Gross–Pitaevskii 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.
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.
In that case, the pseudopotential model Hamiltonian of the system can be written as
The variational method shows that if the single-particle wavefunction satisfies the following Gross–Pitaevskii equation
the total wave-function minimizes the expectation value of the model Hamiltonian under normalization condition
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.
[4][5] In order to study the BEC beyond that limit of weak interactions, one needs to implement the Lee-Huang-Yang (LHY) correction.
is the chemical potential, which is found from the condition that the number of particles is related to the wavefunction by 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).
Both solitons are local disturbances in a condensate with a uniform background density.
This solution represents the dark soliton, since there is a deficit of condensate in a space of nonzero density.
This solution represents the bright soliton, since there is a concentration of condensate in a space of zero density.
The healing length also determines the size of vortices that can form in a superfluid.
It is 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).
In systems where an exact analytical solution may not be feasible, one can make a variational approximation.
This is called the Thomas–Fermi approximation and leads to the single-particle wavefunction And the density profile is 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 treatment of the Gross–Pitaevskii equation is a method that finds the elementary excitations of a Bose–Einstein condensate.
: Then this form is inserted in the time-dependent Gross–Pitaevskii equation and its complex conjugate, and linearized to first order in
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.
[19] The same dispersion relation is found when the condensate is described from a microscopical approach using the formalism of second quantization.
the potential well have a remarkable double-helix geometry:[20] In a reference frame rotating with angular velocity
is a superposition of two phase-conjugated matter–wave vortices: The macroscopically observable momentum of condensate is where
axis with group velocity whose direction is defined by signs of topological charge
:[22] The angular momentum of helically trapped condensate is exactly zero:[21] Numerical modeling of cold atomic ensemble in spiral potential have shown the confinement of individual atomic trajectories within helical potential well.
[23] The Gross–Pitaevskii equation can also be derived as the semi-classical limit of the many body theory of s-wave interacting identical bosons represented in terms of coherent states.
Finite-temperature effects can be treated within a generalised Gross–Pitaevskii equation by including scattering between condensate and noncondensate atoms,[25][26][27][28][29] from which the Gross–Pitaevskii equation may be recovered in the low-temperature limit.
