 Gross-Pitaevskii Equation
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Gross%E2%80%93Pitaevskii 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(-{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial \mathbf {r} _{i}^{2}}+V(\mathbf {r} _{i})\right)+\sum _{i 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 wave-function satisfies the following Gross-Pitaevskii equation:

$\left(-{\frac {\hbar ^{2}}{2m}}{\partial ^{2} \over \partial \mathbf {r} ^{2}}+V(\mathbf {r} )+{4\pi \hbar ^{2}a_{s} \over m}\vert \psi (\mathbf {r} )\vert ^{2}\right)\psi (\mathbf {r} )=\mu \psi (\mathbf {r} ),$ the total wave-function minimizes the expectation value of the model Hamiltonian under normalization condition $\int dV|\Psi |^{2}=N.$ Therefore, such single-particle wave-function 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: When setting the coupling constant of interaction in the Gross-Pitaevskii equation to zero (see the following section): thereby, 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 s-wave scattering length $a_{s}$ of two interacting bosons:

$g={\frac {4\pi \hbar ^{2}a_{s}}{m}}$ ,

where $\hbar$ is the reduced Planck's constant and $m$ is the mass of the boson. The energy density is

${\mathcal {E}}={\frac {\hbar ^{2}}{2m}}\vert \nabla \Psi (\mathbf {r} )\vert ^{2}+V(\mathbf {r} )\vert \Psi (\mathbf {r} )\vert ^{2}+{\frac {1}{2}}g\vert \Psi (\mathbf {r} )\vert ^{4},$ where $\Psi$ is the wavefunction, or order parameter, and $V$ is the external potential (e.g. a harmonic trap). The time-independent Gross-Pitaevskii equation, for a conserved number of particles, is

$\mu \Psi (\mathbf {r} )=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )+g\vert \Psi (\mathbf {r} )\vert ^{2}\right)\Psi (\mathbf {r} )$ 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

$N=\int \vert \Psi (\mathbf {r} )\vert ^{2}\,d^{3}r.$ 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

$i\hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )+g\vert \Psi (\mathbf {r} ,t)\vert ^{2}\right)\Psi (\mathbf {r} ,t).$ 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.

## Solutions

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 $V(\mathbf {r} )=0$ ,

$\Psi (\mathbf {r} )={\sqrt {\frac {N}{V}}}e^{i\mathbf {k} \cdot \mathbf {r} }.$ 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:

$E(\mathbf {k} )=N\left[{\frac {\hbar ^{2}k^{2}}{2m}}+g{\frac {N}{2V}}\right].$ According to the Hugenholtz-Pines theorem,  an interacting bose gas does not exhibit an energy gap (in the case of repulsive interactions).

#### Soliton

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 bright 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 solution of the Gross-Pitaevskii equation is,

$\psi (x)=\psi _{0}\tanh \left({\frac {x}{{\sqrt {2}}\xi }}\right)$ ,

where $\psi _{0}$ is the value of the condensate wavefunction at $\infty$ , and $\xi =\hbar /{\sqrt {2mn_{0}g}}=1/{\sqrt {8\pi a_{s}n_{0}}}$ is the coherence length (a.k.a. the healing length, see below). 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 $g<0$ $\psi (x,t)=\psi (0)e^{-i\mu t/\hbar }{\frac {1}{\cosh \left[{\sqrt {2m\vert \mu \vert /\hbar ^{2}}}x\right]}},$ where the chemical potential is $\mu =g\vert \psi (0)\vert ^{2}/2$ . This solution represents the bright soliton, since there is a concentration of condensate in a space of zero density.

### Healing length

The healing length can be understood as the length scale where the kinetic energy of the boson equals the chemical potential:

${\frac {\hbar ^{2}}{2m\xi ^{2}}}=\mu =gn_{0}\,.$ The healing length gives the shortest distance over which the wavefuction can change; It must be much smaller than any length scale in the solution of the single-particle wavefunction. The healing length also determines the size of vortices that can form in a superfluid; It's the distance over which the wavefunction recovers from zero in the center of the vortex to the value in the bulk of the superfluid (hence the name "healing" length).

### 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.

### Numerical solutions

Several numerical methods, such as the split-step Crank-Nicolson and Fourier spectral methods, have been used for solving GPE. There are also different Fortran and C programs for its solution for the contact interaction and long-range dipolar interaction.

### 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.

$\psi (x,t)={\sqrt {\frac {\mu -V(x)}{Ng}}}$ In a harmonic trap (where the potential energy is quadratic with respect to displacement from the center), this gives a density profile commonly referred to as the "inverted parabola" density profile.

### Bogoliubov approximation

Bogoliubov treatment of the Gross-Pitaevskii equation is a method that finds the elementary excitations of a Bose-Einstein condensate. To that purpose, the condensate wavefunction is approximated by a sum of the equilibrium wavefunction $\psi _{0}={\sqrt {n}}e^{-i\mu t}$ and a small perturbation $\delta \psi$ ,

$\psi =\psi _{0}+\delta \psi$ .

Then this form is inserted in the time dependent Gross-Pitaevskii equation and its complex conjugate, and linearized to first order in $\delta \psi$ $i\hbar {\frac {\partial \delta \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\delta \psi +V\delta \psi +g(2|\psi _{0}|^{2}\delta \psi +\psi _{0}^{2}\delta \psi ^{*})$ $-i\hbar {\frac {\partial \delta \psi ^{*}}{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\delta \psi ^{*}+V\delta \psi ^{*}+g(2|\psi _{0}|^{2}\delta \psi ^{*}+(\psi _{0}^{*})^{2}\delta \psi )$ Assuming the following for $\delta \psi$ $\delta \psi =e^{-i\mu t}(u({\boldsymbol {r}})e^{-i\omega t}-v^{*}({\boldsymbol {r}})e^{i\omega t})$ one finds the following coupled differential equations for $u$ and $v$ by taking the $e^{\pm i\omega t}$ parts as independent components

$\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V+2gn-\hbar \mu -\hbar \omega \right)u-gnv=0$ $\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V+2gn-\hbar \mu +\hbar \omega \right)v-gnu=0$ For a homogeneous system, i.e. for $V({\boldsymbol {r}})=const.$ , one can get $V=\hbar \mu -gn$ from the zeroth order equation. Then we assume $u$ and $v$ to be plane waves of momentum ${\boldsymbol {q}}$ , which leads to the energy spectrum

$\hbar \omega =\epsilon _{\boldsymbol {q}}={\sqrt {{\frac {\hbar ^{2}|{\boldsymbol {q}}|^{2}}{2m}}\left({\frac {\hbar ^{2}|{\boldsymbol {q}}|^{2}}{2m}}+2gn\right)}}$ For large ${\boldsymbol {q}}$ , the dispersion relation is quadratic in ${\boldsymbol {q}}$ as one would expect for usual non interacting single particle excitations. For small ${\boldsymbol {q}}$ , the dispersion relation is linear

$\epsilon _{\boldsymbol {q}}=s\hbar q$ with $s={\sqrt {ng/m}}$ being the speed of sound in the condensate, also known as second sound. The fact that $\epsilon _{\boldsymbol {q}}/(\hbar q)>s$ shows, according to Landau's criterion, that the condensate is a superfluid, meaning that if an object is moved in the condensate at a velocity inferior to s, it will not be energetically favorable to produce excitations and the object will move without dissipation, which is a characteristic of a superfluid. Experiments have been done to prove this superfluidity of the condensate, using a tightly focused blue-detuned laser.  The same dispersion relation is found when the condensate is described from a microscopical approach using the formalism of second quantization.

### Superfluid in rotating helical potential

The optical potential well $V_{\rm {twist}}(\mathbf {r} ,t)=V_{\rm {twist}}(z,r,\theta ,t)$ might be formed by two counter propagating optical vortices with wavelengths $\lambda _{\pm }=2\pi c/\omega _{\pm }$ , effective width $D$ and topological charge $\ell$ :

${E_{\pm }}(\mathbf {r} ,t)\sim \exp \left(-{\frac {r^{2}}{2D^{2}}}\right)r^{|\ell |}\exp(-i\omega _{\pm }t\pm ik_{\pm }z+i\ell \theta ),$ where $\delta \omega =(\omega _{+}-\omega _{-})$ .In cylindrical coordinate system $(z,r,\theta )$ the potential well have a remarkable double helix geometry: 

$V_{\rm {twist}}(\mathbf {r} ,t)\sim V_{0}\exp \left(-{\frac {r^{2}}{D^{2}}}\right)r^{2|\ell |}\left(1+\cos[\delta \omega t+(k_{+}+k_{-})z+2\ell \theta ]\right),$ In a reference frame rotating with angular velocity $\Omega =\delta \omega /2\ell$ , time-dependent Gross-Pitaevskii equation with helical potential is as follows: 

$i\hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{\rm {twist}}(\mathbf {r} )+g\vert \Psi (\mathbf {r} ,t)\vert ^{2}-\Omega {\hat {L}}\right)\Psi (\mathbf {r} ,t),$ where ${\hat {L}}=-i\hbar {\frac {\partial }{\partial \theta }}$ is the angular momentum operator. The solution for condensate wavefunction $\Psi (\mathbf {r} ,t)$ is a superposition of two phase-conjugated matter-wave vortices:

$\Psi (\mathbf {r} ,t)\sim \exp \left(-{\frac {r^{2}}{2D^{2}}}\right)r^{|\ell |}\times$ $\left(\exp(-i\omega _{+}t+ik_{+}z+i\ell \theta )+\exp(-i\omega _{-}t-ik_{-}z-i\ell \theta )\right).$ The macroscopically observable momentum of condensate is :

$\langle \Psi \vert {\hat {P}}\vert \Psi \rangle =N_{\rm {at}}\hbar (k_{+}-k_{-}),$ where $N_{\rm {at}}$ is number of atoms in condensate. This means that atomic ensemble moves coherently along $z-$ axis with group velocity whose direction is defined by signs of topological charge $\ell$ and angular velocity $\Omega$ : 

$V_{z}={\frac {2\Omega \ell }{(k_{+}+k_{-})}}$ The angular momentum of helically trapped condensate is exactly zero:

$\langle \Psi \vert {\hat {L}}\vert \Psi \rangle =N_{\rm {at}}[\ell \hbar -\ell \hbar ]=0.$ Numerical modeling of cold atomic ensemble in spiral potential have shown the confinement of individual atomic trajectories within helical potential well. Vortex dipole trap with topological charge $\ell =2$ loaded by ultracold ensemble.