The MP/SOFT methodology for simulations of quantum dynamics: Model study of the photoisomerization of the retinyl chromophore in visual rhodopsin

Comparison of Matrix Product State and Multiconfiguration Time-Dependent Hartree Methods for Nonadiabatic Dynamics of Exciton Dissociation

The excited-state dynamics of organic molecules, molecular aggregates, and donor-acceptor clusters is typically governed by the interplay of electronic excitations and, due to their flexibility and soft bonding, by the interaction with their vibrations. This interaction in these systems can be characterized by a few relevant electronic states that are coupled to numerous vibrational normal modes, encompassing a vast configurational space of the molecules. The full quantum simulation of these type of systems has been long dominated by the multiconfiguration time-dependent Hartree (MCTDH) approach and its multilayer variants, which are considered the gold standard in the presence of electron-vibration coupling with a large number of modes. Recently, also the matrix product state ansatz (MPS) with appropriate time-evolution schemes has been applied to these types of Hamiltonians. In this article, we provide a numerical comparison of excited-state dynamics between the MCTDH and MPS approaches for two electron-vibration coupled systems. Notably, we consider two models for exciton dissociation at a P3HT:PCBM heterojunction, comprising two electronic states and 100 vibrational modes, and 26 electronic states and 113 vibrational modes, respectively. While both methods agree very well for the first model, more pronounced deviations are found for the second model. We trace back the divergence between the methods to the different way entanglement is treated.

Tensor-Train Split-Operator KSL (TT-SOKSL) Method for Quantum Dynamics Simulations

Numerically exact simulations of quantum reaction dynamics, including nonadiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of molecular spectroscopy. Here, we introduce the tensor-train split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train (TT)/matrix product state (MPS) representations. TT-SOKSL propagates the quantum state as a tensor train using the Trotter expansion of the time-evolution operator, as in the tensor-train split-operator Fourier transform (TT-SOFT) method. However, the exponential operators of the Trotter expansion are applied using a rank-adaptive TT-KSL scheme instead of using the scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and efficiency of TT-SOKSL as applied to simulations of the photoisomerization of the retinal chromophore in rhodopsin, including nonadiabatic dynamics at a conical intersection of potential energy surfaces. The quantum evolution is described in full dimensionality by a time-dependent wavepacket evolving according to a two-state 25-dimensional model Hamiltonian. We find that TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed memory requirement of the tensor-train representation and better preserves the norm of the time-evolving state. When compared to the corresponding simulations based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need to construct the matrix product state Laplacian by exploiting the linear scaling of multidimensional tensor-train Fourier transforms.

Floquet Study of Quantum Control of the Cis-Trans Photoisomerization of Rhodopsin

Understanding how to control reaction dynamics of polyatomic systems by using ultrafast laser technology is a fundamental challenge of great technological interest. Here, we report a Floquet theoretical study of the effect of light-induced potentials on the ultrafast cis-trans photoisomerization dynamics of rhodopsin. The Floquet Hamiltonian involves an empirical 3-state 25-mode model with frequencies and excited-state gradients parametrized to reproduce the rhodopsin electronic vertical excitation energy, the resonance Raman spectrum, and the photoisomerization time and efficiency as probed by ultrafast spectroscopy. We simulate the excited state relaxation dynamics using the time-dependent self-consistent field method, as described by a 3-state 2-mode nuclear wavepacket coupled to a Gaussian ansatz of 23 vibronic modes. We analyze the reaction time and product yield obtained with pulses of various widths and intensity profiles, defining 'dressed states' where the perturbational effect of the pulses is naturally decoupled along the different reaction channels. We find pulses that delay the excited-state photoisomerization for hundreds of femtoseconds, and we gain insights on the underlying control mechanisms. The reported findings provide understanding of quantum control, particularly valuable for the development of ultrafast optical switches based on visual pigments.

Steered quantum dynamics for energy minimization

We introduce a quantum optimal control algorithm for energy minimization that combines the diffeomorphic modulation under observable response preserving homotopy (D-MORPH) gradient and the Broyden Fletcher Goldfarb Shanno (BFGS) iterative scheme for nonlinear optimization. An extended set of controls defining the time-dependent mass, dipole moment, and external perturbational field are optimized to find an effective Hamiltonian that steers the dynamics of the system into the global minimum without getting trapped into local minima. The algorithm is illustrated as applied to energy minimization on rugged surfaces and golf potentials comparable to those previously explored for testing quantum annealing methodologies.

Re-evaluation of rhodopsin's relaxation kinetics determined from femtosecond stimulated Raman lineshapes

This work presents a theoretical treatment of the vibrational line shape generated in a femtosecond stimulated Raman spectroscopy (FSRS) experiment under conditions in which the probed vibration undergoes a significant frequency shift during its free induction decay. This theory is applied to simulate the FSRS lineshapes previously observed in rhodopsin (Kukura et al. Science 2005, 310, 1006). The previously determined relaxation times for formation of the trans-photoproduct of rhodopsin were calculated using an incorrect equation for the time dependence of the observed frequency shifts. Here the data are reanalyzed by calculation of the corrected frequency sweep occurring during the vibrational free induction decay. It is shown that the calculated frequency shifts and general conclusions of the original work are sound but that the coherent vibrational frequency shifts of the C(10), C(11), and C(12) hydrogen-out-of-plane vibrations occur with a 140 fs time constant rather than the previously reported 325 fs time constant. This time constant provides an important constraint for models of the dynamics of the cis to trans isomerization process.

Vibrational fine structures revealed by the frequency-to-time fourier transform of the transient spectrum in bacteriorhodopsin

A vibrational progression that is hidden in a featureless spectrum of induced absorption and stimulated emission was found in time-resolved absorption change spectra. The ultrahigh time resolution of the pump-probe measurement made by using an ultrashort laser pulse localizes the wave packet along the potential multimode hyper surfaces, represented by a vibrational progression. The transition energy of the induced absorption and stimulated emission corresponds to a localized point (space) on the hyper surface, which is visited by the wave packets with fixed phases.

Primary conformation change in bacteriorhodopsin on photoexcitation

Ultrafast dynamics of bacteriorhodopsin (bR) has been extensively studied experimentally and theoretically. However, there are several contradictory results reported, indicating that its detailed dynamics and initial process have not yet been fully clarified. In this work, changes in the amplitude and phase of molecular vibration in the isomerization process of bR were real-time probed simultaneously at 128 different wavelengths through intensity modulation of the electronic transition. Systematic information on the transient change in continuous spectrum extending from 505 nm (2.45 eV) to 675 nm (1.84 eV) showed different dynamics in each spectral region reflecting the difference in the excited states and intermediates dominating the dynamics during the photoisomerization. Careful analysis of the transient spectral changes and spectrograms calculated from the vibrational real-time traces elucidated that the primary event just after photoexcitation is the deformation of the retinal configuration, which decays within 30 fs near the C=N bond in the protonated Schiff base. The intensity of C=N stretching mode starts to decrease before the initiation of the frequency modulation of the C=C stretching mode. The C=C stretching mode frequency was modulated by a coupled torsion around the C13=C14 bond, leading to the photoisomerization around the bond. This study clarified the dynamics of the C=N and C=C stretching modes working as key vibration modes in the photoisomerization of bR. Furthermore, we have elucidated the modulation and decay dynamics of the C=C stretching mode in the photoreaction starting from H (Franck-Condon excited state) followed by I (twisted excited), and J (first intermediate) states.

Polynomial-time quantum algorithm for the simulation of chemical dynamics

The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can be applied only to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and interelectronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born-Oppenheimer approximation but faster and more efficient as well, for all reactions with more than about four atoms. This is the case even though the entire electronic wave function is propagated on a grid with appropriately short time steps. Although the preparation and measurement of arbitrary states on a quantum computer is inefficient, here we demonstrate how to prepare states of chemical interest efficiently. We also show how to efficiently obtain chemically relevant observables, such as state-to-state transition probabilities and thermal reaction rates. Quantum computers using these techniques could outperform current classical computers with 100 qubits.