We derive a rigorous nuclear gradient for a molecule-cavity hybrid system using the quantum electrodynamics Hamiltonian. We treat the electronic–photonic degrees of freedom (DOFs) as the quantum subsystem and the nuclei as the classical subsystem. Using the adiabatic basis for the electronic DOF and the Fock basis for the photonic DOF and requiring the total energy conservation of this mixed quantum–classical (MQC) system, we derived the rigorous nuclear gradient for the molecule–cavity hybrid system, which is naturally connected to the approximate gradient under the Jaynes–Cummings approximation. The nuclear gradient expression can be readily used in any MQC simulations and will allow one to perform the non-adiabatic on-the-fly simulation of polariton quantum dynamics. The theoretical developments in this work could significantly benefit the polariton quantum dynamics community with a rigorous nuclear gradient of the molecule–cavity hybrid system and have a broad impact on the future non-adiabatic simulations of polariton quantum dynamics.
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Madelung transform and probability densities in hybrid quantum–classical dynamics
This paper extends the Madelung–Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid quantum–classical Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint quantum–classical density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.
more » « less- PAR ID:
- 10303699
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 33
- Issue:
- 10
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- p. 5383-5424
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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