Engineering Vibrationally Assisted Energy Transfer in a Trapped-Ion Quantum Simulator

Many important chemical and biochemical processes in the condensed phase are notoriously difficult to simulate numerically. Often, this difficulty arises from the complexity of simulating dynamics resulting from coupling to structured, mesoscopic baths, for which no separation of time scales exists and statistical treatments fail. A prime example of such a process is vibrationally assisted charge or energy transfer. A quantum simulator, capable of implementing a realistic model of the system of interest, could provide insight into these processes in regimes where numerical treatments fail. We take a first step towards modeling such transfer processes using an ion-trap quantum simulator. By implementing a minimal model, we observe vibrationally assisted energy transport between the electronic states of a donor and an acceptor ion augmented by coupling the donor ion to its vibration. We tune our simulator into several parameter regimes and, in particular, investigate the transfer dynamics in the nonperturbative regime often found in biochemical situations.

Popular Summary

Charge and energy transfer are essential to many important processes in chemistry, biology, and emerging nanotechnologies. Such transfer processes often occur in noisy thermal environments that strongly modify the transfer dynamics and, in some cases, even improve the transport efficiency or robustness. A prominent example is the energy transfer within photosynthesis centers of cells from pigments in light-harvesting complexes towards reaction centers, where efficiency is believed to critically depend on the spectral properties of the environment. Such noise- (or environmentally) assisted transport processes are often difficult to study numerically owing to the complexity of their structured, mesoscopic molecular environments. Here, we pursue a quantum simulation approach in which the phenomenon can be isolated and studied under fully controlled conditions.

We encode a noise-assisted transport process in a trapped-ion quantum simulator where energy transfer between ions is enhanced when coupled to their thermal vibrational motion. We observe transfer processes wherein the environment changes by an integer number of motional quanta. With the environment prepared near the ground state, we observe oscillatory energy transfer dynamics, indicating the quantum nature of the environment. We demonstrate the ability to tune our quantum simulator into the nonperturbative parameter regimes often encountered in models of biochemical processes for which approximation methods fail.

While our approach does not yet offer new insights into these processes, it is a first step towards encoding more complex transfer processes relevant to biological systems.

Figure 1

(a) Schematic illustrations of the VAET process and time dynamics of the target state population in various regimes. Without the presence of an environment ($\kappa =0$, left drawings), the transition probability from donor to acceptor states is attenuated in the presence of an energy barrier $\mathrm{\Delta }$. By coupling to an environment ($\kappa >0$, right drawing), an excitation can move between donor and acceptor sites by exchanging energy with a phononic environment. Time traces illustrate the three situations: (1) $\mathrm{\Delta }=0$: black trace (theory) and data points (∘). (2) $\mathrm{\Delta }>J$, without assistance from the environment: blue trace (theory) and data points ($\times$). (3) $\mathrm{\Delta }>J$, with assistance from the environment (VAET process): red trace (theory) and data points (⋄). (b) Schematics of the ion trap and laser beams generating the simulated Hamiltonian. Internal levels of the ions (blue spheres) serve as energy sites. A laser beam illuminating both ions generates the site-site coupling with strength $J$. A localized beam generates the coupling to the environment with strength $\kappa$ and controls the detuning $\mathrm{\Delta }$. (c) Laser tones (orange arrows) generating the simulated Hamiltonian. Vertical black lines represent available transitions, with the atomic resonance of the respective ion in the center. The detuning $\mathrm{\Delta }$ is introduced via an ac-Stark shift by imbalancing the relative power of the two tones of the local beam, as indicated in the figure.

Figure 2

Energy transfer probability to the acceptor $P_{\mathrm{acc}}$ vs simulation time ${\tau }_{\mathrm{sim}}$ and vibrational frequency ${\nu }_{\mathrm{eff}}$. Left (a) [right (b)] plots show the time dynamics at environmental temperature $\overline{n}=5$ ($\overline{n}=0.5$). Upper plots show the time dynamics $P_{\mathrm{acc}}$ with ${\nu }_{\mathrm{eff}}/2\pi \approx +4.56\mathrm{kHz}$ (blue points, $\times$) and ${\nu }_{\mathrm{eff}}/2\pi \approx -4.56\mathrm{kHz}$ (red points, circles). The lower plots show $P_{\mathrm{acc}}$ vs ${\nu }_{\mathrm{eff}}$, where the simulation time ${\tau }_{\mathrm{sim}}$ is fixed to 0.7 ms. For all cases, $\{J,\kappa ,\mathrm{\Delta }\}=2\pi \times \{1.30(1),1.40(4),4.56(2)\}\mathrm{kHz}$. Solid lines are numerical simulations of the system with all parameters determined through independent calibrations. The shaded regions represent the estimated systematic uncertainty on the theoretical curve, obtained from the measurement error in the calibration parameters. Note that the theory estimates include the effect that the detuning $\mathrm{\Delta }$ fluctuates around its mean due to relative intensity noise of 0.02 as well as due to variations of the ion-laser coupling strength via finite-temperature effects. In (a), the standard deviation of the corresponding distribution is $2\pi \times 0.23\mathrm{kHz}$, whereas in (b), it is $2\pi \times 0.1\mathrm{kHz}$. Solid lines in the spectral plots represent a numerical solution where a small frequency offset adjusts for a systematic bias of the vibrational frequency measurements. In all plots, the statistical error is smaller than the markers. The measured data points in the spectral plots are connected with a dashed line to guide the eye.

Figure 3

(a) Energy transfer probability $P_{\mathrm{acc}}$ vs vibrational frequency ${\nu }_{\mathrm{eff}}$ in the small detuning regime with $\{J,\kappa ,\mathrm{\Delta }\}=2\pi \times \{1.22(3),0.63(2),1.226(3)\}\mathrm{kHz}$, ${\tau }_{\mathrm{sim}}=0.7\mathrm{ms}$, and $\overline{n}=2.7$. Direct coupling between the sites exceeds the energy differences between the various multiphonon processes, and only two peaks remain, corresponding to either subtracting or adding energy. (b) $P_{\mathrm{acc}}$ vs simulation time for different environmental couplings $\kappa$ for low environment temperature ($\overline{n}=0.04$). In all three scans, all parameters except $\kappa$ are constant, $\{J,\mathrm{\Delta },{\nu }_{\mathrm{eff}}\}=2\pi \times \{1.27(5),1.27(8),-1.72(4)\}\mathrm{kHz}$. Coupling $\kappa$ increases from top to bottom as indicated. Solid traces are numerically simulated dynamics with all parameters determined by independent measurements. Dashed black traces correspond to the predictions of the perturbative treatment developed in Ref. [24]. (c) $P_{\mathrm{acc}}$ vs simulation time ${\tau }_{\mathrm{sim}}$ at high temperature ($\overline{n}=12$ quanta). Simulation parameters: $\{J,\kappa ,\mathrm{\Delta },{\nu }_{\mathrm{eff}}\}=2\pi \times \{1.17(3),0.63(2),1.59(3),-1.72(6)\}\mathrm{kHz}$. In the time traces, the shaded regions represent the estimated systematic error on the theoretical curve, obtained from the measurement error in the simulation parameters. In all plots, the statistical error is smaller than the markers.