1. Introduction

The development of the laser cooling and trapping methods for neutral atoms were essential to the progress of many fields. The reduction of the random thermal velocities using radioactive forces allowed the study of the atomic collisions and the determination of the atom inner structure with great accuracy [^{7}]. Associated to the magneto-optical trap [^{8}] (MOT), this technique also formed the basis for the achievement of Bose-Einstein condensation in atomic gases [^{9}], opening the way to a deeper understanding of the quantum-physical behaviour of gases at ultra-low temperatures.

In this Letter we describe the standard and the alternated optical molasses techniques, first introducing the concept of radiation force, making use of a semiclassical treatment to consider the interaction of electromagnetic radiation with the atomic system. This force is at the heart of the cooling technique, since it has the effect of dissipating the energy of the atoms and, thus, cooling them.

We determine the evolution of the atomic energy during the cooling process, for the standard and alternated molasses configurations.

By looking at the evolution of energy and its minimum, we can trace parallels between the two techniques and their advantages/disadvantages concerning the experimental process and the atomic densities reached by each one of them.

A gas of bosonic atoms enters in the quantum-degenerate regime and forms a Bose-Einstein condensate (BEC) if its spatial density
^{10}]. Low temperatures in combination with high densities have to be reached to obtain quantum degeneracy.

Since the early days of laser cooling, it has been asked if the quantum degenerate regime could be reached using this efficient method as the only cooling process. Until now, however, laser cooling has to be followed by evaporative cooling procedure [^{5}] to produce quantum degeneracy in cold atomic samples. The re-absorption of photons scattered during laser cooling [^{1}] induces an effective repulsion and heating of the low energetic atoms. That is a severe obstacle to the required density be reached.

As will be shown, the alternated molasses configuration may be a better cooling technique in what concerns the spatial density of the atomic sample.

2. Radiation force

There are basically two kinds of forces exerted on an atomic system due to interaction with radiation: spontaneous and dipole forces [^{7},^{11}]. The spontaneous force originates from the momentum transferred to the atom during the photon absorption or emission processes. Each photon transfers to the atom a momentum

In this paper we used a semiclassical approach to consider the interaction of the radiation with the atomic system [^{12},^{13}], that is, we considered atoms, which will be treated quantum mechanically, interacting with a classical electromagnetic field. This theory, which is based on Ehrenfest's theorem and optical Bloch equations, have the advantage of giving an unified treatment to the radiation force, including effects of spontaneous emission and the induced dipole interaction, maintaining the simplicity of the most elementary theories. The atom considered in this theory has two energy levels (two-level quantum system).

Considering the atom moving sufficiently slow (

where

where
^{11}].

Consider now two laser beams in the same direction (

where

Writing this damping force as

This force has a maximum value
^{6}], since we consider the saturation term to be simply
^{11}], besides the dipole force due to the standing waves [^{14}].

3. Optical molasses

As showed in the previous section, the combination of the opposite laser beams produces a viscous force that dissipates the energy of the atom to the electromagnetic field. To determine the total energy of the atomic system, however, we should consider the quantized nature of electromagnetic field to evaluate the contribution of the spontaneous emission process [^{15}], where an atom initially in an excited state can return to the ground state even in the absence of incident radiation (see figure (2)).

In spite of this random process does not change the mean atomic momentum (

which corresponds to the scattering rate of the laser light by the atom times its average kinetic energy. The latter can be calculated considering the atomic diffusion in the momentum space, with discrete walks of size
^{16}], resulting in the following expression

In equation (6),

We consider an isotropic emission process where

where

With the

The balance between the energy lost due to dissipative cooling force and that acquired in the recoil process allow us to construct the dynamic equation for the atomic kinetic energy

In Eqs. (10), the indices N classifies the 1D, 2D and 3D molasses configuration, for the direction with (

Solving these equations for

where

and it is straightforward to verify that it has a minimum value at the time

Although the energy reaches a minimum value in the direction

Equating the energy rate to zero we obtain a stationary condition, where the energy is given by

Solving equation (14), in terms of the

According to equation (15), the minimum energy value is reached with a laser detuning of
^{7}]

4. Optical molasses with temporal alternated beams

In this section we introduce our new cooling technique, to be compared with the standard one, the alternated molasses configuration. It has one pair of lasers at each direction, which, however, will be alternately turned on during a selected time interval. A natural choice is to apply the characteristic time

Considering now the lasers at

With the pair along the

Finally, after finished one cycle, we obtain

By using recurrence relations for the initial kinetic energy, after the second cycle we will have

With the same procedure

This initial sequence allows us to generalize the energy after

After many cycles, i.e., for large values of

5. Numerical Simulation

In this section we compare the atomic kinetic energy evolution resulted from the previous studied molasses configurations. For the simulation, we chose the atomic strontium species, since it has many advantages in the laser cooling context [^{17}]. We selected the isotope

Figure 3 shows the atomic energy evolution in the standard

As expected by our calculations, in both cases we have the same energy limit which correspondents to a temperature of

6. Standard Optical Molasses *versus* Alternated Optical Molasses

It was shown above that the two techniques reach the same energy limit. While the standard molasses may be experimentally easier to perform when compared to the alternated molasses, since it does not require the alternation of the laser beams, our new configuration can possibly reach a higher phase space density.

The atomic density reached in magneto-optical traps has an upper bond value due to the repulsive forces between the atoms caused by the reabsorption of scattered photons, as stated before.

For a collection of atoms in a optical trap with laser field intensity
^{18}]

where

Due to the intensity gradients, the confining lasers also produce an attenuation force
^{19}], which compress the atomic cloud:

These two forces

Thus, the force generated by the re-absorption of light gives an upper limit to the atomic density that is inversely proportional to the field intensity. Since in the alternated molasses this intensity is three times smaller than in the standard 3D molasses, the maximum density will be higher.

7. Conclusion

In this work we studied and compared two configurations of optical molasses. Starting with the semiclassical treatment of the laser-cooling concept, that is, explaining how neutral atoms interact with the radiation force, we determined and studied the dynamic equation that governs the atomic energy in a molasses configuration. This treatment allows us to shown that the atomic sample reaches the same minimum temperature expected for the standard and alternated molasses, without relevant increase in the cooling time. Remarkably, the alternated configuration has the advantage of reducing the power beam fluctuation during the cooling process. This factor is a problematic issue in the standard molasses configuration, that uses three retro-reflected laser beams. Otherwise, the alternated molasses can be produced with just one beam deflected between the three spatial directions, using acoustic-optic modulators with appropriate time interval (
^{20}]. That reduces considerable this source of fluctuation between laser beams.

The main difference of both configurations, however, can be related to the atomic density reached in the magneto-optical traps, which is constraint due to the repulsive forces between the atoms caused by re-absorption of the scattered photons. As a practical matter, the power of the re-scattered light sets a limit to the number of atoms which can be confined in a magneto-optical trap, since the magnitude of this repulsive force depends on the laser field intensity