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In
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
, a Bose–Einstein condensate (BEC) is a
state of matter In physics, a state of matter is one of the distinct forms in which matter can exist. Four states of matter are observable in everyday life: solid, liquid, gas, and Plasma (physics), plasma. Different states are distinguished by the ways the ...
that is typically formed when a gas of
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s at very low densities is cooled to
temperature Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
s very close to
absolute zero Absolute zero is the lowest possible temperature, a state at which a system's internal energy, and in ideal cases entropy, reach their minimum values. The absolute zero is defined as 0 K on the Kelvin scale, equivalent to −273.15 ° ...
, i.e. . Under such conditions, a large fraction of bosons occupy the lowest
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
, at which microscopic quantum-mechanical phenomena, particularly wavefunction interference, become apparent macroscopically. More generally, condensation refers to the appearance of macroscopic occupation of one or several states: for example, in BCS theory, a superconductor is a condensate of Cooper pairs. As such, condensation can be associated with
phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...
, and the macroscopic occupation of the state is the order parameter. Bose–Einstein condensate was first predicted, generally, in 1924–1925 by
Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
, crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics. In 1995, the Bose–Einstein condensate was created by Eric Cornell and Carl Wieman of the University of Colorado Boulder using rubidium atoms. Later that year, Wolfgang Ketterle of MIT produced a BEC using
sodium Sodium is a chemical element; it has Symbol (chemistry), symbol Na (from Neo-Latin ) and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 element, group 1 of the peri ...
atoms. In 2001 Cornell, Wieman, and Ketterle shared the
Nobel Prize in Physics The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
"for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates".


History

Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s), in which he derived Planck's quantum radiation law without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the '' Zeitschrift für Physik'', which published it in 1924. (The Einstein manuscript, once believed to be lost, was found in a library at
Leiden University Leiden University (abbreviated as ''LEI''; ) is a Public university, public research university in Leiden, Netherlands. Established in 1575 by William the Silent, William, Prince of Orange as a Protestantism, Protestant institution, it holds the d ...
in 2005.) Einstein then extended Bose's ideas to matter in two other papers. The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of
identical particles In quantum mechanics, indistinguishable particles (also called identical or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
with
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
spin, now called
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s. Bosons are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
, resulting in a new form of matter. Bosons include the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
, polaritons, magnons, some
atom Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s and
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s (depending on the number of
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number. Until the 1960s, nucleons were thought to be ele ...
s, see #Isotopes) such as atomic hydrogen,
helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...
, lithium-7, rubidium-87 or strontium-84. In 1938, Fritz London proposed the BEC as a mechanism for
superfluidity Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
in and
superconductivity Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ord ...
. The quest to produce a Bose–Einstein condensate in the laboratory was stimulated by a paper published in 1976 by two program directors at the
National Science Foundation The U.S. National Science Foundation (NSF) is an Independent agencies of the United States government#Examples of independent agencies, independent agency of the Federal government of the United States, United States federal government that su ...
(William Stwalley and Lewis Nosanow), proposing to use spin-polarized atomic
hydrogen Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
to produce a gaseous BEC. This led to the immediate pursuit of the idea by four independent research groups; these were led by Isaac Silvera (
University of Amsterdam The University of Amsterdam (abbreviated as UvA, ) is a public university, public research university located in Amsterdam, Netherlands. Established in 1632 by municipal authorities, it is the fourth-oldest academic institution in the Netherlan ...
), Walter Hardy (
University of British Columbia The University of British Columbia (UBC) is a Public university, public research university with campuses near University of British Columbia Vancouver, Vancouver and University of British Columbia Okanagan, Kelowna, in British Columbia, Canada ...
), Thomas Greytak (
Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of moder ...
) and David Lee (
Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...
). However, cooling atomic hydrogen turned out to be technically difficult, and Bose-Einstein condensation of atomic hydrogen was only realized in 1998. On 5 June 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the
University of Colorado at Boulder The University of Colorado Boulder (CU Boulder, CU, or Colorado) is a Public university, public research university in Boulder, Colorado, United States. Founded in 1876, five months before Colorado became a Federated state, state, it is the fla ...
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
JILA lab, in a gas of rubidium atoms cooled to 170  nanokelvins (nK). Shortly thereafter, Wolfgang Ketterle at MIT produced a Bose–Einstein Condensate in a gas of
sodium Sodium is a chemical element; it has Symbol (chemistry), symbol Na (from Neo-Latin ) and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 element, group 1 of the peri ...
atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001
Nobel Prize in Physics The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
. Bose-Einstein condensation of alkali gases is easier because they can be pre-cooled with
laser cooling Laser cooling includes several techniques where atoms, molecules, and small mechanical systems are cooled with laser light. The directed energy of lasers is often associated with heating materials, e.g. laser cutting, so it can be counterintuit ...
techniques, unlike atomic hydrogen at the time, which give a significant head start when performing the final forced evaporative cooling to cross the condensation threshold. These early studies founded the field of ultracold atoms, and hundreds of research groups around the world now routinely produce BECs of dilute atomic vapors in their labs. Since 1995, many other atomic species have been condensed (see #Isotopes), and BECs have also been realized using molecules, polaritons, and other quasi-particles. BECs of photons can be made, for example, in dye microcavites with wavelength-scale mirror separation, forming a two-dimensional harmonically confined photon gas with tunable chemical potential. BEC of plasmonic quasiparticles (plasmon-exciton polaritons) has been realized in periodic arrays of metal nanoparticles overlaid with dye molecules, exhibiting ultrafast sub-picosecond dynamics and long-range correlations.


Critical temperature

This transition to BEC occurs below a critical temperature, which for a uniform
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by : T_\text = \left(\frac\right)^ \frac \approx 3.3125\,\frac, where: : T_\text is the critical temperature, : n is the particle density, : m is the mass per boson, : \hbar is the reduced
Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, : k_\text is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, : \zeta is the Riemann zeta function (\zeta(3/2) \approx 2.6124). Interactions shift the value, and the corrections can be calculated by mean-field theory. This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. The critical temperature depends on the density. A more concise and experimentally relevant condition involves the phase-space density \mathcal=n \lambda_T^3, where : \lambda_T=\hbar \sqrt is the thermal de Broglie wavelength. It is a dimensionless quantity. The transition to BEC occurs when the phase-space density is greater than critical value: : \mathcal_\text=\zeta(3/2) in 3D uniform space. This is equivalent to the above condition on the temperature. In a 3D harmonic potential, the critical value is instead : \mathcal_\text=\zeta(3)\approx 1.202 where n has to be understood as the peak density.


Derivation


Ideal Bose gas

For an ideal Bose gas we have the equation of state : \frac = \frac g_(f) + \frac\frac, where v = V/N is the per-particle volume, \lambda is the thermal wavelength, f is the fugacity, and : g_\alpha(f) = \sum\limits_^\infty \frac. It is noticeable that g_ is a monotonically growing function of f in f \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, 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 \langle n_0 \rangle, the equation of state can be rewritten as : \frac = \frac g_(f) + \frac \Leftrightarrow \frac \lambda^3 = \frac - g_(f). Because the left term on the second equation must always be positive, \frac > g_(f), and because g_(f) \le g_(1), a stronger condition is : \frac > g_(1), 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: : \lambda_c^3 = g_(1)v = \zeta(3/2) v, : T_\text = \frac, recovering the value indicated on the previous section. The critical values are such that if T < T_\text or \lambda > \lambda_\text, 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 f = 1, obtaining : \frac = 1 - \left(\frac\right)^3 and equivalently \frac = 1 - \left(\frac\right)^. So, if T \ll T_\text, the fraction \frac \approx 1, and if T \gg T_\text, the fraction \frac \approx 0. At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum \vec = 0.


Experimental observation


Superfluid helium-4

In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that
helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...
became a new kind of fluid, now known as a
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortex, vortices that continue to rotate indefinitely. Superfluidity occurs ...
, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero
viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
(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 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 that helium-3, a
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
, also enters a
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortex, vortices that continue to rotate indefinitely. Superfluidity occurs ...
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 gases

The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand
rubidium-87 Rubidium (37Rb) has 36 isotopes, with naturally occurring rubidium being composed of just two isotopes; 85Rb (72.2%) and the radioactive 87Rb (27.8%). 87Rb has a half-life of . It readily substitutes for potassium in minerals, and is therefore ...
atoms to below 170 nK using a combination of
laser cooling Laser cooling includes several techniques where atoms, molecules, and small mechanical systems are cooled with laser light. The directed energy of lasers is often associated with heating materials, e.g. laser cutting, so it can be counterintuit ...
(a technique that won its inventors Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997
Nobel Prize in Physics The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
) and
magnetic evaporative cooling Evaporative cooling is an atomic physics technique to achieve high phase space densities which optical cooling techniques alone typically can not reach. Atoms trapped in optical or magnetic traps can be evaporatively cooled via two primary mecha ...
. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium-23. 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 The Nobel Prize in Physics () is an annual award given by the Royal Swedish Academy of Sciences for those who have made the most outstanding contributions to mankind in the field of physics. It is one of the five Nobel Prizes established by the ...
for their achievements. A group led by Randall Hulet at Rice University announced a condensate of
lithium Lithium (from , , ) is a chemical element; it has chemical symbol, symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard temperature and pressure, standard conditions, it is the least dense metal and the ...
atoms only one month following the JILA work. 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 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 The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
: 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 Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ve ...
of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left.


Quasiparticles

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. 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 , 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 Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...
yttrium-iron-garnet thin film was seen even at room temperature, with optical pumping. Excitons, 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 in 2005 on. Polariton condensation was first detected for exciton-polaritons in a quantum well microcavity kept at 5 K. 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.


In zero gravity

In June 2020, the Cold Atom Laboratory experiment on board the
International Space Station The International Space Station (ISS) is a large space station that was Assembly of the International Space Station, assembled and is maintained in low Earth orbit by a collaboration of five space agencies and their contractors: NASA (United ...
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.


Models


Bose Einstein's non-interacting gas

Consider a collection of ''N'' non-interacting particles, which can each be in one of two
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
s, , 0\rangle and , 1\rangle. If the two states are equal in energy, each different configuration is equally likely. If we can tell which particle is which, there are 2^N different configurations, since each particle can be in , 0\rangle or , 1\rangle independently. In almost all of the configurations, about half the particles are in , 0\rangle and the other half in , 1\rangle. 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 , 1\rangle, there are N-K particles in state , 0\rangle. Whether any particular particle is in state , 0\rangle or in state , 1\rangle cannot be determined, so each value of K determines a unique quantum state for the whole system. Suppose now that the energy of state , 1\rangle is slightly greater than the energy of state , 0\rangle by an amount E. At temperature T, a particle will have a lesser probability to be in state , 1\rangle by e^. In the distinguishable case, the particle distribution will be biased slightly towards state , 0\rangle. 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 , 0\rangle. In the distinguishable case, for large ''N'', the fraction in state , 0\rangle can be computed. It is the same as flipping a coin with probability proportional to \exp 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: :\, P(K)= C e^ = C p^K. 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 N\rightarrow \infty, is equal to : \sum_ C n p^n=p/(1-p) 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 , k\rangle. 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): :\, N = V \int = V \int :\, p(k)= e^. When the integral (also known as Bose–Einstein integral) is evaluated with factors of k_B and \hbar 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 thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
\mu. In Bose–Einstein statistics distribution, \mu is actually still nonzero for BECs; however, \mu is less than the ground state energy. Except when specifically talking about the ground state, \mu can be approximated for most energy or momentum states as \mu \approx 0.


Bogoliubov theory for weakly interacting gas

Nikolay Bogoliubov considered perturbations on the limit of dilute gas, 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): P = gn^2/2. The original interacting system can be converted to a system of non-interacting particles with a dispersion law.


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 \psi(\vec). For a system of this nature, , \psi(\vec), ^2 is interpreted as the particle density, so the total number of atoms is N=\int d\vec, \psi(\vec), ^2 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 \psi(\vec) is: :E=\int d\vec\left \nabla\psi(\vec), ^2+V(\vec), \psi(\vec), ^2+\fracU_0, \psi(\vec), ^4\right/math> Minimizing this energy with respect to infinitesimal variations in \psi(\vec), and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
): :i\hbar\frac = \left(-\frac+V(\vec)+U_0, \psi(\vec), ^2\right)\psi(\vec) where: : In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for \ T= 0): : = \sqrt 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 \ T= 0. 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 Crank–Nicolson and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction and long-range dipolar interaction which can be freely used.


Weaknesses of Gross–Pitaevskii model

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 two-body type and also neglects anomalous contributions to self-energy. These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction 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, effectively lower-dimensional condensates, and dense condensates and
superfluid Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortex, vortices that continue to rotate indefinitely. Superfluidity occurs ...
clusters and droplets. It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term \psi \ln , \psi, ^2 found in the Logarithmic Schrödinger equation must be added to the Gross-Pitaevskii equation along with a Ginzburg–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.


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 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.


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
helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...
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.


Peculiar properties


Quantized vortices

As in many other systems, vortices can exist in BECs. Vortices can be created, for example, by "stirring" the condensate with lasers, rotating the confining trap, or by rapid cooling across the phase transition. The vortex created will be a quantum vortex with core shape determined by the interactions. Fluid circulation around any point is quantized due to the single-valued nature of the order BEC order parameter or wavefunction, that can be written in the form \psi(\vec)=\phi(\rho,z)e^ where \rho, z and \theta are as in the cylindrical coordinate system, and \ell 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, in
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
only \ell=1 vortices can exist in the steady state; Higher-charge vortices will have a tendency to split into \ell=1 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 \phi(\rho,z), the energy of \psi(\vec) must be minimized, according to the constraint \psi(\vec)=\phi(\rho,z)e^. This is usually done computationally, however, in a uniform medium, the following analytic form demonstrates the correct behavior, and is a good approximation: :\phi=\frac\,. Here, n is the density far from the vortex and x=\rho/(\ell \xi), where \xi=1/\sqrt is the healing length of the condensate. A singly charged vortex (\ell=1) is in the ground state, with its energy \epsilon_v given by :\epsilon_v=\pi n \frac\ln\left(1.464\frac\right) where \,b is the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary b.) For multiply charged vortices (\ell >1) the energy is approximated by :\epsilon_v\approx \ell^2\pi n \frac\ln\left(\frac\right) which is greater than that of \ell 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
soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
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.


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 JILA 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 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 Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference 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. 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; 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.


Current research

Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile. 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. 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 of
isotope Isotopes are distinct nuclear species (or ''nuclides'') of the same chemical element. They have the same atomic number (number of protons in their Atomic nucleus, nuclei) and position in the periodic table (and hence belong to the same chemica ...
s have been produced; see below.


Fundamental research

Examples include experiments that have demonstrated interference between condensates due to
wave–particle duality Wave–particle duality is the concept in quantum mechanics that fundamental entities of the universe, like photons and electrons, exhibit particle or wave (physics), wave properties according to the experimental circumstances. It expresses the in ...
, the study of
superfluidity Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
and quantized vortices, the creation of bright matter wave
soliton In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such local ...
s from Bose condensates confined to one dimension, and the slowing of light pulses to very low speeds using electromagnetically induced transparency. Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling
black hole A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
s 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 are used to explore the transition between a superfluid and a Mott insulator. They are also useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Lieb–Liniger model (an the limit of strong interactions, the Tonks–Girardeau gas) in 1D and the Berezinskii–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 dimension 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 has been explored via a tweaking of the primary optical lattice by a secondary weaker one. 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 as well as excitations of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken. 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.


Applications

In 1999, Danish physicist Lene Hau led a team from
Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
which slowed a beam of light to about 17 meters per second using a superfluid. 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. Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision
atom interferometry An atom interferometer uses the wave-like nature of atoms in order to produce interference. In atom interferometers, the roles of matter and light are reversed compared to the laser based interferometers, i.e. the beam splitter and mirrors are lase ...
. 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. The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space and it is also the subject of two upcoming experiments on the
International Space Station The International Space Station (ISS) is a large space station that was Assembly of the International Space Station, assembled and is maintained in low Earth orbit by a collaboration of five space agencies and their contractors: NASA (United ...
. Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates in the emerging quantum technology of matter-wave circuits. In 1970, BECs were proposed by Emmanuel David Tannenbaum for anti-
stealth technology Stealth technology, also termed low observable technology (LO technology), is a sub-discipline of military tactics and passive and active electronic countermeasures. The term covers a range of military technology, methods used to make personnel ...
.


Isotopes

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 of or below, Bose–Einstein condensates had been obtained for a multitude of isotopes with more or less ease, mainly of
alkali metal The alkali metals consist of the chemical elements lithium (Li), sodium (Na), potassium (K),The symbols Na and K for sodium and potassium are derived from their Latin names, ''natrium'' and ''kalium''; these are still the origins of the names ...
,
alkaline earth metal The alkaline earth metals are six chemical elements in group (periodic table), group 2 of the periodic table. They are beryllium (Be), magnesium (Mg), calcium (Ca), strontium (Sr), barium (Ba), and radium (Ra).. The elements have very similar p ...
, and lanthanide atoms (, , , , , , , , , , , , , , , , , , and metastable (orthohelium)). Research was finally successful in atomic hydrogen with the aid of the newly developed method of 'evaporative cooling'. In contrast, the superfluid state of below 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. The
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
ic 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 weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer value. Conversely, alkali isotopes which have an integer nuclear spin (such as and ) are fermions and can form degenerate Fermi gases, also called "Fermi condensates". Cooling
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s 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.
molecule A molecule is a group of two or more atoms that are held together by Force, attractive forces known as chemical bonds; depending on context, the term may or may not include ions that satisfy this criterion. In quantum physics, organic chemi ...
s or Cooper pairs). The first molecular condensates were created in November 2003 by the groups of Rudolf Grimm at the
University of Innsbruck The University of Innsbruck (; ) is a public research university in Innsbruck, the capital of the Austrian federal state of Tyrol (state), Tyrol, founded on October 15, 1669. It is the largest education facility in the Austrian States of Austria, ...
, Deborah S. Jin at the
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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.


Continuous Bose–Einstein condensation

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 with .


In solid state physics

In 2020, researchers reported the development of superconducting BEC and that there appears to be a "smooth transition between" BEC and Bardeen–Cooper–Shrieffer regimes.


Dark matter

P. Sikivie and Q. Yang showed that cold dark matter axions would form a Bose–Einstein condensate by thermalisation because of gravitational self-interactions. 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. In 2014, a potential dibaryon was detected at the Jülich Research Center at about 2380 MeV. The center claimed that the measurements confirm results from 2011, via a more replicable method. The particle existed for 10−23 seconds and was named d*(2380). This particle is hypothesized to consist of three up and three
down quark The down quark (symbol: d) is a type of elementary particle, and a major constituent of matter. The down quark is the second-lightest of all quarks, and combines with other quarks to form composite particles called hadrons. Down quarks are most ...
s. 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 such hexaquarks with trapped electrons could behave like
dark matter In astronomy, dark matter is an invisible and hypothetical form of matter that does not interact with light or other electromagnetic radiation. Dark matter is implied by gravity, gravitational effects that cannot be explained by general relat ...
.


In fiction

* In the 2016 film '' Spectral'', the US military battles mysterious enemy creatures fashioned out of Bose–Einstein condensates. * In the 2003 novel '' Blind Lake'', scientists observe sentient life on a planet 51 light-years away using telescopes powered by Bose–Einstein condensate-based quantum computers.


See also

* 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 phenomena are processes showing Quantum mechanics, quantum behavior at the macroscopic scale, rather than at the Atom, atomic scale where quantum effects are prevalent. The best-known examples of macroscopic quantum phenomena ar ...
* Macroscopic quantum self-trapping * Slow light * Super-heavy atom *
Superconductivity Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ord ...
* Superfluid film * Superfluid helium-4 * Supersolid * Tachyon condensation * Timeline of low-temperature technology * Ultracold atom * Wiener sausage


References


Further reading

* * , * * * * * * . * * . * * * * * . * * * * * 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. * * Monique Combescot and Shiue-Yuan Shiau, "Excitons and Cooper Pairs: Two Composite Bosons in Many-Body Physics", Oxford University Press ().


External links


Bose–Einstein Condensation 2009 Conference
– 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 *
Bose–Einstein condensates at JILA

Atomcool at Rice University



Atom Optics at UQ

Einstein's manuscript on the Bose–Einstein condensate discovered at Leiden University

Bose–Einstein condensate on arxiv.org



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 The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical s ...
resource on BEC {{DEFAULTSORT:Bose-Einstein Condensate Albert Einstein Condensed matter physics Exotic matter Phases of matter Articles containing video clips