Modified Ehrenfest Theorem in 1D and 2D Gross-Pitaevskii Systems

Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.

arXiv:1212.4945SIAM Journal on Mathematical AnalysisJan 1, 201234 citations

Low-dimensional stochastic projected Gross-Pitaevskii equation

Ashton S. Bradley, S. J. Rooney, R. G. McDonald

We present reduced-dimensional stochastic projected Gross-Pitaevskii equations describing regimes of confinement and temperature where a one-dimensional (1D) or 2D superfluid is immersed in a 3D thermal cloud. The projection formalism provides both a formally rigorous and physically natural way to effect the dimensional reduction. The 3D form of the number-damping (growth) terms is unchanged by the dimensional reduction. Projection of the energy-damping (scattering) terms leads to modified stochastic equations of motion describing energy exchange with the thermal reservoir. The regime of validity of the dimensional reduction is investigated via variational analysis. For the 1D case, we validate our variational treatment by comparing numerical simulations of a trapped prolate system in three dimensions with the 1D theory and establish a consistent choice of cutoff for the 1D theory. We briefly discuss the scenario involving two components with different degeneracy, suggesting that a wider regime of validity exists for systems in contact with a buffer-gas reservoir.We present reduced-dimensional stochastic projected Gross-Pitaevskii equations describing regimes of confinement and temperature where a one-dimensional (1D) or 2D superfluid is immersed in a 3D thermal cloud. The projection formalism provides both a formally rigorous and physically natural way to effect the dimensional reduction. The 3D form of the number-damping (growth) terms is unchanged by the dimensional reduction. Projection of the energy-damping (scattering) terms leads to modified stochastic equations of motion describing energy exchange with the thermal reservoir. The regime of validity of the dimensional reduction is investigated via variational analysis. For the 1D case, we validate our variational treatment by comparing numerical simulations of a trapped prolate system in three dimensions with the 1D theory and establish a consistent choice of cutoff for the 1D theory. We briefly discuss the scenario involving two components with different degeneracy, suggesting that a wider regime of validity exists for systems in contact with a buffer-gas reservoir.

arXiv:1507.02023Physical Review ASep 30, 201526 citations

On the Ehrenfest theorem in a one-dimensional box

Vidal Alonso Secades, Salvatore De Vincenzo, L. A. González-Díaz

Il Nuovo Cimento BFeb 1, 20007 citations

On the Zero-Energy-Limit Solution for the Modified Gross–Pitaevskii Equation

Kwang‐Hua W. Chu

International Journal of Theoretical PhysicsJun 5, 20072 citations

(-\Delta)^{1/2}

is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.