Optimal Ensemble Control of Matter-Wave Splitting in Bose-Einstein Condensates

Abstract

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 Schroedinger'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.

Figures (3)

• Figure 1: Illustration of a square pulse sequence control $\Omega(t)$ and parameters.
• Figure 2: Performance of moment ensemble controls compared to optimal square pulse sequences. Designs are for $\epsilon\in[1-\delta,1+\delta]$ with $\delta = 0.1$ and a time horizon of $T = 3$. Desired momentum levels are (from left to right) $n=2$, $n=4$, $n=6$, and $n=8$, corresponding to $C_{2}^{+}$, $C_{4}^{+}$, $C_{6}^{+}$, and $C_{8}^{+}$. The plots show (from top to bottom): terminal state fidelity $1-I_e$ with $I_e$ as in equation \ref{['eq:performance_index']} for the square pulse (SP), the moment dynamics (MD) method, and MD method with positivity requirement (MD POS) as functions of the ensemble parameter $\epsilon$; the optimal control function without amplitude bounds; and optimal controls with strict non-negativity constraints. The times when the skew-rate limit $|\dot{\Omega}(t)|\leq 500$ is binding (Saturated) is indicated.
• Figure 3: Control performance given as terminal fidelity $1-I_e$ with $I_e$ as in equation \ref{['eq:performance_index']}, for the sample-design square pulse method for one (SP), three (SP3), and ten (SP10) samples in ensemble space, as well as the moment dynamics (MD) and positive MD method. The design parameters used are (from left to right): $\delta = 0.1$ and $n = 1$; $\delta = 0.1$ and $n = 4$; and $\delta = 0.4$ and $n = 1$. Observe that the MD method clearly has the best performance.