Section 6: Gaseous Bose-Einstein Condensates
Figure 21: Atomic de Broglie waves overlap as temperatures are lowered.
Source: © Wolfgang Ketterle. More info
What does it take to form a gaseous macroscopic quantum system of bosonic atoms? This involves a set of very tricky hurdles to overcome. We want the atoms, trapped in engineered magnetic or optical fields, to be cooled to temperatures where, as we saw in Unit 5, their relative de Broglie wavelengths are large compared with the mean separation between the gaseous atoms themselves. As these de Broglie waves overlap, a single and coherent quantum object is formed. The ultimate wavelength of this (possibly) macroscopic quantum system is, at low enough temperatures, determined by the size of the trap, as that sets the maximum wavelength for both the individual particles and the whole BEC itself.
Einstein had established this condition in his 1924 paper, but it immediately raises a problem. If the atoms get too close, they may well form molecules: Li2, Rb2, and Na2 molecules are all familiar characters in the laboratory. And, at low temperatures (and even room temperatures), Li, Na, and Rb are all soft metals: Cooling them turns them into dense hard metals, not quantum gases. Thus, experiments had to start with a hot gas (many hundreds of degrees K) and cool the gases in such a way that the atoms didn't simply condense into liquids or solids. This requires keeping the density of the atomic gases very low—about a million times less dense than the density of air at the Earth's surface and more than a billion times less dense than the solid metallic forms of these elements.
Figure 22: Atoms in a Bose condensate at 0 K.
Stating all this in terms of bosonic atoms in quantum energy levels, rather than overlapping de Broglie waves, leads to the picture shown in Figure 22. Here, we see the energy levels corresponding to the quantum motion of the atoms in the magnetic (or optical) trap made in the laboratory for collecting them. As the temperature cools, all of the bosonic atoms end up in the lowest energy level of the trap, just as all the photons in an ideal laser occupy a single quantum state in a trap made of mirrors.
The role of low temperature
The fact that gaseous atoms must be quite far apart to avoid condensing into liquids or solids, and yet closer than their relative de Broglie wavelengths, requires a very large wavelength, indeed. This in turn requires very slow atoms and ultra-cold temperatures. Laser cooling, as discussed in Unit 5, only takes us part of the way, down to about 10-6 (one one-millionth) of a degree Kelvin. To actually achieve the temperature of 10-8 K, or colder, needed to form a BEC, atoms undergo a second stage of cooling, ordinary evaporation, just like our bodies use to cool themselves by sweating. When the first condensates were made, no temperature this low had ever been created before, either in the laboratory or in nature. Figure 23 illustrates, using actual data taken as the first such condensate formed, the role of evaporative cooling and the formation of a BEC.
Figure 23: Three stages of cooling, and a quantum phase transition to a BEC.
Source: © Mike Matthews, JILA. More info
The process, reported in Science in July 1995, required both laser cooling and evaporative cooling of 87Rb to produce a pretty pure condensate. Images revealed a sharp and smoothly defined Bose-Einstein condensate surrounded by many "thermal" atoms as the rubidium gas cooled to about 10-8 K. By "pretty pure," we mean that a cloud of uncondensed, and still thermal, atoms is still visible: These atoms are in many different quantum states, whereas those of the central peak of the velocity distribution shown in Figure 23 are in a single quantum state defined by the trap confining the atoms. Subsequent refinements have led to condensates with temperatures just above 10-12 K—cold, indeed, and with no noticeable cloud of uncondensed atoms. That tells us that all these many thousands to many millions of sodium or rubidium atoms are in a single quantum state.
The behavior of trapped atoms
How do such trapped atoms behave? Do they do anything special? In fact, yes, and quite special. Simulations and experimental observations show that the atoms behave very much like superfluids. An initial shaking of the trap starts the BEC sloshing back and forth, which it continues to do for ever longer-times as colder and colder temperatures are attained. This is just the behavior expected from such a gaseous superfluid, just as would be the case with liquid helium, 4He.
The extent of the oscillating behavior depends on the temperature of the BEC. A theoretical computer simulation of such motion in a harmonic trap at absolute zero—a temperature that the laws of physics prevent us from ever reaching—shows that the oscillations would never damp out. But, if we add heat to the simulation, little by little as time passes, the behavior changes. The addition of increasing amounts of the random energy associated with heat causes the individual atoms to act as waves that get more and more out of phase with one another, and we see that the collective and undamped oscillations now begin to dissipate. Experimental observations readily reveal both the back-and-forth oscillations—collective and macroscopic quantum behavior taking place with tens of thousands or millions of atoms in the identical quantum state—and the dissipation caused by increased temperatures.
Figure 24: Super-oscillations of a quantum gas and their dissipation on heating.
Source: © William P. Reinhardt. More info
This loss of phase coherence leads to the dissipation that is part of the origins of our familiar classical world, where material objects seem to be in one place at a time, not spread out like waves, and certainly don't show interference effects. At high temperatures, quantum effects involving many particles "de-phase" or "de-cohere," and their macroscopic quantum properties simply vanish. Thus, baseballs don't diffract around a bat, much to the disappointment of the pitcher and delight of the batter. This also explains why a macroscopic strand of DNA is a de-cohered, and thus classical, macroscopic molecule, although made of fully quantum atoms.
Gaseous atomic BECs are thus large and fully coherent quantum objects, whereby coherent we imply that many millions of atoms act together as a single quantum system, with all atoms in step, or in phase, with one another. This coherence of phase is responsible for the uniformly spaced parallel interference patterns that we will meet in the next section, similar to the coherence seen in the interference of laser light shown in the introduction to this unit. The difference between these gaseous macroscopic quantum systems and liquid helium superfluids is that the quantum origins of the superfluidity of liquid helium are hidden within the condensed matter structure of the liquid helium itself. So, while experiments like the superfluid sloshing are easily available, interference patterns are not, owing to the "hard core" interactions of the helium atoms at the high density of the liquid.