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 of the system of bosons is taken as a product of single-particle functions :
where is the coordinate of the -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. 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
where is the mass of the boson, is the external potential, is the boson–boson s-wave scattering length, and is the Dirac delta-function.
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. 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.
Form of equation
The equation has the form of the Schrödinger equation with the addition of an interaction term. The coupling constant is proportional to the s-wave scattering length of two interacting bosons:
where is the wavefunction, or order parameter, and is the external potential (e.g. a harmonic trap). The time-independent Gross–Pitaevskii equation, for a conserved number of particles, is
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
From this equation we can look at the dynamics of the Bose–Einstein condensate. It is used to find the collective modes of a trapped gas.
The simplest exact solution is the free-particle solution, with :
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:
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 , then a possible solution of the Gross–Pitaevskii equation is
where is the value of the condensate wavefunction at , and 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 flips between positive and negative values across the origin, corresponding to a phase shift.
For the solution is
where the chemical potential is . This solution represents the bright soliton, since there is a concentration of condensate in a space of zero density.
The healing length can be understood as the length scale where the kinetic energy of the boson equals the chemical potential:
The healing length gives the shortest distance over which the wavefunction 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).
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.
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.
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 in the Gross–Pitaevskii equation. This is called the Thomas–Fermi approximation and leads to
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 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 and a small perturbation :
Then this form is inserted in the time-dependent Gross–Pitaevskii equation and its complex conjugate, and linearized to first order in :
one finds the following coupled differential equations for and by taking the parts as independent components:
For a homogeneous system, i.e. for , one can get from the zeroth-order equation. Then we assume and to be plane waves of momentum , which leads to the energy spectrum
For large , the dispersion relation is quadratic in , as one would expect for usual non-interacting single-particle excitations. For small , the dispersion relation is linear:
with being the speed of sound in the condensate, also known as second sound. The fact that 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 might be formed by two counterpropagating optical vortices with wavelengths , effective width and topological charge :
where . In cylindrical coordinate system the potential well have a remarkable double-helix geometry:
In a reference frame rotating with angular velocity , time-dependent Gross–Pitaevskii equation with helical potential is
where is the angular-momentum operator. The solution for condensate wavefunction is a superposition of two phase-conjugated matter–wave vortices:
The macroscopically observable momentum of condensate is
where is number of atoms in condensate. This means that atomic ensemble moves coherently along axis with group velocity whose direction is defined by signs of topological charge and angular velocity :
The angular momentum of helically trapped condensate is exactly zero:
Numerical modeling of cold atomic ensemble in spiral potential have shown the confinement of individual atomic trajectories within helical potential well.
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