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= '''FREEZY JUICE''' =
{{redirect|Super atom|clusters of atoms that seem to exhibit some of the properties of elemental atoms|Superatom}}
{{Use dmy dates|date=September 2013}}
{{Condensed matter physics|expanded=States of matter}}

]

A '''Bose–Einstein condensate''' ('''BEC''') is a ] of a dilute ] of ]s cooled to ]s very close to ]. Under such conditions, a large fraction of bosons occupy the lowest ], at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent. A BEC is formed by cooling a gas of extremely low density, about one-hundred-thousandth the density of ], to ultra-low temperatures.

This state was first predicted, generally, in 1924–1925 by ] and ].

== History ==
] atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. {{nowrap|Left: just}} before the appearance of a Bose–Einstein condensate. {{nowrap|Center: just}} after the appearance of the condensate. {{nowrap|Right: after}} further evaporation, leaving a sample of nearly pure condensate.]]

] first sent a paper to Einstein on the ] of light quanta (now called ]s), in which he derived ] without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the '']'', which published it in 1924.<ref>{{cite journal |author=S. N. Bose |year=1924 |title=Plancks Gesetz und Lichtquantenhypothese |journal=Zeitschrift für Physik |volume=26 |pages=178–181 |bibcode=1924ZPhy...26..178B |doi=10.1007/BF01327326}}</ref> (The Einstein manuscript, once believed to be lost, was found in a library at ] in 2005.<ref>{{cite web |url=http://www.lorentz.leidenuniv.nl/history/Einstein_archive/ |title=Leiden University Einstein archive |publisher=Lorentz.leidenuniv.nl |date=27 October 1920 |accessdate=23 March 2011}}</ref>). Einstein then extended Bose's ideas to matter in two other papers.<ref>{{cite journal |author=A. Einstein |year=1925 |title=Quantentheorie des einatomigen idealen Gases |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften |volume=1 |page=3}}</ref><ref>{{cite book |first=Ronald W. |last=Clark |title=Einstein: The Life and Times |publisher=Avon Books |year=1971 |pages=408–409 |isbn=0-380-01159-X }}</ref> The result of their efforts is the concept of a ], governed by ], which describes the statistical distribution of ] with ] ], now called ]s. Bosons, which include the photon as well as atoms such as ] (<sup>4</sup>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 ], resulting in a new form of matter.

In 1938 ] proposed BEC as a mechanism for ] in <sup>4</sup>He and ].<ref name=London:1938/><ref>London, F. ''Superfluids'' Vol.I and II, (reprinted New York: Dover 1964)</ref>

On June 5, 1995 the first gaseous condensate was produced by ] and ] at the ] ]–] lab, in a gas of ] atoms cooled to 170 ] (nK).<ref>https://www.nist.gov/public_affairs/releases/bec_background.cfm</ref> Shortly thereafter, ] at ] demonstrated important BEC properties. For their achievements Cornell, Wieman, and Ketterle received the 2001 ].<ref>{{cite web | last = Levi | first = Barbara Goss | title = Cornell, Ketterle, and Wieman Share Nobel Prize for Bose–Einstein Condensates | work = Search & Discovery | publisher = Physics Today online| year = 2001 | url = http://www.physicstoday.org/pt/vol-54/iss-12/p14.html | accessdate = 26 January 2008 |archiveurl =https://archive.is/20071024134547/http://www.physicstoday.org/pt/vol-54/iss-12/p14.html |archivedate = 24 October 2007}}</ref>

Many isotopes were soon condensed, then molecules, quasi-particles, and photons in 2010.<ref name=Klaers:2010/>

== Critical temperature ==
This transition to BEC occurs below a critical temperature, which for a uniform ] gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:

:<math>T_c=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi\hbar^2}{m k_B} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_B} </math>

where:

<dl><dd>
{|cellspacing="0" cellpadding="0"
|-
| <math>\,T_c</math>
| &nbsp;is&nbsp;
| the critical temperature,
|-
| <math>\,n</math>
| &nbsp;is&nbsp;
| the ],
|-
| <math>\,m</math>
| &nbsp;is&nbsp;
| the mass per boson,
|-
| <math>\hbar</math>
| &nbsp;is&nbsp;
| the reduced ],
|-
| <math>\,k_B</math>
| &nbsp;is&nbsp;
| the ], and
|-
| <math>\,\zeta</math>
| &nbsp;is&nbsp;
| the ]; <math>\,\zeta(3/2)\approx 2.6124.</math> <ref>{{OEIS|id=A078434}}</ref>
|}
</dd></dl>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 ] using ].

== Models ==

=== Bose Einstein's non-interacting gas ===
Consider a collection of ''N'' noninteracting particles, which can each be in one of two ]s, <math>\scriptstyle|0\rangle</math> and <math>\scriptstyle|1\rangle</math>. If the two states are equal in energy, each different configuration is equally likely.

If we can tell which particle is which, there are <math>2^N</math> different configurations, since each particle can be in <math>\scriptstyle|0\rangle</math> or <math>\scriptstyle|1\rangle</math> independently. In almost all of the configurations, about half the particles are in <math>\scriptstyle|0\rangle</math> and the other half in <math>\scriptstyle|1\rangle</math>. 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 <math>\scriptstyle|1\rangle</math>, there are {{nowrap|''N − K''}} particles in state <math>\scriptstyle|0\rangle</math>. Whether any particular particle is in state <math>\scriptstyle|0\rangle</math> or in state <math>\scriptstyle|1\rangle</math> cannot be determined, so each value of ''K'' determines a unique quantum state for the whole system.

Suppose now that the energy of state <math>\scriptstyle|1\rangle</math> is slightly greater than the energy of state <math>\scriptstyle|0\rangle</math> by an amount ''E''. At temperature ''T'', a particle will have a lesser probability to be in state <math>\scriptstyle|1\rangle</math> by <math>e^{-E/kT}</math>. In the distinguishable case, the particle distribution will be biased slightly towards state <math>\scriptstyle|0\rangle</math>. But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the most-likely outcome is that most of the particles will collapse into state <math>\scriptstyle|0\rangle</math>.

In the distinguishable case, for large ''N'', the fraction in state <math>\scriptstyle|0\rangle</math> can be computed. It is the same as flipping a coin with probability proportional to ''p''&nbsp;=&nbsp;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:

:<math>\,
P(K)= C e^{-KE/T} = C p^K.
</math>

For large ''N'', the normalization constant ''C'' is {{nowrap|(1 − ''p'')}}. The expected total number of particles not in the lowest energy state, in the limit that <math>\scriptstyle N\rightarrow \infty</math>, is equal to <math>\scriptstyle \sum_{n>0} C n p^n=p/(1-p) </math>. 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 <math>\scriptstyle|k\rangle</math>. 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, {{nowrap|''p''/(1 − ''p'')}}:

:<math>\,
N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} </math>
:<math>\,
p(k)= e^{-k^2\over 2mT}.
</math>

When the integral is evaluated with factors of ''k''<sub>''B''</sub> 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 ]. In ] distribution, μ is actually still nonzero for BECs; however, μ is less than the ground state energy. Except when specifically talking about the ground state, μ can be approximated for most energy or momentum states as&nbsp;μ&nbsp;≈&nbsp;0.

=== Bogoliubov theory for weakly interacting gas ===
] considered perturbations on the limit of dilute gas,<ref name=Bogoliubov:1947/> 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''&nbsp;=&nbsp;0): <math>P = g/2 n^2</math>.

The original interacting system can be converted to a system of non-interacting particles with a dispersion law.

=== Gross–Pitaevskii equation ===
{{Main|Gross–Pitaevskii equation}}

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 <math>\psi(\vec{r})</math>. For a ], <math>|\psi(\vec{r})|^2</math> is interpreted as the particle density, so the total number of atoms is <math>N=\int d\vec{r}|\psi(\vec{r})|^2</math>

Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using ], the energy (E) associated with the state <math>\psi(\vec{r})</math> is:

:<math>E=\int
d\vec{r}\left</math>

Minimizing this energy with respect to infinitesimal variations in <math>\psi(\vec{r})</math>, and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear ]):

:<math>i\hbar\frac{\partial \psi(\vec{r})}{\partial t} = \left(-\frac{\hbar^2\nabla^2}{2m}+V(\vec{r})+U_0|\psi(\vec{r})|^2\right)\psi(\vec{r})</math>

where:

<dl><dd>
{|cellspacing="0" cellpadding="0"
|-
| <math>\,m</math>
| &nbsp;is the mass of the bosons,
|-
| <math>\,V(\vec{r})</math>
| &nbsp;is the external potential,
|-
| <math>\,U_0</math>
| &nbsp;is representative of the inter-particle interactions.
|}
</dd></dl>

In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for <math>\ T= 0</math>):

:<math> {\omega _p} = \sqrt {\frac{{{p^2}}}{{2m}}\left( {\frac{{{p^2}}}{{2m}} + 2{U_0}{n_0}} \right)} </math>

The Gross-Pitaevskii 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 <math>\ T= 0</math>.
It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.

==Numerical Solution==

The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and
different numerical methods, such as split-step
]
<ref>{{cite journal
|author=P. Muruganandam and S. K. Adhikari
|year=2009
|title=Fortran Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
|journal=Comput. Phys. Commun.
|volume=180 |issue=3 |pages=1888-1912
|doi=10.1016/j.cpc.2009.04.015
|bibcode=2009CoPhC.180.1888M|arxiv=0904.3131}}</ref>
and ]
<ref>{{cite journal
|author=P. Muruganandam and S. K. Adhikari
|year=2003
|title=Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods
|journal=J. Phys. B
|volume=36 |issue= |pages=2501-2514
|doi=10.1088/0953-4075/36/12/310
|bibcode=2003JPhB...36.2501M|arxiv=cond-mat/0210177}}</ref> methods, are used for its solution. There are different Fortran and C programs for its solution for ]
<ref>{{cite journal
|author=D. Vudragovic et al
|year=2012
|title=C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
|journal= Comput. Phys. Commun.
|volume=183 |issue=9 |pages=2021-2025
|doi=10.1016/j.cpc.2012.03.022
|bibcode=2012CoPhC.183.2021V|arxiv=1206.1361}}</ref><ref>
{{cite journal
|author=L. E. Young-S. et al.
|year=2016
|title=OpenMP Fortran and C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
|journal= Comput. Phys. Commun.
|volume=204 |issue=9 |pages=209-213
|doi=10.1016/j.cpc.2016.03.015
|bibcode=2016CoPhC.204..209Y|arxiv=1605.03958}}</ref>
and long-range ]
<ref>
{{cite journal
|author=K. Kishor Kumar et al.
|year=2015
|title=Fortran and C Programs for the time-dependent dipolar Gross-Pitaevskii equation in a fully anisotropic trap
|journal= Comput. Phys. Commun.
|volume=195 |issue= |pages=117-128
|doi=10.1016/j.cpc.2015.03.024
|bibcode=2015CoPhC.195..117K|arxiv=1506.03283}}</ref> which can be freely used.
==== Weaknesses of Gross–Pitaevskii model ====
The Gross–Pitaevskii model of BEC is a physical ] valid for certain classes of BECs. By construction, the ] uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions to ].<ref>Beliaev, S. T. Zh. Eksp. Teor. Fiz. 34, 417–432 (1958) ; ibid. 34, 433–446 .</ref> These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate ] acquires the terms containing higher-order powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially non-polynomial. The examples where this could happen are the Bose–Fermi composite condensates,<ref name=Schick:1971/><ref name=Kolomeisky:1992/><ref name=Kolomeisky:2000/><ref name=Chui:2004/> effectively lower-dimensional condensates,<ref>
{{cite journal
|author1=L. Salasnich
|author2=A. Parola
|author3=L. Reatto |lastauthoramp=yes
|year=2002
|title=Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates
|journal=Phys. Rev. A
|volume=65 |issue=4 |page=043614
|arxiv=cond-mat/0201395
|bibcode = 2002PhRvA..65d3614S
|doi=10.1103/PhysRevA.65.043614}}
</ref> and dense condensates and ] clusters and droplets.<ref>
{{cite journal
|author1=A. V. Avdeenkov
|author2=K. G. Zloshchastiev
|year=2011
|title=Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent
|journal=J. Phys. B: At. Mol. Opt. Phys.
|volume=44 |issue=19 |pages=195303
|arxiv=1108.0847
|bibcode=2011JPhB...44s5303A
|doi=10.1088/0953-4075/44/19/195303}}
</ref>

=== Other ===
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 criterion ===
The phenomena of superfluidity of a Bose gas and superconductivity of a strongly-correlated 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 ] 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 observation ==

=== Superfluid He-4 ===

In 1938, ], ] and ] discovered that ] became a new kind of fluid, now known as a ], at temperatures less than 2.17 K (the ]). Superfluid helium has many unusual properties, including zero ] (the ability to flow without dissipating energy) and the existence of ]. 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). ] 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 helium-4. Note that ], a ], also enters a ] phase (at a much lower temperature) which can be explained by the formation of bosonic ] of two atoms (see also ]).

=== Gaseous ===

The first "pure" Bose–Einstein condensate was created by ], ], and co-workers at ] on 5 June 1995. They cooled a dilute vapor of approximately two thousand ] atoms to below 170 nK using a combination of ] (a technique that won its inventors ], ], and ] the 1997 ]) and ]. About four months later, an independent effort led by ] at ] condensed ]. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of ] ] between two different condensates. Cornell, Wieman and Ketterle won the 2001 ] for their achievements.<ref name="nobel">{{cite web |url=http://nobelprize.org/nobel_prizes/physics/laureates/2001/cornellwieman-lecture.pdf |title=Eric A. Cornell and Carl E. Wieman&nbsp;— Nobel Lecture |format=PDF |publisher=nobelprize.org}}</ref>

A group led by Randall Hulet at Rice University announced a condensate of ] atoms only one month following the JILA work.<ref name=Bradley:1995/> 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.

==== Velocity-distribution data graph ====

In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of ] 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 ]: 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 ] of the peak on the right is a purely quantum-mechanical 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.<ref>{{cite book |url=https://books.google.com/?id=fqUU71spbZYC&printsec=frontcover|title=Thermal Physics|author=Baierlein, Ralph |publisher=Cambridge University Press|year=1999|isbn=0-521-65838-1}}</ref>

=== Quasiparticles ===

{{Main|Bose–Einstein condensation of quasiparticles}}Bose–Einstein condensation also applies to ]s in solids. ]s, ], and ] have integer spin which means they are ] that can form condensates.

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 ]]]<sub>3</sub>,<ref name=Nikuni:1999/> at temperatures as large as 14 K. The high transition temperature (relative to atomic gases) is due to the magnons small mass (near an electron) and greater achievable density. In 2006, condensation in a ] yttrium-iron-garnet thin film was seen even at room temperature,<ref name=Demokritov:2006/><ref>. Website of the "Westfählische Wilhelms Universität Münster" Prof.Demokritov. Retrieved 25 June 2012.</ref> with optical pumping.

]s, electron-hole 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 sub-kelvin Cu<sub>2</sub>O in 2005 on.

] was firstly detected for ] in a quantum well microcavity kept at 5 K.<ref name="ReferenceA">{{Cite journal|url = |title = Bose–Einstein condensation of exciton polaritons|date = 28 September 2006|journal = Nature|accessdate = |doi = 10.1038/nature05131|pmid = 17006506|volume=443 |issue = 7110|pages=409–414|bibcode = 2006Natur.443..409K |author=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}}</ref>

== Peculiar properties ==

=== Vortices ===
{{refimprove section|date=September 2011}}
As in many other systems, ] 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 ]. These phenomena are allowed for by the non-linear <math>|\psi(\vec{r})|^2</math> term in the GPE.{{Disputed inline|Talk page section|date=April 2017|reason=I do not believe the non-linear term is necessary; vortices are a simple consequence of the continuity equation the condensate wavefunction satisfies, and this equation is the same without the interaction term because the interaction term is real.}} As the vortices must have quantized ] the wavefunction may have the form <math>\psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}</math> where <math>\rho, z</math> and <math>\theta</math> are as in the ], and <math>\ell</math> 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 <math>\phi(\rho,z)</math>, the energy of <math>\psi(\vec{r})</math> must be minimized, according to the constraint <math>\psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}</math>. This is usually done computationally, however in a uniform medium the analytic form:

:<math>\phi=\frac{nx}{\sqrt{2+x^2}}</math>, where:
{| cellspacing="0" cellpadding="0"
|-
| <math>\,n^2</math>
| &nbsp;is&nbsp;
| density far from the vortex,
|-
| <math>\,x = \frac{\rho}{\ell\xi},</math>
|-
| <math>\,\xi</math>
| &nbsp;is&nbsp;
| healing length of the condensate.
|}demonstrates the correct behavior, and is a good approximation.

A singly charged vortex (<math>\ell=1</math>) is in the ground state, with its energy <math>\epsilon_v</math> given by

:<math>\epsilon_v=\pi n
\frac{\hbar^2}{m}\ln\left(1.464\frac{b}{\xi}\right)</math>

where <math>\,b</math>&nbsp;is&nbsp;the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary <math>b</math>.)

For multiply charged vortices (<math>\ell >1</math>) the energy is approximated by

:<math>\epsilon_v\approx \ell^2\pi n
\frac{\hbar^2}{m}\ln\left(\frac{b}{\xi}\right)</math>

which is greater than that of <math>\ell</math> 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 so-called dark ]s in one-dimensional 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 long-lived dark solitons have been produced and studied extensively.<ref name=Becker:2008/>

=== Attractive interactions ===
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 zero-point 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 ] team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturally ''attracting'' atoms of rubidium-85 (having negative atom–atom ]). Through ] involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which rubidium bonds, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum ] among wave-like 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 two-thirds 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.<ref name=nobel/> ] 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;<ref name=vanPutten:2010/> 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.<ref name='sun-16pra2'>{{cite journal|doi=10.1103/PhysRevA.94.033808|title=Landau-Zener extension of the Tavis-Cummings model: Structure of the solution|author1=C. Sun|author2=N. A. Sinitsyn |journal=]|volume=94|issue=3|year=2016|pages=033808|bibcode=2016PhRvA..94c3808S|arxiv=1606.08430}}</ref>

== Current research ==
{{unsolved|physics|How do we rigorously prove the existence of Bose–Einstein condensates for general interacting systems?}}
Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile.<ref>{{Cite web|url=http://physicsworld.com/cws/article/news/2013/nov/28/how-to-watch-a-bose-einstein-condensate-for-a-very-long-time|title=How to watch a Bose–Einstein condensate for a very long time - physicsworld.com|website=physicsworld.com|language=en-GB|access-date=2018-01-22}}</ref> 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|date=April 2011}}

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 ] between condensates due to ],<ref>{{cite web | author=Gorlitz, Axel | url=http://cua.mit.edu/ketterle_group/Projects_1997/Interference/Interference_BEC.htm | title=Interference of Condensates (BEC@MIT) | publisher=Cua.mit.edu | accessdate=13 October 2009 | deadurl=yes | archiveurl=https://web.archive.org/web/20160304092631/http://cua.mit.edu/ketterle_group/Projects_1997/Interference/Interference_BEC.htm | archivedate=4 March 2016 | df=dmy-all }}</ref> the study of ] and quantized ], the creation of bright matter wave ]s from Bose condensates confined to one dimension, and the ] pulses to very low speeds using ].<ref>
{{cite journal
|author1=Z. Dutton
|author2=N. S. Ginsberg
|author3=C. Slowe
|author4=L. Vestergaard Hau |lastauthoramp=yes
|year=2004
|title=The art of taming light: ultra-slow and stopped light
|journal=Europhysics News
|volume=35 |issue=2 |pages=33–39
|doi=10.1051/epn:2004201
|bibcode=2004ENews..35...33D}}
</ref> Vortices in Bose–Einstein condensates are also currently the subject of ] research, studying the possibility of modeling ]s and their related phenomena in such environments in the laboratory. Experimenters have also realized "]s", where the interference pattern from overlapping lasers provides a ]. These have been used to explore the transition between a superfluid and a ],<ref>{{cite web | url=http://qpt.physics.harvard.edu/qptsi.html | title=From Superfluid to Insulator: Bose–Einstein Condensate Undergoes a Quantum Phase Transition | publisher=Qpt.physics.harvard.edu | accessdate=13 October 2009}}</ref> and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the ].

Bose–Einstein condensates composed of a wide range of ]s have been produced.<ref>{{cite web | url=http://physicsworld.com/cws/article/print/2005/jun/01/ten-of-the-best-for-bec| title=Ten of the best for BEC | publisher=Physicsweb.org | date=1 June 2005 }}</ref>

Cooling ]s to extremely low temperatures has created ] gases, subject to the ]. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form bosonic compound particles (e.g. ]s or ]). The first ] condensates were created in November 2003 by the groups of ] at the ], ] at the ] and ] at ]. Jin quickly went on to create the first ] composed of ]s.<ref>{{cite web | url=http://physicsworld.com/cws/article/news/2004/jan/28/fermionic-condensate-makes-its-debut|title=Fermionic condensate makes its debut| publisher=Physicsweb.org | date=28 January 2004 }}</ref>

In 1999, Danish physicist ] led a team from ] which ] to about 17 meters per second,{{Clarify|date=January 2010|reason=group velocity and not actual velocity?}} using a superfluid.<ref>{{cite news | last = Cromie | first = William J. | title = Physicists Slow Speed of Light | publisher = The Harvard University Gazette | date = 18 February 1999 | url = http://news.harvard.edu/gazette/1999/02.18/light.html | accessdate = 26 January 2008 }}</ref> 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 "slow-light-mediated atomic matter-wave amplification" using Bose–Einstein condensates: details are discussed in '']''.<ref name=Ginsberg:2007/>

Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision ]. The first demonstration of a BEC in weightlessness was achieved in 2008 at a ] in Bremen, Germany by a consortium of researchers led by ] from ].<ref>{{Cite journal|last=Zoest|first=T. van|last2=Gaaloul|first2=N.|last3=Singh|first3=Y.|last4=Ahlers|first4=H.|last5=Herr|first5=W.|last6=Seidel|first6=S. T.|last7=Ertmer|first7=W.|last8=Rasel|first8=E.|last9=Eckart|first9=M.|date=2010-06-18|title=Bose-Einstein Condensation in Microgravity|url=http://science.sciencemag.org/content/328/5985/1540|journal=Science|language=en|volume=328|issue=5985|pages=1540–1543|doi=10.1126/science.1189164|issn=0036-8075|pmid=20558713|bibcode=2010Sci...328.1540V}}</ref> The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space<ref>{{Cite news|url=http://www.dlr.de/dlr/en/desktopdefault.aspx/tabid-10081/151_read-20337/#/gallery/25194|title=MAIUS 1 – First Bose-Einstein condensate generated in space|last=DLR|work=DLR Portal|access-date=2017-05-23|language=en-GB}}</ref> and it is also the subject of two upcoming experiments on the ].<ref>{{Cite web|url=https://coldatomlab.jpl.nasa.gov/|title=Cold Atom Laboratory|last=Laboratory|first=Jet Propulsion|website=coldatomlab.jpl.nasa.gov|access-date=2017-05-23}}</ref><ref>{{Cite web|url=http://www.lpi.usra.edu/planetary_news/2017/03/13/2017-nasa-fundamental-physics-workshop/|title=2017 NASA Fundamental Physics Workshop {{!}} Planetary News|website=www.lpi.usra.edu|language=en-US|access-date=2017-05-23}}</ref>

Researchers in the new field of ] use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.<ref>
{{cite journal
|author=P. Weiss
|date=12 February 2000
|title=Atomtronics may be the new electronics
|journal=Science News Online
|volume=157 |issue=7 |page=104
|accessdate=12 February 2011
|doi=10.2307/4012185
|url=http://www.sciencenews.org/view/generic/id/69786
|jstor=4012185}}
</ref>

In 1970, BECs were proposed by ] for anti-].<ref>{{cite arXiv | last=Tannenbaum| first=Emmanuel David | title=Gravimetric Radar: Gravity-based detection of a point-mass moving in a static background| year=1970| eprint=1208.2377| class=physics.ins-det}}</ref>

=== Isotopes ===
{{refimprove section|date=July 2010}}
The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2012, using ultra-low temperatures of {{nowrap|10<sup>−7</sup> K}} or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of ], ],
and ] atoms (], ], ], ], ], <sup>87</sup>Rb, ], ], ], ], ], ], ], ], and ]). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'.<ref>
{{cite journal
|author1=Dale G. Fried
|author2=Thomas C. Killian
|author3=Lorenz Willmann
|author4=David Landhuis
|author5=Stephen C. Moss
|author6=Daniel Kleppner
|author7=Thomas J. Greytak |lastauthoramp=yes
|year=1998
|title=Bose–Einstein Condensation of Atomic Hydrogen
|journal=Phys. Rev. Lett.
|volume=81 |issue=18 |pages=3811
|doi=10.1103/PhysRevLett.81.3811 |bibcode=1998PhRvL..81.3811F|arxiv = physics/9809017 }}
</ref> In contrast, the superfluid state of ] below {{nowrap|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.<ref>{{cite web | url=https://www.nobelprize.org/nobel_prizes/physics/laureates/2001/advanced-physicsprize2001.pdf| title=Bose–Einstein Condensation in Alkali Gases | publisher=The Royal Swedish Academy of Sciences | date=2001 | accessdate = 17 April 2017 }}</ref>

The ]ic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. It arises from a subtle interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell and half-integer total spin of the nucleus are coupled by a very weak ]. 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 also ==
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== Further reading ==
{{Refbegin|colwidth=60em}}
*{{cite journal |author=S. N. Bose |year=1924 |title=Plancks Gesetz und Lichtquantenhypothese |journal=Zeitschrift für Physik |volume=26 |pages=178–181 |bibcode=1924ZPhy...26..178B |doi=10.1007/BF01327326}}
*{{cite journal |author=A. Einstein |year=1925 |title=Quantentheorie des einatomigen idealen Gases |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften |volume=1 |page=3}},
*{{cite journal |author=L. D. Landau |year=1941 |title=The theory of Superfluity of Helium 111 |journal=J. Phys. USSR |volume=5 |pages=71–90}}
*{{cite journal |author=L. D. Landau |year=1941 |title=Theory of the Superfluidity of Helium II |journal=Physical Review |volume=60 |issue=4 |pages=356–358 |bibcode=1941PhRv...60..356L |doi=10.1103/PhysRev.60.356}}
*{{cite journal |author1=M. H. Anderson |author2=J. R. Ensher |author3=M. R. Matthews |author4=C. E. Wieman |author5=E. A. Cornell |lastauthoramp=yes |year=1995 |title=Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor |journal=Science |volume=269 |issue=5221 |pages=198–201 |jstor=2888436 |doi=10.1126/science.269.5221.198 |bibcode = 1995Sci...269..198A |pmid=17789847}}
*{{cite journal |author1=C. Barcelo |author2=S. Liberati |author3=M. Visser |lastauthoramp=yes |year=2001 |title=Analogue gravity from Bose–Einstein condensates |journal=Classical and Quantum Gravity |volume=18 |issue=6 |pages=1137–1156 |arxiv=gr-qc/0011026 |bibcode=2001CQGra..18.1137B |doi=10.1088/0264-9381/18/6/312}}
*{{cite journal |author1=P. G. Kevrekidis |author2=R. Carretero-González |author3=D. J. Frantzeskakis |author4=I. G. Kevrekidis |lastauthoramp=yes |year=2004 |title=Vortices in Bose–Einstein Condensates: Some Recent Developments |journal=Mod. Phys. Lett. B |volume=18 |issue=30 |pages=1481–1505 |doi=10.1142/S0217984904007967|bibcode=2004MPLB...18.1481K |arxiv=cond-mat/0501030 }}
*{{cite journal |author1=K.B. Davis |author2=M.-O. Mewes |author3=M.R. Andrews |author4=N.J. van Druten |author5=D.S. Durfee |author6=D.M. Kurn |author7=W. Ketterle |lastauthoramp=yes |title=Bose–Einstein condensation in a gas of sodium atoms |journal=Phys. Rev. Lett. |year=1995 |volume=75 |pages=3969–3973 |bibcode=1995PhRvL..75.3969D |doi=10.1103/PhysRevLett.75.3969 |pmid=10059782 |issue=22|url=https://pure.uva.nl/ws/files/2240246/27876_p3969_1 }}.
*{{cite journal |author1=D. S. Jin |author2=J. R. Ensher |author3=M. R. Matthews |author4=C. E. Wieman |author5=E. A. Cornell |lastauthoramp=yes |year=1996 |title=Collective Excitations of a Bose–Einstein Condensate in a Dilute Gas |journal=Phys. Rev. Lett. |volume=77 |issue=3 |pages=420–423 |bibcode=1996PhRvL..77..420J |doi=10.1103/PhysRevLett.77.420 |pmid=10062808}}
*{{cite journal |author1=M. R. Andrews |author2=C. G. Townsend |author3=H.-J. Miesner |author4=D. S. Durfee |author5=D. M. Kurn |author6=W. Ketterle |lastauthoramp=yes |year=1997 |title=Observation of interference between two Bose condensates |journal=Science |volume=275 |issue=5300 |pages=637–641 |doi=10.1126/science.275.5300.637 |pmid=9005843|url=http://amo.mit.edu/~bec/publish/papers/fringes.ps }}.
*{{cite journal |author1=E. A. Cornell |author2=C. E. Wieman |lastauthoramp=yes |year=1998 |title=The Bose–Einstein condensate |journal=Scientific American |volume=278 |issue=3 |pages=40–45 |doi=10.1038/scientificamerican0398-40|bibcode = 1998SciAm.278c..40C }}
*{{cite journal |author1=M. R. Matthews |author2=B. P. Anderson |author3=P. C. Haljan |author4=D. S. Hall |author5=C. E. Wieman |author6=E. A. Cornell |lastauthoramp=yes |year=1999 |title=Vortices in a Bose–Einstein condensate |journal=Phys. Rev. Lett. |volume=83 |issue=13 |pages=2498–2501 |arxiv=cond-mat/9908209 |bibcode=1999PhRvL..83.2498M |doi=10.1103/PhysRevLett.83.2498}}
*{{cite journal |author1=E. A. Donley |author2=N. R. Claussen |author3=S. L. Cornish |author4=J. L. Roberts |author5=E. A. Cornell |author6=C. E. Wieman |lastauthoramp=yes |year=2001 |title=Dynamics of collapsing and exploding Bose–Einstein condensates |journal=Nature |volume=412 |issue=6844 |pages=295–299 |arxiv=cond-mat/0105019 |bibcode=2001Natur.412..295D |doi=10.1038/35085500 |pmid=11460153}}
*{{cite journal |author1=A. G. Truscott |author2=K. E. Strecker |author3=W. I. McAlexander |author4=G. B. Partridge |author5=R. G. Hulet |lastauthoramp=yes |year=2001 |title=Observation of Fermi Pressure in a Gas of Trapped Atoms |journal=Science |volume=291 |issue=5513 |pages=2570–2572 |bibcode=2001Sci...291.2570T |doi=10.1126/science.1059318 |pmid=11283362}}
*{{cite journal |author1=M. Greiner |author2=O. Mandel |author3=T. Esslinger |author4=T. W. Hänsch |author5=I. Bloch |lastauthoramp=yes |year=2002 |title=Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms |journal=Nature |volume=415 |issue=6867 |pages=39–44 |bibcode=2002Natur.415...39G |doi=10.1038/415039a |pmid=11780110}}.
*{{cite journal |author1=S. Jochim |author2=M. Bartenstein |author3=A. Altmeyer |author4=G. Hendl |author5=S. Riedl |author6=C. Chin |author7=J. Hecker Denschlag |author8=R. Grimm |lastauthoramp=yes |year=2003 |title=Bose–Einstein Condensation of Molecules |journal=Science |volume=302 |issue=5653 |pages=2101–2103 |bibcode=2003Sci...302.2101J |doi=10.1126/science.1093280 |pmid=14615548}}
*{{cite journal |author1=M. Greiner |author2=C. A. Regal |author3=D. S. Jin |lastauthoramp=yes |year=2003 |title=Emergence of a molecular Bose−Einstein condensate from a Fermi gas |journal=Nature |volume=426 |issue=6966 |pages=537–540 |bibcode=2003Natur.426..537G |doi=10.1038/nature02199 |pmid=14647340}}
*{{cite journal |author1=M. W. Zwierlein |author2=C. A. Stan |author3=C. H. Schunck |author4=S. M. F. Raupach |author5=S. Gupta |author6=Z. Hadzibabic |author7=W. Ketterle |lastauthoramp=yes |year=2003 |title=Observation of Bose–Einstein Condensation of Molecules |journal=Phys. Rev. Lett. |volume=91 |issue=25 |page=250401 |arxiv=cond-mat/0311617 |bibcode=2003PhRvL..91y0401Z |doi=10.1103/PhysRevLett.91.250401 |pmid=14754098}}
*{{cite journal |author1=C. A. Regal |author2=M. Greiner |author3=D. S. Jin |lastauthoramp=yes |year=2004 |title=Observation of Resonance Condensation of Fermionic Atom Pairs |journal=Phys. Rev. Lett. |volume=92 |issue=4 |page=040403 |arxiv=cond-mat/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.
*{{cite journal |author1=M. Mackie |author2=K. A. Suominen |author3=J. Javanainen |lastauthoramp=yes |year=2002 |title=Mean-field theory of Feshbach-resonant interactions in 85Rb condensates |journal=Phys. Rev. Lett. |volume=89 |issue=18 |page=180403 |doi=10.1103/PhysRevLett.89.180403 |bibcode=2002PhRvL..89r0403M |pmid=12398586|arxiv = cond-mat/0205535 }}
{{Refend}}

== External links ==
* Bose–Einstein Condensation 2009 – Frontiers in Quantum Gases
* General introduction to Bose–Einstein condensation
* – for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates
*
*
*
*
*
*
*
*
* – information on constructing a Bose–Einstein condensate machine.
*
*
* – ] resource on BEC

{{State of matter}}
{{Einstein}}

{{Authority control}}

{{DEFAULTSORT:Bose-Einstein Condensate}}
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