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.
The Gross-Pitaevskii equation is said to be limited to the weakly interacting regime. Nevertheless, it may also fail to reproduce interesting phenomena even within this regime. In order to study the BEC beyond that limit of weak interactions, one needs to implement the Lee-Huang-Yang (LHY) correction. Alternatively, in 1D systems one can use either an exact approach, namely the Lieb-Liniger model, or an extended equation, e.g. the Lieb-Liniger Gross-Pitaevskii equation (sometimes called modified or generalized nonlinear Schrödinger equation).
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 reduced Planck's constant, and is the mass of the boson. The energy density is
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
where 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).
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.
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 :
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).