Pareto-optimality of pulses for robust population transfer in a ladder-type qutrit

TL;DR

This work tackles robust population transfer from $|g\rangle$ to $|e\rangle$ in a ladder-type qutrit, aiming to minimize maximum transient leakage into $|f\rangle$ while maximizing detuning robustness. It adopts frequency-modulated pulses with two envelope families (generalized super-Gaussian and hyperbolic secant) and four detuning shapes, yielding eight models evaluated at $T=50$ and $200$ ns; parameter sets are found via NSGA-II multiobjective optimization to balance $\max_t p_f$ and $\Delta_{rob}$, with Landau–Zener bounds used to contextualize limits. The results show two duration-based clusters in the Pareto fronts, with nonlinear detunings (types 2–4) delivering higher detuning robustness and approaching theoretical limits, especially at longer pulses; envelope choice is less critical than detuning shape. Morris sensitivity analysis identifies $\Omega_0$ as the key driver for reducing $\max_t p_f$ and $k_1$ as the dominant factor for $\Delta_{rob}$, while envelope order $n$ significantly affects amplitude robustness. Overall, the study provides design principles for robust, fast control in ladder-type qutrits and related weakly anharmonic systems, with implications for transmon-like platforms.

Abstract

Frequency-modulation schemes offer an alternative to standard Rabi pulses for realizing robust quantum operations. In this work, we investigate short-duration population transfer between the ground and first excited states of a ladder-type qutrit, with the goal of minimizing leakage into the second excited state. Our multiobjective approach seeks to reduce the maximum transient second-state population and maximize detuning robustness. Inspired by two-state models -- such as the Allen-Eberly and Hioe-Carroll models -- we extend these concepts to our system, exploring a range of pulse families, including those with super-Gaussian envelopes and polynomial detuning functions. We identify Pareto fronts for pulse models constructed from one of two envelope functions paired with one of four detuning functions. We then analyze how each Pareto-optimal pulse parameter influences the two Pareto objectives as well as amplitude robustness.

Figures (9)

• Figure 1: The energy diagram of our frequency-modulation scheme shown for two different detuning functions, one linear (blue) and one quintic (green), for time $t \in [-T/2, T/2]$. The transition frequencies are $\omega_{\rm{ge}}$ and $\omega_{\rm{ef}}$; the modulation depth is denoted by $\Delta_{\rm max}$; and the frequency offset is $\delta$. The two-photon transition frequency $\omega_{\rm{gf}}^{\rm 2ph}$ is exactly halfway between $\omega_{\rm{ge}}$ and $\omega_{\rm{ef}}$.
• Figure 1: The Pareto-optimal amplitudes $\Omega_{0}$ vs detuning robustness in panel (a) and $\max_{t}(p_{\rm{f}})$ in panel (b). Each model is evaluated at pulse durations $T = 50$ ns (solid lines) and $T = 200$ ns (dashed lines). The vertical arrows in (a) indicate that there are jumps in $\Omega_{0}$ corresponding to a change in sign of $k_{1}$.
• Figure 1: A log-log scale plot showing amplitude robustness $\Omega_{\rm rob}$ vs detuning robustness $\Delta_{\rm rob}$ for each model evaluated at pulse durations (a) $T = 50$ ns and (b) $T = 200$ ns. The orange reference lines are the best-fit lines, each of which has a power of 1 and a slope near unity, indicating a near one-to-one relationship between $\Delta_{\rm rob}$ and $\Omega_{\rm rob}$.
• Figure 2: (a) The maximum transient second excited state population $\max_{t}(p_{\rm{f}})$ vs detuning robustness for various models with pulse durations of 50 ns (solid curves) and 200 ns (dashed curves). The shaded regions (cyan and magenta for 50 ns and 200 ns pulses, respectively) represent the range of data constituting the fronts found from using the Landau-Zener (LZ) formula to approximate the theoretical limit of the detuning robustness for each model. These LZ-based bounds are computed at the threshold condition $p_{\rm{e}}^{\rm{thresh}} = 0.99$, but without requiring that $p_{\rm{e}}(T/2) \geq 0.999$ anywhere within the corresponding detuning interval (i.e., between the roots $\delta_{-}$ and $\delta_{+}$, where $p_{\rm{e}}(\delta_{\pm}) = 0.99$). The orange dashed boundary includes this additional constraint. As an example, panel (b) shows the time profiles of the envelope function and detuning function for the SECH3 model at maximum detuning robustness. The upper (lower) tile corresponds to $T = 50$ ($200$) ns.
• Figure 2: The Pareto-optimal detuning prefactors $k_{1}$ as a function of (a) $\max_{t}(p_{\rm{f}})$ and (b) detuning robustness for each model at pulse durations $T = 50$ ns (solid lines) and $T = 200$ ns (dashed lines). Panel (c) shows the Pareto front (PF) with our usual definition of $\Delta(t)$ (solid blue line) and the Pareto front with $\Delta(t) \rightarrow -\Delta(t)$ (blue dotted line) for the SG1 model with a pulse duration of $T = 200$ ns. The right vertical axis shows the values of the corresponding Pareto-optimal $k_{1}$ (solid red line).