Wave-packet dynamics in nonlinear Schrödinger equations

Coherent states play an important role in quantum mechanics because of their unique properties under time evolution. Here we explore this concept for one-dimensional repulsive nonlinear Schrödinger equations, which describe weakly interacting Bose-Einstein condensates or light propagation in a nonlinear medium. It is shown that the dynamics of phase-space translations of the ground state of a harmonic potential is quite simple: The center follows a classical trajectory whereas its shape does not vary in time. The parabolic potential is the only one that satisfies this property. We study the time evolution of these nonlinear coherent states under perturbations of their shape or of the confining potential. A rich variety of effects emerges. In particular, in the presence of anharmonicities, we observe that the packet splits into two distinct components. A fraction of the wavepacket is transferred toward incoherent high-energy modes, while the amplitude of oscillation of the remaining coherent component is damped toward the bottom of the well.

Figure 1

Time evolution of a shifted ground state of the GPE with a harmonic confining potential with the parameters $\gamma =115$ and $d=5$. Husimi representations of the wave function are given at times $\omega t/2\pi =0$ (a), 0.25 (b), 0.5 (c), and 0.75 (d). The solid (red) curve is the classical trajectory of the corresponding linear problem of energy given by the center of the initial packet.

Figure 2

Time evolution of the linear Schrödinger equation with a harmonic potential with the parameters $\gamma =0$ and $d=5$. The initial state is the same as in Fig. 1 (a shifted ground state of the GPE). Husimi representations of the wave function are given at times $\omega t/2\pi =0$ (a), 0.25 (b), 0.5 (c), and 0.75 (d).

Figure 3

Normalized square quadrupole frequency for different values of the nonlinear parameter $\gamma$. The dashed (black) line represents the numerically computed frequency, and the solid (blue) line represents the analytical result obtained using a Gaussian ansatz variational principle.

Figure 4

Time evolution of the GPE with a harmonic confining potential with the parameter $\gamma =115$. The initial state is the shifted Gaussian ground state of the linear problem. Husimi representations of the wave function are given at times $\omega t/2\pi =0$ (a), 0.02 (b), 0.13 (c), 0.25 (d), 0.50 (e), 0.51 (f), 0.53 (g), and 0.75 (h).

Figure 5

Time evolution of the linear Schrödinger equation ($\gamma =0$, top row) and of the GPE ($\gamma =115$, bottom row) in the presence of an anharmonic (quartic) confining potential with the parameters $\alpha =0.01$ and $\beta =0.04$. For each panel, the initial state is the corresponding (linear or nonlinear) coherent state, computed for $\alpha =0$. Husimi representations of the wave functions are given at times $\omega t/2\pi =0$ (a) and (e), 3 (b) and (f), 12 (c) and (g), and $40.5$ (d) and (h).

Figure 6

Time evolution of the GPE in the presence of an anharmonic (quartic) confining potential, with the parameters $\gamma =115$, $\alpha =0.01$, and $\beta =0.5$. Husimi representation of the wave function at times $\omega t/2\pi =0$ (a), $1.5$ (b), 4 (c), and 40 (d).

Figure 7

Fluidity factor, defined by the ratio of the average amplitude of oscillation in a stationary regime and the initial amplitude $d$, versus $\beta =\alpha d^2$ for $\gamma =115$ and $\alpha =0.01$. The initial state is a shifted nonlinear coherent state. The inset shows the same plot for different values of $\alpha$, using, for both axes, the same scale as in the main figure.