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arXiv:1111.5262 · Journal of Mathematical Physics

Mappings of open quantum systems onto chain representations and Markovian embeddings

Mischa P. Woods, Roland Groux, Alex W. Chin, Susana F. Huelga, Martin Bodo Plenio

Published Mar 1, 2014• 120 citations
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Abstract

We study systems coupled linearly to a bath of oscillators. In an iterative process, the bath is transformed into a chain of oscillators with nearest neighbour interactions. A systematic procedure is provided to obtain the spectral density of the residual bath in each step, and it is shown that under general conditions these data converge. That is, the asymptotic part of the chain is universal, translation invariant with semicircular spectral density. The methods are based on orthogonal polynomials, in which we also solve the outstanding so-called “sequence of secondary measures problem” and give them a physical interpretation.

DOI
10.1063/1.4866769

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We study systems coupled linearly to a bath of oscillators. In an iterative process, the bath is transformed into a chain of oscillators with nearest neighbour interactions. A systematic procedure is provided to obtain the spectral density of the residual bath in each step, and it is shown that under general conditions these data converge. That is, the asymptotic part of the chain is universal, translation invariant with semicircular spectral density. The methods are based on orthogonal polynomials, in which we also solve the outstanding so-called “sequence of secondary measures problem” and give them a physical interpretation.
This thesis is concerned with the study of quantum systems coupled linearly to a continuous bath of oscillators examples of which are the spin-boson model and the Nelson model. The main theme throughout is to develop a better understanding of the properties of the bath of oscillators such that more efficient representations of it can be made to facilitate the understanding of the system dynamics. The main difficulty in simulating the system dynamics is that the bath of oscillators composes an infinite number of degrees of freedom. In this thesis, we investigate the mathematical properties of an approach in which the bath modes are written as a semi-infinite chain of nearest neighbour interacting harmonic oscillators such that the efficient time dependent density-matrix renormalisation group (t-DMRG) methods can be applied for simulation. In the first section, we show how there are many different ways to represent the bath as semiinfinite chains and prove that seemingly unrelated methods can all be achieved using the same mathematical formalism. We show that in an iterative process the bath can be transformed into a chain of oscillators with nearest neighbour interactions. This is achieved using the formalism of orthogonal polynomials. This allows one to define a sequence of residual spectral densities at each site along the chain. We show that this sequence of residual spectral densities is provided by the so-called ”sequence of secondary measures”. We derive a systematic procedure to obtain the spectral density of the residual bath in each step. We find that these residual spectral densities are related to an old abstract problem in mathematics known as the ”secondary measures”. We solve this problem from the field of orthogonal polynomials to give an explicit expression for the residual spectral densities and go on to prove that these functions converge under very general conditions. That is, the asymptotic part of the chain is universal, translation invariant with universal spectral density. These results suggest efficient methods for handling the numerical treatment of the residual bath. In the second section, we take a different approach. Rather than studying the properties of residual baths, we look at how system observables are affected by truncating the semi-infinite chain of harmonic oscillators to a finite length chain. By developing locality bounds for the dynamics, we derive an upper bound to the error introduced by such a truncation and show that for all finite times it can be made arbitrarily small by including a sufficient number of harmonic oscillators 3 before truncating. Furthermore, it is shown that the speed at which the system communicates with different harmonic oscillators in the chain is proportional to the maximum frequency of the environment but that this speed also depends on the particular version of the chain. These bounds are given for when the dynamics are calculated in the Interaction picture and in the Schrodinger picture.
This chapter gives a self-contained review of the how standard open quantum system Hamiltonians can be mapped analytically onto representations in which the environments appear as one dimensional harmonic chains with nearest neighbour interactions. This mapping, carried out rigorously using orthogonal polynomial theory, then allows the full evolution of the system and environment to be simulated using time-adaptive density matrix renormalisation group methods. With the combination of these two techniques, numerically-exact results can be obtained for dissipative quantum systems in the presence of arbitrarily complex environmental spectral functions, and the correlations and processes in the environment which drive the effectively irreversible dynamics of the reduced state of the quantum system can be explored in real time. The chain representation also reveals a number of universal features of harmonic environments characterised by a spectral density which are discussed here.
By using the properties of orthogonal polynomials, we present an exact unitary transformation that maps the Hamiltonian of a quantum system coupled linearly to a continuum of bosonic or fermionic modes to a Hamiltonian that describes a one-dimensional chain with only nearest-neighbor interactions. This analytical transformation predicts a simple set of relations between the parameters of the chain and the recurrence coefficients of the orthogonal polynomials used in the transformation and allows the chain parameters to be computed using numerically stable algorithms that have been developed to compute recurrence coefficients. We then prove some general properties of this chain system for a wide range of spectral functions and give examples drawn from physical systems where exact analytic expressions for the chain properties can be obtained. Crucially, the short-range interactions of the effective chain system permit these open-quantum systems to be efficiently simulated by the density matrix renormalization group methods.
In studying open quantum systems, the environment is often approximated as a collection of non-interacting harmonic oscillators, a configuration also known as the star-bath model. It is also well known that the star-bath can be transformed into a nearest-neighbor interacting chain of oscillators. The chain-bath model has been widely used in renormalization group approaches. The transformation can be obtained by recursion relations or orthogonal polynomials. Based on a simple linear algebraic approach, we propose a bath partition strategy to reduce the system-bath coupling strength. As a result, the non-interacting star-bath is transformed into a set of weakly coupled multiple parallel chains. The transformed bath model allows complex problems to be practically implemented on quantum simulators, and it can also be employed in various numerical simulations of open quantum dynamics.
In the first part of this thesis we extensively review the general theory of open quantum systems. We introduce new approaches to describe, in a hopefully more transparent way, some aspects which are usually misleading in a broad range of the literature. In addition, we revise new concepts and methods developed in recent years that complement the general theory mainly developed in the 1970s. In the second part, we focus our attention on the so-called Markovian processes which have been described in the previous part. We study the validity of typical equations used in simple systems such as harmonic oscillators, and propose and check different strategies to describe Markovian dynamics in more complicated interacting systems. Finally we propose a counter-intuitive situation where some level of noise can be actually useful for information tasks. In the third and last part, we approach the problem of the non-Markovian description of open quantum systems. To quantify how non-Markovian an open quantum system is we introduce a computable measure and apply it to some relevant situations. We also propose a way of probing the complexity of an environment by visualizing properties of a pair of qubits embedded in it. In addition, we develop a technique based on orthogonal polynomial theory to transform a wide range of environments into simple chains of particles which can be efficiently simulated with numerical techniques.
We consider a well-known, exactly solvable model of an open quantum system with pure decoherence. The aim of this paper is twofold. Firstly, decoherence is a property of open quantum systems important for both quantum technologies and the fundamental question of the quantum–classical transition. It is worth studying how the long-term rate of decoherence depends on the spectral density characterising the system–bath interaction in this exactly solvable model. Secondly, we address a more general problem of the Markovian embedding of non-Markovian open system dynamics. It is often assumed that a non-Markovian open quantum system can be embedded into a larger Markovian system. However, we show that such embedding is possible only for Ohmic spectral densities (for the case of a positive bath temperature) and is impossible for both sub- and super-Ohmic spectral densities. On the other hand, for Ohmic spectral densities, an asymptotic large-time Markovianity (in terms of the quantum regression formula) takes place.