Gross–Pitaevskii equation

From Wikipedia, the free encyclopedia

The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross[1] and Lev Petrovich Pitaevskii[2]) 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,[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 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.[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 Gross-Pitaevskii equation[9] (sometimes called modified[10] or generalized nonlinear Schrödinger equation[11]).

Form of equation[edit]

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.


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[edit]

Free particle[edit]

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,[12] an interacting Bose gas does not exhibit an energy gap (in the case of repulsive interactions).


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,[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[edit]

The healing length can be understood as the length scale where the kinetic energy of the boson equals the chemical potential:[3]

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[edit]

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.

Numerical solutions[edit]

Several numerical methods, such as the split-step Crank–Nicolson[13] and Fourier spectral[14] methods, have been used for solving GPE. There are also different Fortran and C programs for its solution for the contact interaction[15][16] and long-range dipolar interaction.[17]

Thomas–Fermi approximation[edit]

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.[3]

Bogoliubov approximation[edit]

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. 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.[18] 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[edit]

Vortex dipole trap with topological charge loaded by ultracold ensemble

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:[19]

In a reference frame rotating with angular velocity , time-dependent Gross–Pitaevskii equation with helical potential is[20]

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 :[21]

The angular momentum of helically trapped condensate is exactly zero:[20]

Numerical modeling of cold atomic ensemble in spiral potential have shown the confinement of individual atomic trajectories within helical potential well.[22]

Derivations and Generalisations[edit]

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.[23] 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,[24][25][26][27][28] from which the Gross-Pitaevskii equation may be recovered in the low-temperature limit.[29][30]


  1. ^ E. P. Gross (1961). "Structure of a quantized vortex in boson systems". Il Nuovo Cimento. 20 (3): 454–457. Bibcode:1961NCim...20..454G. doi:10.1007/BF02731494. S2CID 121538191.
  2. ^ L. P. Pitaevskii (1961). "Vortex lines in an imperfect Bose gas". Sov. Phys. JETP. 13 (2): 451–454.
  3. ^ a b c d Foot, C. J. (2005). Atomic physics. Oxford University Press. pp. 231–240. ISBN 978-0-19-850695-9.
  4. ^ Lopes, Raphael; Eigen, Christoph; Navon, Nir; Clément, David; Smith, Robert P.; Hadzibabic, Zoran (2017-11-07). "Quantum Depletion of a Homogeneous Bose-Einstein Condensate". Physical Review Letters. 119 (19): 190404. arXiv:1706.01867. Bibcode:2017PhRvL.119s0404L. doi:10.1103/PhysRevLett.119.190404. ISSN 0031-9007. PMID 29219529. S2CID 206302070.
  5. ^ Chang, R.; Bouton, Q.; Cayla, H.; Qu, C.; Aspect, A.; Westbrook, C. I.; Clément, D. (2016-12-02). "Momentum-Resolved Observation of Thermal and Quantum Depletion in a Bose Gas". Physical Review Letters. 117 (23): 235303. arXiv:1608.04693. Bibcode:2016PhRvL.117w5303C. doi:10.1103/PhysRevLett.117.235303. ISSN 0031-9007. PMID 27982640. S2CID 10967623.
  6. ^ Lee, T. D.; Yang, C. N. (1957-02-01). "Many-Body Problem in Quantum Mechanics and Quantum Statistical Mechanics". Physical Review. 105 (3): 1119–1120. Bibcode:1957PhRv..105.1119L. doi:10.1103/PhysRev.105.1119. ISSN 0031-899X.
  7. ^ Lee, T. D.; Huang, Kerson; Yang, C. N. (1957-06-15). "Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties". Physical Review. 106 (6): 1135–1145. Bibcode:1957PhRv..106.1135L. doi:10.1103/PhysRev.106.1135. ISSN 0031-899X.
  8. ^ Lieb, Elliott H.; Liniger, Werner (1963-05-15). "Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State". Physical Review. 130 (4): 1605–1616. Bibcode:1963PhRv..130.1605L. doi:10.1103/PhysRev.130.1605. ISSN 0031-899X.
  9. ^ Kopyciński, Jakub; Łebek, Maciej; Marciniak, Maciej; Ołdziejewski, Rafał; Górecki, Wojciech; Pawłowski, Krzysztof (2022-01-14). "Beyond Gross-Pitaevskii equation for 1D gas: quasiparticles and solitons". SciPost Physics. 12 (1): 023. arXiv:2106.15289. Bibcode:2022ScPP...12...23K. doi:10.21468/SciPostPhys.12.1.023. ISSN 2542-4653. S2CID 235670023.
  10. ^ Choi, S.; Dunjko, V.; Zhang, Z. D.; Olshanii, M. (2015-09-10). "Monopole Excitations of a Harmonically Trapped One-Dimensional Bose Gas from the Ideal Gas to the Tonks-Girardeau Regime". Physical Review Letters. 115 (11): 115302. arXiv:1412.6855. Bibcode:2015PhRvL.115k5302C. doi:10.1103/PhysRevLett.115.115302. ISSN 0031-9007. PMID 26406838. S2CID 2987641.
  11. ^ Peotta, Sebastiano; Ventra, Massimiliano Di (2014-01-24). "Quantum shock waves and population inversion in collisions of ultracold atomic clouds". Physical Review A. 89 (1): 013621. arXiv:1303.6916. Bibcode:2014PhRvA..89a3621P. doi:10.1103/PhysRevA.89.013621. ISSN 1050-2947. S2CID 119290214.
  12. ^ N. M. Hugenholtz; D. Pines (1959). "Ground-state energy and excitation spectrum of a system of interacting bosons". Physical Review. 116 (3): 489–506. Bibcode:1959PhRv..116..489H. doi:10.1103/PhysRev.116.489.
  13. ^ P. Muruganandam and S. K. Adhikari (2009). "Fortran Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 180 (3): 1888–1912. arXiv:0904.3131. Bibcode:2009CoPhC.180.1888M. doi:10.1016/j.cpc.2009.04.015. S2CID 7403553.
  14. ^ P. Muruganandam and S. K. Adhikari (2003). "Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods". J. Phys. B. 36 (12): 2501–2514. arXiv:cond-mat/0210177. Bibcode:2003JPhB...36.2501M. doi:10.1088/0953-4075/36/12/310. S2CID 250851068.
  15. ^ D. Vudragovic; et al. (2012). "C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 183 (9): 2021–2025. arXiv:1206.1361. Bibcode:2012CoPhC.183.2021V. doi:10.1016/j.cpc.2012.03.022. S2CID 12031850.
  16. ^ L. E. Young-S.; et al. (2016). "OpenMP Fortran and C Programs for the time-dependent Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 204 (9): 209–213. arXiv:1605.03958. Bibcode:2016CoPhC.204..209Y. doi:10.1016/j.cpc.2016.03.015. S2CID 206999817.
  17. ^ R. Kishor Kumar; et al. (2015). "Fortran and C Programs for the time-dependent dipolar Gross-Pitaevskii equation in a fully anisotropic trap". Comput. Phys. Commun. 195 (2015): 117–128. arXiv:1506.03283. Bibcode:2015CoPhC.195..117K. doi:10.1016/j.cpc.2015.03.024. S2CID 18949735.
  18. ^ C. Raman; M. Köhl; R. Onofrio; D. S. Durfee; C. E. Kuklewicz; Z. Hadzibabic; W. Ketterle (1999). "Evidence for a Critical Velocity in a Bose–Einstein Condensed Gas". Phys. Rev. Lett. 83 (13): 2502. arXiv:cond-mat/9909109. Bibcode:1999PhRvL..83.2502R. doi:10.1103/PhysRevLett.83.2502. S2CID 14070421.
  19. ^ A. Yu. Okulov (2008). "Angular momentum of photons and phase conjugation". J. Phys. B: At. Mol. Opt. Phys. 41 (10): 101001. arXiv:0801.2675. Bibcode:2008JPhB...41j1001O. doi:10.1088/0953-4075/41/10/101001. S2CID 13307937.
  20. ^ a b A. Yu. Okulov (2012). "Cold matter trapping via slowly rotating helical potential". Phys. Lett. A. 376 (4): 650–655. arXiv:1005.4213. Bibcode:2012PhLA..376..650O. doi:10.1016/j.physleta.2011.11.033. S2CID 119196009.
  21. ^ A. Yu. Okulov (2013). "Superfluid rotation sensor with helical laser trap". J. Low Temp. Phys. 171 (3): 397–407. arXiv:1207.3537. Bibcode:2013JLTP..171..397O. doi:10.1007/s10909-012-0837-7. S2CID 118601627.
  22. ^ A. Al. Rsheed1, A. Lyras, V. E. Lembessis, O. M. Aldossary (2016). "Guiding of atoms in helical optical potential structures". J. Phys. B: At. Mol. Opt. Phys. 49 (12): 125002. Bibcode:2016JPhB...49l5002R. doi:10.1088/0953-4075/49/12/125002. S2CID 124660886.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  23. ^ Steel, M J; Olsen, M K; Plimak, L I; Drummond, P D; Tan, S M; Collett, M J; Walls, D F; Graham, R (1998). "Dynamical quantum noise in trapped Bose-Einstein condensates". Physical Review A. 58 (6): 4824–4835. arXiv:cond-mat/9807349. doi:10.1103/PhysRevA.58.4824. S2CID 43217083.
  24. ^ Zaremba, E; Nikuni, T; Griffin, A (1999). "Dynamics of Trapped Bose Gases at Finite Temperatures". Journal of Low Temperature Physics. 116 (3–4): 277–345. doi:10.1023/A:1021846002995. S2CID 37753.
  25. ^ Stoof, H T C (1999). "Coherent versus incoherent dynamics during Bose-Einstein condensation in atomic gases". Journal of Low Temperature Physics. 114 (1–2): 11–108. doi:10.1023/A:1021897703053. S2CID 16107086.
  26. ^ Davis, M J; Morgan, S A; Burnett, K (2001). "Simulations of Bose Fields at Finite Temperature". Physical Review Letters. 87 (16): 160402. arXiv:cond-mat/0011431. doi:10.1103/PhysRevLett.87.160402. PMID 11690189. S2CID 14195702.
  27. ^ Gardiner, C W; Davis, M J (2003). "The stochastic Gross–Pitaevskii equation: II". Journal of Physics B: Atomic, Molecular and Optical Physics. 36 (23): 4731–4753. arXiv:cond-mat/0308044. doi:10.1088/0953-4075/36/23/010. S2CID 250874049.
  28. ^ Gardiner, S A; Morgan, S A (2007). "Number-conserving approach to a minimal self-consistent treatment of condensate and noncondensate dynamics in a degenerate Bose gas" (PDF). Physical Review A. 75 (4): 261. doi:10.1103/PhysRevA.75.043621. S2CID 119432906.
  29. ^ Proukakis, Nick P.; Jackson, Brian (2008). "Finite-temperature models of Bose–Einstein condensation". Journal of Physics B: Atomic, Molecular and Optical Physics. 41 (20): 203002. arXiv:0810.0210. doi:10.1088/0953-4075/41/20/203002. ISSN 0953-4075. S2CID 118561792. Retrieved 2022-02-14.
  30. ^ Blakie, P.B.; Bradley, A.S.; Davis, M.J.; Ballagh, R.J.; Gardiner, C.W. (2008-09-01). "Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques". Advances in Physics. 57 (5): 363–455. arXiv:0809.1487. doi:10.1080/00018730802564254. ISSN 0001-8732. S2CID 14999178. Retrieved 2021-12-05.

Further reading[edit]

External links[edit]

  • Trotter-Suzuki-MPI Trotter-Suzuki-MPI is a library for large-scale simulations based on the Trotter-Suzuki decomposition that can also address the Gross–Pitaevskii equation.
  • XMDS XMDS is a spectral partial differential equation library that can be used to solve the Gross-Pitaevskii equation.