A
Bose–Einstein condensate (BEC) is a
state of matter of a dilute gas of weakly
interacting
bosons confined in an external
potential and cooled to
temperatures very near to
absolute zero ( , , or ). Under such
conditions, a large fraction of the bosons occupy the lowest
quantum state of the external
potential, and all
wave functions
overlap each other, at which point quantum effects become apparent
on a
macroscopic scale.
This state of matter was first predicted by
Satyendra Nath Bose and
Albert Einstein in 1924–25. Bose first sent
a paper to Einstein on the
quantum
statistics of light quanta (now called
photons). Einstein was impressed, translated the
paper himself from English to German and submitted it for Bose to
the
Zeitschrift für
Physik 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 was produced by Eric Cornell and Carl Wieman in 1995 at the University of
Colorado at Boulder NISTJILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) ( ). Cornell, Wieman, and
Wolfgang Ketterle at MIT were awarded
the 2001 Nobel Prize in
Physics in Stockholm, Sweden for their
achievements.
Theory
The slowing of atoms by use of cooling apparatus produces 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 of
(massless)
photons to (massive) atoms. (The
Einstein manuscript, once believed to be lost, was found in a
library at
Leiden University in
2005.) The result of the efforts of Bose and Einstein is the
concept of a
Bose gas, governed by
Bose–Einstein statistics,
which describes the statistical distribution of
identical particles with
integer spin, now
known as
bosons. Bosonic particles, which
include the photon as well as atoms such as
helium4, are allowed to share quantum states with
each other. Einsteindemonstrated 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
threedimensional
gas consisting of noninteracting particles with no apparent
internal degrees of freedom is given by:
 T_c=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi \hbar^2}{
m k_B}
where:

\,T_c 
is 
the critical temperature, 
\,n 
is 
the particle density, 
\,m 
is 
the mass per boson, 
\hbar 
is 
the reduced Planck
constant, 
\,k_B 
is 
the Boltzmann constant,
and 
\,\zeta 
is 
the Riemann zeta function;
\,\zeta(3/2)\approx 2.6124. 
Einstein's argument
Consider a collection of N noninteracting particles which can each
be in one of twoquantum states, \scriptstyle0\rangle and
\scriptstyle1\rangle. If the two states are equal in energy, each
different configuration is equally likely.
If we can tell which particle is which, there are 2^N different
configurations, since each particle can be in \scriptstyle0\rangle
or \scriptstyle1\rangle independently. In almost all the
configurations, about half the particles are in
\scriptstyle0\rangle and the other half in \scriptstyle1\rangle.
The balance is a statistical effect, the number of configurations
is largest when the particles are divided equally.
If the particles are indistinguishable, however, there are only N+1
different configurations. If there are K particles in state
\scriptstyle1\rangle, there are NK particles in state
\scriptstyle0\rangle. Whether any particular particle is in state
\scriptstyle0\rangle or in state \scriptstyle1\rangle 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 \scriptstyle0\rangle as for the particles to
be split half and half.
Suppose now that the energy of state \scriptstyle1\rangle is
slightly greater than the energy of state \scriptstyle0\rangle by
an amount E. At temperature T, a particle will have a lesser
probability to be in state \scriptstyle1\rangle by exp(E/T). In
the distinguishable case, the particle distribution will be biased
slightly towards state \scriptstyle0\rangle 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 \scriptstyle0\rangle.
In the distinguishable case, for large N, the fraction in state
\scriptstyle0\rangle can be computed. It is the same as coin
flipping with a coin which has probability p = exp(E/T) to land
tails. The fraction of heads is 1/(1+p), which is a smooth function
of p, 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:
 \,
P(K)= C e^{KE/T} = C p^K.
For large N, the normalization constant C is (1p). The expected
total number of particles which are not in the lowest energy state,
in the limit that \scriptstyle N\rightarrow \infty, is equal to
\scriptstyle \sum_{n>0} C n p^n=p/(1p) . It doesn't 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 labelled \scriptstylek\rangle. If the number of particles
is less than the number of thermally accessible states, for high
temperatures and low densities, the particles will all be in
different states. In this limit the gas is classical. As the
density increases or the temperature decreases, the number of
accessible states per particle becomes smaller, and at some point
more particles will be forced into a single state than the maximum
allowed for that state by statistical weighting. From this point
on, any extra particle added will go into the ground state.
To calculate the transition temperature at any density, integrate
over all momentum states the expression for maximum number of
excited particles p/(1p):
 \,
N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}1}
 \,
p(k)= e^{k^2\over 2mT}.
When the integral is evaluated with the factors of k
_{B}
and restored by dimensional analysis, it gives the critical
temperature formula of thepreceding section. Therefore, this
integral defines the critical temperature and particle number
corresponding to the conditions of zero chemical potential (μ = 0
in the
Bose–Einstein
statistics distribution).
Gross–Pitaevskii equation
The state of the BEC can be described by the wavefunction of the
condensate \psi(\vec{r}). For a
system of this nature,
\psi(\vec{r})^2 is interpreted as the particle density, so the
total number of atoms is N=\int d\vec{r}\psi(\vec{r})^2
Provided essentially all atoms are in the condensate (that is, have
condensed to the ground state), and treating the bosons using
mean field theory, the energy (E)
associated with the state \psi(\vec{r}) is:
 E=\int
d\vec{r}\left[\frac{\hbar^2}{2m}\nabla\psi(\vec{r})^2+V(\vec{r})\psi(\vec{r})^2+\frac{1}{2}U_0\psi(\vec{r})^4\right]
Minimising this energy with respect to infinitesimal variations in
\psi(\vec{r}), and holding the number of atoms constant, yields the
GrossPitaevski equation (GPE) (also a nonlinear
Schrödinger equation):
 i\hbar\frac{\partial \psi(\vec{r})}{\partial t} =
\left(\frac{\hbar^2\nabla^2}{2m}+V(\vec{r})+U_0\psi(\vec{r})^2\right)\psi(\vec{r})
where:

\,m 
is the mass of the bosons, 
\,V(\vec{r}) 
is the external potential, 
\,U_0 
is representative of the interparticle
interactions. 
The GPE provides a good description of the behavior of BEC's and is
thus often applied for theoretical analysis.
Discovery
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 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 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, consisting of
fermions instead of
bosons,
also enters a
superfluid phase at low
temperature, which can be explained by the formation of bosonic
Cooper pairs of two atoms each (see
also
fermionic
condensate).
The first "pure" Bose–Einstein condensate was created by
Eric Cornell,
Carl
Wieman, and coworkers at
JILA on June 5,
1995. They did this by cooling a dilute vapor consisting 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 created a condensate made of sodium23. Ketterle's condensate had about a
hundred times more atoms, allowing him to obtain several 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
achievement.
The Bose–Einstein condensation also applies to quasiparticles in
solids. A
magnon in an
antiferromagnet 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.
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
TlCuCl
_{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 was even shown at room
temperature, where the authors used pumping techniques.
Velocitydistribution data graph
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: 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 of the peak on the right is a purely
quantummechanical effect and does not exist in the thermal
distribution on the left. This famous graph served as the
coverdesign 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.
These phenomena are allowed for by the nonlinear \psi(\vec{r})^2
term in the GPE. As the vortices must have quantised
angular momentum, the wavefunction will be
of the form \psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta} where \rho, z
and \theta are as in the
cylindrical coordinate system,
and \ell is the angular number. To determine \phi(\rho,z), the
energy of \psi(\vec{r}) must be minimised, according to the
constraint \psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}. This is
usually done computationally, however in a uniform medium the
analytic form
 \phi=\frac{nx}{\sqrt{2+x^2}}
where:

\,n^2 
is 
density far from the vortex, 
\,x = \frac{\rho}{\ell\xi}, 
\,\xi 
is 
healing length of the condensate. 
demonstrates the correct behavior, and is a good
approximation.
A singlycharged vortex (\ell=1) is in the ground state, with its
energy \epsilon_v given by
 \epsilon_v=\pi n
\frac{\hbar^2}{m}\ln\left(1.464\frac{b}{\xi}\right)
where:

\,b 
is 
the farthest distance from the vortex considered. 
(To obtain an energy which is well defined it is necessary to
include this boundary b.)
For multiplycharged vortices (\ell >1) the energy is
approximated by
 \epsilon_v\approx \ell^2\pi n
\frac{\hbar^2}{m}\ln\left(\frac{b}{\xi}\right)
which is greater than that of \ell singlycharged vortices,
indicating that these multiplycharged 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.
Unusual characteristics
Further experimentation by the
JILA team in
2000 uncovered a hitherto unknown property of Bose–Einstein
condensates. Cornell, Wieman, and their coworkers originally used
rubidium87, an
isotope whose atoms naturally repel each other,
making a more stable condensate. The JILA team instrumentation now
had better control over the condensate so experimentation was made
on naturally
attracting atoms of another rubidium isotope,
rubidium85 (having negative atomatom
scattering length). Through a process
called
Feshbach resonance
involving a sweep of the magnetic field causing spin flip
collisions, the JILA researchers lowered the characteristic,
discrete energies at which the rubidium atoms bond into molecules,
making their Rb85 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 scientists raised the magnetic field strength still
further, the condensate suddenly reverted back to attraction,
imploded and shrank beyond detection, and then exploded, blowing
off about twothirds 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
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.
Because
supernova explosions are also
preceded by an implosion, the explosion of a collapsing
Bose–Einstein condensate was named "
bosenova", a pun on the musical style
bossa nova.
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
waveparticle duality, the study of
superfluidity and quantized
vortices, and the
slowing
of light pulses to very low speeds using
electromagnetically
induced transparency. Vortices in Bose–Einstein condensates are
also currently the subject of analogue gravity research, studying
the possibility of modeling black holes and their related phenomena
in such environments in the lab. Experimentalists have also
realized "optical lattices", 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,
and may be useful in studying Bose–Einstein condensation in fewer
than three dimensions, for example the
TonksGirardeau gas.
Bose–Einstein condensates composed of a wide range of
isotopes have been produced.
Related experiments in cooling
fermions
rather than
bosons to extremely low
temperatures have created
degenerate gases, where the atoms do not
congregate in a single state due to the
Pauli exclusion principle. To
exhibit Bose–Einstein condensation, the fermions must "pair up" to
form compound particles (e.g.
molecules or
Cooper pairs) that are bosons.
The first
molecular Bose–Einstein 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.
In 1999,
Danish physicist Lene Vestergaard Hau led a
team from Harvard
University which succeeded in slowing a beam of light to about
17 metres per second and, in 2001, was able to momentarily stop a
beam. She was able to achieve this by using a superfluid.
Hau and her associates at Harvard University have since
successfully transformed light into matter and back into light
using Bose–Einstein condensates: details of the experiment are
discussed in an article in the journal
Nature, 8 February 2007.
Subtleties
Up to 2004, using the abovementioned "ultralow temperatures",
Bose–Einstein condensates had been obtained for a multitude of
isotopes involving mainly
alkaline and
alkaline earth atoms (
^{7}Li,
^{23}Na,
^{41}K,
^{52}Cr,
^{84}St,
^{85}Rb,
^{87}Rb,
^{133}Cs and
^{174}Yb). Not astonishingly, condensation
research was finally successful even with hydrogen, although with
the aid of special methods. In contrast, the superfluid state of
the bosonic
^{4}He at temperatures
below the temperature of 2.17 K is
not a good example of
Bose–Einstein condensation, because the interaction between the
^{4}He bosons is simply too strong, so that at zero
temperature, contrary to Bose–Einstein theory, only 8% rather than
100% of the atoms are in the ground state. Even the fact that the
abovementioned alkaline gases show
bosonic,
rather than
fermionic behaviour, as solid
state physicists or chemists would expect, is based on a subtle
interplay of electronic and nuclear spins: at ultralow temperatures
and corresponding excitation energies, the halfinteger (in units
of \hbar) total spin of the electronic shell and the also
halfinteger total spin of the nucleus of the atom are
coupled by the (very weak)
hyperfine interaction to the integer (!)
total spin of the atom. Only the fact that this lastmentioned
total spin is integral causes the ultralowtemperature behaviour of
the atom to be bosonic, whereas the "chemistry" of the systems at
room temperature is determined by the electronic properties, i.e.
is essentially fermionic, since at room temperature thermal
excitations have typical energies which are much higher than the
hyperfine values. (Here one should remember the
spinstatistics theorem of
Wolfgang Pauli, which states that
halfinteger spins lead to fermionic behaviour, e.g., the
Pauli exclusion principle
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 ultralow temperature requirement of Bose–Einstein condensates
of alkali metals does not generalize to all types of Bose–Einstein
condensates. In 2006, physicists under S. Demokritov in Münster,
Germany, found Bose–Einstein condensation of
magnons (i.e. quantized spinwaves) at room
temperature, admittedly by the application of pump processes.
See also
Notes
 Ronald W. Clark, "Einstein: The Life and Times" (Avon Books,
1971) p.4089
 http://www.livescience.com/history/ap_050822_einstein.html,
http://www.lorentz.leidenuniv.nl/history/Einstein_archive/
 See e.g. Becker C., Stellmer S., SoltanPanahi P., Dörscher S.,
Baumert M., Richter E.M., Kronjäger J., Bongs K. & Sengstock,
K.: Nature phys 4 (2008) 496501
 Physics Today Online  Search & Discovery
 See e.g. Demokritov, S; Demidov, V; Dzyapko, O; Melkov, G.;
Serga, A; Hillebrands, B; Slavin, A: Nature 443
(2006) 430433
References
 ,
 .
 .
 .
 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.
 Amandine Aftalion, Vortices in Bose–Einstein
Condensates, PNLDE Vol.67, Birkhauser, 2006.
 Mackie M, Suominen KA, Javanainen J., "Meanfield theory of
Feshbachresonant interactions in 85Rb condensates." Phys Rev Lett.
2002 Oct 28;89(18):180403.
External links