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a. Laser cooling and trapping

Dans le document The DART-Europe E-theses Portal (Page 26-31)

Part I – Abstract

I.1. INTRODUCTION

I.1.2. a. Laser cooling and trapping

Magneto-Optical traps

Laser cooling is based in the fact that the spontaneous emission is homogenous in its direction, which means that when an atom emits a photon spontaneously, this photon can be emitted towards any direction with the same probability. When an atom absorbs or emits a photon, light transfers momentum to the atom, or away from it, which equals

p=ℏk (1.6)

where k is the wave-vector associated to the photon. Let us consider an atom traveling in a direction opposite to the propagation direction of a laser beam resonant to one of its transitions. As the laser radiation is directional, while spontaneous emission is homogenous in space, a repetition of absorption - spontaneous emission circles leads to a constant momentum loss in the direction of the atomic motion. This momentum loss is based to the fact that spontaneous emission is isotropic; if the emission is stimulated by the laser field; the photon emitted is forced to have the direction of the simulating field, leading to zero net momentum exchange.

For the simple one dimensional geometry, let us imagine a counter-propagating pair of beams that are resonant to a particular atomic transition. The force that results such a situation is the sum of the forces F± which correspond to the counter-propagating beams and equals to

F±=k 2

s0

1s0[2∓

D

/2]2 (1.7)

where Γ is the natural line width of the excited state, ωD the Doppler shift of the transition frequency and s0 the saturation factor, as this results from the solution of the excitation of the related two-level system and equals to 22/2 with  being the Rabi frequency. The sum of these forces gives the total force

Ftot≃ 8ℏk2s0

 1s02/ 22 (1.8)

Fig.I.1.3: The radiation force Ftot as a function of the atomic velocity, for the convenient choice of s0 = 2 and Γ = 2 s-1. We note that there is a maximum velocity after which the force is not attractive and the atoms cannot be cooled.

In Fig.I.1.3 we see a plot of Ftot for a convenient choice of parameters. Such a process can lead to more general cooling and trapping if the momentum exchange always opposes atomic motion. In a MOT this is ensured by a simple configuration of laser light polarization and magnetic field gradients. In a MOT there are three pairs of counter-propagating, circularly polarized laser beams lying in the three axis of the Cartesian coordinate system, while the magnetic field gradient configuration that is used is the quadrupole configuration. The magnetic field in a quadrupole trap increases in a linear fashion with respect to the distance from its center. As atoms move in an inhomogeneous magnetic field, they experience a shift in their energy levels which depends in which particular spin projection state m they lie and in the sign of the Lande factor gF. If we consider atoms that lie in the negative m state and that have their energy levels lowered for increasing magnetic field, the positive m atoms will have the opposite effect. If the polarization of the light is chosen to be circular right σR, and is frequency lower with respect to an atomic transition, then negative m state atoms will be driven closer to the resonance and will scatter more light, a fact which results to a radiation force that drives the atoms in their original position. A counter-propagating beam with the opposite polarization ensures that the same effect will take place for atoms that lie in the positive m states and that move in the opposite direction. To this process, we add the fact that once an atom starts interacting with such a laser geometry, a combination of radiation forces and of optical pumping keeps it to the extreme m states, and assures that most of the atoms are found to this situation where a force that opposes their motion is applied to them at all times.

The number of absorption-spontaneous emission circles must be significant, thus atoms must perform transitions between two levels without relaxing to different ones and thus stop being resonant to the laser light. Such isolated, 'closed' transitions, are not easily found in real atoms.

Nevertheless, it is possible to find systems, where the losses take place towards states that can be 'connected' to the initial ones with the use of one additional laser, the so-called re-pumper. Although the need of a re-pumper is definitely a complication with respect to the image described above, placing one repumping laser is experimentally realizable. This is not the case for example, in molecules, where the density of states is much larger. Molecules can relax to such a large number of Fig.I.1.4: (a) Schematic representation of laser excitation and spontaneous emission in laser cooling. (b) Configuration of laser beams and coils in a MOT. Two circular coils with counter-propagating currents create a quadrupole field (anti-Helmholtz configuration).

different states that addressing each of them with a different laser is completely impractical. We will return to this interesting aspect in the second part of this thesis, where the difference between atoms and molecules is going to be viewed in more detail, and where we discuss how our research can perhaps give new solutions to related problems.

A detailed analysis of MOTs is beyond the scope of this text as this can be found in many textbooks, like the one referenced here [Metc99]. The physics that are involved for a complete analysis of the MOT is much more complicated than the simplified picture presented here. For example, the given analysis implies that the minimum temperatures that can be achieved coincide with the one photon limit, which is the temperature that corresponds to the kinetic energy acquired by the emission of one photon in a random direction. However, this is not true, since there have been much lower temperatures reported. There are more mechanisms, beyond the photon absorption, that contribute to cooling in a MOT, and that are related to the creation of optical lattices and optical molasses.

Molasses

The operating principle of laser cooling described previously, does not suffice to describe the molasses phase, and additional mechanisms take place as the well known “Sisyphus”

mechanism. This mechanism can be considered in two different configurations, which we see in Fig.4.15, (a) the Lin-Lin and (b) the σ+σ- configurations. In both configurations, the counter-propagating laser beams interfere, and the atoms 'see' effective potentials and polarization gradients that depend on their position with respect to the laser's standing wave. In the Lin-Lin configuration, the atoms 'see' the light polarization to change from linear at +45°, σ+, -45 and σ- within one laser wavelength, while the potential they 'see' varies with a sinusoidal fashion. The laser light is resonant with a mF → mF+1 transition, which can be realized in the position where the light is σ+ polarized, a position which coincides with the position of the potential 'hill'. After this transition is realized, the atoms are found themselves in a state with increased F, which 'sees' a different potential which has a phase shift of π, and which is shown dotted in the figure. Thus, the atoms are found in the potential 'valley' and loose kinetic energy. This process goes on until the atoms have less kinetic energy than the one that would allow them to 'climb' the potential 'hill'. They cannot then perform any transitions and stay trapped in the 'lattice' produced by the laser beams.

In the σ+σ- configuration, the atoms 'see' the light polarization rotate by 90° in each wavelength. The light is now resonant to a F → F-1 transition and not all transitions are allowed by the angular momentum selection rules, leading to the preparation of coupled ΨC and non coupled ΨNC states. The effect of the light on the atoms is that it gradually transfers all atoms to a non coupled state by optical pumping. Kinetic energy can couple the two states, so the atomic population cycles between the two states, until the kinetic energy cannot further couple them and the process terminates.

In Fig. I.1.5 we see a schematic diagram concerning 'Sisyphus' effect in the Lin-Lin and σ+σ- configuration. In part (a) which concerns the Lin-Lin configuration, the polarization of the standing wave produced by the counter propagating linearly polarized trapping beams turns from linear to circular and in a distance equal to ¼ of the wavelength. The atoms are excited when they are found in the potential “hill” and relax to the “valley” of the phase shifted potential. In part (a) which concerns the 'Sisyphus' effect in a σ+σ- configuration, the polarization of the standing wave produced by the counter propagating linearly polarized trapping beams is linear and rotates 45° in a distance equal to ¼ of the wavelength. In each of the angular momentum projection states mJ there are two possible polarizations that cannot excite the atom, creating the non coupled states. The coupled state is excited in the potential 'hill' and relaxes to the non coupled state which sees a flat potential. This scheme is a composition of schemes taken by [Boir98].This cooling methods is also called 'dark' or 'gray' molasses, since in the end of the process the atoms do not interact with the light, and there is no fluorescence.

Raman-Sideband Cooling

Raman Sideband cooling is a laser cooling technique that permits the preparation of atomic samples with temperatures as low as several hundred nKs. As in the case of molasses, the RSC operating principle is complicated and cannot be described by the simple picture of momentum exchange between the photons and the cold atoms.

Fig. I.1.5: Schematic diagram concerning 'Sisyphus' effect in (a) Lin-Lin and (b) a σ+σ- configuration.

Figure composed by parts adapted from [Boir98].

In the RSC technique, light does not interact with free, but with trapped atoms with relatively high 'trapping frequencies' (or small oscillation periods in the trap). The technique succeeds in performing optical transitions between these vibrational states, and transferring the atomic population in the lowest one. The ability to resolve these vibrational states is essential for the realization of the technique, this is why the it requires either enormous resolution or high confinement. RSC was initially used for the cooling of ions [Nago86], where the electrostatic traps achieve the required confinement. For the realization of the technique in neutral atoms, the required confinement can be achieved only by the use of optical lattice traps in one, two or three dimensions.

In our experiment we realized RSC in a three dimensional lattice following the work presented in [Treut01].

In figure 4.23 we see a scheme that presents the operating principle of RSC (taken by [Kerm00]). We consider atoms in the vicinity of an optical lattice pumped in the |F =3,mF= 3,v>

ground state, where v = 0,1,2,3.. is the vibrational quantum number. An appropriate magnetic field can bring into resonance this state with the |F =3,mF= 2,v-1> and |F =3,mF= 1,v-2> states. Two photon degenerate Raman transitions couple these states, with a result of atomic population to be transferred to lower vibrational states with a simultaneous modification of the polarization. If the atoms are simultaneously polarized, for example by our standard optical pumping scheme described in I.4.1.b, the atoms are pumped again in a state with mF=3 and lower v. The process is reinitialized and goes on until all atomic population is pumped in the |F =3,mF= 3,v=0> absolute ground state of the system, which is a dark state and does not interact with the light.

The RSC technique demands for three different laser fields, one to provide with the optical lattice, one to induce the degenerate Raman transitions and one to perform polarization. However, the work presented in [Treut01] simplified considerably the technique, to the point that only two lasers are required. One is the polarization laser and another provides the lattice by being in far detuned with respect to the 6²S1/2 , F=3 → 6²P3/2 , F'=4 transition, and the Raman transitions shown in Fig. I.1.6 at the same time.

Fig. I.1.6: Operating principle of the RSC technique. A specific magnetic field brings states of vibrational quantum number v into degeneracy with state with vibrational quantum number v – 1. A sequence of Raman transitions between those degenerate states and optical pumping results to the transfer of the atomic population in the |F =3,mF= 3,v=0> dark state.

Dans le document The DART-Europe E-theses Portal (Page 26-31)