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arXiv:1302.3357 · Modern Physics Letters B

NONINTEGRAL FORM OF THE GROSS–PITAEVSKII EQUATION FOR POLARIZED MOLECULES

Pavel A. Andreev

Published May 10, 2013• 7 citations
arXivHTMLPDF

Abstract

The Gross–Pitaevskii equation for polarized molecules is an integro-differential equation, consequently it is complicated for solving. We find a possibility to represent it as a nonintegral nonlinear Schrödinger equation, but this equation should be coupled with two linear equations describing the electric field. These two equations are the Maxwell equations. We recapture the dispersion of collective excitations in the three-dimensional electrically polarized BEC with no evolution of the electric dipole moment directions. We trace the contribution of the electric dipole moment. We explicitly consider the contribution of the electric dipole moment in the interaction constant for the short-range interaction. We show that the spectrum of dipolar BEC does not reveal instability at repulsive short-range interaction. Nonlinear excitations are also considered. We present dependence of the bright soliton characteristics on the electric dipole moment.

DOI
10.1142/s0217984913500966

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The Gross–Pitaevskii equation for polarized molecules is an integro-differential equation, consequently it is complicated for solving. We find a possibility to represent it as a nonintegral nonlinear Schrödinger equation, but this equation should be coupled with two linear equations describing the electric field. These two equations are the Maxwell equations. We recapture the dispersion of collective excitations in the three-dimensional electrically polarized BEC with no evolution of the electric dipole moment directions. We trace the contribution of the electric dipole moment. We explicitly consider the contribution of the electric dipole moment in the interaction constant for the short-range interaction. We show that the spectrum of dipolar BEC does not reveal instability at repulsive short-range interaction. Nonlinear excitations are also considered. We present dependence of the bright soliton characteristics on the electric dipole moment.
We present the Gross–Pitaevskii equation for Bose–Einstein condensates (BECs) possessing the electric dipole and the electric quadrupole moments in a non-integral form. These equations are coupled with the Maxwell equations. The model under consideration includes the dipole–dipole, the dipole–quadrupole, and the quadrupole–quadrupole interactions in terms of the electric field created by the dipoles and quadrupoles. We apply this model to obtain the Bogoliubov spectrum for three-dimensional BECs with a repulsive short-range interaction. We obtain an extra term in the Bogoliubov spectrum in comparison with the dipolar BECs. We show that the quadrupole–quadrupole interaction gives a positive contribution in the Bogoliubov spectrum. Hence, this spectrum is stable.
We obtain equations of quantum hydrodynamic (QHD) for magnetized spin-1 neutral Bose-Einstein condensate (BEC). System of QHD equations contains the equation of magnetic moment evolution (an analog of the Bloch equation). We account spin-spin interaction along with the short range interaction. We consider self-consistent field approximation of QHD equations. Starting from QHD equation we derive the Gross-Pitaevskii equation for magnetized BEC. We show that Gross-Pitaevskii equation exists under condition that the magnetic moment direction is not change. Using obtained QHD equations we study the dispersion of collective excitation. As in electrically polarized BEC [P. A. Andreev, L. S. Kuz'menkov, arXiv: 1201.2440], in the magnetically polarized BEC there is second wave mode (polarization mode or spin wave), in addition to the Bogoliubov's mode. Second wave solution appears due to the magnetic moment evolution. The influence of magnetization on dispersion of Bogoliubov's mode is found. We found strong difference of dispersion properties of waves in magnetized BEC from electrically polarized BEC.
We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.
We propose the construction of a set of quantum hydrodynamics equations for the Bose-Einstein condensate (BEC) where atoms have electric dipole moment (EDM). The contribution of the dipole-dipole interactions (DDI) to the Euler equation is estimated. Quantum equations for the evolution of medium polarization are constructed for the first time. The mathematical method we developed allows studying the effects of interactions on the evolution of polarization. The developed method may be applied to various physical systems in which dynamics is affected by DDI. A problem of elementary excitations in BEC, either affected or not affected by the uniform external electric field, is addressed using our method. We show that the evolution of polarization in BEC leads to the formation of a novel type of elementary excitations. Also, we consider the process of wave generation in polarized BEC by means of monoenergetic beam of neutral polarized particles. We compute the possibilities of the generation of Bogoliubov's modes and polarization modes by the dipole beam.
A one-dimensional (1D) Gross-Pitaevskii (GP) equation which governs the dynamics of Bose-Einstein condensates (BECs) with time-dependent scattering length in a harmonic potential is considered. Through a modified phase imprint transformation into the macroscopic order parameter, we suitably generate a derivative cubic-quintic Gross-Pitaevskii (DCQ-GP) equation that describes the dynamics of condensates in the harmonic potential for both twoand three-body interatomic interactions in the presence of nonlinear dispersion. Employing a modified lens-type transformation, we reduce the DCQ-GP equation into a derivative cubic-quintic Ginzburg-Landau (DCQ-GL) equation that allows us to investigate the modulational instability (MI) of the system and study, through the analytical exact solutions of the considered 1D GP equation, the effects of the three-body interactions and nonlinear dispersion on the dynamics of the BECs. We observe that for BECs with repulsive two-body interatomic interactions, the attractive three-body interactions and the nonlinear dispersion have opposite effects on the condensate: (i) The attractive three-body interactions enhance the instability, while the nonlinear dispersion softens the instability of the system; (ii) the attractive three-body interactions contributes to the ejections of atoms from the condensates while the nonlinear dispersion contributes to the accumulation of atoms in the condensates; (iii) the attractive three-body interactions contributes to the broadening of the matter wave solitons, while the nonlinear dispersion contributes to the compression of the matter wave solitons.
Further development of the method of quantum hydrodynamics in application for Bose-Einstein condensates (BECs) is presented. To consider evolution of polarization direction along with particles movement we have developed corresponding set of quantum hydrodynamic equations. It includes equations of the polarization evolution and the polarization current evolution along with the continuity equation and the Euler equation (the momentum balance equation). Dispersion properties of the transverse waves including the electromagnetic waves propagating through the BECs are considered. To this end we consider full set of the Maxwell equations for description of electromagnetic field dynamics. This approximation gives us possibility to consider the electromagnetic waves along with the matter waves. We find a splitting of the electromagnetic wave dispersion on two branches. As a result we have four solutions, two for the electromagnetic waves and two for the matter waves, the last two are the concentration-polarization waves appearing as a generalization of the Bogoliubov mode. We also obtain that if the matter wave propagate perpendicular to external electric field when dipolar contribution does not disappear (as it follows from our generalization of the Bogoliubov spectrum). In this case exist a small dipolar frequency shift due to transverse electric field of perturbation.