A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero (273.15 °C). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density, about onehundredthousandth the density of normal air, to ultralow temperatures.
This state was first predicted, generally, in 1924–1925 by Satyendra Nath Bose and Albert Einstein.
Satyendra Nath Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons), in which he derived Planck's quantum radiation law without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it in 1924.^{[1]} (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.^{[2]}). Einstein then extended Bose's ideas to matter in two other papers.^{[3]}^{[4]} The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now called bosons. Bosons, which include the photon as well as atoms such as helium4 (^{4}He), are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter.
In 1938 Fritz London proposed BEC as a mechanism for superfluidity in ^{4}He and superconductivity.^{[5]}^{[6]}
On June 5, 1995 the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST–JILA lab, in a gas of rubidium atoms cooled to 170 nanokelvins (nK).^{[7]} Shortly thereafter, Wolfgang Ketterle at MIT demonstrated important BEC properties. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics.^{[8]}
Many isotopes were soon condensed, then molecules, quasiparticles, and photons in 2010.^{[9]}
This transition to BEC occurs below a critical temperature, which for a uniform threedimensional gas consisting of noninteracting particles with no apparent internal degrees of freedom is given by:
where:
is  the critical temperature,  
is  the particle density,  
is  the mass per boson,  
is  the reduced Planck constant,  
is  the Boltzmann constant, and  
is  the Riemann zeta function; ^{[10]} 
Interactions shift the value and the corrections can be calculated by meanfield theory.
This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics.
Consider a collection of N noninteracting particles, which can each be in one of two quantum states, and . If the two states are equal in energy, each different configuration is equally likely.
If we can tell which particle is which, there are different configurations, since each particle can be in or independently. In almost all of the configurations, about half the particles are in and the other half in . The balance is a statistical effect: the number of configurations is largest when the particles are divided equally.
If the particles are indistinguishable, however, there are only N+1 different configurations. If there are K particles in state , there are N − K particles in state . Whether any particular particle is in state or in state cannot be determined, so each value of K determines a unique quantum state for the whole system.
Suppose now that the energy of state is slightly greater than the energy of state by an amount E. At temperature T, a particle will have a lesser probability to be in state by . In the distinguishable case, the particle distribution will be biased slightly towards state . But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the mostlikely outcome is that most of the particles will collapse into state .
In the distinguishable case, for large N, the fraction in state can be computed. It is the same as flipping a coin with probability proportional to p = exp(−E/T) to land tails.
In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:
For large N, the normalization constant C is (1 − p). The expected total number of particles not in the lowest energy state, in the limit that , is equal to . It does not grow when N is large; it just approaches a constant. This will be a negligible fraction of the total number of particles. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference.
Consider now a gas of particles, which can be in different momentum states labeled . If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. In this limit, the gas is classical. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. From this point on, any extra particle added will go into the ground state.
To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, p/(1 − p):
When the integral is evaluated with factors of k_{B} and ℏ restored by dimensional analysis, it gives the critical temperature formula of the preceding section. Therefore, this integral defines the critical temperature and particle number corresponding to the conditions of negligible chemical potential μ. In Bose–Einstein statistics 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 μ ≈ 0.
Nikolay Bogoliubov considered perturbations on the limit of dilute gas,^{[11]} finding a finite pressure at zero temperature and positive chemical potential. This leads to corrections for the ground state. The Bogoliubov state has pressure (T = 0): .
The original interacting system can be converted to a system of noninteracting particles with a dispersion law.
In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments.
This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate . For a system of this nature, is interpreted as the particle density, so the total number of atoms is
Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using mean field theory, the energy (E) associated with the state is:
Minimizing this energy with respect to infinitesimal variations in , and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a nonlinear Schrödinger equation):
where:
is the mass of the bosons,  
is the external potential,  
is representative of the interparticle interactions. 
In the case of zero external potential, the dispersion law of interacting Bose–Einsteincondensed particles is given by socalled Bogoliubov spectrum (for ):
The GrossPitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for . It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.
The GrossPitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and different numerical methods, such as splitstep CrankNicolson ^{[12]} and Fourier spectral ^{[13]} methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction ^{[14]}^{[15]} and longrange dipolar interaction ^{[16]} which can be freely used.
The Gross–Pitaevskii model of BEC is a physical approximation valid for certain classes of BECs. By construction, the GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact twobody type and also neglects anomalous contributions to selfenergy.^{[17]} These assumptions are suitable mostly for the dilute threedimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction acquires the terms containing higherorder powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially nonpolynomial. The examples where this could happen are the Bose–Fermi composite condensates,^{[18]}^{[19]}^{[20]}^{[21]} effectively lowerdimensional condensates,^{[22]} and dense condensates and superfluid clusters and droplets.^{[23]}
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.
The phenomena of superfluidity of a Bose gas and superconductivity of a stronglycorrelated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in helium4 and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model.
In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly believed that the superfluidity was due to partial Bose–Einstein condensation of the liquid. In fact, many properties of superfluid helium also appear in gaseous condensates created by Cornell, Wieman and Ketterle (see below). Superfluid helium4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium4. Note that helium3, a fermion, also enters a superfluid phase (at a much lower temperature) which can be explained by the formation of bosonic Cooper pairs of two atoms (see also fermionic condensate).
The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and coworkers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand rubidium87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude CohenTannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium23. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements.^{[24]}
A group led by Randall Hulet at Rice University announced a condensate of lithium 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 velocitydistribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: spatially confined atoms have a minimum width velocity distribution. This width is given by the curvature of the magnetic potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantummechanical effect and does not exist in the thermal distribution on the left. This graph served as the cover design for the 1999 textbook Thermal Physics by Ralph Baierlein.^{[26]}
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 Tl Cu Cl_{3},^{[27]} 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 ferromagnetic yttriumirongarnet thin film was seen even at room temperature,^{[28]}^{[29]} with optical pumping.
Excitons, electronhole pairs, were predicted to condense at low temperature and high density by Boer et al. in 1961. Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance. Fast optical exciton creation was used to form condensates in subkelvin Cu_{2}O in 2005 on.
Polariton condensation was firstly detected for excitonpolaritons in a quantum well microcavity kept at 5 K.^{[30]}
As in many other systems, vortices can exist in BECs. These can be created, for example, by 'stirring' the condensate with lasers, or rotating the confining trap. The vortex created will be a quantum vortex. These phenomena are allowed for by the nonlinear term in the GPE. As the vortices must have quantized angular momentum the wavefunction may have the form where and are as in the cylindrical coordinate system, and is the angular number. This is particularly likely for an axially symmetric (for instance, harmonic) confining potential, which is commonly used. The notion is easily generalized. To determine , the energy of must be minimized, according to the constraint . This is usually done computationally, however in a uniform medium the analytic form:
is  density far from the vortex,  
is  healing length of the condensate. 
demonstrates the correct behavior, and is a good approximation.
A singly charged vortex () is in the ground state, with its energy given by
where is the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary .)
For multiply charged vortices () the energy is approximated by
which is greater than that of singly charged vortices, indicating that these multiply charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes.
Closely related to the creation of vortices in BECs is the generation of socalled dark solitons in onedimensional BECs. These topological objects feature a phase gradient across their nodal plane, which stabilizes their shape even in propagation and interaction. Although solitons carry no charge and are thus prone to decay, relatively longlived dark solitons have been produced and studied extensively.^{[31]}
Experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist up to a critical atom number. Quench cooling the gas, they observed the condensate to grow, then subsequently collapse as the attraction overwhelmed the zeropoint energy of the confining potential, in a burst reminiscent of a supernova, with an explosion preceded by an implosion.
Further work on attractive condensates was performed in 2000 by the JILA team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturally attracting atoms of rubidium85 (having negative atom–atom scattering length). Through Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which rubidium bonds, making their Rb85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among wavelike condensate atoms.
When the JILA team raised the magnetic field strength further, the condensate suddenly reverted to attraction, imploded and shrank beyond detection, then exploded, expelling about twothirds of its 10,000 atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not seen in the cold remnant or expanding gas cloud.^{[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;^{[32]} 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.^{[33]}
Unsolved problem in physics: How do we rigorously prove the existence of Bose–Einstein condensates for general interacting systems? (more unsolved problems in physics)

Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile.^{[34]} 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. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality,^{[35]} the study of superfluidity and quantized vortices, the creation of bright matter wave solitons from Bose condensates confined to one dimension, and the slowing of light pulses to very low speeds using electromagnetically induced transparency.^{[36]} Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the laboratory. Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential. These have been used to explore the transition between a superfluid and a Mott insulator,^{[37]} and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas. Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow onedimensional optical lattice originally observed by Hallerdisplayauthors=et al^{[38]} has been explored via a tweaking of the primary optical lattice by a secondary weaker one.^{[39]} 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 ^{[40]} as well as excitatons of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken.^{[41]}^{[42]} Within this context, the conditions for order and chaos in the dynamics of a trapped Bose–Einstein condensate have been explored by the application of moving blue and reddetuned laser beams via the timedependent GrossPitaevskii equation.^{[43]}
Bose–Einstein condensates composed of a wide range of isotopes have been produced.^{[44]}
Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form bosonic compound particles (e.g. molecules or Cooper pairs). The first molecular condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate composed of Cooper pairs.^{[45]}
In 1999, Danish physicist Lene Hau led a team from Harvard University which slowed a beam of light to about 17 meters per second, using a superfluid.^{[46]} Hau and her associates have since made a group of condensate atoms recoil from a light pulse such that they recorded the light's phase and amplitude, recovered by a second nearby condensate, in what they term "slowlightmediated atomic matterwave amplification" using Bose–Einstein condensates: details are discussed in Nature.^{[47]}
Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision atom interferometry. The first demonstration of a BEC in weightlessness was achieved in 2008 at a drop tower in Bremen, Germany by a consortium of researchers led by Ernst M. Rasel from Leibniz University of Hanover.^{[48]} The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space^{[49]} and it is also the subject of two upcoming experiments on the International Space Station.^{[50]}^{[51]}
Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.^{[52]}
In 1970, BECs were proposed by Emmanuel David Tannenbaum for antistealth technology.^{[53]}
P. Sikivie and Q. Yang showed that cold dark matter axions form a Bose–Einstein condensate by thermalisation because of gravitational selfinteractions.^{[54]} 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.
The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2012, using ultralow temperatures of 10^{−7} K or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of alkali metal, alkaline earth metal, and lanthanide atoms (^{7}Li, ^{23}Na, ^{39}K, ^{41}K, ^{85}Rb, ^{87}Rb, ^{133}Cs, ^{52}Cr, ^{40}Ca, ^{84}Sr, ^{86}Sr, ^{88}Sr, ^{174}Yb, ^{164}Dy, and ^{168}Er). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'.^{[55]} In contrast, the superfluid state of ^{4}He below 2.17 K is not a good example, because the interaction between the atoms is too strong. Only 8% of atoms are in the ground state near absolute zero, rather than the 100% of a true condensate.^{[56]}
The bosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have halfinteger total spin. It arises from a subtle interplay of electronic and nuclear spins: at ultralow temperatures and corresponding excitation energies, the halfinteger total spin of the electronic shell and halfinteger total spin of the nucleus are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer lower value. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values.
Analog models of gravity are attempts to model various phenomena of general relativity (e.g., black holes or cosmological geometries) using other physical systems such as acoustics in a moving fluid, superfluid helium, or Bose–Einstein condensate; gravity waves in water; and propagation of electromagnetic waves in a dielectric medium. These analogs (or analogies) serve to provide new ways of looking at problems, permit ideas from other realms of science to be applied, and may create opportunities for practical experiments within the analogue that can be applied back to the source phenomena.
Atom laserAn atom laser is a coherent state of propagating atoms. They are created out of a Bose–Einstein condensate of atoms that are output coupled using various techniques. Much like an optical laser, an atom laser is a coherent beam that behaves like a wave. There has been some argument that the term "atom laser" is misleading. Indeed, "laser" stands for "Light Amplification by Stimulated Emission of Radiation" which is not particularly related to the physical object called an atom laser, and perhaps describes more accurately the Bose–Einstein condensate (BEC).
The terminology most widely used in the community today is to distinguish between the BEC, typically obtained by evaporation in a conservative trap, from the atom laser itself, which is a propagating atomic wave obtained by extraction from a previously realized BEC. Some ongoing experimental research tries to obtain directly an atom laser from a "hot" beam of atoms without making a trapped BEC first.[1]
BipolaronIn physics, a bipolaron is a type of quasiparticle consisting of two polarons. In organic chemistry, it is a molecule or a part of a macromolecular chain containing two positive charges in a conjugated system.
Bose gasAn ideal Bose gas is a quantummechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for a photon gas, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose–Einstein condensate.
BosenovaA bosenova or bose supernova is a very small, supernovalike explosion, which can be induced in a Bose–Einstein condensate (BEC) by changing the external magnetic field, so that the "selfscattering" interaction transitions from repulsive to attractive due to the Feshbach resonance, causing the BEC to "collapse and bounce" or "rebound."Although the total energy of the explosion is very small, the "collapse and bounce" scenario qualitatively resembles a condensed matter version of a corecollapse supernova, hence the term bosenova. The nomenclature is a play of words on the Brazilian music style, bossa nova.
Carl WiemanCarl Edwin Wieman (born March 26, 1951) is an American physicist and educationist at Stanford University. In 1995, while at the University of Colorado Boulder, he and Eric Allin Cornell produced the first true Bose–Einstein condensate (BEC) and, in 2001, they and Wolfgang Ketterle (for further BEC studies) were awarded the Nobel Prize in Physics. Wieman currently holds a joint appointment as Professor of Physics and Professor in the Stanford Graduate School of Education, as well as the DRC Professor in the Stanford University School of Engineering.
Eric Allin CornellEric Allin Cornell (born December 19, 1961) is an American physicist who, along with Carl E. Wieman, was able to synthesize the first Bose–Einstein condensate in 1995. For their efforts, Cornell, Wieman, and Wolfgang Ketterle shared the Nobel Prize in Physics in 2001.
Fermionic condensateA fermionic condensate is a superfluid phase formed by fermionic particles at low temperatures. It is closely related to the Bose–Einstein condensate, a superfluid phase formed by bosonic atoms under similar conditions. The earliest recognized fermionic condensate described the state of electrons in a superconductor; the physics of other examples including recent work with fermionic atoms is analogous. The first atomic fermionic condensate was created by a team led by Deborah S. Jin in 2003.
Gross–Pitaevskii equationThe Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii ) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.
In the Hartree–Fock approximation the total wavefunction of the system of bosons is taken as a product of singleparticle functions ,
where is the coordinate of the th boson.
The pseudopotential model Hamiltonian of the system is given as
where is the mass of the boson, is the external potential, is the bosonboson scattering length, and is the Dirac deltafunction.
If the singleparticle wavefunction satisfies the Gross–Pitaevski equation,
the total wavefunction minimizes the expectation value of the model Hamiltonian under normalization condition
It is a model equation for the singleparticle wavefunction in a Bose–Einstein condensate. It is similar in form to the Ginzburg–Landau equation and is sometimes referred to as a nonlinear Schrödinger equation.
A Bose–Einstein condensate (BEC) is a gas of bosons that are in the same quantum state, and thus can be described by the same wavefunction. A free quantum particle is described by a singleparticle Schrödinger equation. Interaction between particles in a real gas is taken into account by a pertinent manybody Schrödinger equation. If the average spacing between the particles in a gas is greater than the scattering length (that is, in the socalled dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. The nonlinearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles. This is made evident by setting the coupling constant of interaction in the Gross–Pitaevskii equation to zero (see the following section): thereby, the singleparticle Schrödinger equation describing a particle inside a trapping potential is recovered.
Mixed dark matterMixed dark matter (MDM) is a theory of dark matter (DM) proposed during the late 1990s.Mixed dark matter is also called hot + cold dark matter. The most abundant form of dark matter is cold dark matter, almost one fourth of the energy contents of the Universe. Neutrinos are the only known particles whose BigBang thermal relic should compose at least a fraction of Hot dark matter (HDM), albeit other candidates are speculated to exist.. In the early 1990s, the power spectrum of fluctuations in the galaxy clustering did not agree entirely with the predictions for a standard cosmology built around pure cold DM. Mixed dark matter with a composition of about 80% cold and 20% hot (neutrinos) was investigated and found to agree better with observations. This large amount of HDM was made obsolete by the discovery in 1998 of the acceleration of universal expansion, which eventually led to the dark energy + dark matter paradigm of this decade.The cosmological effects of cold DM are almost opposite to the hot DM effects. Given that cold DM promotes the growth of large scale structures, it is often believed to be composed of Weakly interacting massive particles (WIMPs). Conversely hot DM suffers of freestreaming for most of the history of the Universe, washingout the formation of small scales. In other words, the mass of hot DM particles is too small to produce the observed gravitationally bounded objects in the Universe. For that reason, the hot DM abundance is constrained by Cosmology to less than one percent of the Universe contents.
The Mixed Dark Matter scenario recovered relevance when DM was proposed to be a thermal relic of a Bose–Einstein condensate made of very light bosonic particles, as light as neutrinos or even lighter like the Axion. This cosmological model predicts that cold DM is made of a large number of condensed particles, while a small fraction of these particles resides in excited energetic states contributing to hot DM.
Optical black holeAn optical black hole is a phenomenon in which slow light is passed through a Bose–Einstein condensate that is itself spinning faster than the local speed of light within to create a vortex capable of trapping the light behind an event horizon just as a gravitational black hole would.Unlike other black hole analogs such as a sonic black hole in a Bose–Einstein condensate, a slow light black hole analog is not expected to mimic the quantum effects of a black hole, and thus not emit Hawking radiation. It does, however, mimic the classical properties of a gravitational black hole, making it potentially useful in studying other properties of black holes. More recently, some physicists have developed a fiber optic based system which they believe will emit Hawking radiation.
Peter EngelsProfessor Peter Engels is a physicist at Washington State University who conducts research in the field of ultracold atomic gases. His group at WSU performs a variety of experiments involving quantum hydrodynamics, spinorbit coupling (See Spinorbit interaction), soliton formation, condensed matter physics, and more using Rb87 (bosonic) and K40 (fermionic). Recently, in collaboration with the theorists Prof. Michael Forbes, Yongping Zhang, and Thomas Busch, his team published research demonstrating negative mass hydrodynamics in a spinorbit coupled BoseEinstein condensate.
Randall G. HuletRandall Gardner Hulet (born April 27, 1956 in Walnut Creek, California) is an American physicist.
Hulet studied at Stanford University with a bachelor's degree in 1978 and received his doctorate in 1984 at the Massachusetts Institute of Technology under Daniel Kleppner. He then joined the National Institute of Standards and Technology (NIST) in Boulder with David Wineland. In 1987 he became assistant professor, 1992 associate professor and 1996 professor of physics at Rice University, where he was appointed Fayez Sarofim Professor in 2000.
He is a pioneer of experiments with ultracold atoms and BoseEinstein condensates (BEC). He is known for the first realization of a BoseEinstein condensate in an atomic gas with attractive interaction (gas of lithium atoms), where the formation of the BoseEinstein condensate competes with the usual condensation due to the attractive interaction. With enough gas atoms the gas collapses (condensed) and Hulet also succeeded in the first observation of formation and collapse of a BoseEinstein condensate.
Hulet also achieved the first observation of polarized degenerate Fermi gas (and mixtures of Bose and Fermi gases, corresponding to bosonic and fermionic isotopes of lithium) and matter wave solitons in BoseEinstein condensates. His group investigates optical grids of ultracold atoms as a model of systems of solidstate physics, including hightemperature superconductors (HTS) via a Hubbard model of ultracold atoms with two spin components (where they could demonstrate antiferromagnetic properties as in HTS) and p Wave superfluid as possible topological superfluid (with Vincent Liu). In addition, his ultracold atoms group investigates smallparticle systems of bosons with resonant interactions.
In 2017 he received the Herbert Walther Prize. He received in 2011 the Willis E. Lamb Award for Laser Science and Quantum Optics. 1995 the I. I. Rabi Prize of the American Physical Society and in 1989 the Presidential Young Investigator Award of the National Science Foundation. He is a Fellow of the American Association for the Advancement of Science and the American Physical Society. Hulet is an honorary doctor of the University of Utrecht.
Rydberg polaronA Rydberg polaron is an exotic state of matter, created at low temperatures, in which a very large atom contains other ordinary atoms in the space between the nucleus and the electrons. For the formation of this atom, scientists had to combine two fields of atomic physics: BoseEinstein condensates and Rydberg atoms. Rydberg atoms are formed by exciting a single atom into a highenergy state, in which the electron is very far from the nucleus. BoseEinstein condensates are a state of matter that is produced at temperatures close to absolute zero.
Polarons are induced by using a laser to excite Rydberg atoms contained as impurities in a BoseEinstein condensate. In those Rydberg atoms, the average distance between the electron and its nucleus can be as large as several hundred nanometres, which is more than a thousand times the radius of a hydrogen atom. Under that circumstances, the distance between the nucleus and the electron of the excited Rydberg atoms is higher than the average distance of the atoms of the condensate. As a result, some atoms lie inside the orbit of the Rydberg atom's electron.
As the atoms don't have an electric charge, they only produce a minimal force on the electron. However, the electron is slightly scattered at the neutral atoms, without even leaving its orbit, and the weak bond that is generated between the Rydberg atom and the atoms inside of it, tying them together, is known as the Rydberg Polaron. The new state of matter was predicted by theorists at Harvard University in 2016 and confirmed in 2018 by spectroscopy in an experiment using a strontium BoseEinstein condensate. Theoretically, up to 170 ordinary strontium atoms could fit closely inside the new orbital of the Rydberg atom, depending on the radius of the Rydberg atom and the density of the BoseEinstein condensate. The theoretical work for the experiment was performed by theorists at Vienna University of Technology and Harvard University, while the actual experiment and observation took place at Rice University in Houston, Texas.
Satyendra Nath BoseSatyendra Nath Bose, (Bengali: সত্যেন্দ্রনাথ বসু Sôtyendronath Bosu, IPA: [ʃotːendronatʰ boʃu]; 1 January 1894 – 4 February 1974) was an Indian physicist specialising in theoretical physics. He is best known for his work on quantum mechanics in the early 1920s, providing the foundation for Bose–Einstein statistics and the theory of the Bose–Einstein condensate. A Fellow of the Royal Society, he was awarded India's second highest civilian award, the Padma Vibhushan in 1954 by the Government of India.The class of particles that obey Bose–Einstein statistics, bosons, was named after Bose by Paul Dirac.A selftaught scholar and a polymath, he had a wide range of interests in varied fields including physics, mathematics, chemistry, biology, mineralogy, philosophy, arts, literature, and music. He served on many research and development committees in sovereign India.
Sonic black holeA sonic black hole, sometimes called a dumb hole, is a phenomenon in which phonons (sound perturbations) are unable to escape from a fluid that is flowing more quickly than the local speed of sound. They are called sonic, or acoustic, black holes because these trapped phonons are analogous to light in astrophysical (gravitational) black holes. Physicists are interested in them because they have many properties similar to astrophysical black holes and, in particular, emit a phononic version of Hawking radiation. The border of a sonic black hole, at which the flow speed changes from being greater than the speed of sound to less than the speed of sound, is called the event horizon. At this point the frequency of phonons approaches zero.Sonic black holes are possible because phonons in perfect fluids exhibit the same properties of motion as fields, such as gravity, in space and time. For this reason, a system in which a sonic black hole can be created is called a gravity analogue. Nearly any fluid can be used to create an acoustic event horizon, but the viscosity of most fluids creates random motion that makes features like Hawking radiation nearly impossible to detect. The complexity of such a system would make it very difficult to gain any knowledge about such features even if they could be detected. Many nearly perfect fluids have been suggested for use in creating sonic black holes, such as superfluid helium, one–dimensional degenerate Fermi gases, and Bose–Einstein condensate. Gravity analogues other than phonons in a fluid, such as slow light and a system of ions, have also been proposed for studying black hole analogues. The fact that so many systems mimic gravity is sometimes used as evidence for the theory of emergent gravity, which could help reconcile relativity and quantum mechanics.Acoustic black holes were first theorized to be useful by William Unruh in 1981. However, the first black hole analogue was not created in a laboratory until 2009. It was created in a rubidium Bose–Einstein condensate using a technique called density inversion. This technique creates a flow by repelling the condensate with a potential minimum. The surface gravity and temperature of the sonic black hole were measured, but no attempt was made to detect Hawking radiation. However, the scientists who created it predicted that the experiment was suitable for detection and suggested a method by which it might be done by lasing the phonons. In 2014, selfamplifying Hawking radiation was observed in an analogue blackhole laser by the same researchers.
State of matterIn 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. Many other states are known to exist, such as glass or liquid crystal, and some only exist under extreme conditions, such as Bose–Einstein condensates, neutrondegenerate matter, and quarkgluon plasma, which only occur, respectively, in situations of extreme cold, extreme density, and extremely highenergy. Some other states are believed to be possible but remain theoretical for now. For a complete list of all exotic states of matter, see the list of states of matter.
Historically, the distinction is made based on qualitative differences in properties. Matter in the solid state maintains a fixed volume and shape, with component particles (atoms, molecules or ions) close together and fixed into place. Matter in the liquid state maintains a fixed volume, but has a variable shape that adapts to fit its container. Its particles are still close together but move freely. Matter in the gaseous state has both variable volume and shape, adapting both to fit its container. Its particles are neither close together nor fixed in place. Matter in the plasma state has variable volume and shape, but as well as neutral atoms, it contains a significant number of ions and electrons, both of which can move around freely.
The term phase is sometimes used as a synonym for state of matter, but a system can contain several immiscible phases of the same state of matter.
Strong confinement limitIn physics, the strong confinement limit, or "festina lente" limit, is a mode of an atom laser in which the frequency of emission of the Bose–Einstein condensate is less than the confinement frequency of the trap.
Wiener sausageIn the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by M. D. Donsker and S. R. Srinivasa Varadhan (1975) because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" means "Viennese" in German.
The Wiener sausage is one of the simplest nonMarkovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by Frank Spitzer (1964), and it was used by Mark Kac and Joaquin Mazdak Luttinger (1973, 1974) to explain results of a Bose–Einstein condensate, with proofs published by M. D. Donsker and S. R. Srinivasa Varadhan (1975).
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