An amended Ehrenfest theorem for the Gross-Pitaevskii equation in one- and two-dimensional potential boxes

TL;DR

The paper addresses the breakdown of the conventional Ehrenfest theorem for quantum particles confined in box potentials and derives an amended form that incorporates edge-generated forces within the Gross-Pitaevskii framework. It extends the amended ET to both 2D square boxes and 1D boxes, showing that the center-of-mass acceleration comprises a quantum-mean force plus an edge contribution, while the cubic nonlinearity affects the dynamics indirectly through the edge term. Numerical simulations in 2D and 1D confirm the amended equations and reveal that nonlinearity can induce irregular or chaotic COM motion in box geometries. The results are directly relevant to experiments on Bose-Einstein condensates in box traps and suggest further extensions to more complex geometries and multi-component condensates.

Abstract

It is known that the usual form of the Ehrenfest theorem (ET), which couples the motion of the center of mass (COM) of the one-dimensional (1D) wave function to the respective classical equation of motion, is not valid in the case of the potential box, confined by the zero boundary conditions. A modified form of the ET was proposed for this case, which includes an effective force originating from the interaction of the 1D quantum particle with the box edges. In this work, we derive an amended ET for the Gross-Pitaevskii equation (GPE), which includes the cubic nonlinear term, as well as for the 2D square-shaped potential box. In the latter case, we derive an amended COM equation of motion with an effective force exerted by the edges of the rectangular box, while the nonlinear term makes no direct contribution to the 1D and 2D versions of the ET. Nonetheless, the nonlinearity affects the amended ET through the edge-generated force. As a result, the nonlinearity of the underlying GPE can make the COM motion in the potential box irregular. The validity of the amended ET for the 1D and 2D GPEs with the respective potential boxes is confirmed by the comparison of numerical simulations of the underlying GPE and the corresponding amended COM equation of motion. The reported findings are relevant to the ongoing experiments carried out for atomic Bose-Einstein condensates trapped in the box potentials.

Figures (3)

• Figure 1: (a) The evolution of the peak (largest) value of $|\psi (x,y,t)|$ for $g=-1$ (the upper solid line), $g=0$ (the flat dashed line), and $g=+1$ (the lower solid line), as produced by the numerical simulations of Eq. (\ref{['GP']}) with the HO potential (\ref{['HO']}) and input (\ref{['0']}). (b) The evolution of $|\psi (x,y,t)|$ in the cross section $y=0$ for $g=-1$. (c) The evolution of the COM coordinates $X$ and $Y$ (the solid green and dashed blue lines, respectively), as produced by the data of the numerical simulations, as per definitions (\ref{['R']}) for $g=+1$, $0$, and $-1$. The ET-predicted dependences (\ref{['XY']}) of the COM coordinates are plotted by the solid green and dashed blue lines, respectively. The parameters of the input (\ref{['0']}) are $N=5.7$ and $a=2$, hence the other parameters are $\aleph \approx 0.60$, $A_{p}\approx 1.14$, and $x_{\mathrm{COM}}=0.4$, see Eqs. (\ref{['N']}), (\ref{['Ap']}), and (\ref{['COM']}).
• Figure 2: The evolution of the COM coordinates $X(t)$ and $Y(t)$ (the solid green and dashed blue lines, respectively), as directly produced by the simulations of the 2D GPE (\ref{['GP']}) with $U=0$, BC (\ref{['Diri']}), and input (\ref{['2D0']}). These dependences are indistingushable from their counterparts produced by the equations of motion (\ref{['X']}) and (\ref{['Y']}), which represent the amended ET for the 2D potential box. Panels (a), (b), and (c) represent, severally, the GPE with $g=+1$, $g=0$, and $g=-1$. The corresponding COM trajectories in the $\left( x,y\right)$ plane are plotted in bottom panels. The parameters are $N=5$, $a=0.4$, and $L=10$.
• Figure 3: The evolution of the COM coordinate $X$ (the solid line), as produced by the direct simulations of Eq. (\ref{['1D']}) with input (\ref{['1D0']}) in the 1D potential box, and its counterpart $X_{r}$ (the dashed line), as reconstructed by means of the amended-ET equation ( \ref{['Ehr2']}), for $g=+1$ (a), $g=0$ (b), and $g=-1$ (c) .