Gross-Pitaevskii Equation
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Gross%E2%80%93Pitaevskii Equation

The Gross-Pitaevskii equation (GPE, named after Eugene P. Gross[1] and Lev Petrovich Pitaevskii[2]) 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 ${\displaystyle \Psi }$ of the system of ${\displaystyle N}$ bosons is taken as a product of single-particle functions ${\displaystyle \psi }$,

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

where ${\displaystyle \mathbf {r} _{i}}$ is the coordinate of the ${\displaystyle i}$-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,[3] the scattering process can be well approximated by the s-wave scattering (i.e. ${\displaystyle \ell =0}$ 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:

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

The variational method shows that if the single-particle wave-function satisfies the following Gross-Pitaevskii equation:

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

the total wave-function minimizes the expectation value of the model Hamiltonian under normalization condition ${\displaystyle \int dV|\Psi |^{2}=N.}$ 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 ${\displaystyle g}$ is proportional to the s-wave scattering length ${\displaystyle a_{s}}$ of two interacting bosons:

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

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

${\displaystyle {\mathcal {E}}={\frac {\hbar ^{2}}{2m}}\vert \nabla \Psi (\mathbf {r} )\vert ^{2}+V(\mathbf {r} )\vert \Psi (\mathbf {r} )\vert ^{2}+{\frac {1}{2}}g\vert \Psi (\mathbf {r} )\vert ^{4},}$

where ${\displaystyle \Psi }$ is the wavefunction, or order parameter, and ${\displaystyle V}$ is the external potential (e.g. a harmonic trap). The time-independent Gross-Pitaevskii equation, for a conserved number of particles, is

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

where ${\displaystyle \mu }$ is the chemical potential. The chemical potential is found from the condition that the number of particles is related to the wavefunction by

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

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

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

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

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

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:

${\displaystyle E(\mathbf {k} )=N\left[{\frac {\hbar ^{2}k^{2}}{2m}}+g{\frac {N}{2V}}\right].}$

According to the Hugenholtz-Pines theorem, [4] 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 ${\displaystyle g>0}$, then a possible solution of the Gross-Pitaevskii equation is,

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

where ${\displaystyle \psi _{0}}$ is the value of the condensate wavefunction at ${\displaystyle \infty }$, and ${\displaystyle \xi =\hbar /{\sqrt {2mn_{0}g}}=1/{\sqrt {8\pi a_{s}n_{0}}}}$ is the coherence length (a.k.a. the healing length,[3] 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 ${\displaystyle \psi }$ flips between positive and negative values across the origin, corresponding to a ${\displaystyle \pi }$ phase shift.

For ${\displaystyle g<0}$

${\displaystyle \psi (x,t)=\psi (0)e^{-i\mu t/\hbar }{\frac {1}{\cosh \left[{\sqrt {2m\vert \mu \vert /\hbar ^{2}}}x\right]}},}$

where the chemical potential is ${\displaystyle \mu =g\vert \psi (0)\vert ^{2}/2}$. 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:[3]

${\displaystyle {\frac {\hbar ^{2}}{2m\xi ^{2}}}=\mu =gn_{0}\,.}$

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[5] and Fourier spectral[6] methods, have been used for solving GPE. There are also different Fortran and C programs for its solution for the contact interaction[7][8] and long-range dipolar interaction.[9]

### 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.

${\displaystyle \psi (x,t)={\sqrt {\frac {\mu -V(x)}{Ng}}}}$

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.[3]

### 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 ${\displaystyle \psi _{0}={\sqrt {n}}e^{-i\mu t}}$ and a small perturbation ${\displaystyle \delta \psi }$,

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

Then this form is inserted in the time dependent Gross-Pitaevskii equation and its complex conjugate, and linearized to first order in ${\displaystyle \delta \psi }$

${\displaystyle i\hbar {\frac {\partial \delta \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\delta \psi +V\delta \psi +g(2|\psi _{0}|^{2}\delta \psi +\psi _{0}^{2}\delta \psi ^{*})}$
${\displaystyle -i\hbar {\frac {\partial \delta \psi ^{*}}{\partial t}}=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\delta \psi ^{*}+V\delta \psi ^{*}+g(2|\psi _{0}|^{2}\delta \psi ^{*}+(\psi _{0}^{*})^{2}\delta \psi )}$

Assuming the following for ${\displaystyle \delta \psi }$

${\displaystyle \delta \psi =e^{-i\mu t}(u({\boldsymbol {r}})e^{-i\omega t}-v^{*}({\boldsymbol {r}})e^{i\omega t})}$

one finds the following coupled differential equations for ${\displaystyle u}$ and ${\displaystyle v}$ by taking the ${\displaystyle e^{\pm i\omega t}}$ parts as independent components

${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V+2gn-\hbar \mu -\hbar \omega \right)u-gnv=0}$
${\displaystyle \left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V+2gn-\hbar \mu +\hbar \omega \right)v-gnu=0}$

For a homogeneous system, i.e. for ${\displaystyle V({\boldsymbol {r}})=const.}$, one can get ${\displaystyle V=\hbar \mu -gn}$ from the zeroth order equation. Then we assume ${\displaystyle u}$ and ${\displaystyle v}$ to be plane waves of momentum ${\displaystyle {\boldsymbol {q}}}$, which leads to the energy spectrum

${\displaystyle \hbar \omega =\epsilon _{\boldsymbol {q}}={\sqrt {{\frac {\hbar ^{2}|{\boldsymbol {q}}|^{2}}{2m}}\left({\frac {\hbar ^{2}|{\boldsymbol {q}}|^{2}}{2m}}+2gn\right)}}}$

For large ${\displaystyle {\boldsymbol {q}}}$, the dispersion relation is quadratic in ${\displaystyle {\boldsymbol {q}}}$ as one would expect for usual non interacting single particle excitations. For small ${\displaystyle {\boldsymbol {q}}}$, the dispersion relation is linear

${\displaystyle \epsilon _{\boldsymbol {q}}=s\hbar q}$

with ${\displaystyle s={\sqrt {ng/m}}}$ being the speed of sound in the condensate, also known as second sound. The fact that ${\displaystyle \epsilon _{\boldsymbol {q}}/(\hbar q)>s}$ 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. [10] 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 ${\displaystyle V_{\rm {twist}}(\mathbf {r} ,t)=V_{\rm {twist}}(z,r,\theta ,t)}$ might be formed by two counter propagating optical vortices with wavelengths ${\displaystyle \lambda _{\pm }=2\pi c/\omega _{\pm }}$, effective width ${\displaystyle D}$ and topological charge ${\displaystyle \ell }$ :

${\displaystyle {E_{\pm }}(\mathbf {r} ,t)\sim \exp \left(-{\frac {r^{2}}{2D^{2}}}\right)r^{|\ell |}\exp(-i\omega _{\pm }t\pm ik_{\pm }z+i\ell \theta ),}$

where ${\displaystyle \delta \omega =(\omega _{+}-\omega _{-})}$.In cylindrical coordinate system ${\displaystyle (z,r,\theta )}$ the potential well have a remarkable double helix geometry: [11]

${\displaystyle V_{\rm {twist}}(\mathbf {r} ,t)\sim V_{0}\exp \left(-{\frac {r^{2}}{D^{2}}}\right)r^{2|\ell |}\left(1+\cos[\delta \omega t+(k_{+}+k_{-})z+2\ell \theta ]\right),}$

In a reference frame rotating with angular velocity ${\displaystyle \Omega =\delta \omega /2\ell }$, time-dependent Gross-Pitaevskii equation with helical potential is as follows: [12]

${\displaystyle i\hbar {\frac {\partial \Psi (\mathbf {r} ,t)}{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{\rm {twist}}(\mathbf {r} )+g\vert \Psi (\mathbf {r} ,t)\vert ^{2}-\Omega {\hat {L}}\right)\Psi (\mathbf {r} ,t),}$

where ${\displaystyle {\hat {L}}=-i\hbar {\frac {\partial }{\partial \theta }}}$ is the angular momentum operator. The solution for condensate wavefunction ${\displaystyle \Psi (\mathbf {r} ,t)}$ is a superposition of two phase-conjugated matter-wave vortices:

${\displaystyle \Psi (\mathbf {r} ,t)\sim \exp \left(-{\frac {r^{2}}{2D^{2}}}\right)r^{|\ell |}\times }$
${\displaystyle \left(\exp(-i\omega _{+}t+ik_{+}z+i\ell \theta )+\exp(-i\omega _{-}t-ik_{-}z-i\ell \theta )\right).}$

The macroscopically observable momentum of condensate is :

${\displaystyle \langle \Psi \vert {\hat {P}}\vert \Psi \rangle =N_{\rm {at}}\hbar (k_{+}-k_{-}),}$

where ${\displaystyle N_{\rm {at}}}$ is number of atoms in condensate. This means that atomic ensemble moves coherently along ${\displaystyle z-}$ axis with group velocity whose direction is defined by signs of topological charge ${\displaystyle \ell }$ and angular velocity ${\displaystyle \Omega }$: [13]

${\displaystyle V_{z}={\frac {2\Omega \ell }{(k_{+}+k_{-})}}}$

The angular momentum of helically trapped condensate is exactly zero:[12]

${\displaystyle \langle \Psi \vert {\hat {L}}\vert \Psi \rangle =N_{\rm {at}}[\ell \hbar -\ell \hbar ]=0.}$

Numerical modeling of cold atomic ensemble in spiral potential have shown the confinement of individual atomic trajectories within helical potential well.[14]

Vortex dipole trap with topological charge ${\displaystyle \ell =2}$ loaded by ultracold ensemble.

## References

1. ^ E. P. Gross (1961). "Structure of a quantized vortex in boson systems" (Submitted manuscript). Il Nuovo Cimento. 20 (3): 454-457. Bibcode:1961NCim...20..454G. doi:10.1007/BF02731494.
2. ^ L. P. Pitaevskii (1961). "Vortex lines in an imperfect Bose gas". Sov. Phys. JETP. 13 (2): 451-454.
3. ^ a b c d Foot, C. J. (2005). Atomic physics. Oxford University Press. pp. 231-240. ISBN 978-0-19-850695-9.
4. ^ N. M. Hugenholtz; D. Pines (1959). "Ground-state energy and excitation spectrum of a system of interacting bosons". Physical Review. 116 (3): 489-506. Bibcode:1959PhRv..116..489H. doi:10.1103/PhysRev.116.489.
5. ^ P. Muruganandam and S. K. Adhikari (2009). "Fortran Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 180 (3): 1888-1912. arXiv:0904.3131. Bibcode:2009CoPhC.180.1888M. doi:10.1016/j.cpc.2009.04.015.
6. ^ P. Muruganandam and S. K. Adhikari (2003). "Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods". J. Phys. B. 36 (12): 2501-2514. arXiv:cond-mat/0210177. Bibcode:2003JPhB...36.2501M. doi:10.1088/0953-4075/36/12/310.
7. ^ D. Vudragovic; et al. (2012). "C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 183 (9): 2021-2025. arXiv:1206.1361. Bibcode:2012CoPhC.183.2021V. doi:10.1016/j.cpc.2012.03.022.
8. ^ L. E. Young-S.; et al. (2016). "OpenMP Fortran and C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 204 (9): 209-213. arXiv:1605.03958. Bibcode:2016CoPhC.204..209Y. doi:10.1016/j.cpc.2016.03.015.
9. ^ R. Kishor Kumar; et al. (2015). "Fortran and C Programs for the time-dependent dipolar Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 195 (2015): 117-128. arXiv:1506.03283. Bibcode:2015CoPhC.195..117K. doi:10.1016/j.cpc.2015.03.024.
10. ^ C. Raman; M. Köhl; R. Onofrio; D. S. Durfee; C. E. Kuklewicz; Z. Hadzibabic; W. Ketterle (1999). "Evidence for a Critical Velocity in a Bose-Einstein Condensed Gas". Phys. Rev. Lett. 83 (13): 2502. arXiv:cond-mat/9909109. Bibcode:1999PhRvL..83.2502R. doi:10.1103/PhysRevLett.83.2502.
11. ^ A.Yu. Okulov (2008). "Angular momentum of photons and phase conjugation". J. Phys. B: At. Mol. Opt. Phys. 41 (10): 101001. arXiv:0801.2675. Bibcode:2008JPhB...41j1001O. doi:10.1088/0953-4075/41/10/101001.
12. ^ a b A. Yu. Okulov (2012). "Cold matter trapping via slowly rotating helical potential". Phys. Lett. A. 376 (4): 650-655. arXiv:1005.4213. Bibcode:2012PhLA..376..650O. doi:10.1016/j.physleta.2011.11.033.
13. ^ A. Yu. Okulov (2013). "Superfluid rotation sensor with helical laser trap". J. Low Temp. Phys. 171 (3): 397-407. arXiv:1207.3537. Bibcode:2013JLTP..171..397O. doi:10.1007/s10909-012-0837-7.
14. ^ A.Al.Rsheed1, A.Lyras, V. E. Lembessis and O. M. Aldossary (2016). "Guiding of atoms in helical optical potential structures". J. Phys. B: At. Mol. Opt. Phys. 49 (12): 125002. doi:10.1088/0953-4075/49/12/125002.CS1 maint: multiple names: authors list (link)