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arXiv:1603.08475 · Physical Review A

Optimal nonlinear coherent mode transitions in Bose-Einstein condensates utilizing spatiotemporal controls

David L. Hocker, Julia Yan, HERSCHEL A. RABITZ

Published May 18, 2016• 20 citations
arXivHTMLPDF

Abstract

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 $99%$ 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.

DOI
10.1103/physreva.93.053612

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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.
Abstract In the present work, we consider the generation of excited nonlinear topological modes of Bose–Einstein condensates. These nonlinear modes present a number of interesting properties and can be used in many applications, such as in matter-wave optics experiments. The resonant generation of nonlinear modes by modulating either the trapping potential or the scattering length have been previously reported. Here, we propose the simultaneous use of both modulations with an adjustable relative phase between them in order to coherently control the transition from the ground state to an excited mode. Within the framework of the Gross–Pitaevskii equation (GPE), we show that the transition probability and the transition duration can be controlled by means of the manipulation of the relative phase. In particular, the acceleration of the transition can be useful to avoid dissipative effects of the condensate with its surroundings. In our analysis, we employ approximate analytical techniques, including a perturbative treatment, and numerical calculations for the GPE.
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.
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.
Abstract This study investigates the formation and dynamics of solitons in Bose–Einstein condensates (BECs) within dark traps generated by two crossed Laguerre–Gaussian (LG) beams with varying azimuthal indices ℓ . As the index ℓ increases, the potential transitions from a harmonic trap when ℓ = 1 to a square-well potential for larger values of ℓ . This transition allows us to study a range of soliton dynamics under different confinement conditions while maintaining the same BEC volume. Through the derivation of the Gross-Pitaevskii equation (GPE) and under these specific conditions in both one-dimensional and two-dimensional configurations, we explore the dynamics of solitons across multiple scenarios. The study examines two primary methods for solitons generation: the temporal modulation of the scattering length and the implementation of an initial potential barrier that is subsequently removed. The results indicate that the trap shape plays a critical role in the generation and interaction dynamics of solitons. In harmonic traps, solitons exhibit a behavior different from those observed in anharmonic traps, where the dynamics is significantly influenced by the azimuthal index of the trap. The ability to control soliton dynamics in BECs holds significant promise for applications in quantum technologies, precision sensing, and the exploration of fundamental quantum phenomena.
We present a controlling potential method for solving the three-dimensional Gross-Pitaevskii equation (GPE), which governs the nonlinear dynamics of the Bose-Einstein condensates (BECs) in an inhomogeneous potential trap. Our method allows one to construct ground and excited matter wave states whose longitudinal profiles can have bright solitons. This method provides the confining potential that filters and controls localized BECs. Moreover, it is predicted that, while the BEC longitudinal soliton profile is controlled and kept unchanged, the transverse profile may exhibit oscillatory breathers (the unmatched case) or move as a rigid body in the form of either coherent states (performing the Lissajous figures) or a Schrödinger cat state (matched case).
We analyze spatiotemporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt's method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits ("modulated amplitude waves") on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.