Identification and optimal control strategies for the transversal splitting of ultra--cold Bose gases

TL;DR

The paper tackles fast, high-fidelity transversal splitting of a BEC by formulating the process as an optimal feedforward control problem. It develops a physically interpretable reduced-order model for the transversal potential V(x,𝒜) with five core parameters, and couples this with a targeted, information-theoretic experimental design (Fisher information, GA-based selection) to calibrate the model from limited TOF data. Using the calibrated model, the authors perform indirect optimal control to design shortcuts to adiabaticity (STA) that drive the system to the ground state of the final double-well while suppressing excitations, achieving orders-of-magnitude reductions in residual energy compared with naive ramps. The approach yields high-fidelity state transfers across multiple configurations, demonstrates robustness and scalability, and sets a foundation for metrology-relevant initialization and quantum-simulation applications, with future work extending to quantum fluctuations and SG-QFT regimes.

Abstract

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.

Figures (14)

• Figure 1: Schematic of the transversal splitting procedure run on the experimental setup. The left column shows the BEC being split by applying RF-dressing controlled via the normalized RF amplitude $\mathcal{A}(t)$, changing from a single-well potential $V_{SW}$ to a double-well potential $V_{DW}$. This potential is then deactivated; in the following time-of-flight phase, expansion of the BEC creates the measured two-dimensional interference pattern. By approximating the influence of the second transversal direction $y$ and averaging over the longitudinal direction $z$, we arrive at the reduced one-dimensional representation along the $x$-direction, shown in the right column.
• Figure 2: Properties of the transversal potential obtained from first-principles-based field simulation (solid blue line) and the reduced-order model (dashed red line). The top panel shows the position of the minima $x_m(\mathcal{A})$ as a function of the normalized RF amplitude $\mathcal{A}$. The middle panel shows the curvature $k_m(\mathcal{A})=\frac{\partial^2}{\partial x^2}V(x_m(\mathcal{A}),\mathcal{A})$. The bottom panel shows the resulting effective potential $V(x,\mathcal{A})$ along $x$.
• Figure 3: Simulation results of a linear change from $\mathcal{A}(0)=0.28$ to $\mathcal{A}(2~ms)=0.6$ and the subsequent hold time using the modeled potential. The top panel shows the evolution of the transversal wavefunction density $|\Psi(x,t)|^2$ over time, the central panel illustrates the input trajectory $\mathcal{A}(t)$ as well as the fitted width $\sigma_\Psi(t)$ and inter-well distance $d(t)$ of the in-situ wavefunction. The bottom panel displays the corresponding transversal momentum of the BEC's mean field, calculated using the Fourier transformation of the transversal wavefunction $\Psi(x,t)$. Simulated using interpolated values of the first-principles model
• Figure 4: Integrated momentum density for TOF with nonlinear interaction ($g_\perp>0$) and no interaction ($g_\perp=0$) compared for the ground-state wavefunction for $V(x,\mathcal{A}=0.5)$, normalized for illustrative purposes. The broadening of the envelope and shift of the minima to higher k resembles a BEC that is split less and confined more tightly.
• Figure 5: The top two panels show the different types of experiments applied; linear ramps (LR) with the ramp time $T_R$ and value $\mathcal{A}_R$ varied in the left plot, and consecutive linear ramps (CLR) where only the end value $\mathcal{A}_R$ is varied in the right. Different combinations of selected experiments are shown in the lower panels. The optimized selection for Trap 1 ($\rho_\mathcal{E}=2.5$) is marked $\circ$. The fittest member of the initial population ($\rho_\mathcal{E}=5.5$) is also shown, marked with $\times$. The optimized selection for Trap 2 ($\rho_\mathcal{E}=1.7$) is marked with $\diamond$. All selectable experiments are marked with gray dots.

Theorems & Definitions (2)

• Remark 1
• Remark 2