Quantum nonlinear Schrödinger equation on a lattice

George Vahala, Linda L. Vahala, Jeffrey Yepez

Shlomo E. Sklarz, David J. Tannor

Using a set of general methods developed by Krotov [A. I. Konnov and V. A. Krotov, Automation and Remote Control 60, 1427 (1999)], we extend the capabilities of optimal control theory to the nonlinear Schr\"odinger equation (NLSE). The paper begins with a general review of the Krotov approach to optimization. Although the linearized version of the method is sufficient for the linear Schr\"odinger equation, the full flexibility of the general method is required for treatment of the nonlinear Schr\"odinger equation. Formal equations for the optimization of the NLSE, as well as a concrete algorithm are presented. As an illustration, we consider a Bose-Einstein condensate (BEC) initially at rest in a harmonic trap. A phase develops across the BEC when an optical lattice potential is turned on. The goal is to counter this effect and keep the phase flat by adjusting the trap strength. The problem is formulated in the language of optimal control theory (OCT) and solved using the above methodology. To our knowledge, this is the first rigorous application of OCT to the nonlinear Schr\"odinger equation, a capability that is bound to have numerous other applications.

Eugène Gutkin

David J. Fernández C., O. Pavón-Torres

By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.

Miloslav Znojil

A Kundu, O Ragnisco

A lattice version of the quantum nonlinear Schrodinger (NLS) equation is considered, which has a significant simple form and fulfils most of the criteria desirable for such lattice variants of field models. Unlike most of the known lattice NLS equations, the present model belongs to a class which does not exhibit the usual symmetry properties. However, this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory, such as the Lax operator, the Hamiltonian and other commuting conserved quantities as well as their spectra. The model allows exact quantum solution through the algebraic Bethe ansatz and also a straightforward and natural generalization to the vector case, thus giving a new exact lattice version of the vector NLS model. A deformation representing a new quantum integrable system involving Tamm-Dancoff-like q-boson operators is constructed.