# Bose-Einstein Condensation

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## Bose-Einstein condensation

When a gas of bosonic particles is cooled below a critical temperature, it condenses into a Bose-Einstein condensate. The condensate consists of a macroscopic number of particles, which are all in the ground state of the system. Bose-Einstein condensation is a phase transition, which does not depend on the specific interactions between particles. It is based on the indistinguishability and wave nature of particles, both of which are at the heart of quantum mechanics.

#### Basic phenomenon in ideal gas

In a simplified picture, particles in a gas may be regarded as quantum-mechanical wavepackets which have a spatial extent on the order of a thermal de Broglie wavelength, given by Eq. (1),

*T*is the temperature,

*m*the mass of the particle,

*k*is the Boltzmann constant, and ℏ is Planck's constant divided by 2&pgr;. The wavelength λ

_{B}_{dB}can be regarded as the position uncertainty associated with the thermal momentum distribution of the particles. At high temperature, λ

_{dB}is small, and the probability of finding two particles within this distance of each other is extremely low. Therefore, the indistinguishability of particles is not important, and a classical description applies (namely, Boltzmann statistics). When the gas is cooled to the point where λ

_{dB}is comparable to the distance between particles, the individual wavepackets start to overlap and the indistinguishability of particles becomes crucial—an identity crisis can be said to occur. For fermions, the Pauli exclusion principle prevents two particles from occupying the same quantum state; whereas for bosons, quantum statistics (in this case, Bose-Einstein statistics) dramatically increases the probability of finding several particles in the same quantum state. The system undergoes a phase transition and forms a Bose-Einstein condensate, where a macroscopic number of particles occupy the lowest-energy quantum state (Fig. 1).

*See*Boltzmann statistics

Bose-Einstein condensation can be described intuitively in the following way: When the quantum-mechanical wave functions of bosonic particles spatially overlap, the matter waves start to oscillate in concert. A coherent matter wave forms that comprises all particles in the ground state of the system. This transition from disordered to coherent matter waves can be compared to the step from incoherent light to laser light. Indeed, atom lasers based on Bose-Einstein condensation have been realized. *See* Coherence, Laser

#### Experimental techniques

The phenomenon of Bose-Einstein condensation is responsible for the superfluidity of helium and for the superconductivity of an electron gas, which involves Bose-condensed electron pairs. However, these phenomena happen at high density, and their understanding requires a detailed treatment of the interactions. *See* Superconductivity, Superfluidity

The quest to realize Bose-Einstein condensation in a dilute weakly interacting gas focused on atomic gases. At ultralow temperatures, all atomic gases liquefy or solidify in thermal equilibrium. Keeping the gas at sufficiently low density can prevent this from occurring. Typical number densities of atoms between 10^{12} and 10^{15} cm^{3} imply transition temperatures for Bose-Einstein condensation in the nanokelvin or microkelvin regime.

The realization of Bose-Einstein condensation in atomic gases required techniques to cool gases to such low temperatures, and atom traps to confine the gases at the required density and keep them away from the much warmer walls of the vacuum chamber. The experiments on alkali vapors (lithium, rubidium, and sodium) use several laser-cooling techniques as precooling, then hold the atoms in a magnetic trap and cool them further by forced evaporative cooling. For atomic hydrogen, the laser-cooling step is replaced by cryogenic cooling.

#### Macroscopic wave function

In superconductors and liquid helium, the existence of coherence and of a macroscopic wave function is impressively demonstrated through the Josephson effect. In the dilute atomic gases, the coherence has been demonstrated even more directly by interfering two Bose condensates (Fig. 2). The interference fringes typically have a spacing of 15 μm, a huge length for matter waves. (In contrast, the matter wavelength of atoms at room temperature is only 0.05 nm, less than the size of the atoms.)

## Bose-Einstein Condensation

a quantum phenomenon in a system of bosons which consists of the fact that at a temperature below a certain critical value (called the degeneration temperature), a portion of the particles is aggregated in a state with zero momentum (if the system as a whole is at rest). The term “Bose-Einstein condensation” is created by analogy with the condensation of molecules of a vapor into a liquid upon cooling. However, condensation in the ordinary sense does not take place; the distribution of particles in space remains as before, and it is a matter of “condensation in momentum space.”

The degeneration temperature for the overwhelming majority of gases is very low, and the substance passes into the solid state long before Bose-Einstein condensation can set in. The exception is helium, which at *T* = 4.2° K (under normal conditions) passes into the liquid state and remains a liquid down to temperatures very close to absolute zero. At *T =* 2.18° K, liquid ^{4}He passes into a special, so-called superfluid state, whose emergence is connected with Bose-Einstein condensation.

V. P. PAVLOV