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The Journal of Chemical Physics

Incorporating Lindblad decay dynamics into mixed quantum-classical simulations

Eric R. Koessler, Arkajit Mandal, Pengfei Huo

Published Jul 4, 2022• 30 citations
Publisher

Abstract

We derive the L-mean-field Ehrenfest (MFE) method to incorporate Lindblad jump operator dynamics into the MFE approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the L-MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories that exactly reproduce Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.

DOI
10.1063/5.0099922

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We derive the L-mean-field Ehrenfest (MFE) method to incorporate Lindblad jump operator dynamics into the MFE approach. We map the density matrix evolution of Lindblad dynamics onto pure state coefficients using trajectory averages. We use simple assumptions to construct the L-MFE method that satisfies this exact mapping. This establishes a method that uses independent trajectories that exactly reproduce Lindblad decay dynamics using a wavefunction description, with deterministic changes of the magnitudes of the quantum expansion coefficients, while only adding on a stochastic phase. We further demonstrate that when including nuclei in the Ehrenfest dynamics, the L-MFE method gives semi-quantitatively accurate results, with the accuracy limited by the accuracy of the approximations present in the semiclassical MFE approach. This work provides a general framework to incorporate Lindblad dynamics into semiclassical or mixed quantum-classical simulations.
Efficient simulation of Markovian open quantum systems remains a central challenge because the density-matrix description grows exponentially with system size. Quantum trajectory methods provide an alternative by replacing mixed-state evolution with stochastic pure-state realizations. Here we investigate this framework for normal Lindblad generators, whose orthogonal eigenoperator decomposition precludes transient amplification. By decomposing the Lindbladian into deterministic smooth and stochastic jump contributions, we derive an exact steady-state balance relation that identifies the interplay between these processes as the mechanism underlying Liouvillian normality. We further show that normal Lindbladians exclude exceptional points and that, although individual quantum trajectories generally exhibit stochastic coupling between Liouvillian eigenmodes, these couplings cancel upon ensemble averaging, recovering independent orthogonal relaxation modes. These results provide a trajectory-level interpretation of Liouvillian normality and clarify how a global property of the Lindblad generator is realized through stochastic quantum dynamics.
In this paper, we introduce a novel and general framework for the variational quantum simulation of Lindblad equations. Building on the close relationship between the unraveled Lindblad dynamics, stochastic Magnus integrators, and variational quantum simulation, we propose a high-order scheme for solving the quantum state diffusion equation using exponential integrators. This formulation facilitates the simulation of wavefunction trajectories within the established framework of variational quantum algorithms for time evolution. Our algorithm significantly enhances robustness in two key aspects: the stability of the simulation with large time steps, and the reduction in the number of quantum trajectories required to accurately simulate the Lindblad dynamics in terms of the ensemble average. We demonstrate the effectiveness of our algorithm through numerical examples in both classical and quantum implementations, including the transverse-field Ising model (TFIM) with damping, the Fenna-Matthews-Olson (FMO) complex, and the radical pair model (RPM). The simulation accuracy can be systematically improved, and the algorithm remains reliable even in highly oscillatory regimes. These methods are expected to be applicable to a broader class of open quantum systems beyond the specific models considered in this study.
Abstract Quantum and classical systems evolving under the same formal Hamiltonian H may exhibit dramatically different behavior after the Ehrenfest timescale tE∼log⁡(ℏ−1)t_E \sim \log (\hbar ^{-1})tE​∼log(ℏ−1) t E ∼ log ( ħ - 1 ) , even as ℏ→0\hbar \rightarrow 0ℏ→0 ħ → 0 . Coupling the system to a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker–Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via the Wigner-Weyl representation. Due to decoherence, they are conjectured to match closely for times far beyond the Ehrenfest timescale as ℏ→0\hbar \rightarrow 0ℏ→0 ħ → 0 . We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian H(x, p) and Lindblad functions Lk(x,p)L_{k}(x,p)Lk​(x,p) L k ( x , p ) . The error is small when the strength of the diffusion D associated to the Lindblad functions satisfies D≫ℏ4/3D \gg \hbar ^{4/3}D≫ℏ4/3 D ≫ ħ 4 / 3 , in particular allowing vanishing noise in the classical limit. Our method uses a time-dependent semiclassical mixture of variably squeezed Gaussian states. The states evolve according to a local harmonic approximation to the Lindblad dynamics constructed from a second-order Taylor expansion of the Lindbladian. Both the exact quantum trajectory and its classical counterpart can be expressed as perturbations of this semiclassical mixture, with the errors bounded using Duhamel’s principle. We present heuristic arguments suggesting the 4/3 exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale. Our presentation aims to be comprehensive and accessible to both mathematicians and physicists. In a shorter companion paper, we treat the special case of Hamiltonians that decompose into kinetic and potential energy with linear Lindblad operators, with explicit bounds that can be applied directly to physical systems.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach that describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin 1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wave packet in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory, or semiclassical approaches.
We propose Ehrenfest Dynamics with Spontaneous Localization (SLED), a decoherence-corrected extension of Ehrenfest dynamics based on the Gisin-Percival quantum-state diffusion equation. In SLED, the electronic wave function evolves stochastically in the adiabatic energy basis, producing trajectory-level localization. The trajectory ensemble reproduces a Lindblad-type propagation of the reduced electronic density matrix. This approach ensures linearity, trace preservation, and complete positivity, providing a physically consistent alternative to ad hoc decoherence corrections commonly adopted in mixed quantum-classical methods. Benchmark simulations on one-dimensional Tully models and multidimensional spin-boson Hamiltonians demonstrate that SLED reproduces electronic populations and captures the essential features of coherence decay. The tests, however, also reveal that accurate treatment will require generalizing the localization kernel controlling the electron-nucleus coupling strength from a constant into a function of time and phase space coordinates. SLED is implemented in the newly developed Skitten program and will be integrated into Newton-X. While the present study serves as a proof of concept, SLED establishes a rigorous and extensible framework that bridges mixed quantum-classical dynamics with open quantum system theory.
A long-standing challenge in mixed quantum-classical trajectory simulations is the treatment of entanglement between the classical and quantal degrees of freedom. We present a novel approach which describes the emergence of entangled states entirely in terms of independent and deterministic Ehrenfest-like classical trajectories. For a two-level quantum system in a classical environment, this is derived by mapping the quantum system onto a path-integral representation of a spin-1/2. We demonstrate that the method correctly accounts for coherence and decoherence and thus reproduces the splitting of a wavepacket in a nonadiabatic scattering problem. This discovery opens up a new class of simulations as an alternative to stochastic surface-hopping, coupled-trajectory or semiclassical approaches.