Realization of Trapped Ion Dynamics in the Strong-Field Regime and Non-Markovianity

TL;DR

This work probes trapped-ion dynamics in the strong-field regime where the Rabi frequency $\Omega$ approaches the vibrational frequency $\nu$, revealing non-Markovian memory effects in the qubit. By reconstructing the reduced qubit state via quantum-state tomography and quantifying non-Markovianity with the trace-distance criterion, the authors link memory effects to the spin–motion coupling in a structured environment. A key finding is the non-monotonic NM dependence on $\Omega$, with NM maxima following a circular pattern in parameter space when $\delta^2+\Omega^2=\nu^2$, where the dynamics map onto a Jaynes-Cummings-like interaction in a transformed frame. The results extend trapped-ion control beyond carrier and sideband regimes, show NM as a sensitive probe of open-system dynamics under extreme driving, and point to new avenues for coherent control and quantum thermodynamics in multi-mode environments.

Abstract

Probing quantum dynamics in the strong-field regime is critical for advancing our understanding of controlled quantum systems and developing robust quantum technologies. In this work, we experimentally investigate the dynamics of a trapped ion where the Rabi frequency (Omega) approaches the vibrational mode frequency (nu), pushing the system beyond the weak-field regime, where non-trivial quantum correlations emerge. We begin by setting the detuning (delta) - the frequency offset between the qubit transition and the driving field - to zero and varying Omega from low to high values, eventually reaching the vibrational frequency. Using quantum state tomography, we reconstruct the density matrix and track its evolution to assess non-Markovianity, revealing significant memory effects governed by the interplay between internal and motional degrees of freedom. Furthermore, by exploring the dynamics across various parameter pairs (Omega, delta), we find that non-Markovianity does not always increase monotonically with Omega for a fixed delta. Strikingly, when the condition delta squared plus Omega squared equals nu squared is met, the non-Markovianity exhibits a circular pattern of maxima. At this parameter combination, the system's Hamiltonian takes a form similar to the Jaynes-Cummings model, enabling the possibility of analytical insights into the observed dynamics. These results go beyond the conventional carrier and sideband regimes, uncovering novel features of strong-field quantum dynamics. Our findings establish a pathway for using trapped-ion platforms to investigate non-Markovianity, coherent control, and the fundamental behavior of open quantum systems in extreme regimes.

Figures (14)

• Figure 1: Schematic diagram of our system. The elliptical trapping potential of the Paul trap gives rise to two radial motional modes with frequencies $\nu_1$ and $\nu_2$, corresponding to vibrations along the $x$ and $y$ directions, respectively. Two electronic (internal) states form a qubit, which is coupled to each vibrational mode via two sets of Raman lasers.
• Figure 2: Dynamics of trace distance and its gradient. (a) Evolution of quantum states $\rho_1 = \ket{+}\bra{+}$ and $\rho_2 = \ket{-}\bra{-}$ on the Bloch sphere, illustrating the gradual loss of distinguishability over time. Here, $\ket{0}$, $\ket{1}$, $\ket{+}$, $\ket{-}$, $\ket{+i}$, and $\ket{-i}$ are eigenstates of the Pauli operators $\sigma_z$, $\sigma_x$, and $\sigma_y$. (c) Time evolution of the trace distance $D(\rho_1, \rho_2)$ (Eq. \ref{['eq:trace_distance']}), revealing its steady decay. (d) Gradient of the trace distance $\sigma(t)$ (Eq. \ref{['eq:time_gradient_trace_distance']}), as a function of time, highlighting the rate at which the states become indistinguishable. The horizontal black dashed line indicates a threshold: the sum of $\sigma(t)>0$ values above this line contributes to non-Markovianity (NM). In all panels, solid curves represent simulation results, and data points correspond to experimental values with corresponding error bars.
• Figure 3: Non-Markovianity (NM) characterization under varying conditions. (a) Dependence of NM on the total evolution time (with $0.5\,\mu s$ resolution). (b) Influence of the number of time steps on NM, for a fixed total evolution time of $100\,\mu s$. (c) Convergence of NM with respect to the number. (d) Numerical simulation showing NM versus the plus-state dephasing rate, with the minus-state dephasing fixed at the experimental value $0.0008\,\nu$. A special marker indicates the NM for the experimental plus-state dephasing.
• Figure 4: Non-Markovianity (NM) behavior versus coupling strength ($\Omega$) of the resonant laser field ($\delta = 0$) at time resolution of $0.5 \, \mu s$. The red curve (top) corresponds to an evolution time of $100~\mu s$. Non-Markovianity (NM), a sensitive probe to unveil strong-field dynamics, initially increases with the coupling strength $\Omega$, indicating a transition from Markovian to non-Markovian behavior due to information backflow from the environment to the system. However, once $\Omega$ exceeds the vibrational mode frequency $\nu_i$, NM begins to decrease, revealing a reversal in this trend. The blue curve (lower) highlights experimental data for $\Omega$ values near to motional mode with an evolution time of $40 \, \mu s$. The reduced evolution time reflects the operational constraints imposed by second ionization and Rabi oscillations power decay, which limit the observation window for non-Markovian dynamics. The curves correspond to simulation results, swith experimental data represented by points.
• Figure 5: Experimental results demonstrating the circular pattern of NM maxima as a function of detuning ($\delta$) and coupling strength ($\Omega$), satisfy the relation $\delta^2 + \Omega^2 = \nu^2$. The insets provide experimental results for specific detuning values: $\delta = 0.25$, $1.37$, $2.32$, and $3.16$, corresponding to points A, B, C, and D, respectively. The smaller circular pattern corresponds to the mode with frequency $\nu_1$, and the larger one to $\nu_2$. These insets show the detailed NM dynamics at each selected point. Experiments were performed with an evolution time of $40\,\mathrm{\mu s}$, sampled at 200 steps for high temporal resolution. The detuning effect was implemented in the interaction frame by applying the laser field, while the quantification of NM was carried out by back-transforming the experimental data to the lab frame.