Illustration of Bose-Einstein condensation: as the temperature of the ensemble of bosons is reduced, the overlap between the particles' wavefunctions increases as the thermal de Broglie wavelength increases. At one point, when the overlap becomes significant, a macroscopic number of particles condense into the ground state.
Velocity-distribution data (3 views) for gas ofrubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate.Left: just before the appearance of a Bose–Einstein condensate.Center: just after the appearance of the condensate.Right: after further evaporation, leaving a sample of nearly pure condensate.
Bose first sent a paper to Einstein on thequantum statistics of light quanta (now calledphotons), in which he derivedPlanck'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 theZeitschrift für Physik, which published it in 1924.[5] Einstein's manuscript, once believed to be lost, was found in a library atLeiden University in 2005.[6] Einstein then extended Bose's ideas to matter in two other papers.[7][8] The result of their efforts is the concept of aBose gas, governed byBose–Einstein statistics, which describes the statistical distribution ofidentical particles withintegerspin, now calledbosons. 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 accessiblequantum state, resulting in a new form of matter. Bosons include thephoton,polaritons,magnons, someatoms andmolecules (depending on the number ofnucleons, see#Isotopes) such as atomic hydrogen,helium-4, lithium-7, rubidium-87 or strontium-84.
The quest to produce a Bose–Einstein condensate in the laboratory was stimulated by a paper published in 1976 by two program directors at theNational Science Foundation (William Stwalley and Lewis Nosanow), proposing to use spin-polarized atomichydrogen to produce a gaseous BEC.[11] This led to the immediate pursuit of the idea by four independent research groups; these were led by Isaac Silvera (University of Amsterdam), Walter Hardy (University of British Columbia), Thomas Greytak (Massachusetts Institute of Technology) and David Lee (Cornell University).[12] However, cooling atomic hydrogen turned out to be technically difficult, and Bose-Einstein condensation of atomic hydrogen was only realized in 1998.[13][14]
On 5 June 1995, the first gaseous condensate was produced byEric Cornell andCarl Wieman at theUniversity of Colorado at BoulderNIST–JILA lab, in a gas ofrubidium atoms cooled to 170 nanokelvins (nK).[15] Shortly thereafter,Wolfgang Ketterle at MIT produced a Bose–Einstein Condensate in a gas ofsodium atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001Nobel Prize in Physics.[16] Bose-Einstein condensation of alkali gases is easier because they can be pre-cooled withlaser cooling 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.[14] These early studies founded the field ofultracold 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.[17] BEC of plasmonic quasiparticles (plasmon-exciton polaritons) has been realized in periodic arrays of metal nanoparticles overlaid with dye molecules,[18] exhibiting ultrafast sub-picosecond dynamics[19] and long-range correlations.[20]
This transition to BEC occurs below a critical temperature, which for a uniformthree-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by
The critical temperature depends on the density. A more concise and experimentally relevant[22] condition involves the phase-space density, where
is thethermal 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
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 right-hand side contains the expression for the average occupation number of the fundamental state, the equation of state can be rewritten as
Because 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 wavelength:
recovering the value indicated on the previous section. The critical values are such that if or, we are in the presence of a Bose–Einstein 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 condense in the fundamental state, which is the state with momentum.
In 1938,Pyotr Kapitsa,John Allen andDon Misener discovered thathelium-4 became a new kind of fluid, now known as asuperfluid, at temperatures less than 2.17 K (thelambda point). Superfluid helium has many unusual properties, including zeroviscosity (the ability to flow without dissipating energy) and the existence ofquantized 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 helium-4 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 thathelium-3, afermion, also enters asuperfluid phase (at a much lower temperature) which can be explained by the formation of bosonicCooper pairs of two atoms (see alsofermionic condensate).
A group led byRandall Hulet at Rice University announced a condensate oflithium atoms only one month following the JILA work.[25] 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.
In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas ofrubidium 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 theHeisenberg 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. Thisanisotropy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left.[26]
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 antiferromagneticTlCuCl 3,[28] 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 aferromagnetic yttrium-iron-garnet thin film was seen even at room temperature,[29][30] with optical pumping.
Excitons, electron–hole pairs, were predicted to condense at low temperature and high density by Boer et al., in 1961.[citation needed] Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance.[31] Fast optical exciton creation was used to form condensates in sub-kelvinCu 2O in 2005 on.[32]
Polariton condensation was first detected forexciton-polaritons in a quantum well microcavity kept at 5 K.[33] Quasiparticle BECs have been achieved at room-temperature, for example, in microcavity-coupled organic semiconductors and plasmon-exciton polaritons in periodic arrays of metal nanoparticles coupled to dye molecules.[18][19][20]
In June 2020, theCold Atom Laboratory experiment on board theInternational Space Station successfully created a BEC of rubidium atoms and observed them for over a second in free-fall. Although initially just a proof of function, early results showed that, in the microgravity environment of the ISS, about half of the atoms formed into a magnetically insensitive halo-like cloud around the main body of the BEC.[34][35]
Consider a collection ofN non-interacting particles, which can each be in one of twoquantum 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 different configurations. If there are particles in state, there are particles in state. Whether any particular particle is in state or in state cannot be determined, so each value of 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. At temperature, 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 most-likely outcome is that most of the particles will collapse into state.
In the distinguishable case, for largeN, the fraction in state can be computed. It is the same as flipping a coin with probability proportional to to land tails.
In the indistinguishable case, each value of is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:
For large, the normalization constant is. The expected total number of particles not in the lowest energy state, in the limit that, is equal to
It does not grow whenN 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,:
When the integral (also known asBose–Einstein integral) is evaluated with factors of and 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 negligiblechemical potential. InBose–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 .
Nikolay Bogolyubov considered perturbations on the limit of dilute gas,[36] finding a finite pressure at zero temperature and positive chemical potential. This leads to corrections for the ground state. The Bogoliubov state has pressure:.
Theweakly interacting Bose gas can be converted to a system of non-interacting particles with a dispersion law.
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 asystem 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 usingmean-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 non-linearSchrödinger equation):
where:
is the mass of the bosons,
is the external potential, and
represents the inter-particle interactions.
In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for):
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.It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.
The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution anddifferent numerical methods, such as split-stepCrank–Nicolson[37]andFourier spectral[38] methods, are used for its solution. There are different Fortran and C programs for its solution forcontact interaction[39][40]and long-rangedipolar interaction[41] which can be freely used.
The Gross–Pitaevskii model of BEC is a physicalapproximation valid for certain classes of BECs. By construction, theGPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions toself-energy.[42] These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensatewavefunction 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,[43][44][45][46] effectively lower-dimensional condensates,[47] and dense condensates andsuperfluid clusters and droplets.[48] It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term found in theLogarithmic Schrödinger equation must be added to the Gross-Pitaevskii equation along with aGinzburg–Sobyanin contribution to correctly determine that the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures in close agreement with experiment.[49]
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 solved 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.
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 inhelium-4 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.
As in many other systems,vortices can exist in BECs.[50]Vortices can be created, for example, by "stirring" the condensate with lasers,[51]rotating the confining trap,[52]or by rapid cooling across the phase transition.[53]The vortex created will be aquantum vortex with core shape determined by the interactions.[54] Fluid circulation around any point is quantized due to the single-valued nature of the order BEC order parameter or wavefunction,[55] that can be written in the form where and are as in thecylindrical coordinate system, and is the angular quantum number (a.k.a. the "charge" of the vortex). Since the energy of a vortex is proportional to the square of its angular momentum, intrivial topology only vortices can exist in thesteady state; Higher-charge vortices will have a tendency to split into vortices, if allowed by the topology of the geometry.
An axially symmetric (for instance, harmonic) confining potential is commonly used for the study of vortices in BEC. To determine, the energy of must be minimized, according to the constraint. This is usually done computationally, however, in a uniform medium, the following analytic form demonstrates the correct behavior, and is a good approximation:
Here, is the density far from the vortex and, where is thehealing length of the condensate.
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 so-called darksolitons 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.[56]
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 theJILA team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturallyattracting atoms of rubidium-85 (having negative atom–atomscattering length). ThroughFeshbach resonance 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 quantuminterference 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.[24]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;[57] 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.[58]
Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile.[59] 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.[60]
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.
Bose–Einstein condensates composed of a wide range ofisotopes have been produced; see below.[61]
Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides aperiodic potential. These are used to explore the transition between a superfluid and aMott insulator.[64]
They are also useful in studying Bose–Einstein condensation in fewer than three dimensions, for example theLieb–Liniger model (an the limit of strong interactions, theTonks–Girardeau gas) in 1D and theBerezinskii–Kosterlitz–Thouless transition in 2D. Indeed, a deep optical lattice allows the experimentalist to freeze the motion of the particles along one or two directions, effectively eliminating one or two dimensions from the system.
Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow one-dimensional optical lattice originally observed by Haller[65] has been explored via a tweaking of the primary optical lattice by a secondary weaker one.[66] Thus for a resulting weak bichromatic optical lattice, it has been found that the pinning transition is robust against theintroduction of the weaker secondary optical lattice.
Studies of vortices in nonuniform Bose–Einstein condensates[67] as well as excitations of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken.[68][69] 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 red-detuned laser beams (hitting frequencies slightly above and below the resonance frequency, respectively) via the time-dependent Gross-Pitaevskii equation.[70]
In 1999, Danish physicistLene Hau led a team fromHarvard University whichslowed a beam of light to about 17 meters per second[clarification needed] using a superfluid.[71] 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.[72]
Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precisionatom interferometry. The first demonstration of a BEC in weightlessness was achieved in 2008 at adrop tower in Bremen, Germany by a consortium of researchers led byErnst M. Rasel fromLeibniz University Hannover.[73] The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space[74] and it is also the subject of two upcoming experiments on theInternational Space Station.[75][76]
Researchers in the new field ofatomtronics use the properties of Bose–Einstein condensates in the emerging quantum technology of matter-wave circuits.[77][78]
Bose-Einstein condensation has mainly been observed on alkaline atoms, some of which have collisional properties particularly suitable for evaporative cooling in traps, and which were the first to be laser-cooled. As of 2021, using ultra-low temperatures of10−7 K or below, Bose–Einstein condensates had been obtained for a multitude of isotopes with more or less ease, mainly ofalkali metal,alkaline earth metal, andlanthanide 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,170 Yb,174 Yb,176 Yb,164 Dy,168 Er,169 Tm, and metastable4 He (orthohelium)).[80][81] Research was finally successful in atomic hydrogen with the aid of the newly developed method of 'evaporative cooling'.[82]
In contrast, the superfluid state of4 He below2.17 K is differs significantly from dilute degenerate atomic gases because the interaction between the atoms is strong. Only 8% of atoms are in the condensed fraction near absolute zero, rather than near 100% of a weakly interacting BEC.[83]
Thebosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. It arises from the interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell (one outer electron) and half-integer total spin of the nucleus are coupled by a very weakhyperfine interaction.[84] The total spin of the atom, arising from this coupling, is an integer value.[85] Conversely, alkali isotopes which have an integer nuclear spin (such as6 Li and40 K) are fermions and can form degenerateFermi gases, also called "Fermi condensates".[86]
Limitations of evaporative cooling have restricted atomic BECs to "pulsed" operation, involving a highly inefficient duty cycle that discards more than 99% of atoms to reach BEC. Achieving continuous BEC has been a major open problem of experimental BEC research, driven by the same motivations as continuous optical laser development: high flux, high coherence matter waves produced continuously would enable new sensing applications.
Continuous BEC was achieved for the first time in 2022 with84 Sr.[88]
P. Sikivie and Q. Yang showed thatcold dark matteraxions would form a Bose–Einstein condensate bythermalisation because of gravitational self-interactions.[91] Axions have not yet been confirmed to exist. However the important search for them has been greatly enhanced with the completion of upgrades to theAxion Dark Matter Experiment (ADMX) at the University of Washington in early 2018.
In 2014, a potential dibaryon was detected at theJülich Research Center at about 2380 MeV. The center claimed that the measurements confirm results from 2011, via a more replicable method.[92][93] The particle existed for 10−23 seconds and was named d*(2380).[94] This particle is hypothesized to consist of threeup quarks and threedown quarks.[95] It is theorized that groups of d* (d-stars) could form Bose–Einstein condensates due to prevailing low temperatures in the early universe, and that BECs made of suchhexaquarks with trapped electrons could behave likedark matter.[96][97][98]
In the 2016 filmSpectral, the US military battles mysterious enemy creatures fashioned out of Bose–Einstein condensates.[99]
In the 2003 novelBlind Lake, scientists observe sentient life on a planet 51 light-years away using telescopes powered by Bose–Einstein condensate-based quantum computers.
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^Producing Bose-Einstein condensates experimentally often involves a change in both density and temperature. This is the case with evaporative cooling of dilute gases, for which the temperature is decreased by loosing particles. The goal is then to reduce the temperature faster than particles are lost, so that the phase-space density increases overall.
^Monique Combescot and Shiue-Yuan Shiau, "Excitons and Cooper Pairs: Two Composite Bosons in Many-Body Physics", Oxford University Press (ISBN9780198753735)
^S. O. Demokritov; V. E. Demidov; O. Dzyapko; G. A. Melkov; A. A. Serga; B. Hillebrands & A. N. Slavin (2006). "Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping".Nature.443 (7110):430–433.Bibcode:2006Natur.443..430D.doi:10.1038/nature05117.PMID17006509.S2CID4421089.
^Elmar Haller; Russell Hart; Manfred J. Mark; Johann G. Danzl; Lukas Reichsoellner; Mattias Gustavsson; Marcello Dalmonte; Guido Pupillo; Hanns-Christoph Naegerl (2010). "Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons".Nature Letters.466 (7306):597–600.arXiv:1004.3168.Bibcode:2010Natur.466..597H.doi:10.1038/nature09259.PMID20671704.S2CID687095.
^Roger R. Sakhel; Asaad R. Sakhel; Humam B. Ghassib (2011). "Self-interfering matter-wave patterns generated by a moving laser obstacle in a two-dimensional Bose–Einstein condensate inside a power trap cut off by box potential boundaries".Physical Review A.84 (3) 033634.arXiv:1107.0369.Bibcode:2011PhRvA..84c3634S.doi:10.1103/PhysRevA.84.033634.S2CID119277418.
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^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.
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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
