The Gross–Pitaevskii equation (GPE, named after
Eugene P. Gross
Eugene Paul Gross (27 June 1926 – 19 January 1991) was a theoretical physicist and Edward and Gertrude Swartz professor at Brandeis University, known for his contribution to the Bhatnagar-Gross-Krook (BGK) collision model used in the Boltzmann ...
and
Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical
boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s using the
Hartree–Fock approximation and the
pseudopotential
In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced ...
interaction model.
A
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
(BEC) is a gas of
bosons
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer ...
that are in the same
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
, and thus can be described by the same
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
. A free quantum particle is described by a single-particle
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
. 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 The scattering length in quantum mechanics describes low-energy scattering. For potentials that decay faster than 1/r^3 as r\to \infty, it is defined as the following low-energy limit (mathematics), limit:
:
\lim_ k\cot\delta(k) =- \frac\;,
wher ...
(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 physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced ...
. At sufficiently low temperature, where the
de Broglie wavelength
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
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
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
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
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
in a
Bose–Einstein condensate
In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero (−273.15 °C or −459.67&n ...
. It is similar in form to the
Ginzburg–Landau equation and is sometimes referred to as the "
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
".
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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...
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
In physics, energy density is the amount of energy stored in a given system or region of space per unit volume. It is sometimes confused with energy per unit mass which is properly called specific energy or .
Often only the ''useful'' or extract ...
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
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
, which is found from the condition that the number of particles is related to the
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
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
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
, 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).
Soliton
A one-dimensional
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
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
A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological ...
, 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.
Healing length
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).
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
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
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 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.
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 in the Gross–Pitaevskii equation. 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.
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
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
:
:
:
Assuming that
:
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
Second sound is a quantum mechanical phenomenon in which heat transfer occurs by wave-like motion, rather than by the more usual mechanism of diffusion. Its presence leads to a very high thermal conductivity. It is known as "second sound" because t ...
. 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
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
. 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
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
.
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.
Derivations and Generalisations
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.
The semi-classical limit is reached for a large number of quanta, expressing the field theory either in the positive-P representation (generalised
Glauber-Sudarshan P representation) or
Wigner representation.
Finite-temperature effects can be treated within a generalised Gross-Pitaevskii equation by including scattering between condensate and noncondensate atoms, from which the Gross-Pitaevskii equation may be recovered in the low-temperature limit.
References
Further reading
*
*
External links
Trotter-Suzuki-MPITrotter-Suzuki-MPI is a library for large-scale simulations based on the
Trotter-Suzuki decomposition that can also address the Gross–Pitaevskii equation.
XMDSXMDS is a spectral partial differential equation library that can be used to solve the Gross-Pitaevskii equation.
{{DEFAULTSORT:Gross-Pitaevskii equation
Bose–Einstein condensates
Superfluidity