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arXiv:2304.01051 · IFAC-PapersOnLine

Optimal control of quasi-1D Bose gases in optical box potentials

Andreas Deutschmann, Katharina Schrom, Nikolaus Würkner, Jörg Schmiedmayer, Sebastian Erne, Andreas Kugi

Published Jan 1, 2023• 0 citations
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Abstract

In this paper, we investigate the manipulation of quasi-1D Bose gases that are trapped in a highly elongated potential by optimal control methods. The effective mean-field dynamics of the gas can be described by a one-dimensional non-polynomial Schrödinger equation. We extend the indirect optimal control method for the Gross-Pitaevskii equation by Winckel and Borzì (2008) to obtain necessary optimality conditions for state and energy cost functionals. This approach is then applied to optimally compress a quasi-1D Bose gase in an (optical) box potential, i.e., to find a so-called short-cut to adiabaticity, by solving the optimality conditions numerically. The behavior of the proposed method is finally analyzed and compared to simple direct optimization strategies using reduced basis functions. Simulations results demonstrate the feasibility of the proposed approach.

DOI
10.1016/j.ifacol.2023.10.1781

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In this paper, we investigate the manipulation of quasi-1D Bose gases that are trapped in a highly elongated potential by optimal control methods. The effective mean-field dynamics of the gas can be described by a one-dimensional non-polynomial Schrödinger equation. We extend the indirect optimal control method for the Gross-Pitaevskii equation by Winckel and Borzì (2008) to obtain necessary optimality conditions for state and energy cost functionals. This approach is then applied to optimally compress a quasi-1D Bose gase in an (optical) box potential, i.e., to find a so-called short-cut to adiabaticity, by solving the optimality conditions numerically. The behavior of the proposed method is finally analyzed and compared to simple direct optimization strategies using reduced basis functions. Simulations results demonstrate the feasibility of the proposed approach.
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.
Ultracold gases promise many applications in quantum metrology, simulation and computation. In this context, optimal control theory (OCT) provides a versatile framework for the efficient preparation of complex quantum states. However, due to the high computational cost, OCT of ultracold gases has so far mostly been applied to one-dimensional (1D) problems. Here, we realize computationally efficient OCT of the Gross–Pitaevskii equation to manipulate Bose–Einstein condensates in all three spatial dimensions. We study various realistic experimental applications where 1D simulations can only be applied approximately or not at all. Moreover, we provide a stringent mathematical footing for our scheme and carefully study the creation of elementary excitations and their minimization using multiple control parameters. The results are directly applicable to recent experiments and might thus be of immediate use in the ongoing effort to employ the properties of the quantum world for technological applications.
Bose-Einstein condensates (BECs) offer the potential to examine quantum behavior at large length and time scales, as well as forming promising candidates for quantum technology applications. Thus, the manipulation of BECs using control fields is a topic of prime interest. We consider BECs in the mean-field model of the Gross-Pitaevskii equation (GPE), which contains linear and nonlinear features, both of which are subject to control. In this work we report successful optimal control simulations of a one-dimensional GPE by modulation of the linear and nonlinear terms to stimulate transitions into excited coherent modes. The linear and nonlinear controls are allowed to freely vary over space and time to seek their optimal forms. The determination of the excited coherent modes targeted for optimization is numerically performed through an adaptive imaginary time propagation method. Numerical simulations are performed for optimal control of mode-to-mode transitions between the ground coherent mode and the excited modes of a BEC trapped in a harmonic well. The results show greater than 9999%99 success for nearly all trials utilizing reasonable initial guesses for the controls, and analysis of the optimal controls reveals primarily direct transitions between initial and target modes. The success of using solely the nonlinearity term as a control opens up further research toward exploring novel control mechanisms inaccessible to linear Schr\"odinger-type systems.
We analyze the localization of a Bose-Einstein condensate in a one-dimensional bichromatic quasi-periodic optical-lattice potential by numerically solving the 1D Gross-Pitaevskii equation (1D GPE).We first derive the 1D Gross-Pitaevskii equation from the dimensional reduction of the 3D quantum field theory of interacting bosons obtaining two coupled differential equations (for axial wave fuction and space-time dependent transverse width) which reduce to the 1D Gross-Pitaevskii equation under strict conditions.Then, by using the 1D Gross-Pitaevskii equation we report the suppression of localization in the interacting Bose-Einstein condensate when the repulsive scattering length between bosonic atoms is sufficiently large.
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.