Gross–Pitaevskii equation

The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

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. In the Hartree–Fock approximation, the total wave-function $\Psi$ of the system of $N$ bosons is taken as a product of single-particle functions $\psi$ :

\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{N})=\psi (\mathbf {r} _{1})\psi (\mathbf {r} _{2})\dots \psi (\mathbf {r} _{N}),

where $\mathbf {r} _{i}$ is the coordinate of the $i$ -th boson. 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. At sufficiently low temperature, where the de Broglie wavelength is much longer than the range of boson–boson interaction, the scattering process can be well approximated by the s-wave scattering (i.e. $\ell =0$ in the partial-wave analysis, a.k.a. the hard-sphere potential) term alone. In that case, the pseudopotential model Hamiltonian of the system can be written as

H=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial \mathbf {r} _{i}^{2}}}+V(\mathbf {r} _{i})\right)+\sum _{i<j}{\frac {4\pi \hbar ^{2}a_{s}}{m}}\delta (\mathbf {r} _{i}-\mathbf {r} _{j}),

where $m$ is the mass of the boson, $V$ is the external potential, $a_{s}$ is the boson–boson s-wave scattering length, and $\delta (\mathbf {r} )$ is the Dirac delta-function.

The variational method shows that if the single-particle wavefunction satisfies the following Gross–Pitaevskii equation

\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}+V(\mathbf {r} )+{\frac {4\pi \hbar ^{2}a_{s}}{m}}|\psi (\mathbf {r} )|^{2}\right)\psi (\mathbf {r} )=\mu \psi (\mathbf {r} ),

the total wave-function minimizes the expectation value of the model Hamiltonian under normalization condition ${\textstyle \int dV\,|\Psi |^{2}=N.}$ Therefore, such single-particle wavefunction describes the ground state of the system.

GPE is a model equation for the ground-state single-particle wavefunction in a Bose–Einstein condensate. It is similar in form to the Ginzburg–Landau equation and is sometimes referred to as the "nonlinear Schrödinger equation".

The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles: setting the coupling constant of interaction in the Gross–Pitaevskii equation to zero (see the following section) recovers the single-particle Schrödinger equation describing a particle inside a trapping potential.

The Gross-Pitaevskii equation is said to be limited to the weakly interacting regime. Nevertheless, it may also fail to reproduce interesting phenomena even within this regime. In order to study the BEC beyond that limit of weak interactions, one needs to implement the Lee-Huang-Yang (LHY) correction. Alternatively, in 1D systems one can use either an exact approach, namely the Lieb-Liniger model, or an extended equation, e.g. the Lieb-Liniger Gross-Pitaevskii equation (sometimes called modified or generalized nonlinear Schrödinger equation).

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