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SIAM Journal on Scientific Computing

Multigrid Optimization Schemes for Solving Bose–Einstein Condensate Control Problems

Alfio Borzı̀, Ulrich Hohenester

Published Jan 1, 2008• 24 citations
Publisher

Abstract

The control of the transport of Bose–Einstein condensates in magnetic microtraps is formulated within the framework of optimal control theory and solved by multigrid optimization schemes. The time evolution of the wave function of Bose–Einstein condensates is governed by the Gross–Pitaevskii equation and can be manipulated through variation of a controllable magnetic confinement potential. In order to define an optimal control strategy, an appropriate cost functional is introduced that must be minimized under the constraint given by the dynamic equation. The resulting optimality system consists of two nonlinear Schrödinger-type equations with opposite time orientation coupled with an elliptic equation for the control function. These equations are approximated by using a time-splitting pseudospectral method and finite differences. To solve the resulting problem a cascadic nonlinear conjugate gradient scheme and a multigrid optimization scheme are considered. The convergence properties of these two schemes are investigated theoretically, and their computational performance is discussed based on results of numerical experiments. It appears that the multigrid optimization scheme provides a robust optimization strategy.

DOI
10.1137/070686135

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The control of the transport of Bose–Einstein condensates in magnetic microtraps is formulated within the framework of optimal control theory and solved by multigrid optimization schemes. The time evolution of the wave function of Bose–Einstein condensates is governed by the Gross–Pitaevskii equation and can be manipulated through variation of a controllable magnetic confinement potential. In order to define an optimal control strategy, an appropriate cost functional is introduced that must be minimized under the constraint given by the dynamic equation. The resulting optimality system consists of two nonlinear Schrödinger-type equations with opposite time orientation coupled with an elliptic equation for the control function. These equations are approximated by using a time-splitting pseudospectral method and finite differences. To solve the resulting problem a cascadic nonlinear conjugate gradient scheme and a multigrid optimization scheme are considered. The convergence properties of these two schemes are investigated theoretically, and their computational performance is discussed based on results of numerical experiments. It appears that the multigrid optimization scheme provides a robust optimization strategy.
Transport of Bose-Einstein condensates in magnetic microtraps, controllable by external parameters such as wire currents or radio-frequency fields, is studied within the framework of optimal control theory (OCT). We derive from the Gross-Pitaevskii equation the optimality system for the OCT fields that allow efficient channeling of the condensate between given initial and desired states. For a variety of magnetic confinement potentials we study transport and wave-function splitting of the condensate, and demonstrate that OCT drastically outperforms simpler schemes for the time variation of the microtrap control parameters.
This thesis develops a numerical optimal-control framework for investigating the manipulation of one-dimensional Bose–Einstein condensates governed by the Gross–Pitaevskii equation. The control problem is formulated as a constrained optimization problem in which time-dependent external potentials are optimized to steer condensate dynamics while minimizing unwanted excitations.The first part considers the compression of a condensate containing a darksoliton confined in a box potential. Optimal trajectories of the trap boundaries are obtained that preserve the solitonic structure while strongly reducing den-sity and phase excitations compared with conventional linear protocols. Thesecond part addresses the opposite problem of soliton absorption. By optimizing the motion of the trapping boundaries, moving gray solitons can be efficiently removed and the condensate driven toward its ground state. The third part investigates the merging of two condensates separated by a potential barrier. Optimal barrier-removal protocols are computed for different initialphase differences between the condensates. The dependence of the optimal control strategy on the phase mismatch is analyzed, including the formation of adark soliton in the case of a PI phase difference and the construction of robustprotocols applicable when the phase difference is unknown.Finally, the optimal-control framework is extended to tunnel-coupled con-densates. The coupled Gross–Pitaevskii equations are formulated as a con-strained optimization problem, and the corresponding adjoint equations arederived. This framework provides the foundation for future investigations of optimal control in coupled condensate systems.Together, these results demonstrate how optimal control can be used both to preserve and to manipulate nonlinear excitations and phase structures in Bose–Einstein condensates, providing tools for quantum-state engineering inultracold atomic systems.
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Applications of Bose-Einstein condensates (BEC) often require that the condensate be prepared in a specific complex state. Optimal control is a reliable framework to prepare such a state while avoiding undesirable excitations, and, when applied to the time-dependent Gross-Pitaevskii equation (GPE) model of BEC in multiple space dimensions, results in a large computational problem. We propose a control method based on first reducing the problem, using a Galerkin expansion, from a partial differential equation to a low-dimensional Hamiltonian ordinary differential equation system. We then apply a two-stage hybrid control strategy. At the first stage, we approximate the control using a second Galerkin-like method known as the chopped random basis to derive a finite-dimensional nonlinear programing problem, which we solve with a differential evolution algorithm. This search method then yields a candidate local minimum which we further refine using a variant of gradient descent. This hybrid strategy allows us to greatly reduce excitations both in the reduced model and the full GPE system.
We theoretically investigate protocols based on optimal control theory (OCT) for manipulating Bose-Einstein condensates in magnetic microtraps, using the framework of the Gross-Pitaevskii equation. In our approach we explicitly account for filter functions that distort the computed optimal control, a situation inherent to many experimental OCT implementations. We apply our scheme to the shakeup process of a condensate from the ground to the first excited state, following a recent experimental and theoretical study, and demonstrate that the fidelity of OCT protocols is not significantly deteriorated by typical filters.
We study optimal quantum control of the dynamics of trapped Bose-Einstein condensates: The targets are to split a condensate, residing initially in a single well, into a double well, without inducing excitation, and to excite a condensate from the ground state to the first-excited state of a single well. The condensate is described in the mean-field approximation of the Gross-Pitaevskii equation. We compare two optimization approaches in terms of their performance and ease of use; namely, gradient-ascent pulse engineering (GRAPE) and Krotov's method. Both approaches are derived from the variational principle but differ in the way the control is updated, additional costs are accounted for, and second-order-derivative information can be included. We find that GRAPE produces smoother control fields and works in a black-box manner, whereas Krotov with a suitably chosen step-size parameter converges faster but can produce sharp features in the control fields.