Open quantum-classical systems: A hybrid MASH master equation

TL;DR

The paper tackles the challenge of modeling open quantum–classical dynamics where a two-level system interacts with both slow classical degrees of freedom and a quantum environment. It introduces a hybrid framework that merges the MASH trajectory method with a stochastic unravelling of secular Redfield theory, enabling a trajectory-based description that preserves correct equilibrium behavior and information flow between quantum and classical subsystems. The authors demonstrate accuracy by benchmarking against HEOM on a spin–boson model with dual baths and by simulating cavity-enhanced spontaneous emission, showing that the hybrid method captures both nonadiabatic transitions and bath-induced dissipation. This approach offers a scalable, interpretable tool for molecular quantum optics and related fields, with potential extensions to non-Markovian baths and more complex environments. The work thus provides a principled route to include dissipative quantum environments in large-scale quantum–classical simulations, enabling realistic modeling of systems in condensed-phase and photonic settings.

Abstract

We propose a method which combines the quantum-classical mapping approach to surface hopping (MASH) with the dissipative quantum dynamics of the Lindblad master equation. Like conventional surface-hopping methods, our approach is based on classical trajectories coupled to the dynamics of a quantum subsystem. However, instead of evolving the subsystem wavefunction according to the time-dependent Schrödinger equation, we use stochastic quantum trajectories derived from secular Redfield theory. This enables the simulation of open quantum systems coupled simultaneously to Markovian quantum baths and anharmonic non-Markovian classical degrees of freedom. Applications to the spin-boson model and to the cavity-enhanced fluorescence of an electronically nonadiabatic molecule show excellent agreement with fully quantum-mechanical benchmarks.

Figures (5)

• Figure 1: Schematic diagram of the hybrid Redfield--MASH method. The two-level quantum system is coupled simultaneously to a large number of classical degrees of freedom and a Markovian quantum bath. We use the MASH framework to treat the nonadiabatic coupling between the system and the classical degrees of freedom, and we use secular Redfield theory to treat the coupling to the quantum bath.
• Figure 2: The two distinct mechanisms for changing active state in the hybrid method. Left: Energy-conserving nonadiabatic "hops" as the spin vector passes the equator, mediated by the coupling to the classical coordinates. Right: Stochastic "jumps" initiated by the Lindblad operators coupling the system to the quantum bath.
• Figure 3: The population of the diabatic $\ket{a}$ state as a function of time using MASH (blue), secular Redfield theory (red), hybrid Redfield--MASH (purple), and HEOM (dashed). 20$\sigma$ error bars indicate a 95% confidence interval for $10^4$ trajectories of the hybrid method. Inset: The spectral density of the slow bath (blue), the fast bath (red) and the total spectral density (purple) of the spin--boson system.
• Figure 4: The adiabatic potentials of Eq. \ref{['eq:electron_transfer']} and the corresponding cavity-enhanced decay rates and nonadiabatic couplings. Red arrows show the energy gap at resonance with the cavity. The initial wavepacket is also plotted in gray.
• Figure 5: Top: Time-dependent population of the upper adiabatic state. The quantum and MASH calculations are for an isolated system, and the quantum Redfield and hybrid Redfield--MASH results are for a coupled system--cavity simulation. Bottom: Probability of photon emission per unit time for both Redfield methods. $2\sqrt{10}\sigma$ error bars indicate a 95% confidence interval for $10^5$ trajectories of the hybrid method.