Control of Open Quantum Systems: A Theoretical Approach to Control of Surface Photochemistry

Transcription

1 Control of Open Quantum Systems: A Theoretical Approach to Control of Surface Photochemistry Von der Fakultät für Mathematik und Naturwissenschaften der Carl von Ossietzky Universität Oldenburg zur Erlangung des Grades und Titels eines Doktors der Naturwissenschaften (Dr. rer. nat.) angenommene Dissertation von Herrn Erik Asplund geboren am in Nynäshamn Oldenburg 2011

2 Erstgutachter: Zweitgutachter: Prof. Dr. T. Klüner Prof. Dr. M. Holthaus Tag der Disputation: 28. März 2011

3 To my wife

4 Erwin kann mit seinem psi kalkulieren wie noch nie. Doch wird jeder leicht einsehen, psi läßt sich nicht recht verstehen. Erich Armand Arthur Joseph Hückel

5 vii Abstract The topic of this thesis was the control of open quantum systems with the emphasis on the control of surface photochemical reactions. An open quantum system is a quantum system which interacts with its surroundings. A quantum system in condensed phase, which undergoes strong dissipative processes, is an open quantum system. The last decades have seen both experimental and theoretical approaches for gaining control over dissipative processes. From the theoretical viewpoint, it is important to model such processes in a rigorous way. In this thesis, the description of open quantum systems was realized within the Surrogate Hamiltonian approach. If systems, for which excitation and deexcitation processes have comparable time-scales, are investigated, a non-markovian approach should be employed. The Surrogate Hamiltonian approach, which is a non-markovian approach, is an effective scheme, not relying on the separation of time-scales, to simulate the dynamics of a system subjected to dissipation. Besides the traditional formulation of dissipative processes through the spectral density, the Surrogate Hamiltonian method enables a microscopic description of excitation and relaxation processes for open quantum systems. An efficient and accurate method to find control fields is optimal control theory (OCT). In this thesis, a control scheme relying on OCT with time-dependent targets was employed to minimize the dissipative behavior of open quantum systems. Furthermore, the pulse design equations were derived and the OCT iteration algorithm was presented. To gain control of open quantum systems, the Surrogate Hamiltonian approach and OCT, with time-dependent targets, were combined. Three open quantum systems were investigated by the combined method, a harmonic oscillator immersed in an ohmic bath, CO adsorbed on a platinum surface and NO adsorbed on a nickel oxide surface. The harmonic oscillator immersed in an ohmic bath served as a benchmark system to investigate the behavior of the combined Surrogate Hamiltonian /OCT approach. The two adsorbate-surface systems were chosen to investigate the applicability of the combined approach to surface photochemistry. To nullify the dissipation modeled within the Surrogate Hamiltonian approach, a freely propagating wave packet was used as a target in the OCT iteration algo-

6 viii rithm, i.e. the aim of the control was to follow a predefined evolution of a wave packet. The controllability of the systems was monitored, i.e. to which degree dissipation could be surpressed. It was observed that dissipation could be subdued to a high degree for all three systems. Furthermore, the influence of the control pulses on the expectation values of the investigated systems was studied.

7 ix Kurzfassung Das Thema dieser Arbeit war die Kontrolle offener Quantensysteme, wobei der Schwerpunkt auf der Kontrolle photochemischer Oberflächenreaktionen lag. Ein offenes Quantensystem ist ein Quantensystem, das mit seiner Umgebung wechselwirkt. Ein quantenmechanisches System in kondensiertem Zustand, das starke dissipative Prozesse erfährt, ist ein offenes quantenmechanisches System. In den letzten Jahrzehnten wurden sowohl experimentelle als auch theoretische Ansätze zur die Kontrolle von dissipativen Prozessen vorgeschlagen. Um dissipative Prozesse theoretisch zu beschreiben ist eine gründliche Annäherung nötig. In dieser Arbeit wurde die Beschreibung offener Quantensysteme innerhalb der,,surrogate Hamiltonian Methode realisiert. Wenn Systeme untersucht werden, bei denen Anregungs- und Relaxations-Prozesse vergleichbare Zeitskalen haben, sollten nicht-markov-ansätze verwendet werden. Die Methode des,,surrogate Hamiltonian, die ein nicht-markov-ansatz ist, ist eine effiziente Beschreibung, die sich nicht auf die Trennung der Zeitskalen verlässt, um die Dynamik eines dissipativen Systems zu beschreiben. Neben der traditionellen Formulierung dissipativer Prozesse durch die Spektraldichte, ermöglicht die Methode des,,surrogate Hamiltonian eine mikroskopische Beschreibung der Anregungs- und Relaxations-Prozesse offener Quantensysteme. Eine effiziente und genaue Methode zur Berechnung von Kontrollfelder, ist die optimale Kontrolltheorie (Opimal Control Theory (OCT)). In dieser Arbeit wurde ein Kontrollkonzept, das auf der optimalen Kontrolltheorie mit zeitabhängiger Zielfunktion beruht, verwendet, um das dissipative Verhalten offener Quantensysteme zu minimieren. Außerdem wurden die Gleichungen für die Berechnung optimaler Pulse bestimmt und der iterative Algorithmus zur Berechnung optimaler Pulse vorgestellt. Um Kontrolle über offene Quantensysteme zu erzielen, wurden der,,surrogate Hamiltonian Ansatz und die optimale Kontrolltheorie mit zeitabhängigen Zielfunktionen miteinander verbunden. Drei offene Quantensysteme wurden mittels der kombinierten Methode untersucht, ein harmonischer Oszillator eingetaucht in ein ohmsches Bad, CO adsorbiert auf einer Platin-Oberfläche und NO adsorbiert auf einer Nickeloxid Oberfläche. Der harmonische Oszillator, eingetaucht in einem ohmschen Bad, diente als ein Benchmark-System, um das Verhalten des kombinierten,,surro-

8 x gate Hamiltonian /OCT Ansatzes zu untersuchen. Die beiden Adsorbat-Substrat- Systeme wurden ausgewählt, um die Anwendbarkeit des kombinierten Ansatzes auf photochemische Oberflächenreaktionen zu untersuchen. Um die Dissipation, modelliert mittels der Methode des,,surrogate Hamiltonian, zu unterdrücken, wurde ein frei propagiertes Wellenpaket als Zielfunktion in dem iterativen Algorithmus verwendet, d.h. das Ziel der Kontrolle war es, einer vordefinierte Zeitentwicklung eines Wellenpakets zu folgen. Des Weiteren wurde die Kontrollierbarkeit der Systeme ermittelt, d.h. es wurde untersucht zu welchem Grad Dissipation unterdrückt werden kann. Für alle drei Systeme wurde festgestellt, dass Dissipation zu einem hohen Maß gedämpft werden kann. Darüber hinaus wurde der Einfluss der Kotrollpulse auf die Erwartungswerte der untersuchten Systeme ermittelt.

9 xi

10 xii

11 Contents 1 Introduction 1 I Theory 5 2 Quantum Dissipation Open Quantum Systems The Markov Approximation/The Quantum Markovian Semigroup Master Equation Non-Markovian Approaches The Surrogate Hamiltonian Method Dissipative Processes Modeled by the Surrogate Hamiltonian Method Numerical Implementation Optical Control of Quantum Dynamics Optimal Control Theory Formulation of the Control Problem Derivation of the Control Equations Numerical Solution of the Control Equations Discrete Implementation of the Iteration Algorithm Target Operators Coherent Control Experiments Pulse Shaping Technologies Coherent Control Experiments xiii

12 xiv CONTENTS Connecting Experiment and Theory II Applications 71 4 A Harmonic Oscillator in an Ohmic Bath 73 5 CO Adsorbed on a Platinum Surface The Model Control Laser Induced Desorption of NO from NiO(100) Experimental Findings Theoretical Calculations of the Potential Energy Surfaces Microscopic Description of the Bath and Interaction Quantum Dynamical Calculations Optimization Calculations Summary and Outlook 153 Appendices 159 A Parallelization of the Surrogate Hamiltonian 161 A.1 Parallel Computer Architectures A.1.1 SISD A.1.2 SIMD A.1.3 MISD A.1.4 MIMD A.2 Shared and Distributed Memory A.2.1 Shared Memory A.2.2 Distributed Memory A.2.3 Hybrid Distributed-Shared Memory A.3 Parallelization Strategy A.3.1 Domain Decomposition

13 CONTENTS xv A.3.2 Compressed Row Storage (CRS) A.3.3 Communication Algorithm B The Representation of Wave Functions and Propagation Schemes 179 B.1 Numerical Representation of Wave Functions B.1.1 The Fourier Method B.1.2 Grid Change B.1.3 Spectral Range B.2 Propagation Schemes B.2.1 The Chebychev Propagator B.2.2 The Split Operator B.3 Numerical Methods for Determining Eigenfunctions and Eigenvalues. 196 B.3.1 Imaginary Time Propagation C The Interaction Hamiltonian for NO/NiO 199 Bibliography 214 List of Figures 223 List of Tables 225

14

15 Chapter 1 Introduction Quantum mechanics is one of the most fundamental physical theories and can hence not be neglected in modern physics and chemistry. Up to date, no underlying theory has been found from which quantum probabilities, i.e. all predictions derived from quatum mechanics are of a probabilistic charactar which implies that predictions are being made about the behavior of ensambles [1], can be deduced. Quantum mechanics must hence be employed when handling problems, which include quantum probabilities. Furthermore, laser techniques have been developed so far that tailored femtosecond laser pulses can be generated in laboratories. On the femtosecond time scale, matter wave packets (particle-type) can be created and their coherent evolution as a single-molecule trajectory can be observed... It also offers new possibilities for the control of reactivity and for structural femtochemistry and femtobiology [2], i.e. light interacting with atoms and molecules is not only a source of information about the atoms and molecules studied, it can also initiate chemical reactions [3, 4]. A field of chemistry, that relies on the interaction of light with atoms and molecules, is surface photochemistry, which is important in many different applications such as molecular photo devices, photochemical vapor deposition of thin layered semiconductors, sensitive optical media, and control of photochemical reaction paths [5]. The chemical reactions, which are important in these applications occur in condensed phase. Many photochemical experiments are also conducted in condensed phase, e.g. on atoms or molecules that are adsorbed on a surface. Molecules on oxide surfaces, such as nickel oxide (NiO) and chromium oxide (Cr 2 O 3 ), have 1

16 2 CHAPTER 1. INTRODUCTION been extensively studied both theoretically and experimentally [6 23]. The surfaces in such experiments constitute an environment for the molecules studied. Due to interactions between the adsorbed molecule and the surface, energy and phase exchange emerge, which in turn perturbs or even destroys quantum coherence. The degradation of quantum coherence limits the applicability of quantum phenomena in proposed technologies, e.g. quantum computing. It is hence of importance to understand and optimally control the processes, which destroy quantum coherence. One approach to gain control would be to apply external fields. The focus of this thesis is on the interactions of quantum systems and the surrounding environment and the control of such systems. If excitation, excited state dynamics and relaxation have comparable time-scales, a theoretical approach is required, which describes the system accurately and does not rely on the separation of time-scales. This can be achieved within the framework of the Surrogate Hamiltonian approach [24]. In this approach, a quantum system is separated into a primary system and a bath. The interactions between the primary system and the bath are introduced by explicitly immersing the primary system into the bath, which acts as dissipative environment. A quantum system in condensed phase is affected by dissipative forces and the coherent evolution of the system is hence degraded. A perfect external field could in principle be used to protect an open quantum system from decoherence. Control of surface chemical reactions by means of external light fields is a challenging problem, both theoretically and experimentally. So far, only few approaches have been employed which typically require certain assumptions and conditions. A general approach to the control of quantum systems is provided by control theory [1, 25]. Recently, Local Control Theory (LCT) was used to protect an open quantum system from decoherence and it was proposed that Optimal Control Theory (OCT) could be used to achieve superior results [26]. OCT is a theoretical tool for the design of external fields to transfer an initial state to a specific final state. Besides the traditional final-time control algorithms [27, 28], methods exist to handle time-dependent control targets [29, 30]. Recently, a hybrid time-dependent/time-independent optimal control algorithm has been developed [31]. To this date, optimal control theory has only been applied to dissipative sys-

17 3 tems in the density matrix formalism of open quantum systems [32 35] or under the adiabatic approximation [36]. The advantage of the Surrogate Hamiltonian method is that it allows a wavefunction description of open quantum systems and a microscopic modeling of the bath. In this thesis, OCT with time-dependent targets is combined with the Surrogate Hamiltonian method for the first time, in order to gain control of open quantum systems. Active control of the evolution of the atoms and molecules involved in surface photochemistry experiments will shed new light in surface photochemical processes. Although many different aspects of control can be thought of, this thesis emphases on the possibility to protect open quantum systems from dissipation, which is of prime importance in quantum computing. This thesis consists of two parts. The first part will cover the theoretical tools required for the treatment and control of open quantum systems, i.e. the Surrogate Hamiltonian approach and Optimal Control Theory. In the second part three different open quantum systems will be investigated with the combined Surrogate Hamiltonian /OCT approach. Chapter 2, where different theoretical approaches to treat open quantum systems are outlined, serves as an introduction to the problem of quantum dissipation. The chapter emphases on the Surrogate Hamiltonian approach, which is a potential method for the treatment of open quantum systems. The general problem of optical control of quantum systems is introduced in chapter 3. An overview of approaches for optical control of quantum systems is presented before Optimal Control Theory is introduced, where OCT with time-dependent targets is highlighted. The chapter is concluded with an overview of experimental coherent control techniques. The second part begins with chapter 4, where the harmonic oscillator serves as a benchmark system to demonstrate how the combined Surrogate Hamiltonian /OCT approach works and to which extent control can be achieved. In chapter 5 the combined Surrogate Hamiltonian /OCT approach is applied to a model system representing carbon monoxide adsorbed on a platinum surface. The degree of freedom considered in the model is the vibrational CO-Pt mode. The system will be investigated regarding the possibility to protect a vibrationally excited system from vibrational relaxation. A Surrogate Hamiltonian treatment on an ab initio level of photodesorption dynamics is presented in chapter 6, where the laser induced desorption of

18 4 CHAPTER 1. INTRODUCTION nitrogen monoxide from a nickel oxide surface is investigated. The controllability of the system and the influence of the control pulse on the desorption observables are investigated. Chapter 7 summarizes this thesis and gives an outlook into future work in the area of control of open quantum systems. In the appendix, the numerical and computational tools necessary for all calculations are rewieved. Throughout this thesis atomic units, i.e. = m e = e = a 0 = 1, have been used unless otherwise stated.

19 Part I Theory 5

20

21 Chapter 2 Quantum Dissipation Since a perfect isolation of a quantum system is impossible [1], all realistic quantum systems are influenced by there surroundings. Since the influence of the surroundings cannot be neglected, any realistic quantum system must be treated as an open system. The problem of open quantum systems has been extensively studied since the early 60 s [37, 38], although the existence of the problem was known much earlier. Quantum fluctuations and quantum dissipation are two phenomena that naturally occur when a quantum system is influenced by its surrounding, and the theory of open quantum systems has to be applied to describe and understand these phenomena. Processes in which a quantum system loses energy and/or phase due to interactions with the surrounding environment are termed quantum dissipation. Energy relaxation does not only occur in quantum systems but is also observed in classical systems, e.g. friction, while a process, in which a system loses phase, termed dephasing, is specific for quantum systems. Examples of quantum dissipation can be found in quantum optics, nuclear magnetic resonance and condensed matter physics. In condensed matter physics dissipation plays a crucial role in areas such as surface photochemistry, photosynthesis and atoms or molecules caged in a cluster [39]. The essential idea behind the theory of open quantum systems is the separation of a global quantum system into two parts. The general approach is then to derive the dynamics of one of the subsystems from a microscopic theory by elimination of the degrees of freedom of the other subsystem. This separation is not only made due 7

22 8 CHAPTER 2. QUANTUM DISSIPATION to the fact that most realistic systems are far to complicated to be described by the fundamental laws of physics, but a microscopic description of a realistic quantum system would also result in an overwhelming amount of information. Most of this information would not contribute to the understanding of the investigated system [1]. The separation of a system into two subsystems is non-trivial, the decomposition must be made in such a way that the Hamiltonian of the system is assured to be well-defined, that the subsystem representing the environment is stable and that the interaction between the two subsystems is non-singular [39]. The subsystem representing the surrounding environment is usually treated implicitly. The surrounding environment is normally represented by an abstract description since only its influence on the other subsystem is of interest and not the surrounding environment itself. There are two classes of commonly used environment descriptions, the environment is usually modeled by an ensamble of either two level systems [40] or harmonic oscillators [41]. The basic idea behind the use of harmonic oscillators in the description of the environment steams from a normal mode analysis together with the assumption of a weak interaction between the two subsystems [41]. There exists a wide range of different methods to treat open quantum systems and many of the methods are formally equivalent. Some of the most prominent methods are the path-integral and projection operator formulations [42]. The following section will deal with the standard approaches and concepts of the theory of open quantum systems. The section will also serve as a prerequisite before the Surrogate Hamiltonian method, which is the method of choice for the handling of open quantum systems in this thesis, is introduced. 2.1 Open Quantum Systems As already stated, the fundamental idea of the theory of open quantum systems is to assume that the total system can be split up into two parts. The notation of composite quantum systems is hence fundamental. The Hilbert space of a composite system is the tensor product space of the Hilbert spaces of the subsystems [1], H = H (1) H (2). (2.1)

23 2.1. OPEN QUANTUM SYSTEMS 9 In the above equation H (1) denotes the Hilbert space of subsystem S (1), H (2) the Hilbert space of subsystem S (2) and H is the Hilbert space of the composite system. Equation 2.1 follows from the postulates of quantum mechanics [1]. Since it is intended to develop a formalism for open quantum systems, it is further assumed that the two subsystems are allowed to interact with each other. In the treatment of open quantum systems one of the subsystems contains one or a few active degrees of freedom, this subsystem is frequently termed primary system, denoted S. The remaining degrees of freedom are taken care of by a second subsystem usually called the environment or bath [43] and is denoted B. The Hilbert space of the compsite system is hence, according to equation 2.1, written as H = H S H B. (2.2) It is usually assumed that this composite system is a closed system, but the composite system can also be coupled to yet another system [44]. The Hamiltonian of the primary system is denoted Ĥ S and the bath Hamiltonian is denoted Ĥ B. The coupling between the primary system and the bath degrees of freedom is taken care of by an interaction Hamiltonian, which is denoted Ĥ SB. The Hamiltonian of the composite system is thus written as [1, 39] Ĥ = Ĥ S Î B + Ĥ B Î S + Ĥ SB, (2.3) where ÎS and ÎB are the identity operators of the primary system and the bath, respectively [45]. Since the evolution of an open quantum system cannot generally be represented in terms of a unitary time evolution, its dynamic is often formulated by an eligible equation of motion for its density matrix, a quantum master equation [1]. The density matrix is given by the expression [39] ˆρ = Ψ Ψ, (2.4) where Ψ is the wave function of the composite system. Since only observables of the primary system are of interest, it is convenient to introduce the reduced density matrix, ˆρ S tr Bˆρ, (2.5)

24 10 CHAPTER 2. QUANTUM DISSIPATION where ˆρ is the density matrix of the composite system and tr B denotes the partial trace over the bath degrees of freedom. The operators of the primary system are of the form  S Î B [1], where  S is an arbitrary operator that acts on the primary system Hilbert space, H S. The expectation value of the operator  S is then determined by [39]  S = tr S { Sˆρ S }, (2.6) where tr S denotes the partial trace over primary system s Hilbert space [1]. The density matrix of the composite system evolves unitary [45] and its time evolution is given by [1] ˆρ(t) = Û t t 0ˆρ(t 0 )(Û t t 0 ), (2.7) where Û t t 0 is the time-evolution operator, also called the propagator. The timeevolution of the reduced density matrix is hence, in accordance with equation 2.5, given by ˆρ S (t) = tr Bˆρ(t) = tr B {Ût t0ˆρ(t 0 )(Û t t 0 ) }. (2.8) By taking the time derivate of equation 2.7 and replacing the time derivative of the wavefunction with the Hamiltonian, the equation of motion for the density matrix is obtained [46], i tˆρ(t) = [Ĥ, ˆρ(t) ]. (2.9) The above equation is commonly referred to as the Liouville-von Neumann equation. Analogically with equation 2.9 the equation of motion of the reduced density matrix is given by i tˆρ S(t) = tr B [Ĥ, ˆρ(t) ]. (2.10) In the next sections, a brief review of the most applied approaches and approximations to the above equation of motion will be given. 2.2 The Markov Approximation/The Quantum Markovian Semigroup Master Equation Assuming that there are no initial correlations between the primary system and the bath at the initial time t = t 0, makes it possible to write the initial state of the

25 2.2. MARKOV APPROXIMATION 11 composite system as ˆρ(t 0 ) = ˆρ S (t 0 ) ˆρ B, (2.11) where ˆρ B tr Sˆρ. Using the above assumption and remembering that the dynamics of the primary system is given by equation 2.8, the transformation describing the evolution of the primary system from the initial time t 0 to some later time t > t 0 may be written as ˆρ S (t) = tr B {Ût t0 (ˆρ S (t 0 ) ˆρ B ) (Û t t 0 ) }. (2.12) The dynamics expressed through the above equation can be viewed as a map, V (t fix ), which maps the initial state of the primary system, ˆρ S (t 0 ), to another state, ˆρ S (t fix ), at a later fixed time t = t fix [1], ˆρ S (t 0 ) ˆρ S (t fix ) = V (t fix )ˆρ S (t 0 ). (2.13) This map is known as a dynamical map. If t fix is allowed to vary, i.e. t fix = t, a oneparameter family of dynamical maps is provided, {V (t) t 0}, which determines the future evolution of the primary system [1]. If it is assumed that the time scale of the evolution of the primary system is much greater than the time scale of the bath correlation function, it is justified to omit memory effects for the dynamics of the primary system [1], i.e. the future of the reduced density matrix is independent of its past history. When memory effects are neglected, the process is a so called Markov process. A Markovian behavior can be formalized by the semigroup property [1]: V (t 1 )V (t 2 ) = V (t 1 + t 2 ), t 1, t 2 0, (2.14) which is a quantum dynamical semigroup. The quantum dynamical semigroup can be represented in exponential form, V (t) = e Lt, (2.15) where L is the generator of the semigroup. The equation of motion for the reduced density matrix is hence given by d dtˆρ S(t) = Lˆρ S (t). (2.16)

26 12 CHAPTER 2. QUANTUM DISSIPATION The above equation is called the Markovian quantum master equation [47] and the generator L for an N-level system in its most general form is given by [48] Lˆρ S (t) = i[ĥ S, ˆρ S (t)] + N 2 1 α=1 γ α (Ĝ αˆρ S (t)ĝ α 1 ) 2 {Ĝ αĝα, ˆρ S (t)}, (2.17) where Ĝ α and Ĝ α are orthonormal system operators, usually referred to as Lindblad operators, with the corresponding relaxation times γ α. Equation 2.16 is mostly called the Lindblad equation. A quantum master equation, whose generator is of the Lindblad form, can be derived for a number of physical problems using a microscopic theory with certain approximations. The low density limit, the singular coupling limit and the weak-coupling limit are some of the most important cases [45]. In some physical problems, non-linear quantum master equations with generators of the Lindblad form are encountered. The coefficients of the Lindblad generators in such problems depend parametrically on the density matrix, which leads to a quantum master equation of the form L L(ˆρ), (2.18) d dtˆρ S(t) = L(ˆρ S (t))ˆρ S (t). (2.19) The mean field master equation and the quantum Boltzmann equation are some examples of non-linear quantum master equations [45]. 2.3 Non-Markovian Approaches In the Markov approach it is assumed that the bath correlations decay on a much shorter timescale than the timescale of the primary system. In many physical and chemical problems this is not the case, e.g. for systems with a moderate to strong system-bath coupling [39]. The starting point for most non-markovian approaches is, as in the Markov approximation, provided by a microscopic theory for the combined system, S +B. The first step in a theoretical treatment of non-markovian processes is to derive an exact equation of motion for the primary system through the elimination of the dynamical variables of the bath [45]. Analytical expressions for the evolution

27 2.3. NON-MARKOVIAN APPROACHES 13 of the reduced density matrix only exist for a few simple problems, e.g. for a damped harmonic oscillator. In most cases, an exact representation of the reduced density matrix is the starting point for a perturbative treatment of the primary system-bath coupling [45]. One approach to obtain the equation of motion for a non-markovian open system is provided by projection operator techniques. The basic idea of projection operator techniques is to regard the operation of tracing over the bath as a projection, P. The density matrix Pρ is called the relevant part of the density matrix. Hence, the aim is to derive an equation of motion for the relevant part, Pρ. The Nakajima-Zwanzig projection operator technique is probably the most prominent of the projection operator techniques. In this approach, a formally exact equation of motion for the primary system, which involves a certain memory kernel, is derived [45]. A disadvantage with this approach is that the perturbative expansion of the memory kernel gives rise to an integro-differential equation, whose numerical solution can be quite demanding [1]. Another strategy is to derive a time-local master equation for the reduced density matrix. In this approach, the equation of motion is given by d dt ρ S(t) = K(t)ρ S (t), (2.20) where K is a time-dependent generator. It can be shown, employing the timeconvolutionless (TCL) projection operator technique [1], that a master equation of the above form exists for weak and intermediate coupling. Equation 2.20 is local in time and the integration over its past is hence needed [45]. A different approach to treat open systems is the path integral formulation of quantum mechanics. The state of the system, the wave function or the density matrix, at a specific time t in this approach is given by the propagator, which is the sum over all possible paths starting at a specified point at time t = 0 [39]. In contrast to the previously mentioned methods, the integration over the bath degrees of freedom gives rise to an influence functional, which contains correlations in time between different paths [49]. Non-Markovian processes can hence, in principle, be accounted for. The influence functional for a linearly coupled bath of harmonic oscillators can be derived analytically and the result is the famous Caldeira-Leggett model [50]. In spite of efforts trying to derive an influence functional for a system non-linearly coupled to a bath of anharmonic oscillators, the strength of the path