[Paper Review] A non-perturbative analysis of spin-boson interactions using the Weyl relations

José de Ramón, Eduardo Martín-Martínez|arXiv (Cornell University)|Feb 5, 2020

Quantum Mechanics and Applications5 references4 citations

TL;DR

This paper presents a non-perturbative analysis of a two-level system (spin) interacting with a bosonic environment via impulsive, delta-function-coupled interactions using the Weyl relations. It derives a closed-form expression for the resulting spin state and demonstrates that time-ordering independence arises due to commuting system observables, revealing non-Markovian memory effects tied to quantum correlations in the environment.

ABSTRACT

We perform a non-perturbative analysis of the dynamics of a two-level quantum system subjected to repeated interactions with a bosonic environment when these interactions are intense and localized in time. We use the Weyl relations to obtain a closed expression for the resulting state of the spin despite the non-perturbative nature of the problem. Furthermore, we study divisibility and memory effects in the dynamics and draw conclusions about the role that the quantum-mechanical features of the environment play on the dynamics of the two-level system.

Motivation & Objective

• To analyze the non-perturbative dynamics of a two-level system coupled to a bosonic environment through intense, localized interactions.
• To avoid standard approximations such as the rotating wave or single-mode approximations, which fail in strong-coupling regimes.
• To establish a rigorous framework using Weyl relations for deriving exact evolution maps in impulsive interaction models.
• To investigate memory effects and divisibility in the dynamics, particularly how quantum features of the environment influence the system’s evolution.
• To show that time-ordering independence in pure dephasing arises from commuting system observables, not the switching function.

Proposed method

• Model the interaction Hamiltonian as a spin-boson coupling modulated by a train of Dirac delta functions, representing impulsive interactions.
• Use the Weyl relations to systematically handle the non-perturbative evolution of the system-environment state under these impulsive couplings.
• Derive a closed-form expression for the final density matrix of the spin by exploiting the unitary evolution generated by the interaction Hamiltonian.
• Apply the Magnus expansion to the time-ordered exponential, showing that commutativity of the system’s coupling observable at different times leads to time-ordering independence.
• Analyze the resulting dynamics in terms of quantum channels, focusing on divisibility and memory effects via entropy and relative entropy arguments.
• Demonstrate that the dynamics reduce to a pure dephasing channel when the system’s coupling observable is unitary and self-adjoint, such as Pauli operators.

Experimental results

Research questions

• RQ1 How can the non-perturbative dynamics of a two-level system interacting with a bosonic environment be exactly solved when interactions are impulsive and strong?
• RQ2 What is the role of the environment’s quantum correlations in inducing memory effects in the system’s evolution?
• RQ3 Why does time-ordering become irrelevant in the dynamics when the system’s coupling observable commutes at all times?
• RQ4 To what extent do standard approximations like the rotating wave approximation fail in this non-perturbative regime?
• RQ5 How do the Weyl relations enable the derivation of exact, closed-form expressions for the system’s final state in impulsive interaction models?

Key findings

• A closed-form expression for the final state of the spin is derived using the Weyl relations, valid even in the non-perturbative regime of strong, impulsive interactions.
• The dynamics are independent of time-ordering when the system’s coupling observable commutes at all times, a property arising from the structure of the Magnus expansion and the commutativity of the observable.
• The resulting evolution is a pure dephasing channel when the system couples via unitary, self-adjoint operators such as Pauli matrices, regardless of the switching function’s form.
• Memory effects in the dynamics are linked to quantum correlations in the environment, with non-Markovianity detectable through entropy monotonicity violations.
• The relative entropy between states remains monotonic under the channel, and unitality implies entropy increase, confirming the role of quantum features in non-Markovian behavior.
• The model is equivalent to a collision model with correlated environments, providing a framework to study memory effects in a controlled, analytically tractable setting.