Bose–Einstein condensate
A Bose–Einstein condensate (BEC) is a state of matter (sometimes called the fifth state of matter) which is typically formed when a gas of bosons at low densities is cooled to temperatures very close to absolute zero (273.15 °C). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density, about onehundredthousandth (1/100,000) the density of normal air, to ultralow temperatures.
This state was first predicted, generally, in 1924–1925 by Albert Einstein^{[1]} following a paper written by Satyendra Nath Bose, although Bose came up with the pioneering paper on the new statistics.^{[2]}
HistoryEdit
Satyendra Nath 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.^{[3]} (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.^{[4]}) Einstein then extended Bose's ideas to matter in two other papers.^{[5]}^{[6]} 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, which include the photon as well as atoms such as helium4 (^{4}
_{}He^{}
_{}), 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.
In 1938, Fritz London proposed the BEC as a mechanism for superfluidity in ^{4}
_{}He^{}
_{} and superconductivity.^{[7]}^{[8]}
On June 5, 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).^{[9]} Shortly thereafter, Wolfgang Ketterle at MIT realized a BEC in a gas of sodium atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics.^{[10]} 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, and BECs have also been realized using molecules, quasiparticles, and photons.^{[11]}
Critical temperatureEdit
This transition to BEC occurs below a critical temperature, which for a uniform threedimensional gas consisting of noninteracting 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, is the Boltzmann constant, and is the Riemann zeta function; ^{[12]}
Interactions shift the value and the corrections can be calculated by meanfield theory. This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics.
DerivationEdit
Ideal Bose gasEdit
For an ideal Bose gas we have the equation of state:
where is the per particle volume, the thermal wavelength, the fugacity and . It is noticeable that is a monotonically growing function of in , which are the only values for which the series converge.
Recognizing that the second term on the righthand side contains the expression for the average ocupation number of the fundamental state , the equation of state can be rewritten as
Because the the left term on the second equation must always be positive, and because , a stronger condition is
which defines a transition between a gas phase and a condensed phase. On the critical region it is possible to define a critical temperature and thermal wavelenght:
recovering the value indicated on the previous section. The critical values are such that if or we are in the presence of a BoseEinstein Condensate.
Understanding what happens with the fraction of particles on the fundamental level is crucial. As so, write the equation of state for , obtaining
and equivalently .
So, if the fraction and if the fraction . At temperatures near to absolute 0, particles tend to condensate in the fundamental state (state with momentum ).
ModelsEdit
Bose Einstein's noninteracting gasEdit
Consider a collection of N noninteracting particles, which can each be in one of two quantum states, and . If the two states are equal in energy, each different configuration is equally likely.
If we can tell which particle is which, there are different configurations, since each particle can be in or independently. In almost all of the configurations, about half the particles are in and the other half in . The balance is a statistical effect: the number of configurations is largest when the particles are divided equally.
If the particles are indistinguishable, however, there are only N+1 different configurations. If there are K particles in state , there are N − K particles in state . Whether any particular particle is in state or in state cannot be determined, so each value of K determines a unique quantum state for the whole system.
Suppose now that the energy of state is slightly greater than the energy of state by an amount E. At temperature T, a particle will have a lesser probability to be in state by . In the distinguishable case, the particle distribution will be biased slightly towards state . But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the mostlikely outcome is that most of the particles will collapse into state .
In the distinguishable case, for large N, the fraction in state can be computed. It is the same as flipping a coin with probability proportional to p = exp(−E/T) to land tails.
In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:
For large N, the normalization constant C is (1 − p). The expected total number of particles not in the lowest energy state, in the limit that , is equal to . It does not grow when N is large; it just approaches a constant. This will be a negligible fraction of the total number of particles. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference.
Consider now a gas of particles, which can be in different momentum states labeled . If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. In this limit, the gas is classical. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. From this point on, any extra particle added will go into the ground state.
To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, p/(1 − p):
When the integral (also known as BoseEinstein integral) is evaluated with factors of and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section. Therefore, this integral defines the critical temperature and particle number corresponding to the conditions of negligible chemical potential . In Bose–Einstein statistics distribution, is actually still nonzero for BECs; however, is less than the ground state energy. Except when specifically talking about the ground state, can be approximated for most energy or momentum states as .
Bogoliubov theory for weakly interacting gasEdit
Nikolay Bogoliubov considered perturbations on the limit of dilute gas,^{[13]} finding a finite pressure at zero temperature and positive chemical potential. This leads to corrections for the ground state. The Bogoliubov state has pressure (T = 0): .
The original interacting system can be converted to a system of noninteracting particles with a dispersion law.
Gross–Pitaevskii equationEdit
In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments.
This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate . For a system of this nature, is interpreted as the particle density, so the total number of atoms is
Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using mean field theory, the energy (E) associated with the state is:
Minimizing this energy with respect to infinitesimal variations in , and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a nonlinear Schrödinger equation):
where:
is the mass of the bosons, is the external potential, is representative of the interparticle interactions.
In the case of zero external potential, the dispersion law of interacting Bose–Einsteincondensed particles is given by socalled Bogoliubov spectrum (for ):
The GrossPitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for . It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.
Numerical SolutionEdit
The GrossPitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and different numerical methods, such as splitstep CrankNicolson ^{[14]} and Fourier spectral ^{[15]} methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction ^{[16]}^{[17]} and longrange dipolar interaction ^{[18]} which can be freely used.
Weaknesses of Gross–Pitaevskii modelEdit
The Gross–Pitaevskii model of BEC is a physical approximation valid for certain classes of BECs. By construction, the GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact twobody type and also neglects anomalous contributions to selfenergy.^{[19]} These assumptions are suitable mostly for the dilute threedimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction acquires the terms containing higherorder powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially nonpolynomial. The examples where this could happen are the Bose–Fermi composite condensates,^{[20]}^{[21]}^{[22]}^{[23]} effectively lowerdimensional condensates,^{[24]} and dense condensates and superfluid clusters and droplets.^{[25]}
OtherEdit
However, it is clear that in a general case the behaviour of Bose–Einstein condensate can be described by coupled evolution equations for condensate density, superfluid velocity and distribution function of elementary excitations. This problem was in 1977 by Peletminskii et al. in microscopical approach. The Peletminskii equations are valid for any finite temperatures below the critical point. Years after, in 1985, Kirkpatrick and Dorfman obtained similar equations using another microscopical approach. The Peletminskii equations also reproduce Khalatnikov hydrodynamical equations for superfluid as a limiting case.
Superfluidity of BEC and Landau criterionEdit
The phenomena of superfluidity of a Bose gas and superconductivity of a stronglycorrelated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in helium4 and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model.
Experimental observationEdit
Superfluid He4Edit
In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly believed that the superfluidity was due to partial Bose–Einstein condensation of the liquid. In fact, many properties of superfluid helium also appear in gaseous condensates created by Cornell, Wieman and Ketterle (see below). Superfluid helium4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium4. Note that helium3, a fermion, also enters a superfluid phase (at a much lower temperature) which can be explained by the formation of bosonic Cooper pairs of two atoms (see also fermionic condensate).
Dilute atomic gasesEdit
The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and coworkers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand rubidium87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude CohenTannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium23. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements.^{[26]}
A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work.^{[27]} Lithium has attractive interactions, causing the condensate to be unstable and collapse for all but a few atoms. Hulet's team subsequently showed the condensate could be stabilized by confinement quantum pressure for up to about 1000 atoms. Various isotopes have since been condensed.
Velocitydistribution data graphEdit
In the image accompanying this article, the velocitydistribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false 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 at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: spatially confined atoms have a minimum width velocity distribution. This width is given by the curvature of the magnetic potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantummechanical effect and does not exist in the thermal distribution on the left. This graph served as the cover design for the 1999 textbook Thermal Physics by Ralph Baierlein.^{[28]}
QuasiparticlesEdit
Bose–Einstein condensation also applies to quasiparticles in solids. Magnons, Excitons, and Polaritons have integer spin which means they are bosons that can form condensates.^{[29]}
Magnons, electron spin waves, can be controlled by a magnetic field. Densities from the limit of a dilute gas to a strongly interacting Bose liquid are possible. Magnetic ordering is the analog of superfluidity. In 1999 condensation was demonstrated in antiferromagnetic TlCuCl^{}
_{3},^{[30]} at temperatures as great as 14 K. The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. In 2006, condensation in a ferromagnetic yttriumirongarnet thin film was seen even at room temperature,^{[31]}^{[32]} with optical pumping.
Excitons, electronhole pairs, were predicted to condense at low temperature and high density by Boer et al., in 1961. Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance. Fast optical exciton creation was used to form condensates in subkelvin Cu^{}
_{2}O in 2005 on.
Polariton condensation was first detected for excitonpolaritons in a quantum well microcavity kept at 5 K.^{[33]}
Peculiar propertiesEdit
VorticesEdit
This section needs additional citations for verification. (September 2011) (Learn how and when to remove this template message) 
As in many other systems, vortices can exist in BECs. These can be created, for example, by 'stirring' the condensate with lasers, or rotating the confining trap. The vortex created will be a quantum vortex. These phenomena are allowed for by the nonlinear term in the GPE.^{[disputed – discuss]} As the vortices must have quantized angular momentum the wavefunction may have the form where and are as in the cylindrical coordinate system, and is the angular number. This is particularly likely for an axially symmetric (for instance, harmonic) confining potential, which is commonly used. The notion is easily generalized. To determine , the energy of must be minimized, according to the constraint . This is usually done computationally, however in a uniform medium the analytic form:
 , where:
is  density far from the vortex,  
is  healing length of the condensate. 
demonstrates the correct behavior, and is a good approximation.
A singly charged vortex ( ) is in the ground state, with its energy given by
where is the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary .)
For multiply charged vortices ( ) the energy is approximated by
which is greater than that of singly charged vortices, indicating that these multiply charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes.
Closely related to the creation of vortices in BECs is the generation of socalled dark solitons in onedimensional BECs. These topological objects feature a phase gradient across their nodal plane, which stabilizes their shape even in propagation and interaction. Although solitons carry no charge and are thus prone to decay, relatively longlived dark solitons have been produced and studied extensively.^{[34]}
Attractive interactionsEdit
Experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist up to a critical atom number. Quench cooling the gas, they observed the condensate to grow, then subsequently collapse as the attraction overwhelmed the zeropoint energy of the confining potential, in a burst reminiscent of a supernova, with an explosion preceded by an implosion.
Further work on attractive condensates was performed in 2000 by the JILA team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturally attracting atoms of rubidium85 (having negative atom–atom scattering length). Through Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which rubidium bonds, making their Rb85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among wavelike condensate atoms.
When the JILA team raised the magnetic field strength further, the condensate suddenly reverted to attraction, imploded and shrank beyond detection, then exploded, expelling about twothirds of its 10,000 atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not seen in the cold remnant or expanding gas cloud.^{[26]} Carl Wieman explained that under current atomic theory this characteristic of Bose–Einstein condensate could not be explained because the energy state of an atom near absolute zero should not be enough to cause an implosion; however, subsequent mean field theories have been proposed to explain it. Most likely they formed molecules of two rubidium atoms;^{[35]} energy gained by this bond imparts velocity sufficient to leave the trap without being detected.
The process of creation of molecular Bose condensate during the sweep of the magnetic field throughout the Feshbach resonance, as well as the reverse process, are described by the exactly solvable model that can explain many experimental observations.^{[36]}
Current researchEdit
Unsolved problem in physics: How do we rigorously prove the existence of Bose–Einstein condensates for general interacting systems? (more unsolved problems in physics)

Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile.^{[37]} The slightest interaction with the external environment can be enough to warm them past the condensation threshold, eliminating their interesting properties and forming a normal gas.^{[citation needed]}
Nevertheless, they have proven useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an increase in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality,^{[38]} the study of superfluidity and quantized vortices, the creation of bright matter wave solitons from Bose condensates confined to one dimension, and the slowing of light pulses to very low speeds using electromagnetically induced transparency.^{[39]} Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the laboratory. Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential. These have been used to explore the transition between a superfluid and a Mott insulator,^{[40]} and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas. Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow onedimensional optical lattice originally observed by Haller^{[41]} has been explored via a tweaking of the primary optical lattice by a secondary weaker one.^{[42]} Thus for a resulting weak bichromatic optical lattice, it has been found that the pinning transition is robust against the introduction of the weaker secondary optical lattice. Studies of vortices in nonuniform Bose–Einstein condensates ^{[43]} as well as excitatons of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken.^{[44]}^{[45]} Within this context, the conditions for order and chaos in the dynamics of a trapped Bose–Einstein condensate have been explored by the application of moving blue and reddetuned laser beams via the timedependent GrossPitaevskii equation.^{[46]}
Bose–Einstein condensates composed of a wide range of isotopes have been produced.^{[47]}
Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form bosonic compound particles (e.g. molecules or Cooper pairs). The first molecular condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate, working with the same system but outside the molecular regime.^{[48]}
In 1999, Danish physicist Lene Hau led a team from Harvard University which slowed a beam of light to about 17 meters per second^{[clarification needed]} using a superfluid.^{[49]} Hau and her associates have since made a group of condensate atoms recoil from a light pulse such that they recorded the light's phase and amplitude, recovered by a second nearby condensate, in what they term "slowlightmediated atomic matterwave amplification" using Bose–Einstein condensates: details are discussed in Nature.^{[50]}
Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision atom interferometry. The first demonstration of a BEC in weightlessness was achieved in 2008 at a drop tower in Bremen, Germany by a consortium of researchers led by Ernst M. Rasel from Leibniz University Hannover.^{[51]} The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space^{[52]} and it is also the subject of two upcoming experiments on the International Space Station.^{[53]}^{[54]}
Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.^{[55]}
In 1970, BECs were proposed by Emmanuel David Tannenbaum for antistealth technology.^{[56]}
Dark matterEdit
P. Sikivie and Q. Yang showed that cold dark matter axions form a Bose–Einstein condensate by thermalisation because of gravitational selfinteractions.^{[57]} Axions have not yet been confirmed to exist. However the important search for them has been greatly enhanced with the completion of upgrades to the Axion Dark Matter Experiment(ADMX) at the University of Washington in early 2018.
IsotopesEdit
This section needs additional citations for verification. (July 2010) (Learn how and when to remove this template message) 
The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2012, using ultralow temperatures of or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of alkali metal, alkaline earth metal,
and lanthanide atoms (^{7}
_{}Li^{}
_{}, ^{23}
_{}Na^{}
_{}, ^{39}
_{}K^{}
_{}, ^{41}
_{}K^{}
_{}, ^{85}
_{}Rb^{}
_{}, ^{87}
_{}Rb^{}
_{}, ^{133}
_{}Cs^{}
_{}, ^{52}
_{}Cr^{}
_{}, ^{40}
_{}Ca^{}
_{}, ^{84}
_{}Sr^{}
_{}, ^{86}
_{}Sr^{}
_{}, ^{88}
_{}Sr^{}
_{}, ^{174}
_{}Yb^{}
_{}, ^{164}
_{}Dy^{}
_{}, and ^{168}
_{}Er^{}
_{}). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'.^{[58]} In contrast, the superfluid state of ^{4}
_{}He^{}
_{} below 2.17 K is not a good example, because the interaction between the atoms is too strong. Only 8% of atoms are in the ground state near absolute zero, rather than the 100% of a true condensate.^{[59]}
The bosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have halfinteger total spin. It arises from a subtle interplay of electronic and nuclear spins: at ultralow temperatures and corresponding excitation energies, the halfinteger total spin of the electronic shell and halfinteger total spin of the nucleus are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer lower value. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values.
See alsoEdit
 Atom laser
 Atomic coherence
 Bose–Einstein correlations
 Bose–Einstein condensation: a network theory approach
 Bose–Einstein condensation of quasiparticles
 Bose–Einstein statistics
 Cold Atom Laboratory
 Electromagnetically induced transparency
 Fermionic condensate
 Gas in a box
 Gross–Pitaevskii equation
 Macroscopic quantum phenomena
 Macroscopic quantum selftrapping
 Slow light
 Superconductivity
 Superfluid film
 Superfluid helium4
 Supersolid
 Tachyon condensation
 Timeline of lowtemperature technology
 Superheavy atom
 Ultracold atom
 Wiener sausage
ReferencesEdit
 ^ Einstein, A (10 July 1924). "Quantentheorie des einatomigen idealen Gases" (PDF). Königliche Preußische Akademie der Wissenschaften. Sitzungsberichte: 261–267.
 ^ A.Douglas Stone, Chapter 24, The Indian Comet, in the book Einstein and the Quantum, Princeton University Press, Princeton, New Jersey, 2013.
 ^ S. N. Bose (1924). "Plancks Gesetz und Lichtquantenhypothese". Zeitschrift für Physik. 26 (1): 178–181. Bibcode:1924ZPhy...26..178B. doi:10.1007/BF01327326.
 ^ "Leiden University Einstein archive". Lorentz.leidenuniv.nl. 27 October 1920. Retrieved 23 March 2011.
 ^ A. Einstein (1925). "Quantentheorie des einatomigen idealen Gases". Sitzungsberichte der Preussischen Akademie der Wissenschaften. 1: 3.
 ^ Clark, Ronald W. (1971). Einstein: The Life and Times. Avon Books. pp. 408–409. ISBN 9780380011599.
 ^ F. London (1938). "The λPhenomenon of liquid Helium and the Bose–Einstein degeneracy". Nature. 141 (3571): 643–644. Bibcode:1938Natur.141..643L. doi:10.1038/141643a0.
 ^ London, F. Superfluids Vol.I and II, (reprinted New York: Dover 1964)
 ^ BoseEinstein Condensate: A New Form of Matter, NIST, 9 October 2001
 ^ Levi, Barbara Goss (2001). "Cornell, Ketterle, and Wieman Share Nobel Prize for Bose–Einstein Condensates". Search & Discovery. Physics Today online. Archived from the original on 24 October 2007. Retrieved 26 January 2008.
 ^ J. Klaers; J. Schmitt; F. Vewinger & M. Weitz. "Bose–Einstein condensation of photons in an optical microcavity/year 2010". Nature. 468 (7323): 545–548. arXiv:1007.4088. Bibcode:2010Natur.468..545K. doi:10.1038/nature09567. PMID 21107426.
 ^ (sequence A078434 in the OEIS)
 ^ N. N. Bogoliubov (1947). "On the theory of superfluidity". J. Phys. (USSR). 11: 23.
 ^ P. Muruganandam and S. K. Adhikari (2009). "Fortran Programs for the timedependent GrossPitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 180 (3): 1888–1912. arXiv:0904.3131. Bibcode:2009CoPhC.180.1888M. doi:10.1016/j.cpc.2009.04.015.
 ^ P. Muruganandam and S. K. Adhikari (2003). "BoseEinstein condensation dynamics in three dimensions by the pseudospectral and finitedifference methods". J. Phys. B. 36 (12): 2501–2514. arXiv:condmat/0210177. Bibcode:2003JPhB...36.2501M. doi:10.1088/09534075/36/12/310.
 ^ D. Vudragovic; et al. (2012). "C Programs for the timedependent GrossPitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 183 (9): 2021–2025. arXiv:1206.1361. Bibcode:2012CoPhC.183.2021V. doi:10.1016/j.cpc.2012.03.022.
 ^ L. E. YoungS.; et al. (2016). "OpenMP Fortran and C Programs for the timedependent GrossPitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 204 (9): 209–213. arXiv:1605.03958. Bibcode:2016CoPhC.204..209Y. doi:10.1016/j.cpc.2016.03.015.
 ^ K. Kishor Kumar; et al. (2015). "Fortran and C Programs for the timedependent dipolar GrossPitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 195: 117–128. arXiv:1506.03283. Bibcode:2015CoPhC.195..117K. doi:10.1016/j.cpc.2015.03.024.
 ^ Beliaev, S. T. Zh. Eksp. Teor. Fiz. 34, 417–432 (1958) [Soviet Phys. JETP 7, 289 (1958)]; ibid. 34, 433–446 [Soviet Phys. JETP 7, 299 (1958)].
 ^ M. Schick (1971). "Twodimensional system of hardcore bosons". Phys. Rev. A. 3 (3): 1067–1073. Bibcode:1971PhRvA...3.1067S. doi:10.1103/PhysRevA.3.1067.
 ^ E. Kolomeisky; J. Straley (1992). "Renormalizationgroup analysis of the groundstate properties of dilute Bose systems in d spatial dimensions". Phys. Rev. B. 46 (18): 11749–11756. Bibcode:1992PhRvB..4611749K. doi:10.1103/PhysRevB.46.11749. PMID 10003067.
 ^ E. B. Kolomeisky; T. J. Newman; J. P. Straley & X. Qi (2000). "Lowdimensional Bose liquids: Beyond the GrossPitaevskii approximation". Phys. Rev. Lett. 85 (6): 1146–1149. arXiv:condmat/0002282. Bibcode:2000PhRvL..85.1146K. doi:10.1103/PhysRevLett.85.1146. PMID 10991498.
 ^ S. Chui; V. Ryzhov (2004). "Collapse transition in mixtures of bosons and fermions". Phys. Rev. A. 69 (4): 043607. arXiv:condmat/0211411. Bibcode:2004PhRvA..69d3607C. doi:10.1103/PhysRevA.69.043607.
 ^ L. Salasnich; A. Parola & L. Reatto (2002). "Effective wave equations for the dynamics of cigarshaped and diskshaped Bose condensates". Phys. Rev. A. 65 (4): 043614. arXiv:condmat/0201395. Bibcode:2002PhRvA..65d3614S. doi:10.1103/PhysRevA.65.043614.
 ^ A. V. Avdeenkov; K. G. Zloshchastiev (2011). "Quantum Bose liquids with logarithmic nonlinearity: Selfsustainability and emergence of spatial extent". J. Phys. B: At. Mol. Opt. Phys. 44 (19): 195303. arXiv:1108.0847. Bibcode:2011JPhB...44s5303A. doi:10.1088/09534075/44/19/195303.
 ^ ^{a} ^{b} "Eric A. Cornell and Carl E. Wieman — Nobel Lecture" (PDF). nobelprize.org.
 ^ C. C. Bradley; C. A. Sackett; J. J. Tollett & R. G. Hulet (1995). "Evidence of Bose–Einstein condensation in an atomic gas with attractive interactions" (PDF). Phys. Rev. Lett. 75 (9): 1687–1690. Bibcode:1995PhRvL..75.1687B. doi:10.1103/PhysRevLett.75.1687. PMID 10060366.
 ^ Baierlein, Ralph (1999). Thermal Physics. Cambridge University Press. ISBN 9780521658386.
 ^ Monique Combescot and ShiueYuan Shiau, "Excitons and Cooper Pairs: Two Composite Bosons in ManyBody Physics", Oxford University Press (ISBN 9780198753735)
 ^ T. Nikuni; M. Oshikawa; A. Oosawa & H. Tanaka (1999). "Bose–Einstein condensation of dilute magnons in TlCuCl_{3}". Phys. Rev. Lett. 84 (25): 5868–71. arXiv:condmat/9908118. Bibcode:2000PhRvL..84.5868N. doi:10.1103/PhysRevLett.84.5868. PMID 10991075.
 ^ S. O. Demokritov; V. E. Demidov; O. Dzyapko; G. A. Melkov; A. A. Serga; B. Hillebrands & A. N. Slavin (2006). "Bose–Einstein condensation of quasiequilibrium magnons at room temperature under pumping". Nature. 443 (7110): 430–433. Bibcode:2006Natur.443..430D. doi:10.1038/nature05117. PMID 17006509.
 ^ Magnon Bose Einstein Condensation made simple. Website of the "Westfählische Wilhelms Universität Münster" Prof.Demokritov. Retrieved 25 June 2012.
 ^ Kasprzak J, Richard M, Kundermann S, Baas A, Jeambrun P, Keeling JM, Marchetti FM, Szymańska MH, André R, Staehli JL, Savona V, Littlewood PB, Deveaud B, Dang (28 September 2006). "Bose–Einstein condensation of exciton polaritons". Nature. 443 (7110): 409–414. Bibcode:2006Natur.443..409K. doi:10.1038/nature05131. PMID 17006506.CS1 maint: multiple names: authors list (link)
 ^ C. Becker; S. Stellmer; P. SoltanPanahi; S. Dörscher; M. Baumert; E.M. Richter; J. Kronjäger; K. Bongs & K. Sengstock (2008). "Oscillations and interactions of dark and dark–bright solitons in Bose–Einstein condensates". Nature Physics. 4 (6): 496–501. arXiv:0804.0544. Bibcode:2008NatPh...4..496B. doi:10.1038/nphys962.
 ^ M. H. P. M. van Putten (2010). "Pair condensates produced in bosenovae". Phys. Lett. A. 374 (33): 3346–3347. Bibcode:2010PhLA..374.3346V. doi:10.1016/j.physleta.2010.06.020.
 ^ C. Sun; N. A. Sinitsyn (2016). "LandauZener extension of the TavisCummings model: Structure of the solution". Phys. Rev. A. 94 (3): 033808. arXiv:1606.08430. Bibcode:2016PhRvA..94c3808S. doi:10.1103/PhysRevA.94.033808.
 ^ "How to watch a Bose–Einstein condensate for a very long time  physicsworld.com". physicsworld.com. Retrieved 22 January 2018.
 ^ Gorlitz, Axel. "Interference of Condensates (BEC@MIT)". Cua.mit.edu. Archived from the original on 4 March 2016. Retrieved 13 October 2009.
 ^ Z. Dutton; N. S. Ginsberg; C. Slowe & L. Vestergaard Hau (2004). "The art of taming light: ultraslow and stopped light". Europhysics News. 35 (2): 33–39. Bibcode:2004ENews..35...33D. doi:10.1051/epn:2004201.
 ^ "From Superfluid to Insulator: Bose–Einstein Condensate Undergoes a Quantum Phase Transition". Qpt.physics.harvard.edu. Retrieved 13 October 2009.
 ^ Elmar Haller; Russell Hart; Manfred J. Mark; Johann G. Danzl; Lukas Reichsoellner; Mattias Gustavsson; Marcello Dalmonte; Guido Pupillo; HannsChristoph Naegerl (2010). "Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons". Nature Letters. 466 (7306): 597–600. arXiv:1004.3168. doi:10.1038/nature09259. PMID 20671704.
 ^ Asaad R. Sakhel (2016). "Properties of bosons in a onedimensional bichromatic optical lattice in the regime of the pinning transition: A worm algorithm Monte Carlo study". Physical Review A. 94 (3): 033622. arXiv:1511.00745. doi:10.1103/PhysRevA.94.033622.
 ^ Roger R. Sakhel; Asaad R. Sakhel (2016). "Elements of VortexDipole Dynamics in a Nonuniform Bose–Einstein Condensate". Journal of Low Temperature Physics. 184 (5–6): 1092–1113. doi:10.1007/s1090901616363.
 ^ Roger R. Sakhel; Asaad R. Sakhel; Humam B. Ghassib (2011). "Selfinterfering matterwave patterns generated by a moving laser obstacle in a twodimensional BoseEinstein condensate inside a power trap cut off by box potential boundaries". Physical Review A. 84 (3): 033634. arXiv:1107.0369. doi:10.1103/PhysRevA.84.033634.
 ^ Roger R. Sakhel; Asaad R. Sakhel; Humam B. Ghassib (2013). "Nonequilibrium Dynamics of a BoseEinstein Condensate Excited by a Red Laser Inside a PowerLaw Trap with Hard Walls". Journal of Low Temperature Physics. 173 (3–4): 177–206. doi:10.1007/s1090901308946.
 ^ Roger R. Sakhel; Asaad R. Sakhel; Humam B. Ghassib; Antun Balaz (2016). "Conditions for order and chaos in the dynamics of a trapped BoseEinstein condensate in coordinate and energy space". European Physical Journal D. 70 (3): 66. arXiv:1604.01349. doi:10.1140/epjd/e2016600852.
 ^ "Ten of the best for BEC". Physicsweb.org. 1 June 2005.
 ^ "Fermionic condensate makes its debut". Physicsweb.org. 28 January 2004.
 ^ Cromie, William J. (18 February 1999). "Physicists Slow Speed of Light". The Harvard University Gazette. Retrieved 26 January 2008.
 ^ N. S. Ginsberg; S. R. Garner & L. V. Hau (2007). "Coherent control of optical information with matter wave dynamics". Nature. 445 (7128): 623–626. doi:10.1038/nature05493. PMID 17287804.
 ^ Zoest, T. van; Gaaloul, N.; Singh, Y.; Ahlers, H.; Herr, W.; Seidel, S. T.; Ertmer, W.; Rasel, E.; Eckart, M. (18 June 2010). "BoseEinstein Condensation in Microgravity". Science. 328 (5985): 1540–1543. Bibcode:2010Sci...328.1540V. doi:10.1126/science.1189164. ISSN 00368075. PMID 20558713.
 ^ DLR. "MAIUS 1 – First BoseEinstein condensate generated in space". DLR Portal. Retrieved 23 May 2017.
 ^ Laboratory, Jet Propulsion. "Cold Atom Laboratory". coldatomlab.jpl.nasa.gov. Retrieved 23 May 2017.
 ^ "2017 NASA Fundamental Physics Workshop  Planetary News". www.lpi.usra.edu. Retrieved 23 May 2017.
 ^ P. Weiss (12 February 2000). "Atomtronics may be the new electronics". Science News Online. 157 (7): 104. doi:10.2307/4012185. JSTOR 4012185. Retrieved 12 February 2011.
 ^ Tannenbaum, Emmanuel David (1970). "Gravimetric Radar: Gravitybased detection of a pointmass moving in a static background". arXiv:1208.2377 [physics.insdet].
 ^ P. Sikivie, Q. Yang; Phys. Rev. Lett.,103:111103; 2009
 ^ Dale G. Fried; Thomas C. Killian; Lorenz Willmann; David Landhuis; Stephen C. Moss; Daniel Kleppner & Thomas J. Greytak (1998). "Bose–Einstein Condensation of Atomic Hydrogen". Phys. Rev. Lett. 81 (18): 3811. arXiv:physics/9809017. Bibcode:1998PhRvL..81.3811F. doi:10.1103/PhysRevLett.81.3811.
 ^ "Bose–Einstein Condensation in Alkali Gases" (PDF). The Royal Swedish Academy of Sciences. 2001. Retrieved 17 April 2017.
Further readingEdit
 S. N. Bose (1924). "Plancks Gesetz und Lichtquantenhypothese". Zeitschrift für Physik. 26 (1): 178–181. Bibcode:1924ZPhy...26..178B. doi:10.1007/BF01327326.
 A. Einstein (1925). "Quantentheorie des einatomigen idealen Gases". Sitzungsberichte der Preussischen Akademie der Wissenschaften. 1: 3.,
 L. D. Landau (1941). "The theory of Superfluity of Helium 111". J. Phys. USSR. 5: 71–90.
 L. D. Landau (1941). "Theory of the Superfluidity of Helium II". Physical Review. 60 (4): 356–358. Bibcode:1941PhRv...60..356L. doi:10.1103/PhysRev.60.356.
 M. H. Anderson; J. R. Ensher; M. R. Matthews; C. E. Wieman & E. A. Cornell (1995). "Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor". Science. 269 (5221): 198–201. Bibcode:1995Sci...269..198A. doi:10.1126/science.269.5221.198. JSTOR 2888436. PMID 17789847.
 C. Barcelo; S. Liberati & M. Visser (2001). "Analogue gravity from Bose–Einstein condensates". Classical and Quantum Gravity. 18 (6): 1137–1156. arXiv:grqc/0011026. Bibcode:2001CQGra..18.1137B. doi:10.1088/02649381/18/6/312.
 P. G. Kevrekidis; R. CarreteroGonzález; D. J. Frantzeskakis & I. G. Kevrekidis (2004). "Vortices in Bose–Einstein Condensates: Some Recent Developments". Mod. Phys. Lett. B. 18 (30): 1481–1505. arXiv:condmat/0501030. Bibcode:2004MPLB...18.1481K. doi:10.1142/S0217984904007967.
 K.B. Davis; M.O. Mewes; M.R. Andrews; N.J. van Druten; D.S. Durfee; D.M. Kurn & W. Ketterle (1995). "Bose–Einstein condensation in a gas of sodium atoms". Phys. Rev. Lett. 75 (22): 3969–3973. Bibcode:1995PhRvL..75.3969D. doi:10.1103/PhysRevLett.75.3969. PMID 10059782..
 D. S. Jin; J. R. Ensher; M. R. Matthews; C. E. Wieman & E. A. Cornell (1996). "Collective Excitations of a Bose–Einstein Condensate in a Dilute Gas". Phys. Rev. Lett. 77 (3): 420–423. Bibcode:1996PhRvL..77..420J. doi:10.1103/PhysRevLett.77.420. PMID 10062808.
 M. R. Andrews; C. G. Townsend; H.J. Miesner; D. S. Durfee; D. M. Kurn & W. Ketterle (1997). "Observation of interference between two Bose condensates". Science. 275 (5300): 637–641. doi:10.1126/science.275.5300.637. PMID 9005843. Archived from the original on 12 October 2000. Retrieved 26 October 2017..
 E. A. Cornell & C. E. Wieman (1998). "The Bose–Einstein condensate". Scientific American. 278 (3): 40–45. Bibcode:1998SciAm.278c..40C. doi:10.1038/scientificamerican039840.
 M. R. Matthews; B. P. Anderson; P. C. Haljan; D. S. Hall; C. E. Wieman & E. A. Cornell (1999). "Vortices in a Bose–Einstein condensate". Phys. Rev. Lett. 83 (13): 2498–2501. arXiv:condmat/9908209. Bibcode:1999PhRvL..83.2498M. doi:10.1103/PhysRevLett.83.2498.
 E. A. Donley; N. R. Claussen; S. L. Cornish; J. L. Roberts; E. A. Cornell & C. E. Wieman (2001). "Dynamics of collapsing and exploding Bose–Einstein condensates". Nature. 412 (6844): 295–299. arXiv:condmat/0105019. Bibcode:2001Natur.412..295D. doi:10.1038/35085500. PMID 11460153.
 A. G. Truscott; K. E. Strecker; W. I. McAlexander; G. B. Partridge & R. G. Hulet (2001). "Observation of Fermi Pressure in a Gas of Trapped Atoms". Science. 291 (5513): 2570–2572. Bibcode:2001Sci...291.2570T. doi:10.1126/science.1059318. PMID 11283362.
 M. Greiner; O. Mandel; T. Esslinger; T. W. Hänsch & I. Bloch (2002). "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms". Nature. 415 (6867): 39–44. Bibcode:2002Natur.415...39G. doi:10.1038/415039a. PMID 11780110..
 S. Jochim; M. Bartenstein; A. Altmeyer; G. Hendl; S. Riedl; C. Chin; J. Hecker Denschlag & R. Grimm (2003). "Bose–Einstein Condensation of Molecules". Science. 302 (5653): 2101–2103. Bibcode:2003Sci...302.2101J. doi:10.1126/science.1093280. PMID 14615548.
 M. Greiner; C. A. Regal & D. S. Jin (2003). "Emergence of a molecular Bose−Einstein condensate from a Fermi gas" (PDF). Nature. 426 (6966): 537–540. Bibcode:2003Natur.426..537G. doi:10.1038/nature02199. PMID 14647340.
 M. W. Zwierlein; C. A. Stan; C. H. Schunck; S. M. F. Raupach; S. Gupta; Z. Hadzibabic & W. Ketterle (2003). "Observation of Bose–Einstein Condensation of Molecules". Phys. Rev. Lett. 91 (25): 250401. arXiv:condmat/0311617. Bibcode:2003PhRvL..91y0401Z. doi:10.1103/PhysRevLett.91.250401. PMID 14754098.
 C. A. Regal; M. Greiner & D. S. Jin (2004). "Observation of Resonance Condensation of Fermionic Atom Pairs". Phys. Rev. Lett. 92 (4): 040403. arXiv:condmat/0401554. Bibcode:2004PhRvL..92d0403R. doi:10.1103/PhysRevLett.92.040403. PMID 14995356.
 C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001.
 Lev P. Pitaevskii and S. Stringari, Bose–Einstein Condensation, Clarendon Press, Oxford, 2003.
 M. Mackie; K. A. Suominen & J. Javanainen (2002). "Meanfield theory of Feshbachresonant interactions in 85Rb condensates". Phys. Rev. Lett. 89 (18): 180403. arXiv:condmat/0205535. Bibcode:2002PhRvL..89r0403M. doi:10.1103/PhysRevLett.89.180403. PMID 12398586.
 Monique Combescot and ShiueYuan Shiau, "Excitons and Cooper Pairs: Two Composite Bosons in ManyBody Physics", Oxford University Press (ISBN 9780198753735).
External linksEdit
 Bose–Einstein Condensation 2009 Conference Bose–Einstein Condensation 2009 – Frontiers in Quantum Gases
 BEC Homepage General introduction to Bose–Einstein condensation
 Nobel Prize in Physics 2001 – for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates
 Physics Today: Cornell, Ketterle, and Wieman Share Nobel Prize for Bose–Einstein Condensates
 Bose–Einstein condensates at JILA
 Atomcool at Rice University
 Alkali Quantum Gases at MIT
 Atom Optics at UQ
 Einstein's manuscript on the Bose–Einstein condensate discovered at Leiden University
 Bose–Einstein condensate on arxiv.org
 Bosons – The Birds That Flock and Sing Together
 Easy BEC machine – information on constructing a Bose–Einstein condensate machine.
 Verging on absolute zero – Cosmos Online
 Lecture by W Ketterle at MIT in 2001
 Bose–Einstein Condensation at NIST – NIST resource on BEC