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:quant-ph/0204046

Center-of-mass motion in many-body theory of BEC

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

Published Apr 9, 2002• 0 citations
arXivHTMLPDF

Abstract

The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schrödinger equation in a harmonic potential proposed recently [ J. J. Garc\'ıa-Ripoll, V. M. Pérez-Garc\'ıa, and V. Vekslerchik, Phys. Rev. E {\bf 64}, 056602 (2001)] is extended to many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics and results in the separation of the center of mass motion from the internal dynamics of many-body systems. The center of mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.

DOI
10.48550/arxiv.quant-ph/0204046

Related papers

Center-of-mass motion in the many-body theory of Bose-Einstein condensates

Iwo Białynicki‐Birula, Zofia Białynicka-Birula

The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schr\"odinger equation in a harmonic potential proposed recently [J. J. Garc\'{\i}a-Ripoll, V. M. P\'erez-Garc\'{\i}a, and V. Vekslerchik, Phys. Rev. E 64, 056602 (2001)] is extended to the many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics, and results in the separation of the center-of-mass motion from the internal dynamics of many-body systems. The center-of-mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.

Physical Review A•Jun 3, 2002•33 citations

Center-of-Mass Motion in Many-Particle Systems

S. Gartenhaus, Charles Schwartz

An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.

Physical Review•Oct 15, 1957•271 citations

Generalized Center of Mass and Relative Motions in Classical Many-Body System: An Example of Solutions of Equations of Collective Submanifold

Masao Yamamura, A. Kuriyama, Shinji Iida

A theory of collective submanifold, recently, developed by the present authors (M. Y. and A. K.) is applied to a classical many-body system, the Hamiltonian of which depends on the relative coordinates. The solution enables the separation of the total degrees of freedom into the collective and the intrinsic ones, which may be called the generalized center of mass and the relative motions. The results can be applied to the large amplitude collective motion and its couplings with the intrinsic degrees of freedom, which cannot be given in the HF plus RPA method.

Progress of Theoretical Physics•Sep 1, 1984•4 citations

Center-of-Mass Motion in Many-Particle Systems. II. Critique of the Gartenhaus-Schwartz Transformation

David J. Ernst, Carl M. Shakin, Roy M. Thaler

We examine the Gartenhaus and Schwartz (G-S) transformation, which generates a translationally invariant wave function from one which does not possess this property. A clarification of the G-S procedure for the calculation of matrix elements is achieved if one takes two independent limits of the parameters $\ensuremath{\Lambda}$, ${\ensuremath{\Lambda}}^{\ensuremath{'}}$ which appear in the expression for a transformed operator ${\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}}^{\ensuremath{'}}=({\mathrm{lim}}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}}^{\ifmmode\dagger\else\textdagger\fi{}})\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}({\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}})$. It is shown that the G-S operator, $U={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}}={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}\mathrm{exp}[\ensuremath{-}\frac{1}{2}i\ensuremath{\Lambda}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}+\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})]$, is equivalent to the operator $U={(2\ensuremath{\pi})}^{3}\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}})\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})$. This operator is seen to be a specific example of the more general class of operators presented earlier.

Physical Review C•Apr 1, 1973•7 citations

Internal oscillations of a dark-bright soliton in a harmonic potential

Majed O D Alotaibi, Lincoln D. Carr

Abstract We investigate the dynamics of a dark-bright soliton in a harmonic potential using a mean-field approach via coupled nonlinear Schrödinger equations appropriate to multi-component Bose–Einstein condensates. We use a modified perturbed dynamical variational Lagrangian approximation, where the perturbation is due to the trap, taken as a Thomas–Fermi profile. The wave function ansatz is taken as the correct hyperbolic tangent and secant solutions in the scalar case for the dark and bright components of the soliton, respectively. We also solve the problem numerically with pseudo-spectral Runge–Kutta methods. We find, analytically and numerically, for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. This result is robust against noise in the initial condition and should, therefore, be experimentally observable.

arXiv:1805.03339•Journal of Physics B Atomic Molecular and Optical Physics•Sep 7, 2018•9 citations

Motion of a condensate in a shaken and vibrating harmonic trap

Y Japha, Y B Band

The dynamics of a Bose-Einstein condensate (BEC) in a time-dependent harmonic trapping potential is determined for arbitrary variations of the position of the center of the trap and its frequencies. The dynamics of the BEC wavepacket is soliton-like. The motion of the center of the wavepacket, and the spatially and temporally dependent phase (which affects the coherence properties of the BEC) multiplying the soliton-like part of the wavepacket, are analytically determined.

arXiv:cond-mat/0204300•Journal of Physics B Atomic Molecular and Optical Physics•May 9, 2002•15 citations
The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schrödinger equation in a harmonic potential proposed recently [ J. J. Garc'ıa-Ripoll, V. M. Pérez-Garc'ıa, and V. Vekslerchik, Phys. Rev. E {\bf 64}, 056602 (2001)] is extended to many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics and results in the separation of the center of mass motion from the internal dynamics of many-body systems. The center of mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.
The method of generating a family of new solutions starting from any wave function satisfying the nonlinear Schr\"odinger equation in a harmonic potential proposed recently [J. J. Garc\'{\i}a-Ripoll, V. M. P\'erez-Garc\'{\i}a, and V. Vekslerchik, Phys. Rev. E 64, 056602 (2001)] is extended to the many-body theory of mutually interacting particles. Our method is based on a generalization of the displacement operator known in quantum optics, and results in the separation of the center-of-mass motion from the internal dynamics of many-body systems. The center-of-mass motion is analyzed for an anisotropic rotating trap and a region of instability for intermediate rotational velocities is predicted.
An explicit construction is found for a unitary operator which insures the free motion of the center of mass of any many-particle wave function on which it is allowed to act. The transformation is used to calculate recoil correction terms for the internal energy and external interactions of nuclei, and some numerical evaluations are given for cases of interest. The many-body harmonic-oscillator problem is exactly soluble when one uses the transformation, and one is thus enabled to give a more general discussion of the spurious states.
A theory of collective submanifold, recently, developed by the present authors (M. Y. and A. K.) is applied to a classical many-body system, the Hamiltonian of which depends on the relative coordinates. The solution enables the separation of the total degrees of freedom into the collective and the intrinsic ones, which may be called the generalized center of mass and the relative motions. The results can be applied to the large amplitude collective motion and its couplings with the intrinsic degrees of freedom, which cannot be given in the HF plus RPA method.
We examine the Gartenhaus and Schwartz (G-S) transformation, which generates a translationally invariant wave function from one which does not possess this property. A clarification of the G-S procedure for the calculation of matrix elements is achieved if one takes two independent limits of the parameters $\ensuremath{\Lambda}$, ${\ensuremath{\Lambda}}^{\ensuremath{'}}$ which appear in the expression for a transformed operator ${\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}}^{\ensuremath{'}}=({\mathrm{lim}}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{{\ensuremath{\Lambda}}^{\ensuremath{'}}}^{\ifmmode\dagger\else\textdagger\fi{}})\stackrel{\ensuremath{\rightarrow}}{\mathfrak{a}}({\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}})$. It is shown that the G-S operator, $U={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}{U}_{\ensuremath{\Lambda}}={\mathrm{lim}}_{\ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty}}\mathrm{exp}[\ensuremath{-}\frac{1}{2}i\ensuremath{\Lambda}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}+\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})]$, is equivalent to the operator $U={(2\ensuremath{\pi})}^{3}\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{P}})\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathcal{R}})$. This operator is seen to be a specific example of the more general class of operators presented earlier.
Abstract We investigate the dynamics of a dark-bright soliton in a harmonic potential using a mean-field approach via coupled nonlinear Schrödinger equations appropriate to multi-component Bose–Einstein condensates. We use a modified perturbed dynamical variational Lagrangian approximation, where the perturbation is due to the trap, taken as a Thomas–Fermi profile. The wave function ansatz is taken as the correct hyperbolic tangent and secant solutions in the scalar case for the dark and bright components of the soliton, respectively. We also solve the problem numerically with pseudo-spectral Runge–Kutta methods. We find, analytically and numerically, for weak trapping the internal modes are nearly independent of center of mass motion of the dark-bright soliton. In contrast, in tighter traps the internal modes couple strongly to the center of mass motion, showing that for dark-bright solitons in a harmonic potential the center of mass and relative degrees of freedom are not independent. This result is robust against noise in the initial condition and should, therefore, be experimentally observable.