Abstract. In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg’s principle, we overcame these difficulties by blending Koopman’s classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz that introduces the trajectories of computational particles—the koopmons—sampling the Lagrangian classical paths in phase space. In the case of Tully’s nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results.
Abstract. In order to alleviate the computational costs of fully quantum nonadiabatic dynamics, we present a mixed quantum-classical (MQC) particle method based on the theory of Koopman wavefunctions. Although conventional MQC models often suffer from consistency issues such as the violation of Heisenberg’s principle, we overcame these difficulties by blending Koopman’s classical mechanics on Hilbert spaces with methods in symplectic geometry. The resulting continuum model enjoys both a variational and a Hamiltonian structure, while its nonlinear character calls for suitable closures. Benefiting from the underlying action principle, here we apply a regularization technique previously developed within our team. This step allows for a singular solution ansatz that introduces the trajectories of computational particles—the koopmons—sampling the Lagrangian classical paths in phase space. In the case of Tully’s nonadiabatic problems, the method reproduces the results of fully quantum simulations with levels of accuracy that are not achieved by standard MQC Ehrenfest simulations. In addition, the koopmon method is computationally advantageous over similar fully quantum approaches, which are also considered in our study. As a further step, we probe the limits of the method by considering the Rabi problem in both the ultrastrong and the deep strong coupling regimes, where MQC treatments appear hardly applicable. In this case, the method succeeds in reproducing parts of the fully quantum results.
Abstract We consider the dynamics of interacting quantum and classical systems in the Heisenberg representation. Unlike the usual construction in standard quantum mechanics, mixed quantum-classical systems involve the interplay of unitary operators acting on the quantum observables and the Lagrangian trajectories sweeping the classical degrees of freedom. This interplay reflects an intricate structure which is made particularly challenging by the backreaction excerpted on the classical trajectories by the quantum degrees of freedom. While the backreaction is underestimated in the common Ehrenfest model, more recent methodologies succeed in capturing this important effect by resorting to Koopman wavefunctions in classical mechanics. Luckily, both Ehrenfest and Koopman models enjoy a variational framework which is exploited here to unfold the geometric structure underlying quantum-classical coupling. A special role is played by the action of the diffeomorphic Lagrangian paths on a non-Abelian pure-gauge potential which comprises statistical correlations. After presenting the treatment in the simple case of Ehrenfest dynamics, we move on to the Koopman model and present the role of the backreaction terms therein. Finally, we compare both models in the context of pure-dephasing systems.
Mixed quantum-classical models are widely used to reduce the computational cost of fully quantum simulations. However, their general applicability across different classes of problems remains an open question. Here, we address this issue for systems featuring spin-orbit coupling. In particular, we study the interaction dynamics of quantum spin-1/2 and classical orbital momentum in one-dimensional models of Rashba nanowires. We tackle this problem by resorting to a new quantum-classical Hamiltonian model that, unlike conventional approaches, retains the Heisenberg principle and captures correlation effects beyond the common Ehrenfest method. Based on Koopman wavefunctions in classical mechanics, the new model was recently implemented numerically via a particle scheme -- the koopmon method -- which is extended here to treat spin-orbit coupling. We apply the koopmon method to study the quantum-classical dynamics of nanowire models, with and without the presence of a harmonic potential and in both Rashba-dominated (strong coupling) and Zeeman-dominated (weak coupling) regimes. Considering realistic semiconductor parameters, the results are contrasted with both fully quantum and quantum-classical Ehrenfest dynamics. In the absence of external potential, the koopmon method qualitatively reproduces the features of the fully quantum evolution for all coupling regimes. While it exhibits a slight loss in spin accuracy compared to Ehrenfest simulations, the latter fail to capture the orbital dynamics. In the presence of a harmonic potential, the koopmon scheme reproduces several features of full quantum dynamics with accuracy levels that are unachievable by the Ehrenfest model. We conclude by presenting a test case that exhibits the formation of cat-like states.
We deal with the reversible dynamics of coupled quantum and classical systems. Based on a recent proposal by the authors, we exploit the theory of hybrid quantum–classical wavefunctions to devise a closure model for the coupled dynamics in which both the quantum density matrix and the classical Liouville distribution retain their initial positive sign. In this way, the evolution allows identifying a classical and a quantum state in interaction at all times, thereby addressing a series of stringent consistency requirements. After combining Koopman's Hilbert-space method in classical mechanics with van Hove's unitary representations in prequantum theory, the closure model is made available by the variational structure underlying a suitable wavefunction factorization. Also, we use Poisson reduction by symmetry to show that the hybrid model possesses a noncanonical Poisson structure that does not seem to have appeared before. As an example, this structure is specialized to the case of quantum two-level systems.
A gauge-invariant wave equation for the dynamics of hybrid quantum–classical systems is formulated by combining the variational setting of Lagrangian paths in continuum theories with Koopman wavefunctions in classical mechanics. We identify gauge transformations with unobservable phase factors in the classical phase-space and we introduce gauge invariance in the variational principle underlying a hybrid wave equation previously proposed by the authors. While the original construction ensures a positive-definite quantum density matrix, the present model also guarantees the same property for the classical Liouville density. After a suitable wavefunction factorization, gauge invariance is achieved by resorting to the classical Lagrangian paths made available by the Madelung transform of Koopman wavefunctions. Due to the appearance of a phase-space analogue of the Berry connection, the new hybrid wave equation is highly nonlinear and it is proposed here as a platform for further developments in quantum–classical dynamics. Indeed, the associated model is Hamiltonian and appears to be the first to ensure a series of consistency properties beyond positivity of quantum and classical densities. For example, the model possesses a quantum–classical Poincaré integral invariant and its special cases include both the mean-field model and the Ehrenfest model from chemical physics.
Starting with the exact factorization of the molecular wavefunction, this paper presents the results from the numerical implementation in ab-initio nonadiabatic dynamics of the recently proposed bohmion method. Within the context of quantum hydrodynamics, the regularized nuclear Bohm potential in the bohmion method admits solutions comprising a train of δ-functions which serve as a finite-dimensional sampling of the hydrodynamic flow paths. In addition, the nonlocal structure of the regularized Bohm potential admits nuclear quantum tunneling events. After reviewing the general theory, the bohmion method is applied to the well-known Tully models, which are used here as benchmark problems for the comparison of the new bohmion method with previous mixed quantum-classical methods.
We consider the numerical propagation of models that combine both quantum and classical degrees of freedom, usually, electrons and nuclei, respectively. We focus, in our computational examples, on the case in which the quantum electrons are modeled with time-dependent density-functional theory, although the methods discussed below can be used with any other level of theory. Often, for these so-called quantum-classical molecular dynamics models, one uses some propagation technique to deal with the quantum part and a different one for the classical equations. While the resulting procedure may, in principle, be consistent, it can however spoil some of the properties of the methods, such as the accuracy order with respect to the time step or the preservation of the geometrical structure of the equations. Few methods have been developed specifically for hybrid quantum-classical models. We propose using the same method for both the quantum and classical particles, in particular, one family of techniques that proves to be very efficient for the propagation of Schrödinger-like equations: the (quasi)-commutator free Magnus expansions. These have been developed, however, for linear systems, yet our problem is nonlinear: formally, the full quantum-classical system can be rewritten as a nonlinear Schrödinger equation, i.e., one in which the Hamiltonian depends on the system itself. The Magnus expansion algorithms for linear systems require the application of the Hamiltonian at intermediate points in a given propagating interval. For nonlinear systems, this poses a problem as this Hamiltonian is unknown due to its dependence on the state. We approximate it by employing a higher order extrapolation using previous steps as input. The resulting technique can then be regarded as a multistep technique or, alternatively, as a predictor corrector formula.