent (at low saturation). Theories have recently been developed
to deal with the case of coherent transport in resonant scatterers. The
scattered wave is characterized by its energy transport velocity vE,
which is predicted to become much smaller than c in highly resonant
samples.
A simplified scheme of the experiment is shown below :
We measure the decay of fluorescence light after the probe light is switched off. The late decay constant, characteristic of the fundamental Holstein mode, scales like the square of the optical thickness as shown below. We extract from these data a diffusion coefficient D = 0.66 m2/s and a transport velocity vE = 3×10-5 c !!!
Despite the low temperature of our sample (~100 mK), Doppler frequency redistribution is not completely negligible. Indeed, the frequency of the scattered light diffuses during the propagation in the sample and its rms value increases as N1/2, where N is the number of scattering events. A manifestation of this residual frequency redistribution can be observed on the figure below.
These results are reported in PRL2003.The two curves correspond respectively to resonant (red) and off resonant (blue) incident light. Although the decay rates are different at early times after switch off (it is faster for the detuned case because the optical thickness is smaller), they become identical at late times indicating that the photons have been frequency shifted closer to resonance.
Multiple scattering of light is
known to limit the spatial density achievable in usual magneto optical traps,
because of the repulsive force created by scattered photons re-absorption.
In the case of an optically thick MOT, an additional compression force arises
from the screening of the usual magneto optical force by the external layers
of the atomic cloud. At equilibrium, the competition between these forces
is responsible for the increase of the cloud's size with number of atoms,
effectively limiting the atomic density in the MOT.
At larger number of atoms, the competition between these various forces can lead to an unstable behavior of the MOT. In a preliminary series of experiments, we observed such instabilities, characterized by "breathing" oscillation modes of the MOT. These instabilities typically start to build-up above a threshold for the considered trap parameter, either the magnetic field gradient, the laser detuning, or the number of atoms in the trap. This is illustrated below, where we show the fluorescence collected from one half of the MOT as a function of loading time.
As can be seen, the MOT spontaniously starts to oscillate once the number of atoms has reached
a certain threshold. Some theoretical efforts are under way in our group
to analyze this novel situation, which may be similar to oscillatory mechanisms
in cepheid stars and plasmas.
Cold atoms are known to exhibit strong non linearities originating from various phenomena : saturation, optical pumping, ... On the other hand, beautiful experiments of "optical feedback" on hot atomic vapors have shown that these non linearities can lead to the apparition of self organized patterns. Such an experiment has never been performed with cold atoms. In addition, the use of cold atoms opens the intriguing possibility of mechanical feedback from the light on the atoms.
In a preliminary experiment, we have observed the nonlinear shaping of a laser beam focussed through the optically thick cold atomic cloud. The figure below shows the transmitted beam far field intensity distribution (after substraction of the incident beam distribution), for low saturation (left, s=50) and intermediate saturation (right, s=1000).
At low saturation the overall transmitted intensity is reduced, while for larger saturation, the transmitted beam exhibits an on-axis enhancement of its far field intensity distribution by ~ 20%. We also observe a smoothing of the intensity distribution in the presence of the cold atoms.
These results, which can be interpreted in terms of non-linear lensing of a Gaussian beam, are reported in EurPhysJD2003 and OPN2003.