# Dynamics of Open Quantum Systems I, Oscillation and Decay

Marco Merkli

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, A1C 5S7, Canada

### Abstract

We develop a framework to analyze the dynamics of a finite-dimensional quantum system $\rm S$ in contact with a reservoir $\rm R$. The full, interacting $\rm SR$ dynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of the $\rm SR$ coupling constant, uniformly for all times $t\ge 0$. The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of the $\rm SR$ dynamics, based on Mourre theory. We allow for $\rm SR$ interactions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.
In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of the $\rm SR$ dynamics. In the companion paper [27] we apply the formalism to the concrete case of an $N$-level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.

We consider a quantum system in contact with an environment – think of a few spins/qubits interacting with a bath of oscillating degrees of freedom. The environment is very large, has a single stationary (equilibrium) state and is dispersive, meaning that local perturbations created around the system's location travel away from it eventually. We analyze the evolution of the system-environment complex, given by the Schrödinger equation.

We show that the evolution is decomposed into a sum of three parts: One first main part describes the system evolving according to a markovian (memory-less) dynamics while the environment is constantly in its equilibrium (“Born approxiation''). A second main part describes the evolution of initial states away from (orthogonal to) the reservoir equilibrium. A third part is a remainder term, which is small in the system-environment coupling parameter, for all times. The markovian part is composed of explicit (exponentially in time) decaying and oscillating terms and the second main part decays as an inverse power of time.

The paper explains the strategy of the approach and contains the details and estimates leading to the main result. This makes it suitable for researchers who want to learn about the relevant mathematical methods developed in the theory of open quantum systems. In a companion paper, the results are applied to the ubiquitous open system model of an $N$ level system coupled to a bath (field) of free bosonic degrees of freedom. We show there that the markovian approximation, given by the Davies generator, holds for all times, even for reservoirs with slowly (polynomially) decaying correlations.

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### Cited by

[1] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders, "Open quantum system dynamics and the mean force Gibbs state", AVS Quantum Science 4 1, 012301 (2022).

[2] Marco Merkli, "Dynamics of Open Quantum Systems II, Markovian Approximation", Quantum 6, 616 (2022).

[3] Marco Merkli, "Correlation Decay and Markovianity in Open Systems", Annales Henri Poincaré (2022).

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