A
BoseāEinstein 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
Schrƶdinger equation. Interaction between particles in a real gas is taken into account by a pertinent many-body Schrƶdinger equation. In the HartreeāFock 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 bosonāboson interaction,[3] 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 bosonāboson s-wave scattering length, and is the
Dirac delta-function.
The
variational method shows that if the single-particle wavefunction satisfies the following GrossāPitaevskii 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.
The non-linearity of the GrossāPitaevskii equation has its origin in the interaction between the particles: setting the coupling constant of interaction in the GrossāPitaevskii equation to zero (see the following section) recovers the single-particle Schrƶdinger equation describing a particle inside a trapping potential.
The GrossāPitaevskii equation is said to be limited to the weakly interacting regime. Nevertheless, it may also fail to reproduce interesting phenomena even within this regime.[4][5] In order to study the BEC beyond that limit of weak interactions, one needs to implement the Lee-Huang-Yang (LHY) correction.[6][7] Alternatively, in 1D systems one can use either an exact approach, namely the
Lieb-Liniger model,[8] or an extended equation, e.g. the Lieb-Liniger GrossāPitaevskii equation[9] (sometimes called modified[10] or generalized nonlinear Schrƶdinger equation[11]).
Form of equation
The equation has the form of the
Schrƶdinger 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 GrossāPitaevskii 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 GrossāPitaevskii equation, we can find the structure of a BoseāEinstein condensate in various external potentials (e.g. a harmonic trap).
The time-dependent GrossāPitaevskii equation is
From this equation we can look at the dynamics of the BoseāEinstein condensate. It is used to find the collective modes of a trapped gas.
Solutions
Since the GrossāPitaevskii equation is a
nonlinearpartial 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 GrossāPitaevskii equation, it leaves a gap in the energy spectrum due to the interaction:
According to the
HugenholtzāPines theorem,[12] 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 BoseāEinstein 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 GrossāPitaevskii equation is
where is the value of the condensate wavefunction at , and is the coherence length (a.k.a. the healing length,[3] 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.
Healing length
The healing length gives the minimum distance over which the
order parameter can heal, which describes how quickly the wave function of the BEC can adjust to changes in the potential. If the condensate density grows from 0 to n within a distance Ī¾, the healing length can calculated by equating the
quantum pressure and the interaction energy:[3][13]
The healing length 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 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).
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.
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 GrossāPitaevskii equation. This is called the
ThomasāFermi 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 approximation
Bogoliubov treatment of the GrossāPitaevskii equation is a method that finds the elementary excitations of a BoseāEinstein 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 GrossāPitaevskii 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. 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.[19] 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:[20]
In a reference frame rotating with angular velocity , time-dependent GrossāPitaevskii equation with helical potential is[21]
where is the angular-momentum operator.
The solution for condensate wavefunction is a superposition of two phase-conjugated matterāwave 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 :[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]
Derivations and Generalisations
The GrossāPitaevskii 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.[24] 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 GrossāPitaevskii equation by including scattering between condensate and noncondensate atoms,[25][26][27][28][29] from which the GrossāPitaevskii equation may be recovered in the low-temperature limit.[30][31]
^Zaremba, E; Nikuni, T; Griffin, A (1999). "Dynamics of Trapped Bose Gases at Finite Temperatures". Journal of Low Temperature Physics. 116 (3ā4): 277ā345.
doi:
10.1023/A:1021846002995.
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^Stoof, H T C (1999). "Coherent versus incoherent dynamics during Bose-Einstein condensation in atomic gases". Journal of Low Temperature Physics. 114 (1ā2): 11ā108.
doi:
10.1023/A:1021897703053.
S2CID16107086.