6.6 Bose-Einstein Condensation

This section examines what happens to a system of noninteracting bosons in a box if the temperature is somewhat greater than absolute zero.

**Figure 6.3:** The system of bosons at a very low temperature.
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**Figure 6.4:** The system of bosons at a relatively low temperature.
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As noted in the second last section, in the ground state all bosons are in the single-particle state of lowest energy. This was indicated by the fat blue dot next to the origin in the wave number space figure 6.2. Nonzero temperature implies that the bosons obtain an additional amount of energy above the ground state. Therefore they will spread out a bit towards states of higher energy. The single fat blue point will become a colored cloud as shown in figures 6.3 and 6.4. So far, that all seems plausible enough.

That is weird because the average thermal energy available to each boson dwarfs the difference in energy between the lowest energy single-particle state and its immediate neighbors. If the energy difference between the lowest energy state and its neighbors is negligibly small, you would reasonably expect that they will hold similar numbers of bosons. And if a lot of states near the origin each hold about the same number of bosons, then that number must be a small fraction of the total, not a finite one. But reasonable or not, it is untrue. For a system of noninteracting bosons below the critical temperature, the lowest energy state holds a finite fraction of the bosons, much more than its immediate neighbors.

If you raise the temperature of the system, you “boil” away the bosons in the lowest energy state into the surrounding cloud. Above the critical temperature, the excess bosons are gone and the lowest energy state now only holds a similar number of bosons as its immediate neighbors. That is illustrated in figure 6.4. Conversely, if you lower the temperature of the system from above to below the critical temperature, the bosons start “condensing” into the lowest energy state. This process is called “Bose-Einstein condensation” after Bose and Einstein who first predicted it.

Bose-Einstein condensation is a pure quantum effect; it is due to the symmetrization requirement for the wave function. It does not occur for fermions, or if each particle in the box is distinguishable from every other particle. “Distinguishable” should here be taken to mean that there are no antisymmetrization requirements, as there are not if each particle in the system is a different type of particle from every other particle.

It should be noted that the given description is simplistic. In particular, it is certainly possible to cool a microscopic system of distinguishable particles down until say about half the particles are in the single-particle state of lowest energy. Based on the above discussion, you would then conclude that Bose-Einstein condensation has occurred. That is not true. The problem is that this supposed “condensation” disappears when you scale up the system to macroscopic dimensions and a corresponding macroscopic number of particles.

Given a microscopic system of distinguishable particles with half in the single-particle ground state, if you hold the temperature constant while increasing the system size, the size of the cloud of occupied states in wave number space remains about the same. However, the bigger macroscopic system has much more energy states, spaced much closer together in wave number space. Distinguishable particles spread out over these additional states, leaving only a vanishingly small fraction in the lowest energy state. This does not happen if you scale up a Bose-Einstein condensate; here the fraction of bosons in the lowest energy state stays finite regardless of system size.

Bose-Einstein condensation was achieved in 1995 by Cornell, Wieman, et al by cooling a dilute gas of rubidium atoms to below about 170 nK (nano Kelvin). Based on the extremely low temperature and fragility of the condensate, practical applications are very likely to be well into the future, and even determination of the condensate’s basic properties will be difficult.

A process similar to Bose-Einstein condensation is also believed to occur in liquid helium when it turns into a superfluid below 2.17 K. However, this case is more tricky, {N.21}. For one, the atoms in liquid helium can hardly be considered to be noninteracting. That makes the entire concept of “single-particle states” poorly defined. Still, it is quite widely believed that for helium below 2.17 K, a finite fraction of the atoms starts accumulating in what is taken to be a single-particle state of zero wave number. Unlike for normal Bose-Einstein condensation, for helium it is believed that the number of atoms in this state remains limited. At absolute zero only about 9% of the atoms end up in the state.

Currently there is a lot of interest in other systems of particles undergoing Bose-Einstein condensation. One example is liquid helium-3. Compared to normal helium, helium-3 misses a neutron in its nucleus. That makes its spin half-integer, so it is not a boson but a fermion. Therefore, it should not turn into a superfluid like normal liquid helium. And it does not. Helium 3 behaves in almost all aspects exactly the same as normal helium. It becomes liquid at a similar temperature, 3.2 K instead of 4.2 K. But it does not become a superfluid like normal helium at any temperature comparable to 2.17 K. That is very strong evidence that the superfluid behavior of normal helium is due to the fact that it is a boson.

Still it turns out that at temperatures three orders of magnitude smaller, helium-3 does turn into a superfluid. That is believed to be due to the fact that the atoms pair up. A composite of two fermions has integer spin, so it is a boson. Similarly, superconductivity of simple solids is due to the fact that the electrons pair up into “Cooper pairs.” They get tied together due to their interaction with the surrounding atoms.

A variety of other particles can pair up to. At the time of writing, there is interest in polariton condensates. A polariton is a quantum mechanical superposition of a photon and an electronic excitation in a solid. It is hoped that these will allow Bose-Einstein condensation to be studied at room temperature. There is still much to be learned about it. For example, while the relationship between superfluidity and Bose-Einstein condensation is quite generally accepted, there are some issues. Snoke & Baym point out, (in the introduction to Bose-Einstein Condensation, Griffin, A., Snoke, D.W., & Stringari, S., Eds, 1995, Cambridge), that examples indicate that Bose-Einstein condensation is neither necessary nor sufficient for superfluidity. With only approximate theoretical models and approximate experimental data, it is often difficult to make solid specific statements.

6.6.1 Rough explanation of the condensation

The reason why bosons show Bose-Einstein condensation while systems of distinguishable particles do not is complex. It is discussed in chapter 11. However, the idea can be explained qualitatively by examining a very simple system

**Figure 6.5:** Ground state system energy eigenfunction for a simple model system. The system has only 6 single-particle states; each of these has one of 3 energy levels. In the specific case shown here, the system contains 3 distinguishable spinless particles. All three are in the single-particle ground state. Left: mathematical form. Right: graphical representation.
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Assume that there are just three different single-particle energy levels, with values ${\vphantom' E}^{\rm p}_1$ , $2{\vphantom' E}^{\rm p}_1$ , and $3{\vphantom' E}^{\rm p}_1$ . Also assume that there is just one single-particle state with energy ${\vphantom' E}^{\rm p}_1$ , but two with energy $2{\vphantom' E}^{\rm p}_1$ and 3 with energy $3{\vphantom' E}^{\rm p}_1$ . That makes a total of 6 single particle-states; they are shown as “boxes” that can hold particles at the right hand side of figure 6.5. Assume also that there are just three particles and for now take them to be distinguishable. Figure 6.5 then shows the system ground state in which every particle is in the single-particle ground state with energy ${\vphantom' E}^{\rm p}_1$ . That makes the total system energy $3{\vphantom' E}^{\rm p}_1$ .

**Figure 6.6:** Example system energy eigenfunction with five times the single-particle ground state energy.
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However, now assume that the system is at a nonzero temperature. In particular, assume that the total system energy is $5{\vphantom' E}^{\rm p}_1$ . An example system energy eigenfunction with that energy is illustrated in figure 6.6.

**Figure 6.7:** For distinguishable particles, there are 9 system energy eigenfunctions that have energy distribution A.
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**Figure 6.8:** For distinguishable particles, there are 12 system energy eigenfunctions that have energy distribution B.
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But there are a lot more system eigenfunctions with energy $5{\vphantom' E}^{\rm p}_1$ . There are two general ways to achieve that energy:

Therefore, all else being the same, energy distribution B is more likely to be observed than A!

Of course, the difference between 9 system eigenfunctions and 12 is minor. Also, everything else is not the same; the eigenfunctions differ. But it turns out that if the system size is increased to macroscopic dimensions, the differences in numbers of energy eigenfunctions become gigantic. There will be one energy distribution for which there are astronomically more system eigenfunctions than for any other energy distribution. Common-sense statistics then says that this energy distribution is the only one that will ever be observed. If there are countless orders of magnitude more eigenfunctions for a distribution B than for a distribution A, what are the chances of A ever being found?

It is curious to think of it: only one energy distribution is observed for a given macroscopic system. And that is not because of any physics; other energy distributions are physically just as good. It is because of a mathematical count; there are just so many more energy eigenfunctions with that distribution.

**Figure 6.9:** For identical bosons, there are only 3 system energy eigenfunctions that have energy distribution A.
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**Figure 6.10:** For identical bosons, there are also only 3 system energy eigenfunctions that have energy distribution B.
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Bose-Einstein condensation has to do with the fact that the count of eigenfunctions is different for identical bosons than for distinguishable particles. The details were worked out in chapter 5.7. The symmetrization requirement for bosons implies that system eigenfunctions that are the same except for exchanges of particles must be combined together into one. In particular for distribution A, in each of the rows of figure 6.7 the eigenfunctions are the same except for such exchanges. Simply put, they merely differ in what number is stamped on each particle. Therefore, for each row, the eigenfunctions must be combined together into a single eigenfunction. That leaves only the three system eigenfunctions shown in figure 6.9.

In the combination eigenfunction, every particle occupies every single-particle state involved equally. Therefore, numbers on the particles would not add any nontrivial information and may as well be left away. Sure, you could put all three numbers 1,2, and 3 in each of the particles in figure 6.9. But what good would that do?

Comparing figures 6.7 and 6.9, you can see why particles satisfying symmetrization requirements are commonly called “indistinguishable.” Classical quantum mechanics may imagine to stamp numbers on the three identical bosons to keep them apart, but you sure do not see the difference between them in the system energy eigenfunctions.

For distribution B of figure 6.8, under the symmetrization requirement the three energy eigenfunctions in the first row must be combined into one, the six in the second and third rows must be combined into one, and the three in the fourth row must be combined into one. That gives a total of 3 system eigenfunctions for distribution B, as shown in figure 6.10.

It follows that the symmetrization requirement reduces the number of eigenfunctions for distribution A, with 2 particles in the ground state, from 9 to 3. However, it reduces the eigenfunctions for distribution B, with 1 particle in the ground state, from 12 to 3. Not only does the symmetrization requirement reduce the number of energy eigenfunctions, but it also tends to shift the balance towards eigenfunctions that have more particles in the ground state.

And so, if the system size is increased under conditions of Bose-Einstein condensation, it turns out that there are astronomically more system eigenfunctions for an energy distribution that keeps a finite number of bosons in the ground state than for anything else.

It may be noted from comparing figures 6.7 and 6.8 with 6.9 and 6.10 that any energy distribution that is physically possible for distinguishable particles is just as possible for identical bosons. Bose-Einstein condensation does not occur because the physics says it must, but because there are so gigantically more system eigenfunctions that have a finite fraction of bosons in the ground state than ones that do not.

It may also be noted that the reduction in the number of system energy eigenfunctions for bosons is believed to be an important factor in superfluidity. It eliminates low-energy eigenfunctions that cannot be interpreted as phonons, traveling particle wave solutions, [37,17]. The lack of alternate eigenfunctions leaves no mechanism for the traveling particles to get scattered by small effects.