Phase evolution of superposition target states in adiabatic population transfer

TL;DR

This paper analyzes STIRAP when the final state is a superposition of two non-degenerate states, focusing on how the relative phase of the final state depends on pulse amplitudes, widths, and timing. By modeling a four-level system and deriving an analytic treatment for the phase evolution, the authors show that the final relative phase arises from the energy-splitting of two near-degenerate quasi-dark states and provide a closed-form expression for the plateau phase in terms of the pulse parameters; they also identify onset corrections and confirm negligible contributions from detuning variations and geometric phases. The main contributions are the explicit phase formula $\phi(\tau)=\epsilon(\tau)\ln{\left[\frac{1-\sqrt{1+\gamma}}{1+\sqrt{1+\gamma}}\frac{1+\sqrt{1+\gamma e^{-\tau}}}{1-\sqrt{1+\gamma e^{-\tau}}}\right]}$, the characterization of how phase plateaus depend on $r=\Omega_{0s}/\Omega_{0p}$ and pulse width $T$, and the demonstration that the STIRAP-induced phase is typically small for precision symmetry-violation measurements like the YbF experiment. This work informs how to manage or mitigate STIRAP-associated phases in high-sensitivity weak-symmetry tests.

Abstract

We consider stimulated Raman adiabatic passage (STIRAP) when the final state is a superposition of two non-degenerate states. The system consists of four states coupled by two light fields. We find the relative phase of the final superposition depends on relative amplitude, width and timing of the adiabatic transfer pulses. We discuss these results in the context of experiments measuring symmetry violation in atomic and molecular systems.

Figures (5)

• Figure 1: Energy levels and coupling in the four-sate system. The splitting $\delta$ is very small compared to $\Delta$ and the $\ket{e}-\ket{g}$ energy difference.
• Figure 2: Numeric calculation of the time evolution of the relative phase between states $\ket{\uparrow}$ and $\ket{\downarrow}$ for a range of different $\delta$ (in units of $10^{-3} T^{-1}$). The envelopes of the Stokes and pump light are shown in gray. The relative phase evolves in two steps. First, a sudden jump to a plateau appears before a switch to linear phase accumulation 2$\delta t$.
• Figure 3: Time evolution of the eigenenergies (a) and components of the two eigenstates $\ket{a}$ (b) and $\ket{b}$ (c). Both states start as a superposition of initial and excited state and then gradually evolve into one of the target states. A fast reorientation from and back to the initial composition occurs as the light turns on and off.
• Figure 4: Comparison of numeric and analytic results for the evolution of the relative phase in the plateau region. There is little dependence on peak Rabi frequency (b) and nearly linear dependence on the detuning $\delta$ (c). The separation of the beams alone has been varied in (d) while both the width $T$ of the pulse envelopes and their separation has been varied in (e) such that the separation is always at 1.5T. All plots use a peak Rabi frequency of 248 $T^{-1}$ and a beam separation of $1.5T$ except ones where these are the variables.
• Figure 5: The dependence of the phase correction term ($t_2$) on beam parameters. Numerical simulations shown in red are compared with the analytical formula (eq. \ref{['eq: phase onset']}) shown in blue. With other parameters fixed, (a) varies the separation of the beams $\mu_s-\mu_p$, and (b) the Rabi envelope width $T$.