Bose–Einstein condensate

In condensed matter physics,[] a Bose–Einstein condensate is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero, i.e.. Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which microscopic quantum-mechanical phenomena, particularly wavefunction interference, become apparent macroscopically.
More generally, condensation refers to the appearance of macroscopic occupation of one or several states: for example, in BCS theory, a superconductor is a condensate of Cooper pairs. As such, condensation can be associated with phase transition, and the macroscopic occupation of the state is the order parameter.
Bose–Einstein condensates were first predicted, generally, in 1924–1925 by Albert Einstein, crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics. In 1995, the Bose–Einstein condensate was created by Eric Cornell and Carl Wieman of the University of Colorado Boulder using rubidium atoms. Later that year, Wolfgang Ketterle of MIT produced a BEC using sodium atoms. In 2001 Cornell, Wieman, and Ketterle shared the Nobel Prize in Physics "for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates".

History

Bose first sent a paper to Einstein on the quantum statistics of light quanta, 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. Einstein's manuscript, once believed to be lost, was found in a library at Leiden University in 2005. Einstein then extended Bose's ideas to matter in two other papers. The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now called bosons. Bosons are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall into the lowest accessible quantum state, resulting in a new form of matter. Bosons include the photon, polaritons, magnons, some atoms and molecules such as atomic hydrogen, helium-4, lithium-7, rubidium-87 or strontium-84.
In 1938, Fritz London proposed the BEC as a mechanism for superfluidity in helium-4 and superconductivity.
The quest to produce a Bose–Einstein condensate in the laboratory was stimulated by a paper published in 1976 by two program directors at the National Science Foundation, proposing to use spin-polarized atomic hydrogen to produce a gaseous BEC. This led to the immediate pursuit of the idea by four independent research groups; these were led by Isaac Silvera, Walter Hardy, Thomas Greytak and David Lee. However, cooling atomic hydrogen turned out to be technically difficult, and Bose-Einstein condensation of atomic hydrogen was only realized in 1998.
On 5 June 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST–JILA lab, in a gas of rubidium atoms cooled to 170 nanokelvins. Shortly thereafter, Wolfgang Ketterle at MIT produced a Bose–Einstein Condensate in a gas of sodium atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics. Bose-Einstein condensation of alkali gases is easier because they can be pre-cooled with laser cooling techniques, unlike atomic hydrogen at the time, which give a significant head start when performing the final forced evaporative cooling to cross the condensation threshold. These early studies founded the field of ultracold atoms, and hundreds of research groups around the world now routinely produce BECs of dilute atomic vapors in their labs.
Since 1995, many other atomic species have been condensed, and BECs have also been realized using molecules, polaritons, and other quasi-particles. BECs of photons can be made, for example, in dye microcavites with wavelength-scale mirror separation, forming a two-dimensional harmonically confined photon gas with tunable chemical potential. BEC of plasmonic quasiparticles has been realized in periodic arrays of metal nanoparticles overlaid with dye molecules, exhibiting ultrafast sub-picosecond dynamics and long-range correlations.

Critical temperature

This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by
where:
Interactions shift the value, and the corrections can be calculated by mean-field theory.
This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics.
The critical temperature depends on the density. A more concise and experimentally relevant condition involves the phase-space density, where
is the thermal de Broglie wavelength. It is a dimensionless quantity. The transition to BEC occurs when the phase-space density is greater than critical value:
in 3D uniform space. This is equivalent to the above condition on the temperature. In a 3D harmonic potential, the critical value is instead
where has to be understood as the peak density.

Derivation

Ideal Bose gas

For an ideal Bose gas we have the equation of state
where is the per-particle volume, is the thermal wavelength, is the fugacity, and
It is noticeable that is a monotonically growing function of in, which are the only values for which the series converge.
Recognizing that the second term on the right-hand side contains the expression for the average occupation number of the fundamental state, the equation of state can be rewritten as
Because the left term on the second equation must always be positive,, and because, a stronger condition is
which defines a transition between a gas phase and a condensed phase. On the critical region it is possible to define a critical temperature and thermal wavelength:
recovering the value indicated on the previous section. The critical values are such that if or, we are in the presence of a Bose–Einstein condensate.
Understanding what happens with the fraction of particles on the fundamental level is crucial. As so, write the equation of state for, obtaining
So, if, the fraction, and if, the fraction. At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum.

Experimental observation

Superfluid helium-4

In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium-4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K. Superfluid helium has many unusual properties, including zero viscosity 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. Superfluid helium-4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium-4. Note that helium-3, a fermion, also enters a superfluid phase which can be explained by the formation of bosonic Cooper pairs of two atoms.

Dilute atomic gases

The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand rubidium-87 atoms to below 170 nK using a combination of laser cooling and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium-23. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements.
A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work. Lithium has attractive interactions, causing the condensate to be unstable and collapse for all but a few atoms. Hulet's team subsequently showed the condensate could be stabilized by confinement quantum pressure for up to about 1000 atoms. Various isotopes have since been condensed.

Velocity-distribution data graph

In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: 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 quantum-mechanical effect and does not exist in the thermal distribution on the left.

Quasiparticles

Bose–Einstein condensation also applies to quasiparticles in solids. Magnons, excitons, and polaritons have integer spin which means they are bosons that can form condensates.
Magnons, electron spin waves, can be controlled by a magnetic field. Densities from the limit of a dilute gas to a strongly interacting Bose liquid are possible. Magnetic ordering is the analog of superfluidity. In 1999 condensation was demonstrated in antiferromagnetic, at temperatures as great as 14 K. The high transition temperature is due to the magnons' small mass and greater achievable density. In 2006, condensation in a ferromagnetic yttrium-iron-garnet thin film was seen even at room temperature, with optical pumping.
Excitons, electron–hole pairs, were predicted to condense at low temperature and high density by Boer et al., in 1961. Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance. Fast optical exciton creation was used to form condensates in sub-kelvin in 2005 on.
Polariton condensation was first detected for exciton-polaritons in a quantum well microcavity kept at 5 K. Quasiparticle BECs have been achieved at room-temperature, for example, in microcavity-coupled organic semiconductors and plasmon-exciton polaritons in periodic arrays of metal nanoparticles coupled to dye molecules.