Non-stochastic matrix Schr\"odinger equation for open systems

TL;DR

This paper introduces a non-stochastic Schr"odinger-like equation for open quantum systems that ensures positive-definite density matrices and is applicable to both Markovian and non-Markovian dynamics, addressing energy conservation issues.

Contribution

It extends the Schr"odinger equation to open systems with a new formalism that guarantees physical density matrices and resolves energy non-conservation problems.

Findings

Preserves positive-definiteness of the density matrix.

Applicable to both Markovian and non-Markovian systems.

Addresses energy non-conservation in variational methods.

Abstract

We propose an extension of the Schr\"odinger equation for a quantum system interacting with environment. This equation describes dynamics of auxiliary wave-functions $\mathbf{m}$, from which the system density matrix can be reconstructed as $\hat{\rho} = \mathbf{m} \mathbf{m}^\dagger$. We formulate a compatibility condition, which ensures that the reconstructed density satisfies a given quantum master equation for the system density. The resulting non-stochastic evolution equation preserves positive-definiteness of the system density and is applicable to both Markovian and non-Markovian system-bath treatments. Our formalism also resolves a long-standing problem of energy non-conservation in the time-dependent variational principle applied to mixed states of closed systems.