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arXiv:2110.02455 · Physical Review A

Capturing non-Markovian dynamics with the reaction coordinate method

Nicholas Anto-Sztrikacs, Dvira Segal

Published Nov 23, 2021• 34 citations
arXivHTMLPDF

Abstract

The reaction coordinate (RC) technique is emerging as a significant tool in the study of quantum dissipative dynamics and quantum thermodynamics. With the objective to further establish this tool, here we explore to what extent the method can capture non-Markovian dynamics of open quantum systems. As a case study, we focus on the pure decoherence model of a spin coupled to a harmonic reservoir. We compare the spin dynamics and measures for non-Markovianity from the exact analytical solution to simulations based on the RC method at the level of a second-order quantum master equation. We find that the RC method can quantitatively capture non-Markovian effects at strong system-bath coupling and for structured baths. This is rationalized by the fact that the collective RC bath mode, which is made part of the system, maintains system-bath correlations. Lastly, we apply our RC method and study the spin-boson model in the non-Markovian regime.

DOI
10.1103/physreva.104.052617

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The reaction coordinate (RC) technique is emerging as a significant tool in the study of quantum dissipative dynamics and quantum thermodynamics. With the objective to further establish this tool, here we explore to what extent the method can capture non-Markovian dynamics of open quantum systems. As a case study, we focus on the pure decoherence model of a spin coupled to a harmonic reservoir. We compare the spin dynamics and measures for non-Markovianity from the exact analytical solution to simulations based on the RC method at the level of a second-order quantum master equation. We find that the RC method can quantitatively capture non-Markovian effects at strong system-bath coupling and for structured baths. This is rationalized by the fact that the collective RC bath mode, which is made part of the system, maintains system-bath correlations. Lastly, we apply our RC method and study the spin-boson model in the non-Markovian regime.
The fundamental building blocks of quantum computers, called qubits, can be physically realized through any quantum system that is restricted to two possible states. The power of qubits arises from their ability to be in a superposition of these two states, allowing for the development of quantum algorithms that are impossible for classical computers. However, interactions with the surrounding environment destroy the superposition in a process called decoherence, which makes it important to find ways to model these interactions and mitigate them. In this thesis we derive a non-Markovian master equation for the spin-boson model, with a time-dependent two-level system, using the reaction coordinate representation. We show numerically that in the superconducting qubit regime this master equation maintains the positivity of the density operator for relevant parameter ranges, and is able to model non-Markovian effects between the system and the environment. We also compare the reaction coordinate master equation to a Markovian master equation with parameters taken from real superconducting qubits. We demonstrate that the Markovian master equation fails to capture the system–bath correlations for short times, and in many cases overestimates relaxation and coherence times. Finally, we test how a time-dependent bias affects the evolution of the two-level system. The bias is assumed to be constant with an additive term arising from an externally applied time-dependent plane wave control field. We show that an amplitude, angular frequency, and phase shift for the plane wave can be chosen such that the control field improves the coherence time of the two-level system.
In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian. The idea of the mesoscopic lead construction and brute force chain mapping techniques has been introduced which can be used to bypass the strong system bath coupling situations.
In this article we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian.
The Hierarchical equations of motion (HEOM) method is an important non-perturbative technique, allowing numerically exact treatment of open quantum systems with strong coupling and non-Markovian memory. However, its encoding of bath memory into auxiliary density operators often limits direct access to detailed bath information. In contrast, the reaction-coordinate (RC) mapping allows direct and transparent access to the dominant collective bath mode, but its perturbative and often Markovian treatment of the residual bath restricts its reliability. In this work, we introduce RC-HEOM, a hybrid method that unifies the strengths of both approaches by combining RC mapping with a fully non-perturbative HEOM description of the residual bath. RC-HEOM simultaneously retains exact non-Markovian memory and access to the RC mode, which enables analysis of system-RC information. Applying this method to the Anderson impurity models, we directly track the emergence of the Kondo singlet from the growth of the Kondo resonance and uncover a nontrivial RC-mediated coherence revival. These results demonstrate that RC-HEOM is a promising method for characterizing open quantum systems in regimes that are difficult to access with conventional master-equation methods.
In this tutorial we want to present a thorough overview about the well known technique known as the Reaction Coordinate Mapping which is one of the most used theoretical tool in the context of Open quantum system and Quantum thermodynamics of quantum systems which is not weakly coupled to the reservoir. Here we discuss different aspects of the Reaction Coordinate mapping along with detailed mathematical derivations for the Bosonic and Fermionic mappings which can be achieved through the Symplectic and unitary transformations. In the later part of this article we discuss the derivation of the Quantum master equation for the extended systems (System+Reaction Coordinate) coupled to the residual reservoir in the weak coupling limit described by the usual Redfield Quantum Master equation. We observe that though the dynamics of the Extended system is Markovian in nature because of the weak coupling between the extended system and the Residual reservoir but the Reaction coordinate transformation enables us to analyses the dynamics of the system which is essentially non-Markovian.
The primary pathway to the investigation of exceptional points (EPs) in open quantum systems has been the use of Markovian master equations. However, recently it has been realised that they can arise in settings that show non-Markovian behaviour. The main approach in this direction has been the use of pseudomodes, which allow the influence of a structured environment to be replaced by new damped auxiliary modes. In this work, we use a Heisenberg-equation approach which relies on the reaction-coordinate formalism to obtain exact analytical dynamics. First, we show how our method considerably simplifies the analysis compared to the use of master equations. Second, we utilise it to discuss non-Markovian revivals in systems with second and higher-order EPs, showing that the notion that EPs demarcate the onset of revivals is not generally true. We then utilise our results to discuss the dynamical effects of higher-order non-Markovian EPs, deriving bounds on the number of revivals and investigating enhanced asymptotic relaxation near higher-order EPs. Finally, we investigate non-Markovian EPs with engineered reservoirs and show how they can be utilised to control the rate of irreversible information loss.