In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero, i.e. 0 K (−273.15 °C; −459.67 °F). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which microscopic quantum-mechanical phenomena, particularly wavefunction interference, become apparent macroscopically.
More generally, condensation refers to the appearance of macroscopic occupation of one or several states: For example, in BCS theory, a superconductor is a condensate of Cooper pairs. As such, condensation can be associated with phase transition, and the macroscopic occupation of the state is the order parameter.
Bose–Einstein condensates were first predicted, generally, in 1924–1925 by Albert Einstein, crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics. In 1995, the Bose–Einstein condensate was created by Eric Cornell and Carl Wieman of the University of Colorado Boulder using rubidium atoms. Later that year, Wolfgang Ketterle of MIT produced a BEC using sodium atoms. In 2001 Cornell, Wieman, and Ketterle shared the Nobel Prize in Physics "for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates".
Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons), in which he derived Planck's quantum radiation law without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it in 1924. Einstein's manuscript, once believed to be lost, was found in a library at Leiden University in 2005. Einstein then extended Bose's ideas to matter in two other papers. The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now called bosons. Bosons are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter. Bosons include the photon, polaritons, magnons, some atoms and molecules (depending on the number of nucleons, see #Isotopes) such as atomic hydrogen, helium-4, lithium-7, rubidium-87 or strontium-84.
In 1938, Fritz London proposed the BEC as a mechanism for superfluidity in helium-4 and superconductivity.
The quest to produce a Bose–Einstein condensate in the laboratory was stimulated by a paper published in 1976 by two program directors at the National Science Foundation (William Stwalley and Lewis Nosanow), proposing to use spin-polarized atomic hydrogen to produce a gaseous BEC. This led to the immediate pursuit of the idea by four independent research groups; these were led by Isaac Silvera (University of Amsterdam), Walter Hardy (University of British Columbia), Thomas Greytak (Massachusetts Institute of Technology) and David Lee (Cornell University). However, cooling atomic hydrogen turned out to be technically difficult, and Bose–Einstein condensation of atomic hydrogen was only realized in 1998.
On 5 June 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST–JILA lab, in a gas of rubidium atoms cooled to 170 nanokelvins (nK). Shortly thereafter, Wolfgang Ketterle at MIT produced a Bose–Einstein condensate in a gas of sodium atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics. Bose–Einstein condensation of alkali gases is easier because they can be pre-cooled with laser cooling techniques, unlike atomic hydrogen at the time, which gave a significant head start when performing the final forced evaporative cooling to cross the condensation threshold. These early studies founded the field of ultracold atoms, and hundreds of research groups around the world now routinely produce BECs of dilute atomic vapors in their labs.
Since 1995, many other atomic species have been condensed (see #Isotopes), and BECs have also been realized using molecules, polaritons, and other quasi-particles. BECs of photons can be made, for example, in dye microcavites with wavelength-scale mirror separation, forming a two-dimensional harmonically confined photon gas with tunable chemical potential. BEC of plasmonic quasiparticles (plasmon–exciton polaritons) has been realized in periodic arrays of metal nanoparticles overlaid with dye molecules, exhibiting ultrafast sub-picosecond dynamics and long-range correlations.
This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by
{\displaystyle T_{\text{c}}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{\text{B}}}}\approx 3.3125\,{\frac {\hbar ^{2}n^{2/3}}{mk_{\text{B}}}},}
is the reduced Planck constant,
is the Riemann zeta function (
{\displaystyle \zeta (3/2)\approx 2.6124}
Interactions shift the value, and the corrections can be calculated by mean-field theory.
This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics.
The critical temperature depends on the density. A more concise and experimentally relevant condition involves the phase-space density
{\displaystyle {\mathcal {D}}=n\lambda _{T}^{3}}
{\displaystyle \lambda _{T}=\hbar {\sqrt {\frac {2\pi }{mk_{\text{B}}T}}}}