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Nonlinear Schrödinger Equation with Dissipation: Two Models for Bose-Einstein Condensates

V. V. Konotop

Springer eBooksApr 6, 20083 citations

Schrödinger equation for the one-particle density matrix of thermal systems: an alternative formulation of Bose–Einstein condensation

Subodha Mishra, Peter Pfeifer

We formulate a linear Schrödinger equation with a temperature-dependent potential for the one-particle density matrix and obtain the condensation temperature of the Bose–Einstein condensate from a bound-state condition for the Schrödinger equation both with and without the confining trap. The results are in very good agreement with those of the full statistical physics treatment. This is an alternative to the Bose–Einstein condensation in the standard ideal Bose-gas treatment.

arXiv:cond-mat/0611506Journal of Physics A Mathematical and TheoreticalFeb 21, 20075 citations

Optimal conversion of an atomic to a molecular Bose-Einstein condensate

Thomas Hornung, S. N. Gordienko, Regina de Vivie‐Riedle +1 more

The work in this article extends the optimal control framework of variational calculus to optimize the conversion of a Bose-Einstein condensate of atoms to one of molecules. It represents the derivation of the closed form optimal control equations for a system governed by a nonlinear Schr\"odinger equation and its successful application. It was necessary to derive a density matrix formulation of the coupled Gross-Pitaevskii equations to optimize STIRAP-like Raman light fields, to overcome dissipation.The work in this article extends the optimal control framework of variational calculus to optimize the conversion of a Bose-Einstein condensate of atoms to one of molecules. It represents the derivation of the closed form optimal control equations for a system governed by a nonlinear Schr\"odinger equation and its successful application. It was necessary to derive a density matrix formulation of the coupled Gross-Pitaevskii equations to optimize STIRAP-like Raman light fields, to overcome dissipation.

Physical Review AOct 11, 200236 citations

Stability of the solution of the nonlinear Schrödinger equation for Bose-Einstein condensation

Yeong E. Kim, Alexander L. Zubarev

arXiv:cond-mat/9803174Physics Letters ASep 1, 199811 citations

1 3D BEC Bright Solitons under Transverse Confinement: Analytical Results with the Nonpolynomial Schrödinger Equation

Luca Salasnich

The Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopicThe Bose-Einstein condensate (BEC) of a dilute gas of bosons is well described by the three-dimensional Gross-Pitaevskii equation (3D GPE), that is a nonlinear Schrödinger equation. By imposing a transverse confinement the BEC can travel only in the cylindrical axial direction. We show that in this case the BEC with attractive interaction admits a 3D bright soliton solution which generalizes the text-book one, that is valid in the one-dimensional limit (1D GPE). Contrary to the 1D case, the 3D bright soliton exists only below a critical number of Bosons that depends on the extent of confinement. Finally, we find that the 3D bright soliton collapses if its density excedes a critical value. Our results are obtained by using a nonpolynomial Schrödinger equation (NPSE), an effective one-dimensional equation derived from the 3D GPE. In a dilute gas of N Bosons at zero temperature practically all particles are in the same single-particle state of the density matrix ρ(r,r ′;t). 1), 2) This macroscopically occupied single-particle state is called Bose-Einstein condensate (BEC) and it is well described by a complex classical field ψ(r,t) (order parameter or macroscopic

Jan 1, 20020 citations

Many-Body Schrödinger Dynamics of Bose-Einstein Condensates

Kaspar Sakmann

Springer eBooksJan 1, 201125 citations

Schrödinger equation for the one-particle density matrix of thermal systems: an alternative formulation of Bose-Einstein condensation

Abstract

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Schrödinger equation for the one-particle density matrix of thermal systems: an alternative formulation of Bose-Einstein condensation

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