Advances in Atomic, Molecular, and Optical Physics (eBook)

7 Inelastic Collisions near a Fermionic Feshbach Resonance

The elastic collisions between atoms discussed in Section 5 are often referred to as good collisions. These collisions allow rethermalization in the gas but do not change the internal state of the atoms or molecules. In atomic gas experiments a constant concern is inelastic collisions between atoms, often referred to as bad collisions. In these collisions the products often are particles in lower-energy internal states. The difference in energy between the incoming particles and the products of the collision must be carried away in the form of kinetic energy. When this energy difference is large compared to other energy scales such as the trap depth particles can be lost from the trap and significant heating of the sample can occur.

Near a Feshbach resonance inelastic collisions can be enhanced along with the elastic collisions. To accomplish the work in this chapter we spent a large fraction of our time understanding the inelastic processes near 40K Feshbach resonances and designing experiments that minimize the effect of inelastic collisions. In this section we will discuss the inelastic collisions near a 40K Feshbach resonance and present measurements of relevant inelastic collision rates. We observe clear evidence of inelastic processes near the fermionic Feshbach resonance but find that despite these inelastic processes the lifetime of the sample is long enough to study BCS-BEC crossover physics.

7.1 EXPECTED INELASTIC DECAY PROCESSES

Let us first consider the stability of free fermionic atoms on the BCS side of a fermionic Feshbach resonance. In particular consider the Feshbach resonance between the f,mf〉 = 9/2,-9/2〉 and 9/2,-7/2〉 states that is used for many of the experiments in 40K. Since these are the two lowest energy states of 40K the only inelastic collision involving two of these fermions that is energetically favorable is

9/2,-9/2〉+|9/2,-7/2〉→|9/2,-9/2〉+|9/2,-9/2〉.

This process however is forbidden due to the fermionic nature of the particles (Bohn, 2000). Thus, any inelastic collision with these states must involve at least three fermions. A three-body inelastic collision in a two-component Fermi gas with components X and Y would take the form

+X+Y→X+(XY)-.

Here the subscript - represents a lower-energy molecular state. Such lower-energy molecular states are always present in these atomic gas systems as there are many vibrational levels of the interatomic potential. To conserve energy and momentum in the collision the products X and XY)- carry away the binding energy of the XY)- molecule in the form of relative kinetic energy. Theory predicts that this three-body collision process will be suppressed for s-wave interactions between fermions because it requires that two identical fermions approach each other (Esry et al., 2001; Petrov, 2003; Petrov et al., 2004; D'Incao and Esry, 2005). However, while the rate of this inelastic collision process is suppressed, it is not forbidden, making it an important process near a Feshbach resonance (Regal et al., 2003b; Suno et al., 2003).

As we cross the Feshbach resonance to the BEC side with a cold 40K Fermi gas we want to consider the stability of a mixture of fermionic atoms and Feshbach molecules. An isolated Feshbach molecule for the f=-7/2,-9/240K resonance will be stable for the same reason as the two fermion mixture is stable. Note that the case in which the two atoms in the molecule are not in the lowest energy internal states, such as molecules created using 85Rb, is quite different (Donley et al., 2002; Hodby et al., 2005). These 85Rb Feshbach molecules will spontaneously dissociate as observed in Ref. (Thompson et al., 2005). For our 40K molecules however we again expect that any decay processes will require more than two fermions, for example, (Petrov et al., 2004; D'Incao and Esry, 2006)

+XY→X+(XY)-,

Y+XY→XY+(XY)-.

The first process is reminiscent of Eq. (22) above. These processes are often referred to as collisional quenching of vibrations (Balakrishnan et al., 1998; Forrey et al., 1999; Soldán et al., 2002). Again, we expect some suppression of these decay channels due to Fermi statistics since two identical fermions must approach each other, as shown schematically in Fig. 34.

7.2 LIFETIME OF FESHBACH MOLECULES

In this section we present experimental data on the stability of a mixture of atoms and Feshbach molecules. To obtain these data we created a molecule sample at the f=-7/2,-9/2 Feshbach resonance in which typically 50% of original atom gas was converted to molecules. We then measured the molecule number as a function of time while holding the molecule/atom mixture in a relatively shallow optical dipole trap (Regal et al., 2004a). Figure 35 shows the result of this measurement at a variety of magnetic fields on the BEC side of the Feshbach resonance. The plot shows ˙/N versus the atom–atom scattering length a. Here N is the number of molecules and ˙ is the initial linear decay rate. We find that far from resonance the molecules decay quickly, but the decay rate changes by orders of magnitude as the Feshbach resonance is approached. Physically this effect is partially related to the overlap between the wave-functions of the XY molecule and the XY)- molecule. As the Feshbach resonance is approached the XY molecules become extremely weakly bound and quite large; hence, they have less overlap with the small XY)- molecules.

A scaling law for the dependence of the molecule decay rate upon the atom–atom scattering length a was found in (Petrov et al., 2004) and later in (D'Incao and Esry, 2005) for both the processes of Eqs. (23) and (24) schematized in Fig. 34. The scaling law is found by solving the full few-body problem in the limit where the molecules are smaller than the interparticle spacing, yet >r0. Physical effects important to the result are the Fermi statistics and the wave-function overlap. The prediction for Eq. (23) (atom–molecule collisions) is that the decay rate should scale with -3.33 and for Eq. (24) (molecule–molecule collisions) with -2.55.

Since our measurement was carried out with thermal molecules the density of the molecule gas remains approximately constant over the =1000a0 to a0 range. (The peak atom density in one spin state in the weakly interacting regime was pk0=7.5×1012cm-3.) Thus, we can measure the power law by fitting the data in Fig. 35 to the functional form a-p, where C and p are constants. We fit only points for which the interatomic spacing at the peak of the cloud is larger than the expected size of a two-body molecule, /2. We find =2.3±0.4, consistent with the predicted power law for molecule–molecule collisions. A similar power law was observed in a gas of Li2 molecules at the 834 G Feshbach resonance (Zhang et al., 2005).

In general we find that the lifetime of the molecules is surprisingly long near the Feshbach resonance. The molecule lifetime for magnetic fields at which >3000a0 is greater than 100 ms. This is much longer than lifetimes observed in bosonic systems for similar densities and internal states (Xu et al., 2003; Dürr et al., 2003). 100 ms is actually a long time compared to many other time scales in our Fermi gas—for example, the time scale for two-body adiabaticity, the mean time between elastic collisions, and the radial trap period. This comparison suggested that it would indeed be possible to study BCS-BEC crossover physics using atomic 40K gases.

7.3 THREE-BODY RECOMBINATION

We also observe inelastic decay of fermionic atoms on the BCS side of the Feshbach resonance, where the decay is due to Eq. (22) (Regal et al., 2003b). These collisions cause both particle loss and heating. The heating can result from a combination of two processes: First, the density dependence of the three-body process results in preferential loss in high density regions of the cloud (Weber et al., 2003)....