- Gross–Pitaevskii equation
The

**Gross–Pitaevskii equation**is a nonlinear model equation for theorder parameter orwavefunction of aBose–Einstein condensate . It is similar in form to theGinzburg–Landau equation and is sometimes referred to as anonlinear Schrödinger equation .A

Bose–Einstein condensate (BEC) is a gas ofbosons that are in the samequantum state , and thus can be described by the samewavefunction . A free quantum particle is described by a single-particleSchrödinger equation . Interaction between particles in a real gas is taken into account by a pertinent many-body Schrödinger equation. If the average spacing between the particles in a gas is greater than thescattering length (that is, in the so-called dilute limit), then one can approximate the true interaction potential that features in this equation by apseudopotential . The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles. This becomes evident by equating the coupling constant of interaction, $g$, in the Gross–Pitaevskii equation with zero (see the following section), on which the single-particle Schrödinger equation describing a particle inside a trapping potential is recovered.**Form of Equation**The equation has the form of the

Schrödinger equation with the addition of an interaction term. The coupling constant, g, is proportional to the scattering length of two interacting bosons::$g=frac\{4pihbar^2\; a\_s\}\{m\}$,

where $hbar$ is

Planck's constant and m is the mass of the boson.Theenergy density is:$mathcal\{E\}=frac\{hbar^2\}\{2m\}vert\; ablaPsi(mathbf\{r\})vert^2\; +\; V(mathbf\{r\})vertPsi(mathbf\{r\})vert^2\; +\; frac\{1\}\{2\}gvertPsi(mathbf\{r\})vert^4,$

where $Psi$ is the wavefunction, or order parameter, and V is an external potential.The time-independent Gross–Pitaevskii equation, for a conserved number of particles, is

:$muPsi(mathbf\{r\})\; =\; left(-frac\{hbar^2\}\{2m\}\; abla^2\; +\; V(mathbf\{r\})\; +\; gvertPsi(mathbf\{r\})vert^2\; ight)Psi(mathbf\{r\})$

where $mu$ is the

chemical potential . Thechemical potential is found from the condition that the number of particles is related to thewavefunction by:$N\; =\; intvertPsi(mathbf\{r\})vert^2\; d^3r$.

From the time-independent Gross–Pitaevskii equation, we can find the structure of a Bose–Einstein condensate in various external potentials (e.g. a harmonic trap).

The time-dependent Gross–Pitaevskii equation is

:$ihbarfrac\{partialPsi(mathbf\{r\})\}\{partial\; t\}\; =\; left(-frac\{hbar^2\}\{2m\}\; abla^2\; +\; V(mathbf\{r\})\; +\; gvertPsi(mathbf\{r\})vert^2\; ight)Psi(mathbf\{r\})$.

From the time-dependent Gross–Pitaevskii equation we can look at the dynamics of the Bose–Einstein condensate. It is used to find the collective modes of a trapped gas.

**olutions**Since the Gross–Pitaevskii equation is a

nonlinear ,partial differential equation , exact solutions are hard to come by. As a result, solutions have to be approximated via myriad techniques.**Exact Solutions****Free Particle**The simplest exact solution is the free particle solution, with $V(mathbf\{r\})\; =0$,

$Psi(mathbf\{r\})\; =\; frac\{1\}\{sqrt\{Ve^\{imathbf\{k\}cdotmathbf\{r$.

This solution is often called the Hartree solution. Although it does satisfy the Gross–Pitaevskii equation, it leaves a gap in the energy spectrum due to the interaction:

$E(mathbf\{k\})\; =\; frac\{Nhbar^2k^2\}\{2m\}+frac\{1\}\{2\}frac\{N^2\}\{V^2\}$.

According to the

Hugenholtz–Pines theorem , [*N. M. Hugenholtz, and D. Pines, "Ground-state energy and excitation spectrum of a system of interacting bosons", Physical Review, Vol.*] an interacting bose gas does not exhibit an energy gap (in the case of repulsive interactions).**116**, No. 3, 489–506 (1959). [*http://prola.aps.org/abstract/PR/v116/i3/p489_1?qid=5294d518f0499616&qseq=1&show=25*]**oliton**A one-dimensional

soliton can form in a Bose–Einstein condensate, and depending upon whether the interaction is attractive or repulsive, there is either a light or dark soliton. Both solitons are local disturbances in a condensate with a uniform background densityIf the BEC is repulsive, so that $g>0$, then a possible solition of the Gross–Pitaevskii equation is,

:$psi(x)\; =\; psi\_0\; anhleft(frac\{x\}\{sqrt\{2\}xi\}\; ight)$,

where $psi\_0$ is the value of the condensate wavefuntion at $infty$, and $xi\; =\; hbar/sqrt\{2mn\_0g\}$, is the coherence length. This solution represents the dark soliton, since there is a deficit of condensate in a space of nonzero density. The dark soliton is also a type of

topological defect , since $psi$ flips between positive and negative values across the origin, corresponding to a $pi$ phase shift.For $g<0$

:$psi(x,t)\; =\; psi(0)e^\{-imu\; t/hbar\}frac\{1\}\{coshleft\; [sqrt\{2mvertmuvert/hbar^2\}x\; ight]\; \},$

where the chemical potential is $mu\; =\; gvertpsi(0)vert^2/2$. This solution represents the bright soliton, since there is a concentration of condensate in a space of zero density.

**1-D Square Well Potential**(r)----$vnruyhft18568572359*6855+9$

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] #REDIRECT format **Variational Solutions**In systems where an exact analytical solution may not be feasible, one can make a variational approximation. The basic idea is to make a variational

ansatz for the wavefunction with free parameters, plug it into the free energy, and minimize the energy with respect to the free parameters.**Thomas–Fermi Approximation**If the number of particles in a gas is very large, the interatomic interaction becomes large so that the kinetic energy term can be neglected from the Gross–Pitaevskii equation. This is called the Thomas–Fermi Approximation.

**Bogoliubov Approximation****Notes****References*** C. J. Pethick and H. Smith, "Bose–Einstein Condensation in Dilute Gases" (Cambridge University Press, Cambridge, 2002). ISBN 0-521-66580-9

* L. P. Pitaevskii and S. Stringari, "Bose–Einstein Condensation" (Clarendon Press, Oxford, 2003). ISBN 0-198-50719-4

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