[Paper Review] Adapted projection operator technique for the treatment of initial correlations

Andrea Trevisan, Andrea Smirne+2 more|arXiv (Cornell University)|Jul 28, 2021

Spectroscopy and Quantum Chemical Studies97 references13 citations

TL;DR

This paper introduces an adapted projection operator technique that treats initial correlations between open quantum systems and their environments in a perturbative framework. By decomposing the initial global state into a convex mixture of product states and defining a family of projections tailored to this decomposition, the method yields a set of uncoupled, homogeneous differential equations for the open system dynamics—limited only by the system's dimensionality, not the environment's. The approach enables accurate long-time dynamics predictions even with fully general initial correlations, outperforming standard projection techniques in benchmark cases like qubit dephasing and damping.

ABSTRACT

The standard theoretical descriptions of the dynamics of open quantum systems rely on the assumption that the correlations with the environment can be neglected at some reference (initial) time. While being reasonable in specific instances, such as when the coupling between the system and the environment is weak or when the interaction starts at a distinguished time, the use of initially uncorrelated states is questionable if one wants to deal with general models, taking into account the mutual influence that the open-system and environmental evolutions perform on each other. Here, we introduce a perturbative method that can be applied to any microscopic modeling of the system-environment interaction, including fully general initial correlations. Extending the standard technique based on projection operators that single out the relevant part of the global dynamics, we define a family of projections adapted to a convenient decomposition of the initial state, which involves a convex mixture of product operators with proper environmental states. This leads us to characterize the open-system dynamics via an uncoupled system of differential equations, which are homogeneous and whose number is limited by the dimensionality of the open system, for any kind of initial correlations. Our method is further illustrated by means of two cases study, for which it reproduces the expected dynamical behavior in the long-time regime more consistently than the standard projection technique.

Motivation & Objective

• To address the limitation of standard open quantum system theories that assume uncorrelated initial states between system and environment.
• To develop a general, perturbative framework applicable to any microscopic model, including those with arbitrary initial correlations.
• To derive a system of uncoupled, homogeneous differential equations for the open system dynamics, independent of environmental dimensionality.
• To improve the accuracy of long-time dynamics predictions in models with initial correlations, especially where standard techniques fail.

Proposed method

• Decompose the initial global state into a convex combination of product operators, where environmental parts are proper states and system parts are not, enabling treatment of entangled initial states.
• Define a family of projection operators into product states, one per component of the decomposition, to isolate the relevant dynamics for the open system.
• Construct a perturbative expansion using these adapted projections, leading to a system of uncoupled, homogeneous differential equations for the system's density matrix elements.
• Express the resulting master equation at second order in the coupling constant, with coefficients involving generalized environmental correlation functions dependent on the chosen decomposition.
• Use the frame-theoretic decomposition to connect the equations to physically relevant environmental correlation functions, enhancing interpretability.
• Apply the method to two paradigmatic models: pure dephasing and damped qubit with a bosonic bath, comparing results to standard techniques.

Experimental results

Research questions

• RQ1 Can a perturbative method be developed that consistently treats initial system-environment correlations in open quantum systems without assuming initial product states?
• RQ2 How can the standard projection operator technique be generalized to handle arbitrary initial correlations while preserving the structure of uncoupled differential equations?
• RQ3 What is the impact of initial correlations on the long-time dynamics of open quantum systems, particularly in non-Markovian regimes?
• RQ4 How does the adapted projection method compare quantitatively to the standard projection technique in predicting dynamics for correlated initial states?
• RQ5 Can the method be systematically applied to general microscopic models, including those with energy exchange and non-Markovian effects?

Key findings

• The adapted projection operator technique successfully generates a system of uncoupled, homogeneous differential equations for the open system dynamics, with the number of equations bounded only by the system's dimensionality, not the environment's.
• The method reproduces the exact solution for pure dephasing with initial correlations more accurately than the standard projection technique, especially in the long-time regime.
• For the damped qubit model with initial correlations, the adapted method yields a consistent second-order master equation that correctly captures non-Markovian features and energy exchange dynamics.
• The derived second-order master equation explicitly depends on generalized environmental correlation functions tied to the initial state decomposition, enabling direct physical interpretation.
• Analytic solutions for the second-order equations in the damped qubit case are derived, showing that the adapted method leads to distinct and more accurate time evolution than the standard approach.
• The method provides a systematic, perturbative framework that can be extended to higher orders and applied to a wide range of open quantum systems with initial correlations, offering a significant improvement over standard techniques.