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:1006.2161 · Physical Review B

Exact results on the two-particle Green’s function of a Bose-Einstein condensate

Takafumi Kita

Published Jun 14, 2010• 18 citations
arXivHTMLPDF

Abstract

Starting from the Dyson-Beliaev and generalized Gross-Pitaevskii equations with an extra nonlocal potential, we derive an exact expression of the two-particle Green's function $\underset{̱}{\mathcal{K}}$ for an interacting Bose-Einstein condensate in terms of unambiguously defined self-energies and vertices. The formula can be a convenient basis for approximate calculations of $\underset{̱}{\mathcal{K}}$. It also tells us that poles of $\underset{̱}{\mathcal{K}}$ are not shared with (i.e., shifted from) those of the single-particle Green's function, contrary to the conclusion of previous studies.

DOI
10.1103/physrevb.81.214513

Related papers

Quantum Field Theory of Correlated Bose–Einstein Condensates: I. Basic Formalism

Takafumi Kita

Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green's functions are found to share poles, i.e., the structure of the two-particle Green's functions predicted by Gavoret and Nozières for a homogeneous condensate at $T=0$ is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.

arXiv:2101.02389•Journal of the Physical Society of Japan•Jan 7, 2021•4 citations

Gross-Pitaevskii equation for Bose particles in a double-well potential: Two-mode models and beyond

D. Ananikian, T. H. Bergeman

In this work, our primary goal has been to explore the range of validity of two-mode models for Bose-Einstein condensates in double-well potentials. Our derivation, like others, uses symmetric and antisymmetric condensate basis functions for the Gross-Pitaevskii equation. In what we call an ``improved two-mode model'' (I2M), the tunneling coupling energy explicitly includes a nonlinear interaction term, which has been given previously in the literature but not widely appreciated. We show that when the atom number (and hence the extent of the wave function) in each well vary appreciably with time, the nonlinear interaction term produces a temporal change in the tunneling energy or rate, which has not previously been considered to our knowledge. In addition, we obtain a parameter, labeled ``interaction tunneling,'' that produces a decrease of the tunneling energy when the wave functions in the two wells overlap to some extent. Especially for larger values of the nonlinear interaction term, results from this model produce better agreement with numerical solutions of the time-dependent Gross-Pitaevskii equation in one and three dimensions, as compared with models that have no interaction term in the tunneling energy. The usefulness of this model is demonstrated by good agreement with recent experimental results for the tunneling oscillation frequency [Albiez et al., Phys. Rev. Lett. 95, 010402 (2005)]. We also present equations and results for a multimode approach, and use the I2M model to obtain modified equations for the second-quantized version of the Bose-Einstein double-well problem.

arXiv:cond-mat/0503220•Physical Review A•Jan 5, 2006•260 citations

Microscopic theory of BEC phase transition in a critical region

Vitaly V. Kocharovsky, Vl. V. Kocharovsky

nd an exact Hamiltonian for Bose-Einstein condensation (BEC) in a mesoscopic system and derive the exact fundamental equations for the condensate wave function and Green’s functions, which are valid both inside and outside critical region. They are reduced to the usual Gross-Pitaevskii and Beliaev-Popov equations in a low-temperature limit outside critical region. The theory is readily extendable to other phase transitions. All these advances come from a correct account, rst, of the Noether’s symmetry con

Bulletin of the American Physical Society•Mar 6, 2015•1 citation

Green function method and the Gross-Pitaevskii equation

Carlos A Trallero-Herrero, Victor Lopez‐Richard, Julio C. Drake-Pérez +1 more

"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."

Revista Mexicana de Física•Jan 1, 2007•0 citations

Overlap of exact and Gross-Pitaevskii wave functions in Bose-Einstein condensates of dilute gases

Shachar Klaiman, L. S. Cederbaum

It has been proven theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the Gross-Pitaevskii equation provides the exact energy and density per particle as does the basic many-particle Schr\"odinger equation [E. H. Lieb and R. Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. Here, we investigate the overlap of the Gross-Pitaevskii and exact ground-state wave functions. It is found that this overlap is always smaller than unity and may even vanish despite the fact that both wave functions provide the same energy and density per particle. Consequences are discussed.

arXiv:1609.05895•Physical Review A•Dec 29, 2016•25 citations

Bose-Einstein condensates with attractive 1∕r interaction: The case of self-trapping

I. Papadopoulos, P. Wagner, G. Wunner +1 more

Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive $1∕r$ interaction, we point out that the instance of self-trapping of the condensate, without an external trap potential, is physically best understood by introducing appropriate ``atomic'' units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters ${N}^{2}a∕{a}_{u}$ and $\ensuremath{\gamma}∕{N}^{2}$, where $N$ is the particle number, $a$ the scattering length, ${a}_{u}$ the ``Bohr'' radius, and $\ensuremath{\gamma}$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, and for energies, sizes, and peak densities, and compare with previous variational results. We point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.

arXiv:0709.2866•Physical Review A•Nov 6, 2007•38 citations
Starting from the Dyson-Beliaev and generalized Gross-Pitaevskii equations with an extra nonlocal potential, we derive an exact expression of the two-particle Green's function \underset{̱}{\mathcal{K}} for an interacting Bose-Einstein condensate in terms of unambiguously defined self-energies and vertices. The formula can be a convenient basis for approximate calculations of \underset{̱}{\mathcal{K}}. It also tells us that poles of \underset{̱}{\mathcal{K}} are not shared with (i.e., shifted from) those of the single-particle Green's function, contrary to the conclusion of previous studies.
Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulated so as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identities among vertices originating from Goldstone's theorem I. The key inputs are irreducible four-point vertices, in terms of which we derive a closed system of equations for Green's functions, three- and four-point vertices, and two-particle Green's functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering any infrared divergence. The single- and two-particle Green's functions are found to share poles, i.e., the structure of the two-particle Green's functions predicted by Gavoret and Nozières for a homogeneous condensate at T=0T=0T=0 is also shown to persist at finite temperatures, in the presence of inhomogeneity, and also in nonequilibrium situations.
In this work, our primary goal has been to explore the range of validity of two-mode models for Bose-Einstein condensates in double-well potentials. Our derivation, like others, uses symmetric and antisymmetric condensate basis functions for the Gross-Pitaevskii equation. In what we call an ``improved two-mode model'' (I2M), the tunneling coupling energy explicitly includes a nonlinear interaction term, which has been given previously in the literature but not widely appreciated. We show that when the atom number (and hence the extent of the wave function) in each well vary appreciably with time, the nonlinear interaction term produces a temporal change in the tunneling energy or rate, which has not previously been considered to our knowledge. In addition, we obtain a parameter, labeled ``interaction tunneling,'' that produces a decrease of the tunneling energy when the wave functions in the two wells overlap to some extent. Especially for larger values of the nonlinear interaction term, results from this model produce better agreement with numerical solutions of the time-dependent Gross-Pitaevskii equation in one and three dimensions, as compared with models that have no interaction term in the tunneling energy. The usefulness of this model is demonstrated by good agreement with recent experimental results for the tunneling oscillation frequency [Albiez et al., Phys. Rev. Lett. 95, 010402 (2005)]. We also present equations and results for a multimode approach, and use the I2M model to obtain modified equations for the second-quantized version of the Bose-Einstein double-well problem.
nd an exact Hamiltonian for Bose-Einstein condensation (BEC) in a mesoscopic system and derive the exact fundamental equations for the condensate wave function and Green’s functions, which are valid both inside and outside critical region. They are reduced to the usual Gross-Pitaevskii and Beliaev-Popov equations in a low-temperature limit outside critical region. The theory is readily extendable to other phase transitions. All these advances come from a correct account, rst, of the Noether’s symmetry con
"The time-independent one-dimensional Gross-Pitaevskii (GP) equation for a Bose-Einstein condensate is solved by using Green functionformalism atT=0 K. From the integral GP equation, the analytical solution for the order parameter,©(x), and the chemical potential,π, has been formally obtained as functions of a single dimensionless parameter. The obtained solutions are valid in the whole range ofnegative and positive values of the non-linear interaction term that is used to characterize the condensates of alkali atoms. A function hasbeen introduced in order to describe the accumulated error that allows a comparative analysis with respect to known methods of solutionsuch as the Thomas-Fermi approximation and the “bright” soliton. This study sets universal ranges of parameters where analytical solutionscan be implemented."
It has been proven theoretically for bosons with two-body repulsive interaction potentials in the dilute limit that the Gross-Pitaevskii equation provides the exact energy and density per particle as does the basic many-particle Schr\"odinger equation [E. H. Lieb and R. Seiringer, Phys. Rev. Lett. 88, 170409 (2002)]. Here, we investigate the overlap of the Gross-Pitaevskii and exact ground-state wave functions. It is found that this overlap is always smaller than unity and may even vanish despite the fact that both wave functions provide the same energy and density per particle. Consequences are discussed.
Amplifying on a proposal by O'Dell et al. for the realization of Bose-Einstein condensates of neutral atoms with attractive 1∕r1∕r1∕r interaction, we point out that the instance of self-trapping of the condensate, without an external trap potential, is physically best understood by introducing appropriate ``atomic'' units. This reveals a remarkable scaling property: the physics of the condensate depends only on the two parameters N2a∕au{N}^{2}a∕{a}_{u}N2a∕au​ and $\ensuremath{\gamma}∕{N}^{2}$, where NNN is the particle number, aaa the scattering length, au{a}_{u}au​ the ``Bohr'' radius, and $\ensuremath{\gamma}$ the trap frequency in atomic units. We calculate accurate numerical results for self-trapping wave functions and potentials, and for energies, sizes, and peak densities, and compare with previous variational results. We point out the existence of a second solution of the extended Gross-Pitaevskii equation for negative scattering lengths, with and without trapping potential, which is born together with the ground state in a tangent bifurcation. This indicates the existence of an unstable collectively excited state of the condensate for negative scattering lengths.