that a non-Markovian master equation describing such a process loses positivity when the inter- action strength gets too large compared to the memory time 72 (see also Ref. 62). However, the master equation can also lose positivity when other interactions in the Hamiltonian are modified.
We demonstrate in this particular case that treating the non-Markovian term as generic results in an equation of motion that is unstable (with respect to positivity) when one turns on addi- tional interactions in the system’s Hamiltonian. Thus, it does not seem likely that one can have a generic non-Markovian term in the master equation, which can be used, even phenomenologically and keeping the system-reservoir interaction constant, to investigate the effects of dissipation as one varies parameters of the system. Having such a generic term, though, would allow one to build reusable tools and approximations for the investigation of non-Markovian effects. Because of this, we suggest an alternative approach for simulating real-time, **open**-system **dynamics** that does not rely on non-Markovian master equations. The methodology is based on performing a sequence of transformations on the reservoir Hamiltonian which create auxiliary **quantum** degrees of freedom that can be included explicitly in the **simulation**. Their inclusion allows one to simulate highly non-Markovian processes without the drawbacks of non-Markovian master equations. Further, for **systems** with strong dissipation or long memory times, such as the spin-boson model, the structure of the reservoir is such that one can use matrix product state techniques. Indeed, this is the basis of the numerical renormalization group, but its implications go well beyond renormalization schemes.

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5.1.2 Self-organization of states in **open** **systems** for both bosonic and fermionic atoms [ 19 , 131 – 134 ].
5.1.2. Self-organization of states in **open** **systems**
Instead of carefully constructing specific **quantum** states in an equilibrium setting, it can be advantageous to design the environment of an **open** **quantum** **systems** in a way, that the steady state is the targeted state acting as an attractor in time [ 135 ]. In this approach, certain perturba- tions to the steady state are corrected exponentially fast in time. Provided that the steady state is unique, the **dynamics** leading to the state can be seen as a self-organization process, vastly independent of the initial state of the system. One example for this is given by the coupling of a photon-leaking cavity to a gas of cold atoms, where the photon field mode effectively medi- ates a global long-range coupling between the atoms causing an accelerated formation of the steady state. Moreover, a feedback loop between atoms and photons can be established, which is able to affect the light field considerably. One remarkable experiment is the observation of the Dicke **quantum** phase transition in the form of a non-equilibrium dynamical phase transi- tion [ 24 , 25 ]. To this end, a Bose-Einstein condensate of atoms is placed in a cavity, coupled to a transverse standing wave pump laser oriented perpendicular to the cavity axis. Increasing the pump strength with time above a certain threshold gives rise to a reordering of the atoms into a checkerboard density pattern via two-photon scattering processes involving both, pump and cavity photons, as predicted in earlier theoretical studies [ 23 , 136 – 139 ]. Moreover, externally controlled optical lattices have been created inside optical resonators, enabling, for example, the experimental realization of a non-equilibrium superfluid to Mott-insulator transition [ 140 , 141 ], which raises the hope for future experimental feasibility of the proposed fermionic model. 5.1.3. Outline of the chapter

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beyond the linear response regime: excess electron injection and trapping in liquids,” Journal of Chemical Physics, 132, 034106 (2010).
Chapter 3 explores the use of RPMD to directly simulate ET reactions between mixed-valence transition metal ions in water. We compare the RPMD approach against benchmark semiclassical and **quantum** **dynamics** methods in both atomistic and system-bath representations for ET in a polar solvent. Without invoking any prior mechanistic or transition state assumptions, RPMD correctly predicts the ET reaction mechanism and quantitatively describes the ET reaction rate over twelve orders of magnitude in the normal and activationless regimes of ET. Detailed analysis of the dynamical trajectories reveals that the accuracy of the method lies in its exact description of statistical fluctuations, with regard to both reorganization of classical nuclear degrees of freedom and the electron tunneling event. The vast majority of the ET reactions in biological and synthetic **systems** occur in the normal and activationless regimes, and this work provides the foundation for future studies of ET and PCET reactions in condensed-phase **systems**. Additionally, this study discovers a shortcoming of the method in the inverted regime of the ET, which arises from the inadequate description of the quantization of the real-time electronic-state **dynamics**, and directly motivates further methodological refinement. This work has been published as Menzeleev, A. R., Ananth, N. and Miller, T. F., “Direct **simulation** of electron transfer using ring polymer molecular **dynamics**: Comparison with semiclassical instanton theory and exact **quantum** methods,” Journal of Chemical Physics, 135, 074106 (2011).

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2
**Open** **quantum** **systems**
2.1 Introduction
An **open** **quantum** system is defined as a **quantum** system coupled to an external bath, leading to the presence of dissipative processes in its **dynamics** [ 59 , 60 ]. In general, all physical **systems** are intrinsically **open**, with the only exception of the universe as a whole. Thus, a description of any realistic system requires accounting for the presence of dissipative channels. This is particularly important in certain areas such as **quantum** information [ 61 ], where it is crucial to understand the impact of losses in order to design robust protocols and efficient experimental platforms. Additionally, dissipation can also serve as a mean to drive a system into particular states of interest, opening the door to the field of reservoir engineering [ 62 ]. Our current understanding of **open** **quantum** **systems** is rudimentary compared to that of closed **systems**. This makes **open** **systems** an active field of research not only from a practical point of view, but also from a fundamental perspective. A representative example is the study of **quantum** many-body **systems** in **open** settings [ 63 , 64 ]. There, great progress is being made along several lines such as classifying many-body phases of matter [ 65 , 66 ], identifying clear signatures for phase transitions [ 67 ] or exploring novel features which emerge when the system is **open** [ 68 ].

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To acheive this, we introduce here a computational strategy based on tree tensor network states (TTNS), and give a step-by- step demonstration of how we can manipulate ab initio data with machine-learning techniques to formulate the problem in a form suitable for TTNS **simulation**. To evolve the resulting TTNS representation of the full many-body wave function, we employ a recent variational principle that allows fast and efﬁcient processing 26 , 31 , 36 , and—in a critical step—we show how an analysis of the entanglement ‘topology’ of the tensor network allows for a ‘rewiring’ of the network that can compress the memory requirements by up to six orders of magnitude. We shall then show that the formalism of TTNS enables a completely general ‘on the ﬂy’ method to discover the dominant many-body/ multidimensional conﬁgurations of the environment that drive **open** **dynamics**, and how this allows us to visualise non- equilibrium processes on just a few low-dimensional environ- mental energy surfaces. With this new capability to identify and distinguish effective ‘reaction coordinates’ according to their evolving relevance, we pinpoint the sequence of ultrafast vibra- tional motions that drive singlet ﬁssion in an ab initio para- metreised dimer molecule known to undergo efﬁcient SF, and highlight the potential utility of our technique for unravelling complex non-Markovian **dynamics** in a wide range of **open** **quantum** **systems** across physics and chemistry.

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Florian A.Y.N. Schröder 1 , David H.P. Turban 1 , Andrew J. Musser 2 , Nicholas D.M. Hine 3 & Alex W. Chin 4
The **simulation** of **open** **quantum** **dynamics** is a critical tool for understanding how the non- classical properties of matter might be functionalised in future devices. However, unlocking the enormous potential of molecular **quantum** processes is highly challenging due to the very strong and non-Markovian coupling of ‘environmental’ molecular vibrations to the electronic ‘system’ degrees of freedom. Here, we present an advanced but general computational strategy that allows tensor network methods to effectively compute the non-perturbative, real-time **dynamics** of exponentially large vibronic wave functions of real molecules. We demonstrate how ab initio modelling, machine learning and entanglement analysis can enable simulations which provide real-time insight and direct visualisation of dissipative photo- physics, and illustrate this with an example based on the ultrafast process known as singlet ﬁssion.

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The simple algorithm for the **simulation** and visualization of non relativistic **quantum** **dynamics** is proposed that is based on a collective behavior of classical particles. Any **quantum** particle is represented as the swarm of its classical sam- ples which interact by simple rules including emission and absorption of samples of tied photons. The **quantum** **dynamics** results from the collective behavior of such a swarm where the eigenstates are treated as the equilibrium states relatively to emission-absorption of photons. The entanglement is treated as a correlation be- tween samples of the different swarms that is stored in the space-time part of the model inaccessible for a user. The amplitude is always grained. The Coulomb field between **quantum** particles is simulated, analogously to free flow of **quantum** pack- age, by the point wise interaction between its samples and scalar photon samples which propagate by diffusion. It gives square root speedup in comparison to each- with-each method. This method obviously includes decoherence and admits the natural generalization on the QED of many particles with the linear computational cost.

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These results point out that quantiﬁers of a speciﬁc kind of correlations, based on different distance measures, exhibit not only quantitative but also qualitative differences. For example, while a quantiﬁer of a speciﬁc kind of correlations, based on a given distance measure, has a dynamical behavior (e.g., constant or decreasing), another quantiﬁer behaves differently (e.g., increasing). These ﬁndings seem to have a counterpart in the relativity of entanglement measures as a result of physical processes [ 28 ]: in this case, however, the relativity of measures shows up when one compares two different states as they evolve from different initial conditions. Here, instead, different qualitative behaviors appear in the evolution of **quantum** correlation quantiﬁers that are used to represent physical **dynamics** of the same kind of correlations present during the evolution of a single state. As a further point, the relationship among geometric total, **quantum**, and classical correlations has been investigated, ﬁnding that they do not satisfy, in general, a closed additivity relation [ 14 ], as happens instead for REB correlation quantiﬁers [ 6 ]. Entropic and geometric **quantum** discords could thus individuate themselves genuinely inequivalent characterizations of nonclassical correlations, as also appears to be corroborated by recent analyses [ 29 , 30 ]. The above results indicate that appropriate quantiﬁcation of the

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Thus for every value y 3 f ∈ [z − , z + ] exists a time t f ∈ [0, max{t − , t + }] such that y 3 (t f ) = y 3 f if u(t) = (0, u 2 , 0) for t ∈ [0, t f ]. Now defining u(t) = (0, 0, E) for t > t f the **systems** converges exponentially fast to (0, 0, y f 3 ).
The result we obtained is partially satisfying. However, it could be a first step to study more general models of **open** **quantum** system. In particular, the main issue of our approach is the effectiveness of the Lindblad equation in the description of adiabatic **open** **quantum** **systems**. As we have seen in our analysis, the **dynamics** described by (4.28) is almost always a motion that converges exponentially fast to a unique equilibrium. In this framework the adiabatic theory cannot be effective because the convergence rate is accelerated when the time is slowed.

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Application : Radiative Forces
**Dynamics** of an atom in a laser field
Within a semi-classical approach, if the atom is sufficiently slow, its internal states adiabatically adapts to the intensity and the phase of the laser locally, it is subjected to the force

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The story started with §II by identifying which **quantum** channels could not be reached by a system with a fixed generator of dissipation using any possible control Hamiltonians. As well as being of interest for purely theoretical reasons, it is of use for simulating **open** **quantum** **systems** using an existing source of noise. An alternative way of achieving the same simulations is to use a closed system with purely unitary **dynamics**, as well as the ability to add and remove ancillary **systems**. This was studied in §III where it was shown that any analytic family of **quantum** channels can be replicated with a time-dependent Hamiltonian and a finite ancillary space. In addition to being a novel way to achieve time-continuous simulations, it is also a lift of Stinespring’s dilation theorem from the level of propagators to generators as the required conditions for such a Hamiltonian to be continuous and bounded were derived. It was also proved that the dilation can always be well approximated to have those properties. Furthermore, it provides a testing ground to see how adding control Hamiltonians on the microscopic system-bath level can modify the generator for the reduced system. This question, which was a key assumption in §II, was specifically addressed in §IV in the case where the controls and the Lindbladian on the reduced system commute. It was conjectured that in such a case the Hamiltonian does not modify the noise no matter what the underlying system-bath interaction is. Although a definitive proof either way was not forthcoming, it was nevertheless showed that there are some interesting constraints on the **dynamics**, and that the conjecture does not hold in a specific example of non-Markovian noise.

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Given this broad range of problems, it would be highly desirable to have a unified theory of **quantum** metastabil- ity. In this paper we lay the ground for such a theory for the case of **open** **quantum** **systems** evolving with Marko- vian **dynamics**. Our starting point is a well-established approach for metastability in classical stochastic **systems** [25–29]. We develop an analogous method for **quantum** Markovian **systems** based on the spectral properties of the generator of the **dynamics**. Separation of timescales implies a splitting in the spectrum, and this spectral di- vision allows us to construct metastable states from the low-lying eigenmatrices of the generator. Based on per- turbative calculations for finite **systems**, we argue that the manifold of metastable states is in general composed of disjoint states, noiseless subsystems and decoherence-

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The study of **open** **quantum** **systems** - microscopic **systems** exhibiting **quantum** coherence that are coupled to their environment - has become increasingly important in the past years, as the ability to control **quantum** coherence on a single particle level has been developed in a wide variety of physical **systems**. In **quantum** optics, the study of **open** **systems** goes well beyond understanding the breakdown of **quantum** coherence. There, the coupling to the environment is sufficiently well understood that it can be manipulated to drive the system into desired **quantum** states, or to project the system onto known states via feedback in **quantum** measurements. Many mathematical frameworks have been developed to describe such **systems**, which for atomic, molecular, and optical (AMO) **systems** generally provide a very accurate description of the **open** **quantum** system on a microscopic level. In recent years, AMO **systems** including cold atomic and molecular gases and trapped ions have been applied heavily to the study of many-body physics, and it has become important to extend previous understanding of **open** system **dynamics** in single- and few-body **systems** to this many-body context. A key formalism that has already proven very useful in this context is the **quantum** trajectories technique. This method was developed in **quantum** optics as a numerical tool for studying **dynamics** in **open** **quantum** **systems**, and falls within a broader framework of continuous measurement theory as a way to understand the **dynamics** of large classes of **open** **quantum** **systems**. In this article, we review the progress that has been made in studying **open** many- body **systems** in the AMO context, focussing on the application of ideas from **quantum** optics, and on the implementation and applications of **quantum** trajectories methods in these **systems**. Control over dissipative processes promises many further tools to prepare interesting and important states in strongly interacting **systems**, including the realisation of parameter regimes in **quantum** simulators that are inaccessible via current techniques.

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We investigate a possibility to generate non-classical states in light-matter coupled noisy quan- tum **systems**, namely the anisotropic Rabi and Dicke models. In these hybrid **quantum** **systems** a competing influence of coherent internal **dynamics** and environment induced dissipation drives the system into non-equilibrium steady states (NESSs). Explicitly, for the anisotropic Rabi model the steady state is given by an incoherent mixture of two states of opposite parities, but as each parity state displays light-matter entanglement we also find that the full state is entangled. Furthermore, as a natural extension of the anisotropic Rabi model to an infinite spin subsystem, we next explored the NESS of the anisotropic Dicke model. The NESS of this linearized Dicke model is also an in- separable state of light and matter. With an aim to enrich the **dynamics** beyond the sustainable entanglement found for the NESS of these hybrid **quantum** **systems**, we also propose to combine an all-optical feedback strategy for **quantum** state protection and for establishing **quantum** control in these **systems**. Our present work further elucidates the relevance of such hybrid **open** **quantum** **systems** for potential applications in **quantum** architectures.

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much larger than the frequency scales of **dynamics** induced by the coupling (Γ).
[27, 28]. The study of **open** **quantum** **systems** also has importance in cosmology [29], and **quantum**-optical approaches have recently been applied to treat pion decay in high-energy physics [30, 31].
While many studies seek to characterise and reduce the destruction of **quantum** coherence in an **open** system, a key guiding aim in **quantum** optics over the last thirty years has been the use of controlled coupling to the environment to control and manipulate **quantum** coherence with high precision. This includes manipulating the environment in such a way as to drive the system into desired **quantum** states, increasing **quantum** coherence in the sample. In the laboratory, this philosophy began with optical pumping in atomic physics [32, 33], whereby using laser driving and spontaneous emission processes, atoms can be driven into a single atomic state with extraordinarily high ﬁdelities approaching 100%. This has lead on to techniques for laser cooling of trapped ions [20, 34, 35] and of atomic and molecular samples [36], which has allowed the production of atomic gases with temperatures of the order of 1 µK. This, in turn, set the stage for realising Bose-Einstein condensation [37–39] and degenerate Fermi gases [40, 41], as well as providing a level of high-precision control necessary for **quantum** computing with trapped ions [17, 20, 42]. Such driving processes, making use of the coupling of the system to its environment, have been extended to algorithmic cooling in Nuclear Magnetic Resonance (NMR) **systems** [43, 44], and are recently being applied to cool the motional modes of macroscopic oscillators to near their **quantum** ground state [45]. A detailed understanding of the back-action on the **quantum** state of the system from coupling to the environment has also been useful in the context of high-ﬁdelity state detection (e.g., in electron shelving [46–51]), or in **quantum** feedback [52–56], and have also been exploited for control over atoms and photons in Cavity QED [57, 58].

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As yet, there have been no experiments aiming to create PREs (involving non-orthogonal pure states). It certainly would be difficult to implement the high-efficiency adaptive mea- surement schemes required, though a suggestion has been made [5], involving **quantum** transport with feedback, which alleviates some of the difficulties. However, PREs might have applications, irrespective of experimental realization, in **quantum** simulations. A generic **quantum** trajectory, which involves periods of non-trivial continuous evolution, will occupy an infinite sized ensemble of states, though this is reduced to a finite number when simulated using finite precision. Despite this, the memory required will still be expo- nentially large in the system dimension, D. In contrast, a finite PRE with only K states, will have memory requirements scaling generically only as D 2 (see Eq. (18)). The catch, as the reader of this paper will appreciate, is that there is a one-time large resource cost associated with identifying a PRE (and adaptive measurement scheme) applicable to the ME. This cost is dependent upon the applied algorithm, with the monodromy extension to polynomial homotopy continuation providing the greatest potential of allowing PREs to be found with sub-exponential (or a very small constant) complexity. If the finding of PREs can be made tractable for the ME in question, then it is possible that trajectory **simulation** could be most efficiently undertaken using the PRE ME unraveling.

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Adiabatic **quantum** algorithms represent a promising approach to universal **quantum** computation. In isolated **systems**, a key limitation to such algorithms is the presence of avoided level crossings, where gaps become extremely small. In **open** **quantum** **systems**, the fundamental robustness of adiabatic algorithms remains unresolved. Here, we study the **dynamics** near an avoided level crossing associated with the adiabatic **quantum** search algorithm, when the system is coupled to a generic environment. At zero temperature, we find that the algorithm remains scalable provided the noise spectral density of the environment decays sufficiently fast at low frequencies. By contrast, higher order scattering processes render the algorithm inefficient at any finite temperature regardless of the spectral density, implying that no **quantum** speedup can be achieved.

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Accurate numerical or analytical description of general **open** **quantum** **systems** **dynamics** ap- pears, prima facie, to be extremely difficult due to the large (often infinite) number of bath and system variables which need to be accounted for. When the environmental degrees of freedom are modeled as a bath of harmonic oscillators exact path integral solutions can be available but are rarely of practical use [10, 1, 2]. Hence assumptions such as weak system-environment coup- ling and vanishing correlation times of the environment, i.e. the Born-Markov approximation, are often invoked to obtain compact and efficiently solvable equations. These approaches suffer the drawback that their accuracy is hard to certify and that they become simply incorrect in many important situations. Indeed, our increasing ability to observe and control **quantum** **systems** on ever shorter time and length scales is constantly revealing new roles of noise and **quantum** coher- ence in important biological and chemical processes [4, 5, 6, 7, 8] and requires an accurate but efficient description of the system-environment interaction that go well beyond the Born-Markov approximation [8, 9] in order to understand the interaction of intrinsic **quantum** **dynamics** and environmental noise. In many biological, chemical and solid-state **systems**, deviations from strict Markovianity, which can be explicitly quantified [12, 13, 14], are significant and methods beyond standard perturbative expansions are required for their efficient description. A number of tech- niques have been developed to operate in this regime. Those include polaron approaches [15], the quasi-adiabatic path-integral (QUAPI) method [16], the hierarchical equation of motion approach [17] and extensions of the **quantum** state diffusion description to non-Markovian regimes [18].

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By tracing out the degrees of freedom of the reservoir, we obtain the **dynamics** of the small system.
Over the last decade and a half, a perturbation theory based on **quantum** resonance methods has been developed in the isolated resonances regime. The perturbation theory developed so far permits a mathematically rigorous treatment of the **dynamics** for fixed, small system-reservoir coupling parameter λ. However, in complex **quantum** **systems**, e.g. when the dimension of small system is large, the problem belongs to the overlapping resonances regime. The theory mentioned above is not applicable in this regime. In this thesis, we adapt the perturbation theory for the treatment of such a regime. We first obtain a representation formula of the reduced **dynamics** involving

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and scenarios in which ladders can outperform chains. In considerations of missing spins and imperfect placement we can see ladders might be as, or even more robust than chains to increasing error rates, but often the reduced fidelity associated with more decay pathways often makes ladder geometries appear to perform worse than their chain counterparts. However, given a sufficiently high missing spin rate, lad- ders do become the better candidate. It would be very interesting to combine studies of missing spin rates and positioning accuracy. This sort of study is especially com- pelling since longer channels seem more robust against coupling variation and there was the observation of ladders outperforming longer chains when spins are (possi- bly) absent. Unfortunately the current computation limitations make any study into combining these effects impossible; vast resources would be required to consider all configurations of missing spins and sample each of them hundreds or thousands of times to converge distributed coupling **dynamics**.

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