Continuously Parametrized Quantum Simulation of Molecular Electron-Transfer Reactions

A comprehensive description of molecular electron-transfer reactions is essential for our understanding of fundamental phenomena in bioenergetics and molecular electronics. However, experimental studies of molecular systems in condensed-phase environments face difficulties in independently controlling the parameters that govern the transfer mechanism with high precision. We show that, instead, trapped-ion experiments allow us to reproduce and continuously connect vastly different regimes of molecular charge transfer through precise tuning of, e.g., the phonon temperature, electron-phonon interaction, and electronic coupling. Such a setting does not only allow us to reproduce widely used transport models, such as Marcus theory. It also provides access to transfer regimes that are unattainable for molecular experiments, while controlling and measuring the relevant observables on the level of individual quanta. Our numerical simulations predict an unconventional quantum transfer regime, featuring a transition from quantum adiabatic to resonance-assisted transfer as a function of the donor-acceptor energy gap, which can be reached by increasing the electronic coupling at low temperatures. Trapped-ion-based quantum simulations thus promise to enhance our microscopic understanding of molecular electron-transfer processes and may help to reveal efficient design principles for synthetic devices.

Popular Summary

Molecular electron-transfer reactions are the fundamental steps of diverse phenomena such as photosynthesis, biological signaling, or energy conversion. Experimental studies of electron transfer in molecular systems face extreme difficulties when aiming to control, independently and continuously, individual parameters that govern the transfer mechanisms, which limits our microscopic understanding of these processes. We show how trapped-ion quantum simulators can provide a highly versatile platform to study electron-transfer phenomena in molecules and beyond.

A laser-cooled system of two trapped ions with strong laser-induced spin-phonon coupling reproduces the electron-transfer dynamics of molecular donor-acceptor systems. Our numerical simulations of the ion system reveal that in a classical high-temperature regime, the incoherent transfer rates match the predictions of Marcus theory, which was recognized with the Nobel Prize in Chemistry in 1992. We then systematically explore new parameter regimes that are hard or even impossible to observe in molecular systems. By reducing the temperature, we enter a quantum nonadiabatic transfer regime, dominated by vibronic resonances. By increasing the electronic coupling, we observe a continuous crossover to quantum adiabatic and exotic resonance-assisted transfer regimes.

Trapped-ion quantum simulations allow great flexibility and precise controllability of the temperature, the coupling strengths, and the trap geometries. They thus provide a powerful test bed for electron-transfer models to enhance our microscopic understanding of molecular systems, benchmark semiclassical approximations, and reveal efficient design principles for synthetic electron-transfer devices. Our work reveals the interesting physics that is hidden in the incoherent decay rates of quantum simulators.

Figure 1

Electron-transfer theory. (a) The vibrational-energy landscape deduced from the Hamiltonian (1), composed of the adiabatic Born-Oppenheimer (BO) surfaces for the electronic states $D$ and $A$, as a function of the reaction coordinate defined by the dimensionless harmonic oscillator position variable $z$. The diabatic BO surfaces, $E_D(z)$ (left parabola) and $E_A(z)$ (right parabola), respectively, are identified by dashed gray lines in the inset. For the weak coherent coupling assumed here, adiabatic and diabatic surfaces are indiscernible on the scale of the main plot. The $D$-$A$ energy gap $\mathrm{\Delta }E=E_A-E_D$ [which is negative given the electronic level structure depicted in (b) below], the reorganization energy $\lambda$, the activation energy $U$, and the coherent coupling $V$ (inset) are indicated. The horizontal lines indicate vibrational levels $\nu$ (${\nu }^{\prime }$) of the D (A) BO surface upon quantization. The employed parameters are $|\mathrm{\Delta }E|=2.5\hslash {\omega }_0$, $V=0.01\hslash {\omega }_0$, and $g=2.5{\omega }_0$, where ${\omega }_0$ denotes the (effective) trap frequency. (b) A “system” ion (blue) simulates the donor-acceptor system with the two electronic states $|D⟩$ and $|A⟩$ (akin to a spin $1/2$). Laser-driven Rabi oscillations (blue upward arrow) induce coherent coupling of strength $V$ between the donor and acceptor states. A second laser (blue downward arrow) induces spin-phonon coupling between the electronic degree of freedom of the “system” ion and a collective motional mode of both ions. This mode is sympathetically cooled by laser interaction (red arrow) through a “cooling” ion (red) of a different species.

Figure 2

Classical nonadiabatic transfer. (a) Diabatic BO energy surfaces and their vibrational levels in the classical nonadiabatic regime. The parameters are chosen as $V=0.2\hslash {\omega }_0$, $g=10{\omega }_0$, $\gamma =0.4{\omega }_0/2\pi$, and $\overline{n}=10$, corresponding to $k_{B}T\approx 10.5\hslash {\omega }_0$ and $\lambda =100\hslash {\omega }_0$. The panels show $|\mathrm{\Delta }E|=0\hslash {\omega }_0$ (A), $|\mathrm{\Delta }E|=50\hslash {\omega }_0$ (B), $|\mathrm{\Delta }E|=100\hslash {\omega }_0$ (C), $|\mathrm{\Delta }E|=150\hslash {\omega }_0$ (D), and $|\mathrm{\Delta }E|=200\hslash {\omega }_0$ (E). Vibrational states are indicated by horizontal gray lines and red (donor) and blue (acceptor) shaded areas indicate the thermal populations. (b) The exponential fit (dashed lines) of the time evolution of the donor-state population $p_D(t)$ (continuous lines) as generated by Eqs. (1) and (3), for the parameters of panels A (yellow), B (pink), and C (violet) in (a). (c) Transfer rates extracted from the exponential fits to $p_D(t)$ as a function of $|\mathrm{\Delta }E|$ (dark blue circles). The orange line is the prediction of classical nonadiabatic transport theory, according to the Marcus formula (4).

Figure 3

Quantum nonadiabatic (QN) transfer. (a) Diabatic BO energy surfaces and their vibrational levels for the quantum nonadiabatic transfer regime. The system parameters are set to $V=0.01\hslash {\omega }_0$, $g=2.5{\omega }_0$, $\gamma =0.05{\omega }_0/2\pi$, and $\overline{n}=0.01$. Consequently, the initial thermal population beyond the ground state amounts to less than 1%, corresponding to $k_{B}T\approx 0.217\hslash {\omega }_0$ and we further have $\lambda =6.25\hslash {\omega }_0$. The panels show $|\mathrm{\Delta }E|=\hslash {\omega }_0$ (A), $|\mathrm{\Delta }E|=3.5\hslash {\omega }_0$ (B), $|\mathrm{\Delta }E|=5\hslash {\omega }_0$ (C), $|\mathrm{\Delta }E|=6.5\hslash {\omega }_0$ (D), and $|\mathrm{\Delta }E|=9\hslash {\omega }_0$ (E). (b) The dynamical evolution of the donor population (continuous lines), with exponential fits (dashed lines), for the parameters of panels A (yellow), B (pink), and C (violet). (c) The transfer rates (dark blue circles) are extracted from the exponential fits in (b). The orange line displays the prediction given in Eq. (5) of the quantum nonadiabatic theory, with resonances of finite width determined by the cooling rate $\gamma$.

Figure 4

The crossover toward quantum adiabatic transfer and population trapping. (a) The BO surfaces split into lower and upper adiabatic surfaces, separated by an avoided crossing of size $2|V|$. The diabatic picture is no longer adequate due to the formation of hybridized vibronic states. (b) The time evolution of the donor-state population as generated by Eqs. (1) and (3), for $V=0.25\hslash {\omega }_0$ (left panels), $V=0.50\hslash {\omega }_0$ (center), and $V=0.75\hslash {\omega }_0$ (right). All remaining parameters are chosen as in Fig. 3. The plots show the evolution at $|\mathrm{\Delta }E|=\hslash {\omega }_0$ (A,D,G; yellow lines), $|\mathrm{\Delta }E|=3.5\hslash {\omega }_0$ (B,E,H; pink lines), and $|\mathrm{\Delta }E|=7\hslash {\omega }_0$ (C,F,I; violet lines). The corresponding dashed lines show exponential fits after a transient evolution time of ${\omega }_{0}t/2\pi =20$. (c) The transfer rates extracted from the exponential fits of the simulated quantum dynamics, as a function of the donor-acceptor energy gap $\mathrm{\Delta }E$.

Figure 5

The properties of the transfer rate in the quantum adiabatic limit. (a) The quantum ET rates as a function of the electronic coupling $V$ for $|\mathrm{\Delta }E|=2\hslash {\omega }_0$ (yellow circles), $|\mathrm{\Delta }E|=4\hslash {\omega }_0$ (pink squares), $|\mathrm{\Delta }E|=6\hslash {\omega }_0$ (violet lower triangles), and $|\mathrm{\Delta }E|=8\hslash {\omega }_0$ (black upper triangles). (b) The quantum ET rates divided by the cooling rate $\gamma$, as a function of the donor-acceptor energy gap $\mathrm{\Delta }E$, for $V=0.75\hslash {\omega }_0$ and $\gamma =0.025{\omega }_0/2\pi$ (yellow circles), $\gamma =0.05{\omega }_0/2\pi$ (pink squares), and $\gamma =0.075{\omega }_0/2\pi$ (violet lower triangles). The rates in the normal regime ($|\mathrm{\Delta }E|<5\hslash {\omega }_0$) approximately follow the quantum adiabatic limit given in Eq. (6), which predicts a linear dependence on $|\mathrm{\Delta }E|$ (since the latter controls the resonantly coupled phonon number $\nu$). For $|\mathrm{\Delta }E|>5\hslash {\omega }_0$, the process is no longer purely quantum adiabatic due to the trapping of part of the initial population in the upper adiabatic BO surface (see the text). The dashed line shows a linear fit, $a|\mathrm{\Delta }E|/(\hslash {\omega }_0)+b$, with $a=2.4\times {10}^{-4}\gamma {\omega }_0$ and $b=5.2\times {10}^{-6}\gamma {\omega }_0$. All remaining parameters are chosen as in Fig. 3.

Figure 6

The eigenstate width of the initial state according to Eq. (B6) for $V=0.25\hslash {\omega }_0$ (yellow, dot-dashed), $0.5\hslash {\omega }_0$ (red, dashed), and $0.75\hslash {\omega }_0$ (blue, solid line). All remaining parameters are identical to those in Fig. 4.