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Section 8: Making BECs from Fermi Gases

We have discussed the fact that a neutral atom of 7Li is a boson. That's because the neutral atom's three protons, three electrons, and four neutrons add up to an even number of spin-1/2 fermions, thus giving the atom an integral total spin. 7Li is the predominant isotope of lithium that occurs naturally on Earth. However, the isotope 6Li accounts for about 7 percent of naturally occurring lithium. This isotope has an odd number of spin-1/2 fermion constituents: three protons, three neutrons, and three electrons, and is thus a fermionic atom. What happens to this fermionic isotope of Li if you trap it and cool it down? Rather than Bose condensing like 7Li, it fills in the energy levels in the trap just like electrons fill in the energy levels of an atom. A fermionic gas in which the lowest energy levels are occupied with one particle in each level is called "degenerate." One can thus envisage making an ultra-cold degenerate Fermi gas from 6Li, and researchers in the laboratory have actually done so.

Atoms in a trap at 0 K: bosons form a BEC (left) and fermions form a degenerate Fermi gas (right).

Figure 30: Atoms in a trap at 0 K: bosons form a BEC (left) and fermions form a degenerate Fermi gas (right).

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Cooling Fermi gases in that way is far more difficult than making an atomic gaseous BEC. Once physicists learned to make them, gaseous BECs immediately became commonplace. Part of the advantage of making a BEC is the bosonic amplification—the effect of sucking all the bosons into the same quantum state that we encountered in our earlier discussion of lasers. The word "laser" is actually an acronym for "light amplification through stimulated emission of radiation." Once a single quantum state begins to fill up with bosons, others, miraculously, want to join them.

This is not at all the case with fermions, as their exclusion principle dictates entirely the opposite behavior. In fact, once a cold and partly degenerate Fermi gas begins to form, many of the energy levels are occupied but a (hopefully) small fraction are not. It then becomes very difficult to further cool the Fermi system. As most levels are already full, only the few empty ones are available to accept another atomic fermion. If these are few and far between, it takes a lot of time and luck to have a Fermi particle lose—through evaporative cooling, say—just the right amount of energy to land in one of the unoccupied energy levels. The other levels are blocked by the exclusion principle. Unsurprisingly called "Pauli blocking," this is a real impediment to making gaseous macroscopic, fully degenerate cold Fermi gases from fermionic atoms. Experimentalists often co-condense 6Li with 7Li and allow the 7Li BEC to act as a refrigerator to cool the recalcitrant 6Li atoms into behaving.

Pairing fermions to make bosons

In the style of earlier parts of this unit, we can represent fermionic 6Li as 6Li, with the now representing the atom's outer unpaired electron. Here we can, at last, illustrate the pairing of fermions to give bosons—the analog of the Cooper pairing of electrons in a superconducting metal that Unit 8 will develop fully. Combine two fermions and you get a boson. This is simple numerics: Doubling an odd number produces an even number. So, for our example, the molecule 6Li2 must be a boson whether it exists in the higher energy level 6Li6Li or the lower energy 6Li6Li. Now you can see why, in a chapter about macroscopic quantum states, we started off with a discussion of simple molecules. We should note, conversely, that combining two bosons just gives another boson, as the number of spin-1/2 particles is still even.

Interference pattern created by the overlap of two clouds of molecular BECs, each composed of 6Li2 diatomic molecules.

Figure 31: Interference pattern created by the overlap of two clouds of molecular BECs, each composed of 6Li2 diatomic molecules.

Source: © C. Kohstall & R. Grimm. More info

Figure 31 indeed shows that fermionic atoms can pair and become molecular bosons, which can then condense into a molecular BEC. This is evident from the striking similarity of the interference patterns shown in Figures 26 and 31. Two overlapping degenerate Fermi gases would not create such macroscopic interference patterns because those wavefunctions have no intrinsic phase relationships, in contrast to the BEC wavefunctions. Molecular BECs, especially if the molecules are polar and can communicate with one anther through long-range forces, are of special interest as being quantum information storage devices, as individual molecules can potentially be addressed via their many internal degrees of freedom.