An Adaptable Route to Fast Coherent State Transport via Bang-Bang-Bang Protocols

TL;DR

This work tackles the challenge of rapid, high-fidelity coherent-state transport in quantum platforms by introducing the bang-bang-bang (BBB) protocol, which leverages backward potential shifts in addition to forward moves to beat the traditional forward-only speed limit. It develops a phase-space framework for piecewise-constant trap trajectories and derives the BBB transport time $ au_{ m BBB}( ho)=rac{2}{\omega}\cos^{-1}\left(1-rac{D}{2\rho}\right)$, showing that for $\rho>D/2$ the BBB protocol can surpass forward-only schemes and approach the quantum speed limit. The authors further extend the scheme with squeezed-state evolution, proposing the DSBBB protocol that uses a deeper then weaker trap frequency to accelerate the angular rotation of the squeezed state, potentially reducing transport time further. A quantum-speed-limit analysis demonstrates that BBB does not violate fundamental bounds, while the DSBBB approach can approach or surpass existing limits under appropriate squeezing and timing. Experimental feasibility is discussed for trapped ions and optical tweezer platforms, highlighting the potential for significantly faster state transport and preparation in quantum information processing and simulation contexts.

Abstract

Fast coherent state transport is essential to quantum computation and quantum information processing. While an adiabatic transport of atomic qubits guarantees a high fidelity of the state preparation, it requires a long timescale that defies efficient quantum operations. Here, we propose an adaptable and fast bang-bang-bang (BBB) protocol, utilizing a combination of forwardand backward-moving trap potentials, to expedite the coherent state transport. This protocol approaches the quantum speed limit under a harmonic trap potential, surpassing the performance by the forward-moving-only potential protocols. We further showcase the advantage of applying squeezed coherent state evolution under a deeper potential followed by a weaker one, where a design of symmetric squeezing potential transports promotes an even shorter timescale for genuine state preparation. Our protocols outperform conventional forward-moving-only methods, providing new insights and opportunities for rapid state transport and preparation, ultimately advancing the capabilities of quantum control and quantum operations.

Figures (7)

• Figure 1: A comparison of potential shift directions, forward or backward, on the phase-space evolution. Starting from an initial state at an angle $\theta_1$ (when the potential is at $X_1$), (a) a forward potential shift to $X_2$ results in a new, smaller angle $\theta_2$. (b) By contrast, a backward shift to $X_2'$ results in a new, larger angle $\theta_2'$. This illustrates the key finding that $\theta_2 < \theta_1 < \theta_2'$, demonstrating that forward movements inhibit while backward movements accelerate the angular evolution toward $\pi$. (c) Phase-space evolution and (d) the corresponding potential trajectory for the standard BB protocol, which demonstrate the forward-moving speed limit $\tau_{\rm for}=\pi/\omega_0$. Dashed (filled) circles indicate the state of the particle at the beginning (end) of each free-evolution step.
• Figure 2: The schematic of the BBB protocol. (a) The evolution of a coherent state in the space under the displaced potential with its center at $R$ and $D-R$, respectively, in the first two bang processes of the BBB protocol, where the last bang process involves a shifted and holding potential at $D$. (b) The corresponding trajectory of the moving potential. The total transport time $T\equiv\tau_\text{BBB}(\omega_0, R)$. Dashed (filled) circles indicate the state at the beginning (end) of each free-evolution step.
• Figure 3: Schematic of two squeezed-BBB (SBBB) protocols. (a) A single-frequency SBBB protocol using $\omega_1$. While the spacial transport (time $\tau_{\rm tr}$) can be made faster, the final squeezed state's orientation is mismatched with the target, requiring an extra wait time $\tau_{\rm ex}$ before the final re-squeeze. (b) The double-squeezed-BBB (DSBBB) protocol. Under the second squeeze ($\omega_1 \to \omega_2$),the orientation angle becomes larger ($\theta_2 > \theta_1$) when projected onto the trap potential with a smaller $\omega_2<\omega_1$, accelerating the angular evolution. Dashed (filled) circles indicate the state at the beginning (end) of each free-evolution step. We define the dimensionless position, momentum, and destination with respect to the frequency $\omega_i$ as $\hat{X}^{(i)}\equiv (\sqrt{m\omega_i/2\hbar})\hat{x}$, $\hat{P}^{(i)}\equiv \hat{p}/\sqrt{2\hbar m\omega_i}$, and $\hat{D}^{(i)}\equiv (\sqrt{m\omega_i/2\hbar})d$, respectively.
• Figure 4: Phase diagram of the time advantage, $\tau_\text{DSBBB}-\pi/\omega_1$ (in units of $\pi/\omega_1$), for the parameters of $\omega_2$ and the first-squeeze time $t_1$ at $\omega_1=2\omega_0$. The blue area shows where the DSBBB evolves faster than a straightforward SBBB protocol with a single $\omega_1$. The parameter space is plotted for $\omega_2$ in the region $(0,\omega_1]$ and $t_2$ in the region $(0,\pi/(2\omega_1)]$. The range of $t_2$ is bounded by $\pi/(2\omega_1)$ in a symmetrical process within the time $\pi/\omega_1$.
• Figure 5: Phase-space evolution of a coherent state under an $N$-step moving potential. The state begins at the origin (the ground state $\ket{0_{\rm init}}$). At $t_1=0$, the potential shifts to $X_1$, causing the state to rotate around $(X_1, 0)$ until $t_2$. At $t_2$, the potential shifts to $X_2$, and the state's center of rotation switches to $(X_2, 0)$, continuing until $t_3$. This process repeats for all subsequent steps until $t=T$. Dashed (filled) circles indicate the coherent state at the beginning (end) of each free-evolution step.