Optimizing Schroedinger functionals using Sobolev gradients: Applications to Quantum Mechanics and Nonlinear Optics

TL;DR

The paper develops and implements Sobolev-gradient preconditioning for minimizing Schrödinger-type functionals in quantum mechanics and nonlinear optics. By replacing the ordinary gradient with $\nabla_S E = (1-\Delta)^{-1} \nabla E$ and implementing it efficiently in discrete Fourier spaces via $\hat{s}_m = \hat{e}_m/(1+k_m^2)$, the authors achieve substantially faster convergence under norm constraints, enabling robust discovery of ground, excited, and nontrivial stationary states. The approach is demonstrated on (i) Bose-Einstein condensates in rotating traps using imaginary-time evolution and free-energy minimization, and (ii) a two-beam saturable nonlinear optical model yielding vector solitons and vortex/dipole-type states, including non-symmetric configurations. The work provides a practical FFT-based framework that improves convergence for constrained variational problems across quantum and nonlinear-optics contexts and expands access to challenging stationary solutions.

Abstract

In this paper we study the application of the Sobolev gradients technique to the problem of minimizing several Schrödinger functionals related to timely and difficult nonlinear problems in Quantum Mechanics and Nonlinear Optics. We show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete Fourier basis and a Fast Fourier Transform. We show that the Sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators.

Figures (4)

• Figure 1: Evolution of error, $\varepsilon_2\equiv\Vert\psi-\psi_{exact}\Vert_2$, through different minimization processes for continuous steepest descent with Sobolev preconditioning (lower solid line) and without it (dashed line), and for imaginary time evolution with Sobolev preconditioning (upper solid line) and without it (dotted line). Plots (a) to (c) correspond respectively to the cases A,B, and C described in the text. Both axes, error and number of iterations, are in a logarithmic scale.
• Figure 2: (a) Density profile of a ground state for $N_u=N_w=30$. (b) Norms $N_u=N_w=N$, as a function of the nonlinear Lagrange multipliers $\lambda_u=\lambda_w=\lambda$.
• Figure 3: (a) Vortex-mode vector solitons and (b) their norms $N_u$, $N_w$, $N_{total}=N_u+N_w$, as a function of $\mu_u$ for $\mu_w=1$. (d) Dipole-mode vector solitons and (c) their norms $N_u$, $N_w$, $N_{total}=N_u+N_w$, as a function of $\mu_u$ for $\mu_w=1$.
• Figure :