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.
Contents
- Form of equation
- Solutions
- Free particle
- Soliton
- Variational solutions
- ThomasFermi approximation
- Bogoliubov approximation
- Superfluid in rotating helical potential
- References
In the Hartree–Fock approximation the total wave-function
where
The pseudopotential model Hamiltonian of the system is given as
where
If the single-particle wave-function satisfies the Gross–Pitaevski equation,
the total wave-function minimizes the expectation value of the model Hamiltonian under normalization condition
It is a model equation for the single-particle wavefunction in a Bose–Einstein condensate. It is similar in form to the Ginzburg–Landau equation and is sometimes referred to as a nonlinear Schrödinger equation.
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. The non-linearity of the Gross–Pitaevskii equation has its origin in the interaction between the particles. This is made evident by 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 scattering length
where
where
where
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 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.
Free particle
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:
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
where
For
where the chemical potential is
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.
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.
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
Then this form is inserted in the time dependent Gross–Pitaevskii equation and its complex conjugate, and linearized to first order in
Assuming the following for
one finds the following coupled differential equations for
For a homogeneous system, i.e. for
For large
with
Superfluid in rotating helical potential
The optical potential well
where
In cylindrical coordinate system
In a reference frame rotating with angular velocity
where
The solution for condensate wavefunction
The macroscopically observable momentum of condensate is :
where
The angular momentum of helically trapped condensate is exactly zero: