Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0301023 · Physical Review Letters

Derivation of the Gross-Pitaevskii Equation for Condensed Bosons from the Bogoliubov–de Gennes Equations for Superfluid Fermions

P. Pieri, Giancarlo Calvanese Strinati

Published Jul 15, 2003• 123 citations
arXivHTMLPDF

Abstract

We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that describe superfluid fermions in the presence of an external potential. Three-body corrections to the Gross-Pitaevskii equation are also obtained by our approach. Our results are relevant to the recent advances with ultracold fermionic atoms in a trap.

DOI
10.1103/physrevlett.91.030401

Related papers

Self-Consistent Derivation of the Modified Gross–Pitaevskii Equation with Lee–Huang–Yang Correction

Luca Salasnich

We consider a dilute and ultracold bosonic gas of weakly-interacting atoms. Within the framework of quantum field theory, we derive a zero-temperature modified Gross–Pitaevskii equation with beyond-mean-field corrections due to quantum depletion and anomalous density. This result is obtained from the stationary equation of the Bose–Einstein order parameter coupled to the Bogoliubov–de Gennes equations of the out-of-condensate field operator. We show that, in the presence of a generic external trapping potential, the key steps to get the modified Gross–Pitaevskii equation are the semiclassical approximation for the Bogoliubov–de Gennes equations, a slowly-varying order parameter and a small quantum depletion. In the uniform case, from the modified Gross–Pitaevskii equation, we get the familiar equation of state with Lee–Huang–Yang correction.

arXiv:1810.02636•Applied Sciences•Oct 21, 2018•20 citations

The Gross-Pitaevskii equations and beyond for inhomogeneous condensed bosons

G. G. N. Angilella, Saverio Bartalini, F. S. Cataliotti +3 more

A simple derivation of the static Gross-Pitaevskii (GP) equation is given from an energy variational principle. The result is then generalized heuristically to the time-dependent GP form. With this as background, a number of different experimental areas explored very recently are reviewed, in each case contact being established between the measurements and the predictions of the GP equations. The various limitations of these equations as used on dilute inhomogeneous condensed Boson atomic gases are then summarized, reference also being made to the fact that there is no many-body wave function underlying the GP formulation. This then leads into a discussion of a recently proposed integral equation, derived by taking the Bogoliubov-de Gennes equation as starting point. Some limitations of the static GP differential equation are thereby removed, though it is a matter of further study to determine whether a correlated wave function exists as underpinning for the integral equation formulation.

arXiv:cond-mat/0410287•Oct 12, 2004•0 citations

Some current issues concerning statics and dynamics of inhomogeneous superfluid gases

Norman Henry March, Monica Tosi

After a brief introduction to the static Gross–Pitaevskii (GP) differential equation for the ground state of a Bose–Einstein condensed gas and an example of its use in describing a vortex array in a trapped condensate, a recent generalization is reviewed. By starting from the Bogoliubov–de Gennes equations for superfluid fermions, an integral equation results. This is shown to contain the static GP equation as a limiting case, by means of a low-order gradient expansion of the order parameter. The time-dependent GP approximation is then presented and illustrated by a numerical example concerning drop emission under gravity from a gaseous condensate in an optical lattice. Further selected topics of present interest relating more generally to the dynamics of trapped superfluid gases concern the effects of temperature on collective excitations in a Bose gas and the fingerprints that these excitations in a gas of paired fermions contain of the crossover from a Bardeen–Cooper–Schrieffer correlated assembly to a Bose–Einstein condensate of composite bosons. Finally, some directions are pointed in which interaction of first-principles theory with experiment should prove fruitful.

Physics and Chemistry of Liquids•Oct 1, 2004•0 citations

Gross-Pitaevskii Equation for a Strongly Interacting Superfluid Boson System

Donald H. Kobe

The Gross-Pitaevskii equation for the order parameter of a superfluid boson system is derived in a form which is applicable to a system in which the two-body potential has a strong repulsive core, as is the case for liquid helium. The equation of motion for the order parameter, defined as the matrix element of the field annihilation operator between the states of an ($N\ensuremath{-}1$)-particle and an $N$-particle system, is obtained. The term describing the average potential due to the other particles is expanded in a perturbation series, in which the coherentstate representation is used. The resulting expansion is the same as using the Bogoliubov approximation and Wick's theorem, and shows that the Bogoliubov expansion is exact in the limit of an infinite number of particles. The perturbation expansion is partially resummed to give a Hartree field in which the potential has been replaced by the $T$ matrix, which gives a finite average effective field.

Physical Review A•Feb 1, 1972•5 citations

Bose representation for a strongly coupled nonequilibrium fermionic superfluid in a time-dependent trap

I. V. Tokatly

Using the functional integral formulation of nonequilibrium quantum many-body theory we develop a regular description of a Fermi system with a strong attractive interaction in the presence of an external time-dependent potential. In the strong coupling limit this fermionic system is equivalent to a nonequilibrium dilute Bose gas of diatomic molecules. We also consider nonequilibrium strongly coupled Bardeen-Cooper-Schrieffer theory and show that it reduces to the full nonlinear time-dependent Gross-Pitaevski equation, which determines the evolution of the condensate wave function.

arXiv:cond-mat/0401603•Physical Review A•Oct 4, 2004•2 citations

Temperature dependence of a vortex in a superfluid Fermi gas

Stefano Simonucci, P. Pieri, Giancarlo Calvanese Strinati

The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.

arXiv:1303.5229•Physical Review B•Jun 10, 2013•30 citations
We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that describe superfluid fermions in the presence of an external potential. Three-body corrections to the Gross-Pitaevskii equation are also obtained by our approach. Our results are relevant to the recent advances with ultracold fermionic atoms in a trap.
We consider a dilute and ultracold bosonic gas of weakly-interacting atoms. Within the framework of quantum field theory, we derive a zero-temperature modified Gross–Pitaevskii equation with beyond-mean-field corrections due to quantum depletion and anomalous density. This result is obtained from the stationary equation of the Bose–Einstein order parameter coupled to the Bogoliubov–de Gennes equations of the out-of-condensate field operator. We show that, in the presence of a generic external trapping potential, the key steps to get the modified Gross–Pitaevskii equation are the semiclassical approximation for the Bogoliubov–de Gennes equations, a slowly-varying order parameter and a small quantum depletion. In the uniform case, from the modified Gross–Pitaevskii equation, we get the familiar equation of state with Lee–Huang–Yang correction.
A simple derivation of the static Gross-Pitaevskii (GP) equation is given from an energy variational principle. The result is then generalized heuristically to the time-dependent GP form. With this as background, a number of different experimental areas explored very recently are reviewed, in each case contact being established between the measurements and the predictions of the GP equations. The various limitations of these equations as used on dilute inhomogeneous condensed Boson atomic gases are then summarized, reference also being made to the fact that there is no many-body wave function underlying the GP formulation. This then leads into a discussion of a recently proposed integral equation, derived by taking the Bogoliubov-de Gennes equation as starting point. Some limitations of the static GP differential equation are thereby removed, though it is a matter of further study to determine whether a correlated wave function exists as underpinning for the integral equation formulation.
After a brief introduction to the static Gross–Pitaevskii (GP) differential equation for the ground state of a Bose–Einstein condensed gas and an example of its use in describing a vortex array in a trapped condensate, a recent generalization is reviewed. By starting from the Bogoliubov–de Gennes equations for superfluid fermions, an integral equation results. This is shown to contain the static GP equation as a limiting case, by means of a low-order gradient expansion of the order parameter. The time-dependent GP approximation is then presented and illustrated by a numerical example concerning drop emission under gravity from a gaseous condensate in an optical lattice. Further selected topics of present interest relating more generally to the dynamics of trapped superfluid gases concern the effects of temperature on collective excitations in a Bose gas and the fingerprints that these excitations in a gas of paired fermions contain of the crossover from a Bardeen–Cooper–Schrieffer correlated assembly to a Bose–Einstein condensate of composite bosons. Finally, some directions are pointed in which interaction of first-principles theory with experiment should prove fruitful.
The Gross-Pitaevskii equation for the order parameter of a superfluid boson system is derived in a form which is applicable to a system in which the two-body potential has a strong repulsive core, as is the case for liquid helium. The equation of motion for the order parameter, defined as the matrix element of the field annihilation operator between the states of an ($N\ensuremath{-}1$)-particle and an NNN-particle system, is obtained. The term describing the average potential due to the other particles is expanded in a perturbation series, in which the coherentstate representation is used. The resulting expansion is the same as using the Bogoliubov approximation and Wick's theorem, and shows that the Bogoliubov expansion is exact in the limit of an infinite number of particles. The perturbation expansion is partially resummed to give a Hartree field in which the potential has been replaced by the TTT matrix, which gives a finite average effective field.
Using the functional integral formulation of nonequilibrium quantum many-body theory we develop a regular description of a Fermi system with a strong attractive interaction in the presence of an external time-dependent potential. In the strong coupling limit this fermionic system is equivalent to a nonequilibrium dilute Bose gas of diatomic molecules. We also consider nonequilibrium strongly coupled Bardeen-Cooper-Schrieffer theory and show that it reduces to the full nonlinear time-dependent Gross-Pitaevski equation, which determines the evolution of the condensate wave function.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.