Condensed matter physics
Phases Phase transition QCP
States of matter Solid Liquid Gas Plasma Bose–Einstein condensate Bose gas Fermionic condensate Fermi gas Fermi liquid Supersolid Superfluidity Luttinger liquid Time crystal
Phase phenomena Order parameter Phase transition QCP
Electronic phases Electronic band structure Plasma Insulator Mott insulator Semiconductor Semimetal Conductor Superconductor Thermoelectric Piezoelectric Ferroelectric Topological insulator Spin gapless semiconductor
Electronic phenomena Quantum Hall effect Spin Hall effect Kondo effect
Magnetic phases Diamagnet Superdiamagnet Paramagnet Superparamagnet Ferromagnet Antiferromagnet Metamagnet Spin glass
Quasiparticles Phonon Exciton Plasmon Polariton Polaron Magnon Roton
Soft matter Amorphous solid Colloid Granular material Liquid crystal Polymer
Scientists Van der Waals Onnes von Laue Bragg Debye Bloch Onsager Mott Peierls Landau Luttinger Anderson Van Vleck Hubbard Shockley Bardeen Cooper Schrieffer Josephson Louis Néel Esaki Giaever Kohn Kadanoff Fisher Wilson von Klitzing Binnig Rohrer Bednorz Müller Laughlin Störmer Yang Tsui Abrikosov Ginzburg Leggett Parisi
Physics portal  Category
v t e

In condensed matter physics, a Bose–Einstein condensate (BEC) 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 (−273.15 °C or −459.67 °F or 0 K). 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.^[1] As such, condensation can be associated with phase transition, and the macroscopic occupation of the state is the order parameter.

Bose–Einstein condensate was first predicted, generally, in 1924–1925 by Albert Einstein,^[2] crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics.^[3] 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".^[4]

History

Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons), in which he derived Planck's quantum radiation law without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it in 1924.^[5] (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.^[6]) Einstein then extended Bose's ideas to matter in two other papers.^[7][8] 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, particles that include the photon and atoms with an even number of neutrons (such as helium-4 (⁴
He
)), are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter.

In 1938, Fritz London proposed the BEC as a mechanism for superfluidity in ⁴
He
and superconductivity.^[9][10]

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 (William Stwalley and Lewis Nosanow).^[11] This led to the immediate pursuit of the idea by four independent research groups; these were led by Isaac Silvera (University of Amsterdam), Walter Hardy (University of British Columbia), Thomas Greytak (Massachusetts Institute of Technology) and David Lee (Cornell University).^[12]

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 (nK).^[13] 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.^[14] 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, quasi-particles, and photons.^[15]

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

${\displaystyle T_{\text{c}}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{\text{B}}}}\approx 3.3125\,{\frac {\hbar ^{2}n^{2/3}}{mk_{\text{B}}}},}$

where:

${\displaystyle T_{\text{c}}}$ is the critical temperature,

${\displaystyle n}$ is the particle density,

${\displaystyle m}$ is the mass per boson,

${\displaystyle \hbar }$ is the reduced Planck constant,

${\displaystyle k_{\text{B}}}$ is the Boltzmann constant,

${\displaystyle \zeta }$ is the Riemann zeta function (${\displaystyle \zeta (3/2)\approx 2.6124}$^[16]).

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.

Derivation

Ideal Bose gas

For an ideal Bose gas we have the equation of state

${\displaystyle {\frac {1}{v}}={\frac {1}{\lambda ^{3}}}g_{3/2}(f)+{\frac {1}{V}}{\frac {f}{1-f}},}$

where ${\displaystyle v=V/N}$ is the per-particle volume, ${\displaystyle \lambda }$ is the thermal wavelength, ${\displaystyle f}$ is the fugacity, and

${\displaystyle g_{\alpha }(f)=\sum \limits _{n=1}^{\infty }{\frac {f^{n}}{n^{\alpha }}}.}$

It is noticeable that ${\displaystyle g_{3/2}}$ is a monotonically growing function of ${\displaystyle f}$ in ${\displaystyle f\in }$, 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 ${\displaystyle \langle n_{0}\rangle }$, the equation of state can be rewritten as

${\displaystyle \frac{1}{v}=\frac{1}{{\mathrm{\lambda <!--\; \lambda \; -->}}^3}}$Zdroj:https://en.wikipedia.org?pojem=Bose_einstein_condensates
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---