Editing Bose–Einstein condensate (section)

== Models ==

=== Bose Einstein's non-interacting gas ===
{{Main|Bose gas}}
Consider a collection of ''N'' non-interacting particles, which can each be in one of two [[quantum state]]s, <math>|0\rangle</math> and <math>|1\rangle</math>. If the two states are equal in energy, each different configuration is equally likely.

If we can tell which particle is which, there are <math>2^N</math> different configurations, since each particle can be in <math>|0\rangle</math> or <math>|1\rangle</math> independently. In almost all of the configurations, about half the particles are in <math>|0\rangle</math> and the other half in <math>|1\rangle</math>. 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 <math>N+1</math> different configurations. If there are <math>K</math> particles in state <math>|1\rangle</math>, there are <math>N-K</math> particles in state <math>|0\rangle</math>. Whether any particular particle is in state <math>|0\rangle</math> or in state <math>|1\rangle</math> cannot be determined, so each value of <math>K</math> determines a unique quantum state for the whole system.

Suppose now that the energy of state <math>|1\rangle</math> is slightly greater than the energy of state <math>|0\rangle</math> by an amount <math>E</math>. At temperature <math>T</math>, a particle will have a lesser probability to be in state <math>|1\rangle</math> by <math>e^{-E/kT}</math>. In the distinguishable case, the particle distribution will be biased slightly towards state <math>|0\rangle</math>. 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 <math>|0\rangle</math>.

In the distinguishable case, for large ''N'', the fraction in state <math>|0\rangle</math> can be computed. It is the same as flipping a coin with probability proportional to <math>\exp{(-E/T)}</math> to land tails.

In the indistinguishable case, each value of <math>K</math> is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:

:<math>\,
P(K)= C e^{-KE/T} = C p^K.
</math>

For large <math>N</math>, the normalization constant <math>C</math> is <math>1-p</math>. The expected total number of particles not in the lowest energy state, in the limit that <math>N\rightarrow \infty</math>, is equal to
: <math>\sum_{n>0} C n p^n=p/(1-p) </math>
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 <math>|k\rangle</math>. 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, <math>p/(1-p)</math>:

:<math>\,
 N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1} </math>
:<math>\,
p(k)= e^{-k^2\over 2mT}.
</math>

When the integral (also known as [[Bose–Einstein integral]]) is evaluated with factors of <math>k_B</math> and <math>\hbar</math> 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]] <math>\mu</math>. In [[Bose–Einstein statistics]] distribution, <math>\mu</math> is actually still nonzero for BECs; however, <math>\mu</math> is less than the ground state energy. Except when specifically talking about the ground state, <math>\mu</math> can be approximated for most energy or momentum states as&nbsp;<math>\mu \approx 0</math>.

=== Bogoliubov theory for weakly interacting gas ===
[[Nikolay Bogolyubov|Nikolay Bogoliubov]] considered perturbations on the limit of dilute gas,<ref name=Bogoliubov:1947/> finding a finite pressure at zero temperature and positive chemical potential. This leads to corrections for the ground state. The Bogoliubov state has pressure <math>(T=0)</math>: <math>P = gn^2/2</math>.

The original interacting system can be converted to a system of non-interacting particles with a dispersion law.

=== Gross–Pitaevskii equation ===
{{Main|Gross–Pitaevskii equation}}

In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments.

This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate <math>\psi(\vec{r})</math>. For a [[Schrödinger field|system of this nature]], <math>|\psi(\vec{r})|^2</math> is interpreted as the particle density, so the total number of atoms is <math>N=\int d\vec{r}|\psi(\vec{r})|^2</math>

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 <math>\psi(\vec{r})</math> is:

:<math>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]</math>

Minimizing this energy with respect to infinitesimal variations in <math>\psi(\vec{r})</math>, and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear [[Schrödinger equation]]):

:<math>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})</math>

where:

:{|cellspacing="0" cellpadding="0"
|-
| <math>\,m</math>
| &nbsp;is the mass of the bosons,
|-
| <math>\,V(\vec{r})</math>
| &nbsp;is the external potential, and
|-
| <math>\,U_0</math>
| &nbsp;represents the inter-particle interactions.
|}

In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for <math>\ T= 0</math>):

:<math> {\omega _p} = \sqrt {\frac{{{p^2}}}{{2m}}\left( {\frac{{{p^2}}}{{2m}} + 2{U_0}{n_0}} \right)} </math>

The Gross-Pitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for <math>\ T= 0</math>.
It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.

==== Numerical solution ====
The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and
different numerical methods, such as split-step
[[Crank–Nicolson method|Crank–Nicolson]]<ref>{{cite journal
 |author=P. Muruganandam and S. K. Adhikari
 |year=2009
 |title=Fortran Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal=Comput. Phys. Commun.
 |volume=180 |issue=3 |pages=1888–1912
 |doi=10.1016/j.cpc.2009.04.015
|bibcode=2009CoPhC.180.1888M|arxiv=0904.3131|s2cid=7403553
 }}</ref>
and [[Fourier spectral]]<ref>{{cite journal
 |author=P. Muruganandam and S. K. Adhikari
 |year=2003
 |title=Bose–Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods
 |journal=J. Phys. B
 |volume=36 |issue= 12|pages=2501–2514
 |doi=10.1088/0953-4075/36/12/310
 |bibcode=2003JPhB...36.2501M|arxiv=cond-mat/0210177|s2cid=13180020
 }}</ref> methods, are used for its solution. There are different Fortran and C programs for its solution for [[contact interaction]]<ref>{{cite journal
 |author=D. Vudragovic|display-authors=et al
 |year=2012
 |title=C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=183 |issue=9 |pages=2021–2025
 |doi=10.1016/j.cpc.2012.03.022
  |bibcode=2012CoPhC.183.2021V|arxiv=1206.1361|s2cid=12031850
 }}</ref><ref>
{{cite journal
 |author=L. E. Young-S.|display-authors=et al
 |year=2016
 |title=OpenMP Fortran and C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=204 |issue=9 |pages=209–213
 |doi=10.1016/j.cpc.2016.03.015
 |bibcode=2016CoPhC.204..209Y|arxiv=1605.03958|s2cid=206999817
 }}</ref>
and long-range [[dipolar interaction]]<ref>
{{cite journal
 |author=K. Kishor Kumar|display-authors=et al
 |year=2015
 |title=Fortran and C Programs for the time-dependent dipolar Gross-Pitaevskii equation in a fully anisotropic trap
 |journal= Comput. Phys. Commun.
 |volume=195 |pages=117–128
 |doi=10.1016/j.cpc.2015.03.024
 |bibcode=2015CoPhC.195..117K|arxiv=1506.03283|s2cid=18949735
 }}</ref> which can be freely used.

==== Weaknesses of Gross–Pitaevskii model ====
The Gross–Pitaevskii model of BEC is a physical [[approximation]] valid for certain classes of BECs. By construction, the [[Gross–Pitaevskii equation|GPE]] uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions to [[self-energy]].<ref>Beliaev, S. T. Zh. Eksp. Teor. Fiz. 34, 417–432 (1958) [Soviet Phys. JETP 7, 289 (1958)]; ibid. 34, 433–446 [Soviet Phys. JETP 7, 299 (1958)].</ref> These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate [[wavefunction]] 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,<ref name=Schick:1971/><ref name=Kolomeisky:1992/><ref name=Kolomeisky:2000/><ref name=Chui:2004/> effectively lower-dimensional condensates,<ref>
{{cite journal
 |author1=L. Salasnich
 |author2=A. Parola
 |author3=L. Reatto |name-list-style=amp
 |year=2002
 |title=Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates
 |journal=Phys. Rev. A
 |volume=65 |issue=4 |page=043614
 |arxiv=cond-mat/0201395
 |bibcode = 2002PhRvA..65d3614S
 |doi=10.1103/PhysRevA.65.043614|s2cid=119376582
 }}
</ref> and dense condensates and [[superfluid]] clusters and droplets.<ref>
{{cite journal
 |author1=A. V. Avdeenkov
 |author2=K. G. Zloshchastiev
 |year=2011
 |title=Quantum Bose liquids with logarithmic nonlinearity: Self-sustainability and emergence of spatial extent
 |journal=J. Phys. B: At. Mol. Opt. Phys.
 |volume=44 |issue=19 |pages=195303
 |arxiv=1108.0847
 |bibcode=2011JPhB...44s5303A
 |doi=10.1088/0953-4075/44/19/195303|s2cid=119248001
 }}
</ref> It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term <math>\psi \ln |\psi|^2 </math> found in the [[Logarithmic Schrödinger equation]] must be added to the Gross-Pitaevskii equation along with a [[Vitaly Ginzburg|Ginzburg]]–Sobyanin contribution to correctly determine that the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures in close agreement with experiment.<ref>{{cite journal|author1=T.C Scott|author2=K. G. Zloshchastiev|title=Resolving the puzzle of sound propagation in liquid helium at low temperatures|journal=Low Temperature Physics|volume=45|issue=12|year=2019|pages=1231–1236|doi=10.1063/10.0000200|arxiv=2006.08981|bibcode=2019LTP....45.1231S|s2cid=213962795}}</ref>

==== Other ====
However, it is clear that in a general case the behaviour of Bose–Einstein condensate can be described by coupled evolution equations for condensate density, superfluid velocity and distribution function of elementary excitations. This problem was solved in 1977 by Peletminskii et al. in microscopical approach. The Peletminskii equations are valid for any finite temperatures below the critical point. Years after, in 1985, Kirkpatrick and Dorfman obtained similar equations using another microscopical approach. The Peletminskii equations also reproduce Khalatnikov hydrodynamical equations for superfluid as a limiting case.

=== Superfluidity of BEC and Landau criterion ===
The phenomena of superfluidity of a Bose gas and superconductivity of a strongly-correlated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in [[helium-4]] and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model.