Gross–Pitaevskii equation

The Gross–Pitaevskii equation is a nonlinear model equation for the order parameter or wavefunction of a Bose–Einstein condensate. It is similar in form to the Ginzburg–Landau equation and is sometimes referred to as a nonlinear Schrödinger equation.

A Bose–Einstein condensate (BEC) is a gas of bosons that are in the same quantum state, and thus can be described by the same wavefunction. A free quantum particle is described by a single-particle Schrödinger 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 the scattering length (that is, in the so-called dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. 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.The energy 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. The chemical potential is found from the condition that the number of particles is related to the wavefunction 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. 116, No. 3, 489–506 (1959). [http://prola.aps.org/abstract/PR/v116/i3/p489_1?qid=5294d518f0499616&qseq=1&show=25] ] an interacting bose gas does not exhibit an energy gap (in the case of repulsive interactions).

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 density

If 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

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

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

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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