Three-body losses induced by the resonant bound state

The bound-state admixture is not only decisive for the creation of molecules, e.g. via PA (see Sec. 2.2), but also for the lifetime of the atoms in the harmonic trap. In the presence of a single open channel two-body collisions are usually elastic, i.e. the kinetic energy is conserved during the collision. Atoms are then predominantly lost from the trap by undergoing inelastic three-body collisions. In the presence of a third atom the state of two unbound or weakly bound atoms can be coupled to a deeply bound molecular state. Due to the large REL motion binding energy of the molecule the state of the molecule and the third atom must have a large COM energy to be in resonance with the scattering state of the three atoms. Since the trapping potentials are much shallower than the molecular binging energy, the molecule and the third atom are lost from the trap at such a three-body resonance.

Usually, atom-loss processes are attributed to a resonance of the scattering length, which leads to a higher probability of finding three atoms within close distance [9].

3. Two-channel solution in a harmonic trap

1065 1066 1067 1068 1069

0.001

1065 1066 1067 1068 1069

B@units of GaussD

Figure 3.8: Top: Energy spectrum of ⁶Li-⁸⁷Rb as a function of the magnetic field B in a trap with ω = 2π ×20 kHz (left) and ω = 2π ×200 kHz (right).

Dots indicate MC calculations while lines indicate solutions of Eq. (3.20).

Bottom: Corresponding admixtures |A|² of the RBS for each energy level as a function of the magnetic field B. Dots indicate MC calculations while lines indicate results of Eq. (3.30).

1064 1066 1068 1070

ÈETC-EMCÈ@unitsofÑΩD 1^stenergy level

2^ndenergy level

Figure 3.9:Left: Absolute difference between TC model and complete MC calculations of the eigenenergies of ⁶Li-⁸⁷Rb in a trap with ω = 2π ×200 kHz (see right column in Fig. 3.8). Right: Corresponding differences of the RBS admixtures. Apart from energies well below zero, where the lowest state forms a bound state, the energy differences are smaller than 0.002~ω and the difference of the RBS admixtures are smaller than <0.1%.

40

3.3. Comparison with experimental results

However, the losses are also influenced by the admixture of the RBS [66]. Since the RBS is already a bound state one can assume that it is stronger coupled to deeply bound states than the unbound open-channel wave function. Unfortunately, in free space the resonance of the scattering length and the maximal RBS admixture coincide such that it is hard to determine whether the value of the scattering length of the open channel or the admixture of the RBS leads dominantly to losses.

In the trap, however, according to Eq. (3.31) the bound-state admixture is shifted from the resonance ifa_bg/a_ho and E/(~ω) are large. Accordingly, the shift should be well observable in a system of two species of Fermionic atoms with a large background scattering length that fill up many trap states. Such a system was, e.g., regarded in an experiment performed by Bourdel et al. [67] with 2N = 7×10^{4 6}Li atoms in two different hyperfine states with a large mutual zero-energy background scattering length ofa0 =−1405 a.u. The atoms were confined in a harmonic trap withωz= 2π×0.78 kHz and ωx ≈ ωy ≈ 2π ×2.2 kHz. The atom loss was determined as a function of the magnetic field. Alocal maximum of atom loss was found close to the resonance position BR of the scattering length. However, the global loss maximum was observed at a surprisingly large shift of about−80 G fromB_R (see Fig. 3.10).

Figure 3.10: Bourdel et al. [67] determined the atom loss of initially 4×10^{4 6}Li atoms in a harmonic trap. While a local maximum of atom-loss is found at the free resonance position, the global maximum appears at the maximal admixture of the RBS,−80 Gauss shifted from the resonance (plot derived from Ref. [67]).

Ultracold atoms scatter predominantly at the Fermi edge where they have a relative-motion energy equal to the Fermi energy. In a harmonic trap with ω_x = ω_y = ηω_z one has EF = ~ωz(6N η²)^1/3 [68]. Employing Eq. (3.30) for an anisotropic trap with η≈3 one can determine at which energyE_maxclose toE_F the RBS admixture reaches its maximum. Knowing E_max and the corresponding scattering length of the open-channel wave functionamax one can determine the magnetic field of the maximal RBS admixtureB_max at whicha(E_max, B_max) =a_max [see Eq. (3.20)].

Since the Fermi energy is large, the energy dependence of the background scattering length in Eq. (3.30) has to be taken into account, which is determined by the van der

3. Two-channel solution in a harmonic trap

Waals coefficient C6 [see Eqs. (3.14) and (3.15)]. For C6 = 1393.4 a.u., ∆B|_E=0 =

−300 G and σ = 2µ_B [46] the model predicts a maximal RBS admixture at B_max = B_R−80.8 G.

This agrees well with the maximum loss position, which can be an indication that the RBS admixture enhances transitions to deeper bound states and thereby influences atom-loss processes. However, the MFR is broad the RBS admixture|A|² itself is only on the order of 10⁻⁶, which raises the question if such a small admixture can influence loss processes. On the other hand, the ordinary loss process of a three-body scattering event in the open channel is strongly suppressed. Because only two Fermionic species of

6Li are present, two of the three scattering atoms must be identical Fermions such that s-wave scattering between them is impossible while the scattering of higher partial waves is hindered by the centrifugal barrier (see Sec. 1.2). The RBS consist on the other hand of two atoms in a superposition of hyperfine states that differ from the open channel such that s-wave scattering of the third atom with the RBS should be possible. Consequently, despite its small magnitude the RBS admixture could be the dominant reason for atom losses.

There exists yet another qualitative explanation of the off-resonant loss. In Ref. [67]

it is argued that for some scattering length a ≥ 0 a weakly bound molecule can be formed, whose binding energy is just sufficient to lead to a loss of the molecule and the third atom. A decisive answer on the question whether the RBS admixture or the formation of a weakly bound molecule leads to the loss could be given experimentally.

If the experiment in [67] would be performed in a deeper trap the gain of binding energy from the weakly bound molecule would not suffice to overcome the trap. In this case the persistence of the atoms loss would be a strong indication that even at a broad MFR the relatively small RBS admixture can be responsible for losses.