Bose-Einstein Condensation

In 1924 the Indian physicist Satyendra Bose derived the quantum statistics of photons by assuming them as indistinguishable particles. Albert Einstein was fascinated by this idea and applied it to atoms. The fundamental difference is that for atoms the particle number is conserved (if they are trapped in a box or a magnetic field) whereas photons can randomly be emitted and absorbed (e.g. from the walls of a box). Particles behaving in accordance to Bose's statistics are today called bosons (in contrast to fermions, which are described by "Fermi-Dirac statistics").

Einstein discovered that the particle number conservation of trapped gas of bosons leads to a very remarkable effect: When the gas is cooled below a certain temperature, the bosons begin to "condense" in the lowest energy state like water vapour condenses to water drops below temperatures of 100°C. This phenomenon is known as "Bose-Einstein condensation". In real experiments, condensation is observed at temperatures of only few µK, so very effective cooling methods are necessary!

Why does Bose-Einstein condensation occur?

The following java-applet simulates the Bose statistics of an ideal Bose gas in a trap (where "ideal" means, that there is no interaction between the bosons, i.e. they don't "feel" each other). There are three freely adjustable parameters: The temperature, the total density of particles (i.e. the number of particles divided by the volume of the trap) and the diameter of the trap (assuming a spherical symmetric harmonic trap potential).

The temperature has a value of "6". As you can see from the bars, most particles are in the ground state and only a few occupy the exited states. The higher the energy, the less is the particle density, because the probability, that a particle gains enough energy to jump to a higher excited state is smaller. Increase the temperature to the maximum value of "10" and press the "New"-Button. You can see that more particles jump to the exited states whereas the ground state particle density decreases. When you set the temperature to "0", all bosons go to the ground state, because they have no energy at all.

Now set the temperature to, say, "8" and change to trap diameter from "2" to "4". We observe, that the distance between two energy levels decreases. This is a quantum mechanical effect: The larger the free space in which an atom can move, the more energy levels exist. But when the bosons can spread over more states, the particle density in a single state becomes smaller. Therefore the bars shrink in their length.

At a trap diameter of "10", the ground state isn't the state with the most particles any more. The shape of the particle density has a "belly" at the low exited states and decreases again at the higher exited states. How can that be? Isn't the probability for a single particle greater to be in a state with low energy than beeing in a state with higher energy? Well, it is. But higher energy levels have a greater degeneracy. Degeneracy means, that there exists more than just one quantum state with the same energy. The ground state is non-degenerate and is therefore one single quantum state. The first exited energy level has a degeneracy of 3, the second level one of 6. The formula for the degeneracy of the n-th energy level is (n+1)*(n+2)/2. Therefore, the particle density in an energy level is the particle density in a single quantum state with the given energy times the degeneracy. That explains why the particle density increases at lower energies. For high energies the occupation probability of a state becomes so low, that in spite of the high degeneracy there are only very few bosons.

Try greater trap diameters. At the maximum value of "100" the energy levels are very close together and the shape of the density curve becomes very flat.

Now let's see the Bose-Einstein condensation! Reduce the temperature to "6" again. When the trap diameter is small, there is no interesting difference. But when you blow the trap up, finally to a diameter of "100", you can see, that the ground state density is significantly larger than the density in any of the excited states. This is the condensate! The applet always tells you whether a condensate exists or not. When yes, you can only see it when the trap diameter is large (so that the energy levels are close together). Check that the condensate can be destroyed again by taking particles out of the system (this can be done by lowering the total particle density).

Try other values for the three parameters.

The source code of the java-applet can be found here.