Bose–Einstein condensate wiki

 In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter (also called the fifth state of matter) which is typically formed when a gas of bosons at low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67 °F). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum mechanical phenomena, particularly wavefunction interference, become apparent macroscopically. A BEC is formed by cooling a gas of extremely low density (about one-hundred-thousandth (1/100,000) the density of normal air) to ultra-low temperatures.

This state was first predicted, generally, in 1924–1925 by Albert Einstein following and crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics.

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:

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

where:

${\displaystyle \,T_{\rm {c}}}$	is the critical temperature,
${\displaystyle \,n}$	the particle density,
${\displaystyle \,m}$	the mass per boson,
${\displaystyle \hbar }$	the reduced Planck constant,
${\displaystyle \,k_{\rm {B}}}$	the Boltzmann constant and
${\displaystyle \,\zeta }$	the Riemann zeta function; ${\displaystyle \,\zeta (3/2)\approx 2.6124.}$ [14]

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 ${\displaystyle v={\frac {V}{N}}}$ is the per particle volume, ${\displaystyle \lambda }$ the thermal wavelength, ${\displaystyle f}$ 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 ${\displaystyle f\in [0,1]}$, 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

${\displaystyle {\frac {1}{v}}={\frac {1}{\lambda ^{3}}}g_{3/2}(f)+{\frac {\langle n_{0}\rangle }{V}}\Leftrightarrow {\frac {\langle n_{0}\rangle }{V}}\lambda ^{3}={\frac {\lambda ^{3}}{v}}-g_{3/2}(f)}$

Because the left term on the second equation must always be positive, ${\displaystyle {\frac {\lambda ^{3}}{v}}>g_{3/2}(f)}$ and because ${\displaystyle g_{3/2}(f)\leq g_{3/2}(1)}$, a stronger condition is

${\displaystyle {\frac {\lambda ^{3}}{v}}>g_{3/2}(1)}$

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:

${\displaystyle \lambda _{c}^{3}=g_{3/2}(1)v=\zeta (3/2)v}$

${\displaystyle T_{c}={\frac {2\pi \hbar ^{2}}{mk_{B}\lambda _{c}^{2}}}}$

recovering the value indicated on the previous section. The critical values are such that if ${\displaystyle T<T_{c}}$ or ${\displaystyle \lambda >\lambda _{c}}$ 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 ${\displaystyle f=1}$, obtaining

${\displaystyle {\frac {\langle n_{0}\rangle }{N}}=1-\left({\frac {\lambda _{c}}{\lambda }}\right)^{3}}$ and equivalently ${\displaystyle {\frac {\langle n_{0}\rangle }{N}}=1-\left({\frac {T}{T_{c}}}\right)^{3/2}}$.

So, if ${\displaystyle T\ll T_{c}}$ the fraction ${\displaystyle {\frac {\langle n_{0}\rangle }{N}}\approx 1}$ and if ${\displaystyle T\gg T_{c}}$ the fraction ${\displaystyle {\frac {\langle n_{0}\rangle }{N}}\approx 0}$. At temperatures near to absolute 0, particles tend to condensate in the fundamental state (state with momentum ${\displaystyle {\vec {p}}=0}$).

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Velocity-distribution data (3 views) for a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.