Ehrenfest Dynamics with Spontaneous Localization (SLED)

• SLED is a hybrid quantum–classical method that integrates quantum-state diffusion to enforce spontaneous localization and rigorous decoherence.
• It employs a stochastic Schrödinger equation with a localization kernel in the adiabatic basis to correct persistent electronic superpositions.
• Benchmarking on Tully models and spin–boson Hamiltonians shows that SLED accurately captures population dynamics and conserves energy through velocity corrections.

Ehrenfest Dynamics with Spontaneous Localization (SLED) is a hybrid quantum–classical dynamics methodology that corrects the shortcomings of conventional Ehrenfest approaches by incorporating trajectory-level stochastic localization of the electronic wavefunction. SLED employs quantum-state diffusion (QSD) to stochastically collapse the electronic state in the adiabatic energy basis, forming an ensemble of trajectories whose averaged density matrix evolution is governed by a Lindblad-type completely positive, trace-preserving map. This provides a rigorous decoherence framework that aligns mixed quantum–classical dynamics with open quantum system theory, and is extensible through a localization kernel controlling the electron–nucleus coupling.

1. Theoretical Foundation: Ehrenfest Dynamics and Quantum-State Diffusion

SLED builds its core on the quantum-state diffusion equation of Gisin and Percival, integrating it into the mixed quantum–classical dynamics arena. In traditional Ehrenfest dynamics, the nuclear degrees of freedom are propagated classically, while the electronic wavefunction evolves coherently along the classical trajectory—typically leading to persistent, unphysical superpositions and failing to capture dynamical decoherence and population relaxation.

SLED reforms this by introducing a stochastic Schrödinger equation,

$$d|\psi(t)\rangle = -iH|\psi(t)\rangle dt - \frac{1}{2} (L^\dagger L - \langle L^\dagger\rangle \langle L\rangle) |\psi(t)\rangle dt + (L - \langle L\rangle) |\psi(t)\rangle dW$$

where $dW$ is a complex-valued Wiener process and $L$ is the localization operator. In SLED, $L$ acts in the adiabatic basis and is parametrized by a kernel $K$ reflecting electron–nucleus coupling strength. The electronic state is expanded as

$$|\psi(t)\rangle = \sum_n c_n(t) e^{-i\gamma_n(t)} |n(t)\rangle$$

Each trajectory, subject to stochastic diffusion, collapses onto a particular adiabatic state; ensemble averaging over many such trajectories produces a Lindblad-type evolution for the reduced density matrix:

$\partial_t \rho = -i[H, \rho] + \mathcal{D}[\rho]$

where $\mathcal{D}[\rho]$ has the usual Lindblad dissipator structure. Linearity, trace preservation, and complete positivity are rigorously ensured, distinguishing SLED from heuristic decoherence corrections such as energy-based rescaling or post hoc ensemble damping.

2. Decoherence and Spontaneous Localization Mechanism

In SLED, decoherence is not imposed post hoc but emerges from the stochastic QSD terms, which drive trajectory-level localization in the adiabatic basis. This corrects the principal failure of Ehrenfest dynamics, which keeps the electron in a continuous superposition and tends to misrepresent non-adiabatic transition probabilities and off-diagonal coherence decay.

At the single trajectory level, collapse occurs stochastically according to the instantaneous realization of the Wiener increment. The strength and rate of localization are regulated by the kernel $K$, which ideally encodes state-specific, time- and coordinate-dependent information about electron–nucleus coupling. The ensemble average of localized trajectories reconstructs the physical decoherence process, ensuring that off-diagonal density matrix elements decay consistently with open quantum system theory.

By design, the SLED protocol avoids ad hoc population transfer and arbitrary "branching" protocols. Instead, all decoherence and relaxation phenomena are dictated by the statistics of the underlying QSD process.

3. Localization Kernel and Parameterization

Central to the efficacy of SLED is the form of the localization kernel $K$. In initial tests, $K$ is taken as constant, acting uniformly in the adiabatic basis for all nuclear configurations and momenta. However, benchmarking revealed that this uniform implementation does not adequately capture all dynamical regimes—optimal reproduction of population dynamics and coherence decay requires $K$ to depend explicitly on phase space coordinates and time, i.e., $K(t, z)$.

This sensitivity arises because electron–nucleus coupling strength is fundamentally topography dependent; regions near avoided crossings, conical intersections, or strong vibronic coupling demand adaptive localization rates. The proposal is to generalize $K$ into a functional form capable of responding to local surface structure and kinetic data, thus enabling SLED to maintain physical transfer rates and decoherence scales across different model complexities.

4. Numerical Implementation and Energy Conservation

SLED is realized in the Skitten program (Stochastic Schrödinger Cats, Fortran), which propagates nuclear coordinates with velocity Verlet and electronic coefficients with a fourth-order Runge-Kutta integrator. After each time step, a rescaling protocol—based on nonadiabatic coupling directions—ensures total energy conservation. Specifically, velocity corrections are performed to absorb energy shifts induced by stochastic collapse in the electronic sector, maintaining consistency with the classical Hamiltonian flow.

The modular Skitten framework allows straightforward integration into Newton-X for more extensive mixed quantum–classical applications, including nonadiabatic photochemistry and condensed-phase dynamics.

5. Benchmarking: Tully Models and Spin-Boson Hamiltonians

SLED was benchmarked on canonical one-dimensional Tully models (three variants), as well as multidimensional spin–boson Hamiltonians with multiple vibrational modes. In Tully model 2, for instance, SLED stochastic trajectories successfully localize into electronic state 1 or 2, yielding ensemble electronic populations and coherence decay rates in close correspondence with numerically exact quantum solutions. Comparable fidelity is observed for the population evolution in extended spin–boson systems.

A limitation, however, is that coherence revival (off-diagonal ensemble density matrix elements) can be underestimated due to the efficient loss of coherence via independent trajectory approximation. This suggests that further refinements, possibly introducing trajectory–trajectory correlations or non-Markovian localization kernels, may be needed for fully quantitative treatment in strongly correlated environments.

6. Connections to Geometric, Statistical, and Stochastic Frameworks

The SLED formalism connects directly with geometric and statistical approaches to mixed quantum–classical dynamics. The underlying QSD equation and the resulting Lindblad master equation are statistically equivalent to an ensemble Fokker–Planck equation on the quantum–classical Poisson manifold $\mathcal{S} = M_C \times M_Q$ (Alonso et al., 2011). This links SLED to generalized stochastic extensions such as Langevin–Ehrenfest and Nosé–Ehrenfest approaches, where stochasticity, ergodicity, and equilibrium distributions naturally emerge from the canonical phase space measure.

SLED also aligns conceptually with recent objective collapse theories, such as continuous spontaneous localization (CSL) and stochastic unitarity violation approaches in the white-noise limit (Mukherjee et al., 2 May 2024). In these theories, spontaneous localization arises dynamically through physical noise sources, ensuring energy conservation and Born rule statistics. SLED implements these ideas algorithmically for molecular quantum–classical systems.

7. Future Perspectives and Limitations

As a proof of concept, SLED establishes a linearly evolving, trace-preserving, completely positive ensemble for electronic populations and decoherence, improving significantly over conventional mixed quantum–classical methods. Future directions include:

• Development of state- and time-dependent localization kernels $K(t, z)$ optimized for variable surface topographies and dynamical regimes.
• Introduction of trajectory–trajectory interaction or memory effects to mitigate inaccuracies in coherence revival and strong vibronic coupling scenarios.
• Broad integration with quantum–classical platforms for widespread application in photochemistry, nonadiabatic condensed-phase processes, and cavity QED.

The method bridges the gap between ensemble-averaged decoherence theory and practical stochastic quantum–classical simulation, providing a systematic and extensible solution to the limitations of traditional Ehrenfest dynamics.

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Key Mathematical and Algorithmic Summary

• Electronic stochastic Schrödinger equation:

$$d|\psi(t)\rangle = -i H |\psi(t)\rangle dt - \frac{1}{2}[L^\dagger L - \langle L^\dagger\rangle \langle L\rangle] |\psi(t)\rangle dt + [L - \langle L\rangle] |\psi(t)\rangle dW$$

• Lindblad-type ensemble evolution:

$$\partial_t \rho = -i [H, \rho] + L \rho L^\dagger - \frac{1}{2} \{L^\dagger L, \rho\}$$

• Nuclear dynamics propagated via velocity Verlet, with periodic velocity corrections for energy conservation after each electronic collapse.

SLED positions itself as a rigorous framework bridging stochastic open quantum system theory and scalable quantum–classical molecular dynamics, subject to further refinement in kernel parameterization and trajectory ensemble treatment.