Other quantum scattering effects

Beside the typical two-body elastic and inelastic collision processes described in the previous sections, several other quantum scattering effects occur in cold and

ultra-cold gas clouds. Some of these effects, such as the three-body collisions and radiative heating are inelastic in character and thus largely undesirable, leading to increased trap loss rates. Others, such as scattering resonances have great experimental potential and can be exploited to an advantage.

In the following, we will look at a few processes, which are subject of present research. However, most of these processes are not of immediate importance to the subjects of the present work.

Inelastic three-body collisions

Trapped cold or ultra-cold atomic and molecular gases exist in a regime of extremely low density. This can be expressed as na³ 1, where n is the particle number density and a is the scattering length. In this regime, binary interaction is dominant and hardly any three-body collisions will occur. So few, in fact, that three-body collisions can safely be neglected. However, it is now possible to fine-tune particle interaction using Feshbach resonances (see further below), allowing realisations of arbitrarily large scattering lengths, so that three-body effects can be studied in more detail.

Particle loss from a trapped cloud due to inelastic two-body and inelastic three-body collisions can be expressed in terms of the two-body loss rate K₂ and three-body loss rate K₃ as

∂n

∂t =−K₂n²−K₃n³, (2.50) With increasing density nthe three-body loss rate will become significant and rapidly bypass the two-body loss rate. Note that three-body collisions are mostly inelastic leading to trap loss, as these collisions are the only way the undercooled gas can reach the solid phase mandated thermodynamically by the low ensemble temperature. In the recombination process, two particles aggregate in the presence of a third particle, transferring a large amount of (kinetic) energy to the third. Thus most often all three participating particles are lost from the trap– the aggregated two since they end up in an untrapped state and the third due to its large kinetic energy, which is orders of magnitude larger than that of the remaining ensemble particles.

In ¹³³Cs, where three-body effects cannot be neglected, Weber et. al. [42]

found aK₃ ∼a⁴dependence of the three-body loss rate coefficient on the elastic scattering length a, which can be controlled precisely by magnetic tuning of a Feshbach resonance.

Feshbach resonances

Exploiting the energetic coupling of different elastic and inelastic scattering channels, the collisional cross section of atoms in specific spin states with specific collision energies can in some cases be tuned on a wide parameter range using an offset magnetic field in an effect known as Feshbach resonance [43].

As we have already shown in section (2.1.4), quasi-bound states can have a tremendous effect on the scattering properties. One must distinguish between

2.1. COLD COLLISIONS 29 different types of resonances. Simple scattering resonances, as discussed earlier, are called shape resonances. They can arise due to a potential barrier and the bound wavefunction belongs to the same system internal state as the continuum wavefunction. No internal state transitions are required.

Feshbach resonances arise when an incoming channel is resonantly coupled to a different channel in such a way that the wavefunction for the incoming continuum state and the wavefunction for the quasi-bound state belong to a different internal state of the particle. For a Feshbach resonance, the bound state is semi-stable and cannot decay into any channel other than the incoming channel.

In magnetically trapped alkali gases and BEC, the continuum and bound internal states relevant for the Feshbach resonance effects are states of the trap magnetic field. In some cases their Zeeman energy changes at a different rate for a change in magnetic field magnitudeB. Thus, using a simple homogeneous offset magnetic field, the Feshbach scattering resonance can be tuned, allowing a precise adjustment of the particle s-wave scattering length in the BEC over an extremely wide range.

Scattering in molecular gases is more strongly influenced by such scattering resonances than is the case in atomic samples, due to the large number of coupled channels and internal rotational molecular states. Feshbach resonances involving changes in one or both molecules’ rotational quantum numbers can be very long-lived and lead to resonance lines, which are broad in temperature and thus relevant over a wide range of collision energies.

Optical Shielding of cold collisions

It is possible to effectively shield collisions of particles in specific states by means of a laser field. The laser is tuned to resonance with a transition of the entrance channel to a state with a repulsive interaction [44, 45]. Particles approaching each other, attracted by the long range part of the inter-particle potential, are coupled into an upper state by a laser beam resonant with the transition at a specific particle separationR_c. This separation R_c marks the Condon-point for the transition, where the potential difference matches hν, where ν is the laser frequency. Further approach is quickly halted and reversed by the particle repulsion in the upper state. Outbound, the particles couple back into the initial state when they reach the resonance distanceRc again. Upper state life times are important for this effect. Rapid decay times make it less efficient.

Partial upper state survival causes the optical shielding process to have a slightly inelastic character. Outbound particles remaining in the upper state will transform the optical excitation energy into kinetic energy heating the trapped particle ensemble. This can be considered to be a radiative heating process.

Radiative Heating

All cold atom and molecule experiments require the use of optical traps at least during some of their stages. In low temperature collisions, the collision

time scales, that is the time the collision partners spend at ranges of separation, where they experience the interaction potential, is roughly comparable with the time scales of electronic excitation and relaxation. Thus a certain amount of

“interferences” between the two processes can be expected, even in non-resonant conditions. This has been shown experimentally and studied computationally in [46, 47].

The optical trapping laser field drives transitions of a fraction of the colliding particles into an excited quasi-molecular state during the collision process. A fraction of these atoms, just as in the optical shielding resonant coupling case, will not be coupled back and will spontaneously decay back into the ground state at a later time, likely after “rolling down” the excited state potential [46]. Thus these particle transfer optical excitation energy into kinetic energy, dispersing it in subsequent collisions and thus heating the trapped ensemble. This effect is called “radiative heating” and puts a lower limit on the temperatures that can be reached in optical cooling processes. Radiative heating effects are most significant at temperatures below the Doppler laser cooling limit T_D.

Photoassociation

Optical transitions can also be used to transfer atoms into molecules while they are undergoing a collision process. In such a case, at a specific interparticle separation Rc both participating atoms absorb a laser photon and couple to an excited molecular state. This effect has been used experimentally in [48] to measure detuning dependent transition rate modulations in the sodium ground state, so-called Condon modulations, which allow precise derivation of s-wave scattering lengths [49].

2.1.8 Semi-classical approach to quantum scattering theory