In
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, a Bose–Einstein condensate (BEC) is a
state of matter
In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and Plasma (physics), plasma.
Different states are distinguished by the ways the ...
that is typically formed when a
gas of
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s at very low
densities is cooled to
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
s very close to
absolute zero
Absolute zero is the lowest possible temperature, a state at which a system's internal energy, and in ideal cases entropy, reach their minimum values. The absolute zero is defined as 0 K on the Kelvin scale, equivalent to −273.15 ° ...
, i.e. . Under such conditions, a large fraction of bosons occupy the lowest
quantum state
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
, 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
In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...
, and the macroscopic occupation of the state is the
order parameter.
Bose–Einstein condensate was first predicted, generally, in 1924–1925 by
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, 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
Sodium is a chemical element; it has Symbol (chemistry), symbol Na (from Neo-Latin ) and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 element, group 1 of the peri ...
atoms. In 2001 Cornell, Wieman, and Ketterle shared the
Nobel Prize in Physics
The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
"for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates".
History

Bose first sent a paper to Einstein on the
quantum statistics of light quanta (now called
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s), 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. (The Einstein manuscript, once believed to be lost, was found in a library at
Leiden University
Leiden University (abbreviated as ''LEI''; ) is a Public university, public research university in Leiden, Netherlands. Established in 1575 by William the Silent, William, Prince of Orange as a Protestantism, Protestant institution, it holds the d ...
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
In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
with
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
spin, now called
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s. 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
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
, resulting in a new form of matter. Bosons include the
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
,
polaritons,
magnons, some
atom
Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s and
molecule
A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s (depending on the number of
nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number.
Until the 1960s, nucleons were thought to be ele ...
s, see
#Isotopes) such as atomic hydrogen,
helium-4
Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...
, lithium-7, rubidium-87 or strontium-84.
In 1938,
Fritz London proposed the BEC as a mechanism for
superfluidity
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
in and
superconductivity
Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ord ...
.
[
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
The U.S. National Science Foundation (NSF) is an Independent agencies of the United States government#Examples of independent agencies, independent agency of the Federal government of the United States, United States federal government that su ...
(William Stwalley and Lewis Nosanow), proposing to use spin-polarized atomic hydrogen
Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
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
The University of Amsterdam (abbreviated as UvA, ) is a public university, public research university located in Amsterdam, Netherlands. Established in 1632 by municipal authorities, it is the fourth-oldest academic institution in the Netherlan ...
), Walter Hardy (University of British Columbia
The University of British Columbia (UBC) is a Public university, public research university with campuses near University of British Columbia Vancouver, Vancouver and University of British Columbia Okanagan, Kelowna, in British Columbia, Canada ...
), Thomas Greytak (Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
) and David Lee (Cornell University
Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...
). 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
The University of Colorado Boulder (CU Boulder, CU, or Colorado) is a Public university, public research university in Boulder, Colorado, United States. Founded in 1876, five months before Colorado became a Federated state, state, it is the fla ...
NIST
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
– 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
Sodium is a chemical element; it has Symbol (chemistry), symbol Na (from Neo-Latin ) and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 element, group 1 of the peri ...
atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics
The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
. Bose-Einstein condensation of alkali gases is easier because they can be pre-cooled with laser cooling
Laser cooling includes several techniques where atoms, molecules, and small mechanical systems are cooled with laser light. The directed energy of lasers is often associated with heating materials, e.g. laser cutting, so it can be counterintuit ...
techniques, unlike atomic hydrogen at the time, which give 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.][
]
Critical temperature
This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by
:
where:
: is the critical temperature,
: is the particle density,
: is the mass per boson,
: is the reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
,
: is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
,
: is the Riemann zeta function ().
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 , where
:
is the thermal de Broglie wavelength. It is a dimensionless quantity. The transition to BEC occurs when the phase-space density is greater than critical value:
:
in 3D uniform space. This is equivalent to the above condition on the temperature. In a 3D harmonic potential, the critical value is instead
:
where has to be understood as the peak density.
Derivation
Ideal Bose gas
For an ideal Bose gas we have the equation of state
:
where is the per-particle volume, is the thermal wavelength, is the fugacity, and
:
It is noticeable that is a monotonically growing function of in