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Bose-Einstein condensate is a state of matter(i.e. solid, gas, liquid) of a dilute gas of weakly interacting bosons(subatomic particles) that are confined in an external potential(a defined field in space) and cooled to temperatures very near to absolute zero. Given these conditions, you will find a large fraction of the bosons occupy the lowest quantum state(a mathematical object that fully describes a quantum system) of the external potential, and all wave functions will overlap each other. This makes the quantum effects measurable and observable to the unaided eye. The concept was first proposed in 1925, but it wasn’t until seventy years later that a team of scientists was able to prove it.
Bose Einstein condensate was first proposed by Satyendra Nath Bose in 1925. There was one small problem, no one could understand the hypothesis and he was unable to get the work published. In a last ditch effort, he sent the body of his work to Albert Einstein who was immediately able to grasp the sheer enormity of the work and helped to get it published. He eventually expanded on the idea in two future papers. Although the concept was generally accepted by the scientific community, it wasn’t until 1995 that it was proven in the laboratory. Eric Cornell and Carl Wieman, at the University of Colorado at Boulder NIST-JILA lab, used a gas of rubidium atoms cooled to 170 nanokelvin. They received the 2001 Nobel Prize of Physics for the work.
The basics of the theory are that the cooling of atoms will produce a single quantum state called the Bose-Einstein condensate. The theory was first put forth for the statistical mechanics(the application of the probability theory to study the thermodynamic behavior of a system of a large number of particles) of massless photons. Einstein generalized the initial theory to atoms, which have mass. The result would be a Bose gas, which is the quantum mechanics version of an ideal gas, governed by Bose-Einstein statistics that describes the statistical distribution of identical particles with integer spin. These particles are now known as bosons. Bosons, which include photons, are allowed to share quantum states with each other. Einstein was able to demonstrate that cooling bosons to a very low temperature would cause them to condense(fall) into the lowest accessible quantum state. This condensing would result in a new form of matter. By supercooling helium-4, scientists were able to create a liquid version of the Bose-Einstein condensate. The true result was the first superfluid. While this was not a true proof of the Bose-Einstein theory, it was an amazing first step towards the work of Cornell and Wieman.
Bose-Einstein condensate is difficult to measure. The graph at the top of this article shows that the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium-87 atoms. The colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are also at the lowest velocities. Since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width. This width is given by the curvature of the magnetic trapping potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotrophy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left.
There is a good article about Bose-Einstein condensate here. It contains many of the mathematical equations needed to work your way through the concept. Here on Universe Today we have a great article about the results of some research done at MIT on Bose-Einstein condensate. Astronomy Cast offers a good episode about the possible results of this theory leading to an understanding of dark matter.