introduction to optical lattices

Domesticating atoms

In a gas, the atoms are always moving; their distribution is then anarchistic and always changing. To the contrary, in a crystal solid, the ions or the atoms are well ordered. However, as the crystal finds its cohesion in the forces which bind the atoms, these lasts lose their individuality and appear only in the global behaviour of the crystal: they are not independent any more. Thus, creating a perfect crystal in which the atoms would weakly interact seemed a difficult dream to reach. However, this ideal situation of almost motionless and well ordered atoms, isolated from their neighbours, had been obtained by using trapping laser beams. These trapping beams make possible to lower the temperature of the atomic cloud (the temperature of an atomic cloud represents its agitation, the more it is important, the more the temperature is high) and to create a confining potential at the same time.

The Sisyphus effect

Since the middle of the 1970s, the efforts of physicists to cool atomic gas were concretized by decisive theoretical and experimental results. Lower and lower temperatures were obtained thanks to increasingly sophisticated process and techniques. One of them is the Sisyphus mechanism discovered at the end of the 1980s jointly in the ultracold atoms group of our laboratory and in the S. Chu group at the Bell Labs (USA) and which leads at the same time to atoms cooling and trapping.

The state of an atom is divided in two components: one is external, characterized by the position or the speed of the nucleus, the other is internal, which describes the state of the atomic electrons turning around the nucleus. The light acts on these two components by creating a force which permanently modifies the speed of the atom but also by inducing changes of internal state. By using a well-suited laser beam configuration, one can combine these two actions and handle atoms. To form crystal structures in one dimension the most common configuration uses two counterpropagating beams (laid out one opposite to the other) as shown in figure 1. The beams create, for an atom, a landscape of hills and valleys whose topography depends on the internal state of the atom (if the atom is in a given internal state, it will see a given landscape, if it is in another internal state, it will see another one). Figure 1 shows two of these landscapes (blue sinusoid corresponding at the internal state |+> and red sinusoid corresponding to the state |->). When an atom (in a given internal state, let us say the state |->) moves (let us say from left to right), it will meet a hill and thus will be slowed down by loosing energy (1). When it arrives at the top of the hill, the action of the light on its internal state will induce the passage from the internal state |-> into the internal state |+> (2). This transition being instantaneous, the position and the speed of the atom do not change appreciably. The atom is then found in the valley corresponding to the state |+> and continues its way to the right (3) travelling again up hill, still being slowed down. Thus, the process continues until the atom does not have enough (kinetic) energy to climb a hill. He is then trapped in a valley and it oscillates (4) with a small speed at the bottom of this one. The atom is trapped and cooled!

Figure 1 : Sisyphus cooling mechanism: an atom is constrained to go up a hill without going down until it looses its kinetic energy and it is trapped in a potential well.

This mechanism is called Sisyphus effect referring to the Greek hero, king of Corinthum condemned by the gods to transport on the slope of a hill a heavy rock which unrelentingly went down again to the bottom of the hill once arriving at the top. The condemned king had to begin again this painful work until exhaustion.

Optical lattices

As we have seen, since the atoms are distributed at the bottom of the valley, they form a reguler pattern, so an atomic crystal. However, contrary to what occurs in a solid, not every site is filled and every site can contain several atoms. In fact, the atomic density of the optical lattice is primarily determined by the initial density of the cloud (before the application of the laser beams). Moreover, the atoms are much more distant from each other than in a solid so that they practically do not interact. The landscape of hills and valleys has a great regularity due to the coherence of the light emitted by the lasers. The trapped atoms in the potential well demonstrate this periodicity of the landscape. One thus creates an " optical lattice " in one dimension.

To obtain a similar regularity of the atomic pattern in two- and three-dimensional situations, one has to use several crossing beams at the place where are located the atoms. If one wishes to create a two-dimensional structure, it is necessary to use three beams and one needs four for a three-dimensional structure as it is shown in figure 2a. The superposition of these laser beams creates for the atoms a landscape of valleys separated by hills similar to that of figure 2. In fact, as the figure shows, one always finds the same pattern and the landscape consists on the repetition of this pattern, as in a tiling of a bathroom. Furthermore it is possible to carry out a large variety of two- and three-dimensional landscapes, only by changing the directions of the laser beams. One can also add laser beams and then get a new variety of landscapes.

Figure 2: Four beam configuration (a) creating a landscape of valleys and hills (b). Thanks to the Sisyphus cooling, the majority of the atoms are trapped in the valleys where they stay a very long time.

The behaviour of the atoms in this landscape is completely identical to the 1D situation: the atoms move from a well to another by crossing the hills. The rises of these hills exhaust their energy and after approximately one millisecond, the atoms are trapped in one of the valley, not having enough energy to climb a new hill; an equilibrium state is then reached where the majority of the atoms are trapped in these wells. In this equilibrium state, the temperature of the atoms is extremely low, and differs from the absolute zero only by one millionth degree.

Optical lattices, a model for crystal lattices

"Optical lattices" can be seen as "super-models" of the traditional crystal lattices where one would have changed the scales length (the micron instead of the angström), temperature (millionth of Kelvin degree instead of ten degrees) and mass (the atomic mass instead of the mass of the electron, 100 000 times weaker). It is because of these scalings, that mechanisms, as elementary as those leading to the great mobility of electrons in the solids, are not found in these optical lattices: atoms located in a well generally remain there for very long times. The use of coherent light to produce these super-models has the advantage of creating structures without defects; i.e. that the pattern of the landscape is repeated at long distances without the defects (fractures, defects of periodicity due to impurities) that one finds in real crystals. Moreover optical lattices have the advantage that the pattern of the landscape is perfectly determined by the physicist who can change it as he wishes (it can thus dig the valley, increase them or narrow them) whereas in the solids, the form of the landscape results from the forces acting between atoms (which are given by nature and which one can hardly modify). It follows that the optical lattices are the ideal medium to test models which will find applications in real crystals.

Some physics about optical lattices

This analogy with solid crystals constitutes an apparently inexhaustible source of experimental studies because any effect observed in solids has an equivalent in optical lattices. However, the properties observed in optical lattices can be very close or extremely different from those in solids: properties related to the symmetry of the landscape often lead to results similar to those obtained in the experiments on crystal structure, whereas properties using scale parameters, such atomic transport, give very different and original results.

The most common method to study the properties of atoms trapped in optical lattices consists in probing the medium by sending in the lattice an additional laser beam (probe) of low intensity. This beam deforms in a controlled way the interference pattern seen by the atoms; this deformation appears by a displacement of the wells and thus of the trapped atoms. The atoms react to this excitation of their equilibrium state by absorbing or emitting light, which results in a modification of the intensity of the additional beam measured after the lattice. The analysis of these variations of intensity gives information on the movement of the atoms and their spatial distribution.

Some applications of optical lattices

Among the potential applications of this field of research, currently undertaken by many laboratories, two could experience a significant development. Firstly, these structures seem to constitute a promising way to carry out a " boser " which would be for the matter what is the laser for the light, i.e. a coherent beam of atoms. Furthermore, these lattices may allow, the realization of regular micro-patterns of atoms, and this would have lots of applications in micro-electronics.

Broad audience papers

G. Grynberg, "Une matrice de lumière pour ranger des atomes", La Recherche 256, vol. 24, 896 (1993) (in French)

S. Chu, "Laser trapping of neutral particles", Scientific American vol. 266, N.2, pagg. 48-, (1992)

A. Aspect et J. Dalibard, "Le refroidissement des atomes par laser", La Recherche 261, vol. 25, 30 (1994) (in French)

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