Optimal Transport Displacement

• Optimal Transport Displacement is a mathematical framework that designs time-dependent protocols to optimally shift atoms or mass between configurations without final excitation.
• It utilizes inverse engineering with Lewis-Riesenfeld invariants and optimal control theory to create trajectories that meet strict boundary and displacement constraints.
• By balancing speed, displacement, and energy costs, this framework offers robust strategies for experimental applications in cold-atom physics and quantum information systems.

Optimal transport displacement is a conceptual and mathematical framework for designing, analyzing, and implementing time-dependent protocols that transfer mass (or, in the quantum context, cold atoms or particles) between two configurations in such a way that certain physical optimality and constraint conditions are met. In the context of atomic transport in harmonic traps, displacement refers specifically to the position difference between the center of mass of atoms and the moving potential minimum. The design of optimal protocols for this displacement, subject to constraints reflecting real experimental limitations, forms the core focus of the work by Xi Chen et al. on "Optimal trajectories for efficient atomic transport without final excitation" (1108.1703).

1. Inverse Engineering and the Role of Lewis-Riesenfeld Invariants

The methodology is anchored in the application of Lewis-Riesenfeld invariants for systems governed by time-dependent harmonic Hamiltonians: $$H(t) = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega_0^2 [\hat{q} - q_0(t)]^2$$ The Lewis-Riesenfeld dynamical invariant,

$$I(t) = \frac{1}{2m} (\hat{p} - m\dot{q}_c)^2 + \frac{1}{2} m \omega_0^2 [\hat{q}-q_c(t)]^2,$$

is constructed so that its eigenstates evolve only by phase and thereby ensure that if the system starts in an eigenstate (e.g., ground state), it stays in the corresponding instantaneous eigenstate of the moving potential, if appropriate boundary conditions are imposed on $q_c(t)$: $$q_c(0)=0,\quad q_c(t_f)=d,\quad \dot{q}_c(0)=\dot{q}_c(t_f)=0,\quad \ddot{q}_c(0)=\ddot{q}_c(t_f)=0$$ Here, $q_c(t)$ describes the "center of mass" (COM) trajectory, indirectly dictating the required history $q_0(t)$ for the trap center via the forced oscillator equation: $\ddot{q}_c + \omega_0^2 (q_c - q_0) = 0$ Inverse engineering thus reduces the control problem to designing $q_c(t)$ subject to boundary and optimality criteria.

2. Optimal Control Formulation of Displacement Trajectories

The optimal protocol is selected among all possible $q_c(t)$ using optimal control theory, recasting the system as: $$x_1 = q_c, \quad x_2 = \dot{q}_c, \quad u(t) = q_c - q_0$$

$\dot{x}_1 = x_2;\qquad \dot{x}_2 = -\omega_0^2 u$

The control variable $u(t)$ quantifies the instantaneous relative displacement—the “transport displacement”—between the atom's COM and the trap’s potential minimum. The optimization targets are:

• Minimal time ($t_f$): Fastest possible transport.
• Minimal integrated displacement ($\int |u(t)|\,dt$): Least total deviation.
• Minimal transient energy ($\int u^2(t)\,dt$): Least excess energy imparted.

The controls are subject to a hard constraint: $|u(t)| \leq \delta$ where $\delta$ is the allowed maximal relative displacement, a parameter dictated by the requirement to avoid excitations due to trap anharmonicity and to ensure the atom remains well-coupled to the trap.

Pontryagin’s Maximum Principle is used to derive optimal protocols. These have distinct structural forms depending on the cost function.

3. Displacement Constraints and Realistic Trap Physics

The displacement bound $|u(t)| \leq \delta$ is central: it models the physical reality that deviations from the harmonic center rapidly degrade the quality of adiabatic transport due to trap imperfections or anharmonicities. This constraint has sharp consequences:

• Minimum achievable time for a given displacement amplitude:

$$t_f \geq \frac{2}{\omega_0} \sqrt{\frac{d}{\delta}}$$

• Restriction on protocol shapes: Sharp, instantaneous switches ("bang-bang" for minimum time) or periods of zero separation interrupted by discrete switchings ("bang-off-bang" for minimum displacement).
• Physical feasibility: Ensures that energy, acceleration, and transient excitations remain within experimentally achievable bounds.

4. Optimal Protocols: Bang-Bang and Bang-Off-Bang Solutions

Minimum-time ("bang-bang") protocol: $$u(t) = \begin{cases} -\delta & 0 < t < t_1 \ +\delta & t_1 < t < t_f \end{cases}, \quad t_1 = t_f/2$$ Gives the fastest protocol allowed by the constraint, with maximal amplitude throughout.

Integrated minimum-displacement ("bang-off-bang") protocol: $$u(t) = \begin{cases} -\delta & 0 < t < t_1 \ 0 & t_1 < t < t_1 + t_2 \ +\delta & t_1 + t_2 < t < t_f \end{cases}$$ Switching times are determined by the allowed velocity and total distance, providing periods of zero relative displacement to reduce overall excitation.

Minimum-energy trajectory (for transient energy cost), when unbounded: $$u(t) = \frac{6d}{\omega_0^2 t_f^2} \left(2\frac{t}{t_f} - 1\right)$$ With the bound $\delta$, the solution interpolates between "bang" and smooth ("off") regions.

5. Trade-off Analysis and Physical Implications

Each optimality criterion leads to a distinct protocol, resulting in trade-offs:

• Minimum time: Maximizes speed and instantaneous energy, may sacrifice energy efficiency and robustness.
• Minimum displacement: Reduces energy transfer to higher modes, avoids large deviations, at the cost of longer total process time.
• Minimum energy: Minimizes transient excitation but may require longer duration and/or larger allowed displacement.

The displacement constraint ensures that protocols remain robust against anharmonic corrections and experimental imperfections. The boundary conditions imposed on $q_c$ and $q_0$ guarantee no net final excitation, crucial in cold atom and quantum information applications.

6. Synthesis, Applications, and Broader Significance

This framework synthesizes shortcut-to-adiabaticity methodologies with optimal control, yielding robust, physically realizable atomic transport strategies. Key consequences include:

• Rigorous quantification of speed limits and energy costs for non-adiabatic atomic shuttling.
• Protocol designs systematically extend to higher-dimensional and more general trapping systems by adapting the state/control formulation and cost functions.
• Theoretical guidance for experimenters in cold-atom physics, quantum information, and precision measurement, where rapid, controlled displacement without excitation is essential.

A plausible implication is that this formalism, ensuring arbitrary control over the displacement profile, can be generalized to more complex constraints or cost functions as found in advanced quantum technology deployments. The classification and analysis of displacement-optimal protocols form a foundation for robust system engineering in non-equilibrium and open quantum systems.