We present a framework for designing optimal optical pulses for the matter-wave splitting of a Bose-Einstein Condensate (BEC) under the influence of experimental inhomogeneities, so that the sample is transferred from an initial rest position into a singular higher diffraction order. To represent the evolution of the population of atoms, the Schrödinger's equation is reinterpreted as a parameterized ensemble of dynamical units that are disparately impacted by the beam light-shift potential in a continuous manner. The derived infinite-dimensional coupled Raman-Nath equations are truncated to a finite system of diffraction levels, and we suppose that the parameter that defines the inhomogeneity in the control applied to the ensemble system is restricted to a compact interval. We first design baseline square pulse sequences for the excitation of BEC beam-splitter states following a previous study, subject to dynamic constraints for either a nominal system assuming no inhomogeneity or for several samples of the uncertain parameter. We then approximate the continuum state-space of the ensemble of dynamics using a spectral approach based on Legendre moments, which is truncated at a finite order. Control functions that steer the BEC system from an equivalent rest position to a desired final excitation are designed using a constrained optimal control approach developed for handling nonlinear dynamics. This representation results in a minimal dimension of the computational problem and is shown to be highly robust to inhomogeneity in comparison to the baseline approach. Our method accomplishes the BEC-splitting state transfer for each subsystem in the ensemble, and is promising for precise excitation in experimental settings where robustness to environmental and intrinsic noise is paramount.
We present a framework for designing optimal optical pulses for the matter-wave splitting of a Bose-Einstein Condensate (BEC) under the influence of experimental inhomogeneities, so that the sample is transferred from an initial rest position into a singular higher diffraction order. To represent the evolution of the population of atoms, the Schrödinger's equation is reinterpreted as a parameterized ensemble of dynamical units that are disparately impacted by the beam light-shift potential in a continuous manner. The derived infinite-dimensional coupled Raman-Nath equations are truncated to a finite system of diffraction levels, and we suppose that the parameter that defines the inhomogeneity in the control applied to the ensemble system is restricted to a compact interval. We first design baseline square pulse sequences for the excitation of BEC beam-splitter states following a previous study, subject to dynamic constraints for either a nominal system assuming no inhomogeneity or for several samples of the uncertain parameter. We then approximate the continuum state-space of the ensemble of dynamics using a spectral approach based on Legendre moments, which is truncated at a finite order. Control functions that steer the BEC system from an equivalent rest position to a desired final excitation are designed using a constrained optimal control approach developed for handling nonlinear dynamics. This representation results in a minimal dimension of the computational problem and is shown to be highly robust to inhomogeneity in comparison to the baseline approach. Our method accomplishes the BEC-splitting state transfer for each subsystem in the ensemble, and is promising for precise excitation in experimental settings where robustness to environmental and intrinsic noise is paramount.
Splitting a Bose--Einstein condensate (BEC) is a key operation in fundamental physics experiments and emerging quantum technologies, where precise preparation of well--defined initial states requires fast yet coherent control of the condensate's nonlinear dynamics. This work formulates the BEC splitting process as an optimal feedforward control problem based on a physically interpretable, reduced--order model identified from limited experimental data. We introduce a systematic calibration strategy that combines optimal experiment selection and constrained nonlinear parameter estimation, enabling accurate system identification with minimal experimental overhead. Using this calibrated model, we compute energy--optimal trajectories via indirect optimal control to realize shortcuts to adiabaticity (STAs), achieving rapid transitions to the ground state of a double--well potential while suppressing excitations. Experiments confirm that the proposed control framework yields high--fidelity state transfers across multiple configurations, demonstrating its robustness and scalability for quantum control applications.
In a recent experiment (Wang et al., e-print cond-mat/0407689), it was observed that a sequence of two standing-wave square pulses can split a Bose-Einstein Condensate at rest into $\ifmmode\pm\else\textpm\fi{}2\ensuremath{\hbar}\mathbf{k}$ diffraction orders with almost 100% efficiency. By truncating the Raman-Nath equations to a two-state model, we provide an intuitive picture that explains this double-square-pulse beam-splitter scheme. We further show it is possible to optimize a standing-wave multiple-square-pulse sequence to efficiently diffract an atom at rest to a symmetric superposition of $\ifmmode\pm\else\textpm\fi{}2n\ensuremath{\hbar}\mathbf{k}$ diffraction orders with n>1. The approach is considered to be qualitatively different from the traditional light-pulse schemes in the Bragg or the Raman-Nath region, and can be extended to more complex atomic optical elements that produce various tailored output momentum states from a cold atom source.
Bose–Einstein condensate (BEC)-based atom interferometry exploits low temperatures and long coherence lengths to facilitate high-precision measurements. Progress in atom interferometry promises improvements in navigational devices like gyroscopes and accelerometers, as well as applications in fundamental physics such as accurate determination of physical constants. Previous work demonstrates that beam splitters and mirrors for coherent manipulation of dilute BEC momentum in atom interferometers can be implemented with sequences of non-resonant standing-wave light pulses. While previous work focuses on the optimization of the optical pulses’ amplitude and duration to produce high-order momentum states with high fidelity, we explore how varying the shape of the optical pulses affects optimal beam-splitter performance, as well as the effect of pulse shape on the sensitivity of optimized parameters in achieving high fidelity in high-momentum states. In simulations of two-pulse beam splitters utilizing optimized square, triangle, and sinc-squared pulse shapes applied to dilute BECs, we, in some cases, reduce parameter sensitivity by an order of magnitude while maintaining fidelity.
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.
Bose-Einstein condensates (BECs), a quantum state of matter formed when bosonic atoms are cooled close to absolute zero, have become the premier platform for investigating fundamental physics with atomic vapours. Experiments on Bose-Einstein condensates now achieve exquisite control over many aspects of the system, including interactions, trapping potential, and dynamics. This has precipitated a new wave of research into many-body quantum phenomena and, in particular, solitons. These structures are fundamental excitations of an interacting non-linear medium, of interest to a multitude of scientific disciplines from non-linear optics to financial markets. The highly controllable environment of BECs form an attractive playground for the study of solitons allowing the non-linearity to be dynamically tuned, facilitating deeper investigations into these structures. Consistently generating and analysing solitons in BEC experiments continues to be problematic. In particular, the non-linear dynamics of BECs, though required for the generation of solitons, produce particularly challenging control and optimization problems. These control problems must be solved before further investigations into the fundamental physics of soliton dynamics can be answered. This thesis makes three important advances in the control and measurement of BECs that will lead to better generation and observation of solitons. (1) a theoretical model for a control scheme capable of highly precise wavefunction engineering, (2) the experimental implementation of a machine learning algorithm for online optimisation, and (3) a continuous non-destructive imaging system capable of directly observing soliton dynamics in real-time. Together, these advances provide a suite of tools for manipulating and exploiting solitons in Bose-Einstein condensates. A novel technique was developed theoretically, offering control of the macroscopic wavefunction of a Bose-Einstein condensate with unprecedented spatial resolution and speed. The ability to control the atomic wavefunction at the fundamental length scale is key to the advancement of many quantum technologies such as quantum simulators. The magnetic resonance control scheme is demonstrated through simulation of a 87Rb condensate with the exemplar model generating a single dark soliton with corresponding π phase kink. The soliton represents a structure at the fundamental length scale of the system, and demonstrates the potential of the scheme for precision state engineering. The scheme is extended to generate higher-order soliton modes which are yet to be experimentally realised. A machine learning algorithm based on Gaussian processes was developed and implemented on the evaporative cooling stage of the production of a 87Rb Bose-Einstein condensate, successfully demonstrating fast optimisation to condensation. The Gaussian process develops a statistical model based on the data that enables the characterisation of the relationship between the experimental controls and resultant quality of the BEC. This relationship is often obfuscated through technical details of the apparatus, frustrating the use of theoretical models to design optimal evaporation ramps. These models often only consider ergodic dynamics with two-body s-wave interactions and no other loss rates with better ramps likely exploiting more complex interactions. The internal model generated from the Gaussian process utilised uncertainty in measured data, making the optimisation more robust to experimental noise than alternate methods. The algorithm is shown to produce high quality Bose-Einstein condensates in 10 times fewer experimental iterations than previously used online optimisation techniques. By exploiting information on the sensitivity of each control, the model can be used to aid experimental design. The convergence of the optimisation is further improved by eliminating a superfluous parameter identified by the model. The general usefulness of machine learning compared with bespoke optimisation algorithms has seen machine learning approach ubiquity. Finally, an experimentally straightforward technique for continuous non-destructive imaging of matter-wave solitons was developed and implemented, facilitating measurements of stochastic phenomena. The technique is readily practicable on any ultracold atom experiment with an existing absorption imaging system, simply requiring the probe laser be far-detuned from resonance. With a signal-to-noise of ∼ 33 at 1.25 GHz detuning, the technique is capable of producing 100 images with no observable heating or atom loss. Coupled with a fast optical phase locked loop, the technique can be used in conjunction with absorption imaging to generate a series of non-destructive images followed by a final high signal-to-noise absorption image solely through moving the laser on resonance for the final image. The high performance and utility of this imaging setup make it a powerful tool for ultra-cold atom experiments.
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.