Bose–Einstein condensate

Bose–Einstein condensate

Overview

A Bose–Einstein condensate (BEC) is a state of matter
State of matter
States of matter are the distinct forms that different phases of matter take on. Solid, liquid and gas are the most common states of matter on Earth. However, much of the baryonic matter of the universe is in the form of hot plasma, both as rarefied interstellar medium and as dense...

 of a dilute gas of weakly interacting boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s confined in an external potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

 and cooled to temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

s very near absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

 ( or ). Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

 effects become apparent on a macroscopic scale.
Discussion
Ask a question about 'Bose–Einstein condensate'
Start a new discussion about 'Bose–Einstein condensate'
Answer questions from other users
Full Discussion Forum
 
Unanswered Questions
Encyclopedia

A Bose–Einstein condensate (BEC) is a state of matter
State of matter
States of matter are the distinct forms that different phases of matter take on. Solid, liquid and gas are the most common states of matter on Earth. However, much of the baryonic matter of the universe is in the form of hot plasma, both as rarefied interstellar medium and as dense...

 of a dilute gas of weakly interacting boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s confined in an external potential
Potential
*In linguistics, the potential mood*The mathematical study of potentials is known as potential theory; it is the study of harmonic functions on manifolds...

 and cooled to temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

s very near absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

 ( or ). Under such conditions, a large fraction of the bosons occupy the lowest quantum state of the external potential, at which point quantum
Quantum
In physics, a quantum is the minimum amount of any physical entity involved in an interaction. Behind this, one finds the fundamental notion that a physical property may be "quantized," referred to as "the hypothesis of quantization". This means that the magnitude can take on only certain discrete...

 effects become apparent on a macroscopic scale.

This state of matter was first predicted by Satyendra Nath Bose
Satyendra Nath Bose
Satyendra Nath Bose FRS was an Indian mathematician and physicist noted for his collaboration with Albert Einstein in developing a theory regarding the gaslike qualities of electromagnetic radiation. He is best known for his work on quantum mechanics in the early 1920s, providing the foundation...

 and Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s). Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik
Zeitschrift für Physik
The European Physical Journal is a joint publication of EDP Sciences, Springer Science+Business Media, and the Società Italiana di Fisica...

, which published it. Einstein then extended Bose's ideas to material particles (or matter) in two other papers.

Seventy years later, the first gaseous condensate
Condensate
Condensate may refer to:* The liquid phase produced by the condensation of steam or any other gas* The product of a chemical condensation reaction, other than water* Natural gas condensate, in the natural gas industry- Quantum physics :...

 was produced by Eric Cornell
Eric Allin Cornell
Eric Allin Cornell is an American physicist who, along with Carl E. Wieman, was able to synthesize the first Bose–Einstein condensate in 1995...

 and Carl Wieman
Carl Wieman
Carl Edwin Wieman is an American physicist at the University of British Columbia and recipient of the Nobel Prize in Physics for the production, in 1995 with Eric Allin Cornell, of the first true Bose–Einstein condensate.-Biography:...

 in 1995 at the University of Colorado at Boulder
University of Colorado at Boulder
The University of Colorado Boulder is a public research university located in Boulder, Colorado...

 NIST
National Institute of Standards and Technology
The National Institute of Standards and Technology , known between 1901 and 1988 as the National Bureau of Standards , is a measurement standards laboratory, otherwise known as a National Metrological Institute , which is a non-regulatory agency of the United States Department of Commerce...

-JILA lab, using a gas of rubidium
Rubidium
Rubidium is a chemical element with the symbol Rb and atomic number 37. Rubidium is a soft, silvery-white metallic element of the alkali metal group. Its atomic mass is 85.4678. Elemental rubidium is highly reactive, with properties similar to those of other elements in group 1, such as very rapid...

 atoms cooled to 170 nanokelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...

 (nK) . For their achievements Cornell, Wieman, and Wolfgang Ketterle
Wolfgang Ketterle
Wolfgang Ketterle is a German physicist and professor of physics at the Massachusetts Institute of Technology . His research has focused on experiments that trap and cool atoms to temperatures close to absolute zero, and he led one of the first groups to realize Bose-Einstein condensation in these...

 at MIT received the 2001 Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

. In November 2010 the first photon BEC was observed.

The slowing of atoms by the use of cooling apparatus produced a singular quantum state known as a Bose condensate or Bose–Einstein condensate. This phenomenon was predicted in 1925 by generalizing Satyendra Nath Bose's work on the statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

 of (massless) photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

s to (massive) atoms. (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University
Leiden University
Leiden University , located in the city of Leiden, is the oldest university in the Netherlands. The university was founded in 1575 by William, Prince of Orange, leader of the Dutch Revolt in the Eighty Years' War. The royal Dutch House of Orange-Nassau and Leiden University still have a close...

 in 2005.) The result of the efforts of Bose and Einstein is the concept of a Bose gas
Bose gas
An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integer value of spin, and obey Bose–Einstein statistics...

, governed by Bose–Einstein statistics
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...

, which describes the statistical distribution of identical particles
Identical particles
Identical particles, or indistinguishable particles, are particles that cannot be distinguished from one another, even in principle. Species of identical particles include elementary particles such as electrons, and, with some clauses, composite particles such as atoms and molecules.There are two...

 with integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

 spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

, are allowed to share quantum states with each other. Einstein
demonstrated 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.

This transition occurs below a critical temperature, which for a uniform three-dimensional
Three-dimensional space
Three-dimensional space is a geometric 3-parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...

 gas consisting of non-interacting 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
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...

,
 is  the Boltzmann constant, and
 is  the Riemann zeta function;


Einstein's argument


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. If all these states are equally likely, there is no statistical spreading out; it is just as likely for all the particles to sit in as for the particles to be split half and half.

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 exp(−E/T). In the distinguishable case, the particle distribution will be biased slightly towards state and the distribution will be slightly different from half and half. But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the most likely outcome is that most of the particles will collapse into state .

In the distinguishable case, for 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. The probability to land heads is 1/(1 + p), which is a smooth function of p, and thus of the energy.

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 the factors of kB and restored by dimensional analysis, it gives the critical temperature formula of the preceding section. Therefore, this integral defines the critical temperature and particle number corresponding to the conditions of negligible chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...

. In Bose–Einstein statistics
Bose–Einstein statistics
In statistical mechanics, Bose–Einstein statistics determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.-Concept:...

 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 consequently be approximated for most energy or momentum states as μ ≈ 0.

Gross–Pitaevskii equation


The state of the BEC can be described by the wavefunction of the condensate . For a system of this nature
Schrödinger field
In quantum mechanics and quantum field theory, a Schrödinger field, named after Erwin Schrödinger, is a quantum field which obeys the Schrödinger equation...

, 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
Mean field theory
Mean field theory is a method to analyse physical systems with multiple bodies. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases . The n-body system is replaced by a 1-body problem with a chosen good external field...

, the energy (E) associated with the state is:


Minimizing this energy with respect to infinitesimal variations in , and holding the number of atoms constant, yields the Gross-Pitaevski equation (GPE) (also a non-linear Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

):


where:

 is the mass of the bosons,
 is the external potential,
 is representative of the inter-particle interactions.



The GPE provides a good description of the behavior of BEC's and is thus often applied for theoretical analysis.

Models beyond Gross–Pitaevskii


The Gross–Pitaevskii model of BEC is the physical approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

 valid for certain classes of BEC's only.
By construction, GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and
also it neglects anomalous contributions to self-energy
Self-energy
In theoretical physics and quantum field theory a particle's self-energy \Sigma represents the contribution to the particle's energy, or effective mass, due to interactions between the particle and the system it is part of...

. These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction
Wavefunction
Not to be confused with the related concept of the Wave equationA wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of...

 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
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...

 clusters and droplets.

Discovery



In 1938, Pyotr Kapitsa, John Allen and Don Misener
Don Misener
Don Misener was a physicist. Along with Pyotr Leonidovich Kapitsa and John F. Allen, Misener discovered the superfluid phase of matter in 1937....

 discovered that helium-4
Helium-4
Helium-4 is a non-radioactive isotope of helium. It is by far the most abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on earth. Its nucleus is the same as an alpha particle, consisting of two protons and two neutrons. Alpha decay of heavy...

 became a new kind of fluid, now known as a superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...

, at temperatures less than 2.17 K (the lambda point
Lambda point
The Lambda point is the temperature below which normal fluid helium transitions to superfluid helium II. More precisely, there is a lower lambda point at 2.172 K, 0.0497 atm, and an upper one at 1.76 K, 29.8 atm....

). Superfluid helium has many unusual properties, including zero viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

 (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly realized that the superfluidity was due to partial Bose–Einstein condensation of the liquid. In fact, many of the properties of superfluid helium also appear in the gaseous Bose–Einstein 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
Helium-3
Helium-3 is a light, non-radioactive isotope of helium with two protons and one neutron. It is rare on Earth, and is sought for use in nuclear fusion research...

, consisting of fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

s instead of boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

s, also enters a superfluid
Superfluid
Superfluidity is a state of matter in which the matter behaves like a fluid without viscosity and with extremely high thermal conductivity. The substance, which appears to be a normal liquid, will flow without friction past any surface, which allows it to continue to circulate over obstructions and...

 phase at low temperature, which can be explained by the formation of bosonic Cooper pairs of two atoms each (see also fermionic condensate
Fermionic condensate
A 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. Unlike the Bose–Einstein condensates, fermionic condensates are formed using...

).

The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman
Carl Wieman
Carl Edwin Wieman is an American physicist at the University of British Columbia and recipient of the Nobel Prize in Physics for the production, in 1995 with Eric Allin Cornell, of the first true Bose–Einstein condensate.-Biography:...

, and co-workers at JILA on June 5, 1995. They did this by cooling a dilute vapor consisting of approximately two thousand rubidium-87
Rubidium
Rubidium is a chemical element with the symbol Rb and atomic number 37. Rubidium is a soft, silvery-white metallic element of the alkali metal group. Its atomic mass is 85.4678. Elemental rubidium is highly reactive, with properties similar to those of other elements in group 1, such as very rapid...

 atoms to below 170 nK using a combination of laser cooling
Laser cooling
Laser cooling refers to the number of techniques in which atomic and molecular samples are cooled through the interaction with one or more laser light fields...

 (a technique that won its inventors Steven Chu
Steven Chu
Steven Chu is an American physicist and the 12th United States Secretary of Energy. Chu is known for his research at Bell Labs in cooling and trapping of atoms with laser light, which won him the Nobel Prize in Physics in 1997, along with his scientific colleagues Claude Cohen-Tannoudji and...

, Claude Cohen-Tannoudji
Claude Cohen-Tannoudji
Claude Cohen-Tannoudji is a French physicist and Nobel Laureate. He shared the 1997 Nobel Prize in Physics with Steven Chu and William Daniel Phillips for research in methods of laser cooling and trapping atoms...

, and William D. Phillips the 1997 Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

) and magnetic evaporative cooling
Magnetic evaporative cooling
Magnetic evaporative cooling is a technique for lowering the temperature of a group of atoms. The process uses a magnetic field to put atoms in a magnetic trap, a flask-shaped magnetic field...

. About four months later, an independent effort led by Wolfgang Ketterle
Wolfgang Ketterle
Wolfgang Ketterle is a German physicist and professor of physics at the Massachusetts Institute of Technology . His research has focused on experiments that trap and cool atoms to temperatures close to absolute zero, and he led one of the first groups to realize Bose-Einstein condensation in these...

 at MIT
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

 created a condensate made of sodium-23
Sodium
Sodium is a chemical element with the symbol Na and atomic number 11. It is a soft, silvery-white, highly reactive metal and is a member of the alkali metals; its only stable isotope is 23Na. It is an abundant element that exists in numerous minerals, most commonly as sodium chloride...

. Ketterle's condensate had about a hundred times more atoms, allowing him to obtain several important results such as the observation of quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics
Nobel Prize in Physics
The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in Chemistry, Nobel Prize in Literature, Nobel Peace Prize, and...

 for their achievements. A group led by Randall Hulet at Rice University announced the creation of a condensate of lithium
Lithium
Lithium is a soft, silver-white metal that belongs to the alkali metal group of chemical elements. It is represented by the symbol Li, and it has the atomic number 3. Under standard conditions it is the lightest metal and the least dense solid element. Like all alkali metals, lithium is highly...

 atoms only one month following the JILA work. Lithium has attractive interactions which causes the condensate to be unstable and to collapse for all but a few atoms. Hulet and co-workers showed in a subsequent experiment that the condensate could be stabilized by the quantum pressure from trap confinement for up to about 1000 atoms.

The Bose–Einstein condensation also applies to quasiparticles in solids. A magnon
Magnon
A magnon is a collective excitation of the electrons' spin structure in a crystal lattice. In contrast, a phonon is a collective excitation of the crystal lattice atoms or ions. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. As a...

 in an antiferromagnet
Antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usuallyrelated to the spins of electrons, align in a regular pattern with neighboring spins pointing in opposite directions. This is, like ferromagnetism and ferrimagnetism, a manifestation of ordered magnetism...

 carries spin 1 and thus obeys Bose–Einstein statistics. The density of magnons is controlled by an external magnetic field, which plays the role of the magnon chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...

. This technique provides access to a wide range of boson densities from the limit of a dilute Bose gas to that of a strongly interacting Bose liquid. A magnetic ordering observed at the point of condensation is the analog of superfluidity. In 1999 Bose condensation of magnons was demonstrated in the antiferromagnet Tl
Thallium
Thallium is a chemical element with the symbol Tl and atomic number 81. This soft gray poor metal resembles tin but discolors when exposed to air. The two chemists William Crookes and Claude-Auguste Lamy discovered thallium independently in 1861 by the newly developed method of flame spectroscopy...

Cu
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...

Cl
Chlorine
Chlorine is the chemical element with atomic number 17 and symbol Cl. It is the second lightest halogen, found in the periodic table in group 17. The element forms diatomic molecules under standard conditions, called dichlorine...

3. The condensation was observed at temperatures as large as 14 K. Such a high transition temperature (relative to that of atomic gases) is due to the greater density achievable with magnons and the smaller mass (roughly equal to the mass of an electron). In 2006, condensation of magnons in ferromagnets
Ferromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...

 was even shown at room temperature, where the authors used pumping techniques.

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
Rubidium
Rubidium is a chemical element with the symbol Rb and atomic number 37. Rubidium is a soft, silvery-white metallic element of the alkali metal group. Its atomic mass is 85.4678. Elemental rubidium is highly reactive, with properties similar to those of other elements in group 1, such as very rapid...

 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
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...

: since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width. This width is given by the curvature of the magnetic trapping potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...

 of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left. This famous graph served as the cover design for 1999 textbook Thermal Physics by Ralph Baierlein.

Vortices



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
Quantum vortex
In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. Superfluids and superconductors are states of matter without friction. They exist only at very low temperatures. The existence of these quantum vortices was independently predicted by Richard Feynman...

. These phenomena are allowed for by the non-linear term in the GPE. As the vortices must have quantized angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

 the wavefunction may have the form where and are as in the cylindrical coordinate system
Cylindrical coordinate system
A cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...

, and is the angular number. This is particularly likely for an axially symmetric (for instance, harmonic) confining potential, which is commonly used. The notion is easily generalized. To determine , the energy of must be minimized, according to the constraint . This is usually done computationally, however in a uniform medium the analytic form


where:

 is  density far from the vortex,
 is  healing length of the condensate.



demonstrates the correct behavior, and is a good approximation.

A singly charged vortex () is in the ground state, with its energy given by


where:

 is  the farthest distance from the vortex considered.


(To obtain an energy which is well defined it is necessary to include this boundary .)

For multiply charged vortices () the energy is approximated by


which is greater than that of singly charged vortices, indicating that these multiply charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes.

Closely related to the creation of vortices in BECs is the generation of so-called dark soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

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


The experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist, but only up to a certain critical atom number. Beyond this critical number, the attraction overwhelmed the zero-point energy of the harmonic confining potential, causing the condensate to collapse in a burst reminiscent of a supernova explosion where an explosion is preceded by an implosion. By quench cooling the gas of lithium atoms, they observed the condensate to first grow, and subsequently collapse when the critical number was exceeded.

Further experimentation on attractive condensates was performed in 2000 by the JILA team, consisting of Cornell, Wieman, and coworkers. They originally used rubidium
Rubidium
Rubidium is a chemical element with the symbol Rb and atomic number 37. Rubidium is a soft, silvery-white metallic element of the alkali metal group. Its atomic mass is 85.4678. Elemental rubidium is highly reactive, with properties similar to those of other elements in group 1, such as very rapid...

-87, an isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...

 whose atoms naturally repel each other, making a more stable condensate. Their instrumentation now had better control over the condensate so experimentation was made on naturally attracting atoms of another rubidium isotope, rubidium-85 (having negative atom-atom scattering length). Through a process called Feshbach resonance
Feshbach resonance
In physics, Feshbach resonance, named after Herman Feshbach, is a resonance of a many-body system in which a bound state is achieved if the coupling between an internal degree of freedom and the reaction coordinates which lead to dissociation vanish...

 involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which the rubidium atoms bond into molecules, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among condensate atoms which behave as waves.

When the JILA team raised the magnetic field strength still further, the condensate suddenly reverted back to attraction, imploded and shrank beyond detection, and then exploded, expelling off about two-thirds of its 10,000 or so atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not being seen either in the cold remnant or the expanding gas cloud. Carl Wieman
Carl Wieman
Carl Edwin Wieman is an American physicist at the University of British Columbia and recipient of the Nobel Prize in Physics for the production, in 1995 with Eric Allin Cornell, of the first true Bose–Einstein condensate.-Biography:...

 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. The atoms that seem to have disappeared almost certainly still exist in some form, just not in a form that could be accounted for in that experiment. Most likely they formed molecules consisting of two bonded rubidium atoms. The energy gained by making this transition imparts a velocity sufficient for them to leave the trap without being detected.

Current research


Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile. The slightest interaction with the outside world can be enough to warm them past the condensation threshold, eliminating their interesting properties and forming a normal gas. It is likely to be some time before any practical applications are developed.

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 explosion in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave-particle duality, the study of superfluidity and quantized vortices
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

, the creation of bright matter wave soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...

s from Bose condensates confined to one dimension, and the slowing of light
Slow light
Slow light is the propagation of an optical pulse or other modulation of an optical carrier at a very low group velocity. Slow light occurs when a propagating pulse is substantially slowed down by the interaction with the medium in which the propagation take place.Researchers at the Rowland...

 pulses to very low speeds using electromagnetically induced transparency
Electromagnetically induced transparency
Electromagnetically induced transparency is a coherent optical nonlinearity which renders a medium transparent over a narrow spectral range within an absorption line...

. 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 lab. Experimentalists have also realized "optical lattice
Optical lattice
An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate in the locations of potential minima...

s", where the interference pattern from overlapping lasers provides a periodic potential for the condensate. These have been used to explore the transition between a superfluid and a Mott insulator
Mott insulator
Mott insulators are a class of materials that should conduct electricity under conventional band theories, but are insulators when measured...

, and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks-Girardeau gas
Tonks-Girardeau gas
In physics, a Tonks–Girardeau gas is a quantum degenerate Bose Gas in which the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. It is named after physicists Marvin D. Girardeau and Lewi Tonks...

.

Bose–Einstein condensates composed of a wide range of isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...

s have been produced.

Related experiments in cooling fermions rather than bosons to extremely low temperatures have created degenerate
Degenerate matter
Degenerate matter is matter that has such extraordinarily high density that the dominant contribution to its pressure is attributable to the Pauli exclusion principle. The pressure maintained by a body of degenerate matter is called the degeneracy pressure, and arises because the Pauli principle...

 gases, where the atoms do not congregate in a single state due to the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form compound particles (e.g. molecules or Cooper pairs
BCS theory
BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...

) that are bosons. The first molecular
Molecule
A molecule is an electrically neutral group of at least two atoms held together by covalent chemical bonds. Molecules are distinguished from ions by their electrical charge...

 Bose–Einstein condensates were created in November 2003 by the groups of Rudolf Grimm
Rudolf Grimm
Rudolf Grimm is an experimental physicist from Austria. His work centres on ultracold atoms and quantum gases. He was the first scientist worldwide who, with his team, succeeded in realizing a Bose-Einstein condensation from molecules.-Career:Grimm graduated in physics from the University of...

 at the University of Innsbruck, Deborah S. Jin
Deborah S. Jin
Deborah S. Jin is a physicist with the National Institute of Standards and Technology ; Professor Adjoint, Department of Physics at the University of Colorado; a fellow of the JILA, a NIST joint laboratory with the University of Colorado. In 2003, Dr. Jin's team at JILA made the first fermionic...

 at the University of Colorado at Boulder
University of Colorado at Boulder
The University of Colorado Boulder is a public research university located in Boulder, Colorado...

 and Wolfgang Ketterle
Wolfgang Ketterle
Wolfgang Ketterle is a German physicist and professor of physics at the Massachusetts Institute of Technology . His research has focused on experiments that trap and cool atoms to temperatures close to absolute zero, and he led one of the first groups to realize Bose-Einstein condensation in these...

 at MIT
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...

. Jin quickly went on to create the first fermionic condensate
Fermionic condensate
A 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. Unlike the Bose–Einstein condensates, fermionic condensates are formed using...

 composed of Cooper pair
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...

s.

In 1999, Danish physicist Lene Vestergaard Hau
Lene Hau
Lene Vestergaard Hau is a Danish physicist. In 1999, she led a Harvard University team who, by use of a superfluid, succeeded in slowing a beam of light to about 17 metres per second, and, in 2001, was able to momentarily stop a beam.In 1989, Hau accepted a two-year appointment as a postdoctoral...

 led a team from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...

 which succeeded in slowing a beam of light to about 17 metres per second. She was able to achieve this by using a superfluid. Hau and her associates at Harvard University have since successfully made a group of condensate atoms recoil from a "light pulse" such that they recorded the light's phase and amplitude, which was recovered by a second nearby condensate, by what they term "slow-light-mediated atomic matter-wave amplification" using Bose–Einstein condensates: details of the experiment are discussed in an article in the journal Nature
Nature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...

, 8 February 2007.

Researchers in the new field of atomtronics
Atomtronics
Atomtronics is the branch of science, engineering and technology that deals with the creation of analogues to electronic circuits and devices by the use of atoms...

 use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.

Isotopes



The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2010, using ultra-low temperatures of or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of alkaline
Alkali metal
The alkali metals are a series of chemical elements in the periodic table. In the modern IUPAC nomenclature, the alkali metals comprise the group 1 elements, along with hydrogen. The alkali metals are lithium , sodium , potassium , rubidium , caesium , and francium...

 and alkaline earth
Alkaline earth metal
The alkaline earth metals are a group in the periodic table. In the modern IUPAC nomenclature, the alkaline earth metals are called the group 2 elements. Previously, they were called the Group IIA elements . The alkaline earth metals contain beryllium , magnesium , calcium , strontium , barium and...

 atoms (7Li
Lithium
Lithium is a soft, silver-white metal that belongs to the alkali metal group of chemical elements. It is represented by the symbol Li, and it has the atomic number 3. Under standard conditions it is the lightest metal and the least dense solid element. Like all alkali metals, lithium is highly...

, 23Na
Sodium
Sodium is a chemical element with the symbol Na and atomic number 11. It is a soft, silvery-white, highly reactive metal and is a member of the alkali metals; its only stable isotope is 23Na. It is an abundant element that exists in numerous minerals, most commonly as sodium chloride...

, 39K
Potassium
Potassium is the chemical element with the symbol K and atomic number 19. Elemental potassium is a soft silvery-white alkali metal that oxidizes rapidly in air and is very reactive with water, generating sufficient heat to ignite the hydrogen emitted in the reaction.Potassium and sodium are...

, 41K
Potassium
Potassium is the chemical element with the symbol K and atomic number 19. Elemental potassium is a soft silvery-white alkali metal that oxidizes rapidly in air and is very reactive with water, generating sufficient heat to ignite the hydrogen emitted in the reaction.Potassium and sodium are...

, 85Rb
Rubidium
Rubidium is a chemical element with the symbol Rb and atomic number 37. Rubidium is a soft, silvery-white metallic element of the alkali metal group. Its atomic mass is 85.4678. Elemental rubidium is highly reactive, with properties similar to those of other elements in group 1, such as very rapid...

, 87Rb, 133Cs
Caesium
Caesium or cesium is the chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-gold alkali metal with a melting point of 28 °C , which makes it one of only five elemental metals that are liquid at room temperature...

, 52Cr
Chromium
Chromium is a chemical element which has the symbol Cr and atomic number 24. It is the first element in Group 6. It is a steely-gray, lustrous, hard metal that takes a high polish and has a high melting point. It is also odorless, tasteless, and malleable...

, 40Ca
Calcium
Calcium is the chemical element with the symbol Ca and atomic number 20. It has an atomic mass of 40.078 amu. Calcium is a soft gray alkaline earth metal, and is the fifth-most-abundant element by mass in the Earth's crust...

, 84Sr
Strontium
Strontium is a chemical element with the symbol Sr and the atomic number 38. An alkaline earth metal, strontium is a soft silver-white or yellowish metallic element that is highly reactive chemically. The metal turns yellow when exposed to air. It occurs naturally in the minerals celestine and...

, 86Sr
Strontium
Strontium is a chemical element with the symbol Sr and the atomic number 38. An alkaline earth metal, strontium is a soft silver-white or yellowish metallic element that is highly reactive chemically. The metal turns yellow when exposed to air. It occurs naturally in the minerals celestine and...

, 88Sr
Strontium
Strontium is a chemical element with the symbol Sr and the atomic number 38. An alkaline earth metal, strontium is a soft silver-white or yellowish metallic element that is highly reactive chemically. The metal turns yellow when exposed to air. It occurs naturally in the minerals celestine and...

, and 174Yb
Ytterbium
Ytterbium is a chemical element with the symbol Yb and atomic number 70. A soft silvery metallic element, ytterbium is a rare earth element of the lanthanide series and is found in the minerals gadolinite, monazite, and xenotime. The element is sometimes associated with yttrium or other related...

). Condensation research was finally successful even with hydrogen with the aid of special methods. In contrast, the superfluid state of the bosonic 4He
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

 at temperatures below is not a good example of Bose–Einstein condensation, because the interaction between the 4He bosons is too strong. Only 8% of the atoms are in the single-particle ground state near zero temperature, rather than the 100% expected of a true Bose–Einstein condensate.

The spin-statistics theorem
Spin-statistics theorem
In quantum mechanics, the spin-statistics theorem relates the spin of a particle to the particle statistics it obeys. The spin of a particle is its intrinsic angular momentum...

 of Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...

 states that half-integer spins (in units of \scriptstyle \hbar) lead to fermionic behaviour, e.g., the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...

 forbidding that more than two electrons possess the same energy, whereas integer spins lead to bosonic behaviour, e.g., condensation of identical bosonic particles in a common ground state.

The boson
Boson
In particle physics, bosons are subatomic particles that obey Bose–Einstein statistics. Several bosons can occupy the same quantum state. The word boson derives from the name of Satyendra Nath Bose....

ic, rather than fermion
Fermion
In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

ic, behaviour of some of these alkaline gases appears odd at first sight since their nuclei have half-integer total spin. The bosonic behaviour arises from a subtle interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell and the half-integer total spin of the nucleus of the atom are coupled by a very weak hyperfine interaction. The total spin of the atom arising from this coupling is an integer value leading to the bosonic ultra-low temperature behaviour of the atom. The chemistry of the systems at room temperature is determined by the electronic properties, which is essentially fermionic, since at room temperature thermal excitations have typical energies much higher than the hyperfine values.

See also



  • Atom laser
    Atom laser
    An 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...

  • Atomic coherence
    Atomic coherence
    In physics, atomic coherence is the induced coherence between levels of a multi-level atomic system sometimes observed when it interacts with a coherent electromagnetic field....

  • Bose–Einstein correlations
  • Bose–Einstein condensation: a network theory approach
  • Electromagnetically induced transparency
    Electromagnetically induced transparency
    Electromagnetically induced transparency is a coherent optical nonlinearity which renders a medium transparent over a narrow spectral range within an absorption line...

  • Fermionic condensate
    Fermionic condensate
    A 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. Unlike the Bose–Einstein condensates, fermionic condensates are formed using...

  • Gas in a box
    Gas in a box
    In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions...

  • Gross–Pitaevskii equation
  • Macroscopic quantum self-trapping
    Macroscopic quantum self-trapping
    In quantum mechanics, macroscopic quantum self-trapping is a phenomenon occurring in the state of matter called the Bose-Einstein condensate between two superconductors linked by a non-conducting barrier known as a Josephson junction....



  • Slow light
    Slow light
    Slow light is the propagation of an optical pulse or other modulation of an optical carrier at a very low group velocity. Slow light occurs when a propagating pulse is substantially slowed down by the interaction with the medium in which the propagation take place.Researchers at the Rowland...

  • Superconductivity
    Superconductivity
    Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...

  • Superfluid film
    Superfluid film
    Superfluidity and superconductivity are macroscopic manifestations of quantum mechanics. There is considerable interest, both theoretical and practical, in these quantum phase transitions. There has been a tremendous amount of work done in the field of phase transitions and critical phenomenon in...

  • Supersolid
    Supersolid
    A supersolid is a spatially ordered material with superfluid properties. Superfluidity is a special quantum state of matter in which a substance flows with zero viscosity.-Background:...

  • Tachyon condensation
    Tachyon condensation
    In particle physics, theoretical processes that eliminate or resolve particles or fields into better understood phenomena are called, by extension and metaphor with the macroscopic process, "condensation"...

  • Timeline of low-temperature technology
    Timeline of low-temperature technology
    The following is a timeline of low-temperature technology and cryogenic technology .-16th century BCE – 17th century CE :...

  • Super-heavy atom
  • Wiener sausage
    Wiener sausage
    In 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...



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.
  • Mackie M, Suominen KA, Javanainen J., "Mean-field theory of Feshbach-resonant interactions in 85Rb condensates." Phys Rev Lett. 2002 Oct 28;89(18):180403.


External links